id stringlengths 1 6 | url stringlengths 16 1.82k | content stringlengths 37 9.64M |
|---|---|---|
13000 | https://byjus.com/maths/pascals-triangle/ | Pascal’s Triangle is a kind of number pattern. Pascal’s Triangle is the triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression. The numbers are so arranged that they reflect as a triangle. Firstly, 1 is placed at the top, and then we start putting the numbers in a triangular pattern. The numbers which we get in each step are the addition of the above two numbers. It is similar to the concept of triangular numbers. In this article, we are going to learn Pascal’s triangle history, definition, properties, patterns, formulas and examples with a complete explanation.
Table of contents:
History
Definition
Construction
Formula
Expansion
How to use
Patterns
Properties
Video Lesson
Solved Examples
Practice Questions
FAQs
Pascal’s Triangle History
Blaise Pascal was born at Clermont-Ferrand, in the Auvergne region of France on June 19, 1623. In 1653 he wrote the Treatise on the Arithmetical Triangle which today is known as Pascal’s Triangle. Although other mathematicians in Persia and China had independently discovered the triangle in the eleventh century, most of the properties and applications of the triangle were discovered by Pascal.
This triangle was among many of Pascal’s contributions to mathematics. He also came up with significant theorems in geometry, discovered the foundations of probability and calculus and also invented the Pascaline-calculator. Still, he is best known for his contributions to the Pascal triangle.
Pascal’s Triangle Definition
Most people are introduced to Pascal’s triangle through an arbitrary-seeming set of rules. Begin with 1 on the top and with 1’s running down the two sides of a triangle. Each new number lies between two numbers and below them, and its value is the sum of the two numbers above it. The theoretical triangle is infinite and continues downward forever, but only the first 6 lines appear in figure 1. More rows of Pascal’s triangle are listed in the last figure of this article. A different way to describe the triangle is to view the first line is an infinite sequence of zeros except for a single 1. To obtain successive lines, add every adjacent pair of numbers and write the sum between and below them. The non-zero part is Pascal’s triangle.
Pascal’s Triangle Construction
The easiest way to construct the triangle is to start at row zero and write only the number one. From there, to obtain the numbers in the following rows, add the number directly above and to the left of the number with the number above and to the right of it. If there are no numbers on the left or right side, replace a zero for that missing number and proceed with the addition. Here is an illustration of rows zero to five.
From the above figure, if we see diagonally, the first diagonal line is the list of ones, the second line is the list of counting numbers, the third diagonal is the list of triangular numbers and so on.
Pascal’s Triangle Formula
The formula to find the entry of an element in the nth row and kth column of a pascal’s triangle is given by:
(\begin{array}{l}i.e.,{n \choose k}\end{array} )
The elements of the following rows and columns can be found using the formula given below:
(\begin{array}{l}Pascal’s\ Triangle\ Formula = {n \choose k}= {n-1 \choose k-1}+ {n-1 \choose k}\end{array} )
Here, n is any non-negative integer and 0 ≤ k ≤ n.
The above notation can be written as:
(\begin{array}{l}{n \choose k} (i.e., n\ choose\ k) = C(n, k) = \ ^{n}C_{k} = \frac{n!}{[k!(n-k)!]}\end{array} )
This pattern of getting binomial coefficients is called Pascal’s rule.
Pascal’s Triangle Binomial Expansion
Pascal’s triangle defines the coefficients which appear in binomial expansions. That means the nth row of Pascal’s triangle comprises the coefficients of the expanded expression of the polynomial (x + y)n.
The expansion of (x + y)n is:
(x + y)n = a0xn + a1xn-1y + a2xn-2y2 + … + an-1xyn-1 + anyn
where the coefficients of the form ak are precisely the numbers in the nth row of Pascal’s triangle. This can be expressed as:
(\begin{array}{l}a_{k}= {n \choose k}\end{array} )
For example, let us expand the expression (x + y)n for n = 3.
(x + y)3 = 3C0x3 + 3C1 x2y + 3C2 xy2 + 3C3 x0y3
= (1)x3 + (3)x2y + (3)xy2 + (1)y3
Here, the coefficients 1, 3, 3, 1 represent elements in the 3rd row of the pascal’s triangle.
How to Use Pascal’s Triangle?
Pascal’s triangle can be used in various probability conditions. Suppose if we are tossing the coin one time, then there are only two possibilities of getting outcomes, either Head (H) or Tail (T).
If we toss it two times, then there are one possibility of getting both heads HH and both as tails TT, but there are two possibilities of getting at least a Head or a Tail, i.e. HT or TH.
Now you may consider how Pascal’s triangle will help here. So let’s see the table given here based on the number of tosses and outcomes.
| | | |
---
| Number of Tosses | Number of Outcomes | Pascal’s Triangle |
| 1 | H T | 1,1 |
| 2 | HH HT TH TT | 1, 2, 1 |
| 3 | HHH HHT, HTH, THH HTT, THT, TTH TTT | 1,3,3,1 |
We can also extend it by increasing the number of tosses.
Pascal’s Triangle Patterns
Addition of the Rows
One of the interesting properties of the triangle is that the sum of numbers in a row is equal to 2n
where n corresponds to the number of the row:
1 = 1 = 20
1 + 1 = 2 = 21
1 + 2 + 1 = 4 = 22
1 + 3 + 3 + 1 = 8 = 23
1 + 4 + 6 + 4 + 1 = 16 = 24
Prime Numbers in the Triangle
Another pattern visible in the triangle deals with prime numbers. If a row starts with a prime number or is a prime numbered row, all the numbers that are in that row (not counting the 1’s) are divisible by that prime. If we look at row 5 (1 5 10 10 51), we can see that 5 and 10 are divisible by 5. However, for a composite numbered row, such as row 8 (1 8 28 56 70 56 28 8 1), 28 and 70 are not divisible by 8.
Fibonacci Sequence in the Triangle
By adding the numbers in the diagonals of the Pascal triangle the Fibonacci sequence can be obtained as seen in the figure given below.
There are various ways to show the Fibonacci numbers on the Pascal triangle. R. Knott was able to find the Fibonacci appearing as sums of “rows” in the Pascal triangle. He moved all the rows over by one place and here the sums of the columns would represent the Fibonacci numbers.
Pascal’s Triangle Properties
Each number is the sum of the two numbers above it.
The outside numbers are all 1.
The triangle is symmetric.
The first diagonal shows the counting numbers.
The sums of the rows give the powers of 2.
Each row gives the digits of the powers of 11.
Each entry is an appropriate “choose number.”
And those are the “binomial coefficients.”
The Fibonacci numbers are there along diagonals.
Here is an 18-lined version of pascal’s triangle;
Video Lesson
Pascals Triangle
Pascal’s Triangle Examples
Example 1:
Find the third element in the fourth row of Pascal’s triangle.
Solution:
To find: 3rd element in 4th row of Pascal’s triangle.
As we know that the nth row of Pascal’s triangle is given as nC0, nC1, nC2, nC3, and so on.
Thus, the formula for Pascal’s triangle is given by:
nCk = n-1Ck-1 + n-1Ck
Here, nCkrepresnts (k+1)th element in the nth row.
Now, to determine the 3rd element in the 4th row, we have to calculate 4C2.
Therefore, 4C2= 4-1C2-1 + 4-1C2
4C2= 3C1 + 3C2
4C2= 3 + 3 [Since 3C1= 3, 3C2= 3]
4C2= 6.
Therefore, the third element in the fourth row of Pascal’s triangle is 6.
Example 2:
Determine the coefficients of expansions of (x+y)2 using Pascal’s triangle.
Solution:
As we know that the coefficients of expansion of (x+y)2should be the elements in the second row of Pascal’s triangle.
Since the elements in the 2nd row of Pascal’s triangle are 1, 2, 1, the coefficients of the expansion of (x+y)2should be 1, 2, 1.
Practice Questions on Pascal’s Triangle
Solve the following problems based on Pascal’s triangle:
Determine the fifth element in the 6th row of Pascal’s triangle using Pascal’s formula.
What is the sum of elements in the 10th row of Pascal’s triangle?
Find the coefficients of the expansions of (a+b)5using Pascal’s triangle.
Frequently Asked Questions on Pascal’s Triangle
Q1
What is Pascal’s Triangle?
Pascal’s triangle is the triangular array of numbers that begins with 1 on the top and with 1’s running down the two sides of a triangle. Each new number lies between two numbers and below them, and its value is the sum of the two numbers above it.
Q2
What are the applications of Pascal’s Triangle?
Pascal’s triangle has various applications in Mathematics, such as in algebra, probability theory, combinatorics, statistics, and so on. Pascal’s triangle can be used to calculate the combinations.
Q3
What are the patterns found in Pascal’s Triangle?
The patterns found in Pascal’s triangle are:Triangular PatternOdd and even patternFibonacci patternSymmetrical pattern
Q4
What is the 5th row of Pascal’s triangle?
The fifth row of Pascal’s triangle is 1 5 10 10 5 1. The sum of the elements in the fifth row of Pascal’s triangle is 32, which can be verified using the formula, 2n. (i.e) 2n = 32.
Q5
Does Pascal’s triangle have a symmetric pattern?
Yes, Pascal’s triangle has a symmetric pattern. The numbers on the left side of the triangle have identical matching numbers on the right side. Hence, we can say that Pascal’s triangle is symmetrical.
Comments
Leave a Comment Cancel reply
Register with BYJU'S & Download Free PDFs
Register with BYJU'S & Watch Live Videos |
13001 | https://johnstonmd.wordpress.com/wp-content/uploads/2015/09/math133a-09-15-15.pdf | Math 133A, September 15: Applications Section 1: Inflow / Outflow Models One application of first-order linear differential equations differential equa-tions is modeling the amount (or concentration) of a substance in a well-stirred tank/vessel subject to constant in-flow and out-flow. Common simple applications are: • an industrial mixing tank with an entry pipe (pumping the chemical of interest in) and an exit pipe; • a lake with a inflow (say, a river) feeding a pollutant from upstream and an outflow (also, a river) flowing downstream; • a tub or sink with a steady inflow (say, a faucet) and a steady outflow (say, a drain).
Tank Inflow Outflow mixing In all cases, we are interested in the modeling the amount (or concentration) of substance x in the mixing vessel over time.
To model this problem, we first translate the description into mathemat-ical equations. The fundamental equation we will use to model the change of substance in the mixing vessel will be [rate of change] = [rate in] −[rate out].
That is to say, at each instance in time, we believe that the rate of change of the overall amount of the quantity of interest to equal the amount that is flowing in minus the amount that is flowing out.
1 To characterize the inflow rate, we need to know the overall flow rate in and the concentration of the quantity of interest within that in-flowing mixture. This could be the amount of pollutant in an inflowing stream, or the amount of chemical diluted in an inflowing pipe. The rate of substance flowing in is given by [rate in] = [volume rate in] × [concentration] since [volume rate in] × [concentration] = volume time × amount volume = amount time The outflow is slightly different. We will assume for simplicity that the vessel is well-mixed so that that the concentration of the quantity of interest is the same everywhere. It is reasonable to assume that, regardless of where the outflow is located and no matter how quickly it is flowing, we have [rate out] = [concentration]×[volume rate out] = [amount] [volume] ×[volume rate out] since [amount] [volume] × [volume rate out] = amount volume × volume time = amount time .
Letting A denote the amount of substance in the tank and V (t) denote the current volume, we have the combined model dA dt = [volume rate in] × [concentration in] − A V (t) × [volume rate out].
Note: The key difference between the inflow and outflow rates is that the amount and volume in the outflow rate depend upon the current amount and volume in the mixing vessel.
In the inflow tank, these quantities are either controlled (for mixing tanks) or known (for rivers and streams). Notice also that if the volume of the in-flow and the volume of the out-flow do not balance, the volume of the tank may be a dynamic function of time (imagine filling a bathtub, or emptying a mixing tank).
2 Example 1 Suppose that there is a factory built upstream of a lake with a volume of 0.5 km3. The factory introduces a new pollutant to a stream which pumps 1 km3 of water into the lake every year. Suppose that the net outflow from the lake is also 1 km3 per year and that the concentration of the pollutant in the inflow stream is 200 kg/km3.
(a) Set up an initial value problem for the amount of pollutant in the lake and solve it.
(b) Assuming there is initially no pollutant in the lake, how much pollutant is there are one month?
(c) What is the limiting pollutant level?
Solution: We need to set up the model in the form [rate of change]=[rate in]−[rate out]. If we let A denote the amount of the pollutant (in kg), we have [rate of change] = dA dt .
In order to determine the rate in, we notice that the amount (in kg) coming from the inflow can be given by [rate in] = [volume rate in] × [concentration in] = (1 km3/year)(200 kg/km3) = 200 kg/year.
The rate out is given by [rate out] = [volume rate out] × [concentration out] = (1 km3/year) A 0.5 kg/km3 = 2A kg/year.
We can see the units have worked as desired. We can drop them and just focus on the initial value problem dA dt = 200 −2A, A(0) = A0.
3 This is a first-order linear differential equation which in standard form is given by dA dt + 2A = 200.
We can see that we have p(x) = 2 and q(x) = 200.
The necessary integration factor is ρ(t) = e R 2 dt = e2t so that we have e2t dA dt + 2e2tA = 200e2t = ⇒ d dt e2tA = 200e2t = ⇒ e2tA = Z 200e−2t dt = 100e2t + C = ⇒ A(t) = 100 + Ce−2t.
In order to solve for C, we use A(0) = A0 to get A(0) = A0 = 100 + C = ⇒ C = A0 −100.
This gives the solution A(t) = 100 + (A0 −100)e−2t.
For this form, we can easily answer part (b). Given an initial pollutant level of zero (i.e. A0 = 0), we have A(t) = 100 −100e−2t.
After one month has passed, we have t = 1/12 so that the amount of pollutant is given by A(1/12) = 100 −100e−2(1/12) ≈15.3528 kg.
We can also easily determine the limiting pollutant level by evaluating lim t→∞A(t) = lim t→∞ 100 + (A0 −100)e−2t = 100.
4 In other words, no matter what the initial amount is in the lake, we will always converge toward 100 kg of pollutant distributed throughout the lake. This makes sense. The limiting level is going to be when the rate in and the rate out are balanced. That occurs for this model when 200 = 2A which implies A = 100.
Example 2 Consider a 50 gallon tank which is initial filled with 20 gallons of brine (salt/water mixture) with a concentration of 1/4 lbs/gallon of salt. Sup-pose that there is an inflow tube which infuses 3 gallons of brine into the tank per minute with a concentration of 1 lbs/gallon. Suppose that there is an outflow tube which flows at a rate of 2 gallons per minute.
(a) Set up and solve a differential equation for the amount of salt in the tank.
(b) How much salt is in the tank when the tank is full?
Solution: This is slightly different than the previous example because the volume of mixture in the tank changes because the inflow and out-flow volume rates are different. There is more mixture flowing into the tank than flowing out. Nevertheless, we can incorporate this into our model by noting that the volume of the tank at time t can be given by V (t) = 20 + (3 −2)t = 20 + t.
We can now complete the model as before. We have dA dt = (3)(1) −(2) A 20 + t = 3 − 2A 20 + t, A(0) = 20(1/4) = 5.
Again, this is a first-order linear differential equation. We can solve it by rewriting dA dt + 2 20 + t A = 3 and determining the integrating factor ρ(t) = e R 2/(20+t) dt = e2 ln(20+t) = (20 + t)2.
5 This gives (20 + t)2 dA dt + 2(20 + t)A = 3(20 + t)2 = ⇒ d dt (20 + t)2A = 3(20 + t)2 = ⇒ (20 + t)2A = (20 + t)3 + C = ⇒ A(t) = (20 + t) + C (20 + t)2 .
Using the initial condition A(0) = 5, we have A(0) = 5 = 20 + C 400 = ⇒ C = −6000 so that the particular solution is A(t) = (20 + t) − 6000 (20 + t)2 .
To answer the question of how much salt will be in the tank when the tank is full, we notice that the tank will be full when V (t) = 20+t = 50, which implies t = 30 (i.e. it will take thirty minutes). This gives A(30) = (20 + 30) − 6000 (20 + 30)2 = 50 −6000 2500 = 47.6.
It follows that there will be 47.6 lbs of salt in the tank when it is full.
Suggested Problems 1. Consider a mixing tank with a total volume of 20 gallons, initially filled with 10 gallons of pure water. Suppose there is an inflow pipe which pumps in a 0.5 lb/gallon brine (salt/water) mixture at a rate of 4 gallons per minute, and there is an outflow pipe which removes the mixture from the tank at a rate of 2 gallon per minute.
(a) Use the given information to derive a differential equation which 6 models the amount of salt in the tank.
(b) Find the general solution of the differential equation derived in part (a).
(c) How much salt is in the tank when it is full?
2. Consider a filled mixing tank with a volume of 20 gallons. Suppose there is an inflow pipe which pumps in a 0.5 lb/gallon brine (salt/water mixture) at a rate of 2 gallons per minute, and there is an outflow pipe which removes the mixture from the tank at a rate of 2 gallons per minute.
(a) Use the given information to derive a differential equation which models the amount of salt in the tank.
(b) Find the general solution of the differential equation derived in part (a).
(c) Suppose there is initially no salt in the tank. How much salt is in the tank after ten minutes?
3. Suppose a factory is built upstream of a lake with a volume of 0.5 km3.
The factory introduces a pollutant into the upstream water system.
Suppose the affected water system pumps 0.25 km3 of water into the lake each year and the downstream water system removes water from the lake at the same rate. Suppose the concentration of pollutant in the feeding water system is 40 kg/km3.
(a) Set up a first-order differential equation which models the amount of pollutant (in kg) in the lake.
(b) Suppose that there is initially no pollutant in the lake. How much pollutant is in the lake after (i) one month; (ii) seven months; (iii) five years? What is the limiting amount of pollutant in the lake?
(c) Suppose now that the inflow and outflow rates of the upstream and downstream water systems vary based on the seasons. Sup-pose this variance can be modeled by the form [volume rate in] = [volume rate out] = 0.25 + 0.25 cos(2πt). Derive the correspond-ing first-order differential equation which models the amount of pollutant (in kg) in Lake Mendota.
(d) Suppose that there is initially no pollutant in the lake. Under the assumptions of part (c), determine how much pollutant is in the lake after (i) one month; (ii) seven months; (iii) five years. What 7 is the limiting amount of pollutant in the lake? Is it the same as in part (b)? Does it converge to this value faster or slower than part (b)? Provide a brief explanation for the observed differences.
8 |
13002 | https://www.guideline-genius.com/article/Index?id=175 | Index
Login
Search
Login or signup
Home
Categories
Contact Us
About
Login/signup
Search
MENU
Home
Categories
Contact Us
About
Login/signup
Search
Recent
#### Pneumothorax NICE Guideline [NG39] Major trauma: assessment and initial management. British Thoracic Society (BTS) Guideline for Pleural Disease 2023
#### Warfarin NICE CKS Anticoagulation – oral: Scenario: Warfarin. Last revised: April 2024BNF Treatment Summaries: oral anticoagulantsBSH Guidelines Oral Anticoagulation with Warfarin - 4th Edition. Last revised: Oct 2016.
#### Infective Endocarditis (IE) NICE clinical guideline [CG64] Prophylaxis against infective endocarditis. Last updated: Jul 2016BNF treatment summary: cardiovascular system infections, antibacterial therapy
#### Familial Hypercholesterolemia (FH) NICE guideline [CG71] Familial hypercholesterolaemia: identification and management. Published: Aug 2008. Last updated: Oct 2019.
#### Lipid Lowering Therapy and Cardiovascular Risk Reduction NICE guideline [NG238] Cardiovascular disease: risk assessment and reduction, including lipid modification. Published: Dec 2023.This article does NOT include information regarding familial hypercholesteremia.
#### Stable Angina NICE Clinical guideline [CG95] Recent-onset chest pain of suspected cardiac origin: assessment and diagnosis. Last updated: Nov 2016NICE Clinical guideline [CG126] Stable angina: management. Last updated: Aug 2016
#### Abdominal Aortic Aneurysm (AAA) NICE guideline [NG156] Abdominal aortic aneurysm: diagnosis and management. Published Mar 2020. NICE CKS Abdominal aortic aneurysm screening. Last revised: May 2024.
#### Hypertension (Primary) NICE guideline [NG136] Hypertension in adults: diagnosis and management. Last updated: Nov 2023
#### Pericarditis (Acute) RCEM Learning: Acute PericarditisACC Guidelines 2025 Concise Clinical Guidance: An ACC Expert Consensus Statement on the Diagnosis and Management of Pericarditis: A Report of the American College of Cardiology Solution Set Oversight Committee
#### Head Injury NICE Guideline [NG232] Head injury: assessment and early management. Published: May 2023The author would like to clarify that only recommendations for 16 y/o are included. Recommendations regarding indications of CT cervical spine are not included.
#### Subarachnoid haemorrhage (SAH) NICE guideline [NG228] Subarachnoid haemorrhage caused by a ruptured aneurysm: diagnosis and management. Published Nov 2022NICE states that the guideline only covers the scope of aneurysmal SAH (i.e. SAH secondary to a ruptured aneurysm).
#### Acute Coronary Syndrome (ACS) NICE guideline [NG185] Acute coronary syndromes. Published: 18 November 2020
#### Aortic Dissection RCEM Learning: Aortic Dissection2014 ESC Guidelines on the diagnosis and treatment of aortic diseases: Document covering acute and chronic aortic diseases of the thoracic and abdominal aorta of the adult2022 ACC/AHA Guideline for the Diagnosis and Management of Aortic Disease: A Report of the American Heart Association/American College of Cardiology Joint Committee on Clinical Practice Guidelines
#### Asthma (Chronic) NICE guideline [NG245] Asthma: diagnosis, monitoring and chronic asthma management (BTS, NICE, SIGN). Published: 27 Nov 2024
#### Tachycardia (Peri-Arrest) Management Resuscitation Council UK Adult Advanced Life Support Guidelines. Published: May 2021.
#### Peripheral Arterial Disease (PAD) NICE clinical guideline [CG147] Peripheral arterial disease: diagnosis and management. Last updated: Dec 2020
#### Acute Limb Ischaemia RCEM Learning Acute limb Ischaemia ESVS 2020 Clinical Practice Guidelines on the Management of Acute Limb Ischaemia
#### Chronic Heart Failure NICE guidelines [NG106] Chronic heart failure: diagnosis and management. Last updated: Sep 2025.
#### Varicose Veins and Venous Ulcers NICE CKS Varicose Veins. Last revised: Sep 2024.NICE CKS Venous leg ulcer. Last revised: Oct 2024.
#### Bradycardia (Peri-Arrest) Resuscitation Council UK Adult Advanced Life Support Guidelines. Published: May 2021.
#### Cardiac Arrest and Advanced Life Support (ALS) Resuscitation Council UK Adult Advanced Life Support Guidelines (2021)
#### Red Eye Referral NICE CKS Red Eye. Last revised: May 2021
#### Angle Closure Glaucoma The Royal College of Ophthalmologists (RCO) Management of Angle Closure Glaucoma Guideline Jun 2022NICE CKS Glaucoma Last Revised Fed 2023
#### Vitamin D Deficiency NICE CKS Vitamin D deficiency in adults. Last revised Jan 2022NICE Public Health Guideline [PH56] Vitamin D: supplement use in specific population groups. Last updated Aug 2017
#### Bisphosphonates NICE CKS Osteoporosis - prevention of fragility fractures, Prescribing information, Bisphosphonates. Last revised Jan 2025.NICE BNF Treatment summary OsteoporosisNICE BNF Alendronic acid, Risedronate sodium, Zoledronic acid
#### Amiodarone NICE BNF Amiodarone
#### Chickenpox / Shingles in Pregnant Women RCOG Chickenpox in Pregnancy (Green-top Guideline No.13). Last reviewed 2024.
#### Anaphylaxis Resuscitation Council UK 2021 Emergency treatment of anaphylaxis: Guidelines for healthcare providers
#### Reduced Fetal Movement (RFM) RCOG Reduced Fetal Movements (Green-top Guideline No. 57) Feb 2011.
#### Ischaemic Stroke National Clinical Guideline for Stroke 2023 Edition.NICE Guideline [NG128] Stroke and transient ischaemic attack in over 16s: diagnosis and initial management. Last updated: Apr 2022
#### Perineal Tears RCOG Green-top Guideline No.29 The Management of Third- and Fourth-Degree Perineal Tears. Jun 2015.
#### Fetal Anomaly Screening Programme NHS England Screening NHS Fetal Anomaly Screening Programme (FASP)NICE guideline [NG201] Antenatal care. Aug 2021.
#### Lyme Disease NICE Guideline [NG95] Lyme disease. Last updated Oct 2018.
#### Impetigo NICE Guideline [NG153] Impetigo: antimicrobial prescribing. Published: Feb 2020.NICE CKS Impetigo. Last revised: Jul 2024.
#### Heavy Menstrual Bleeding (HMB) This article only covers HMB as a presentation; a separate article on fibroids is available to improve clarity. Note that NICE made a single guideline that covers HMB with no identified pathology, fibroid and adenomyosis.NICE Guideline [NG88] Heavy menstrual bleeding: assessment and management. Last updated: May 2021.NICE CKS Menorrhagia (heavy menstrual bleeding)
#### Dysmenorrhoea NICE CKS Dysmenorrhoea. Last Revised Oct 2023.
#### Preterm Labour and PPROM and PROM NICE Guideline [NG25] Preterm Labour and Birth. Last Updated: Jun 2022RCOG Green-top Guideline No. 73 Care of Women Presenting with Suspected Preterm Prelabour Rupture of Membranes from 24+0 Weeks of GestationNICE guideline [NG235] Intrapartum care 1.7 Prelabour rupture of membranes at term. Last updated: Jun 2025.
#### Acne Vulgaris NICE Clinical Guideline [NG198] Acne vulgaris: management. Last Updated: Dec 2023.NICE CKS Acne Vulgaris. Last Updated: Nov 2023.
#### Antidepressants MHRA Guidance Selective serotonin reuptake inhibitors (SSRIs) and serotonin and noradrenaline reuptake inhibitors (SNRIs): use and safety. Published Dec 2014.NICE BNF Drugs: Sertraline, ParoxetineNICE Clinical Guideline [CG192] Antenatal and postnatal mental health: clinical management and service guidance. Last updated: Fed 2020. NICE BNF Treatment summaries Depression. NICE CKS Depression - antenatal and postnatal. Last revised: Nov 2023.NICE CKS Depression. Last revised Feb 2025.NICE BNF Treatment summaries Antidepressant drugs.
#### Genital Herpes in Pregnancy Joint BASHH and RCOG National UK Guideline for the Management of Herpes Simplex Virus (HSV) in Pregnancy and the Neonate (2024 Update).NICE CKS Herpes Simplex - Genital. Last revised May 2024.
#### Genital Herpes BASHH Guidelines for the Management of Anogenital Herpes. Last Updated: Oct 2024.NICE CKS Herpes Simplex - Genital. Last Revised: May 2024.
#### Birth After Previous Caesarean Birth RCOG Green-top Guideline No. 45 Birth After Previous Caesarean Birth. Last reviewed Oct 2015.
#### Pelvic Inflammatory Disease (PID) NICE CKS Pelvic inflammatory disease. Last revised Jun 2024.BASHH PID 2019. Last updated Jan 2019.
#### Diabetes in Pregnancy NICE Guideline NG3 Diabetes in pregnancy: management from preconception to the postnatal period. Last updated: Dec 2020.This article covers BOTH gestational diabetes and pre-existing diabetes in pregnancy.
#### Breech Presentation NICE guideline [NG201] Antenatal care. Published: Aug 2021.RCOG Green-top Guideline No. 20a External Cephalic Version and Reducing the Incidence of Term Breech Presentation. Last reviewed Mar 2017.RCOG Green-top Guideline No. 20b Management of Breech Presentation. Last reviewed Mar 2017.
#### HIV in Pregnancy BHIVA guidelines on the management of HIV in pregnancy and the postpartum period 2025
#### Uterine Fibroids (Leiomyoma) NICE guideline NG88 Heavy menstrual bleeding: assessment and management. Last updated: May 2021NICE CKS Fibroids. Last revised Apr 2023.
#### Seborrhoeic Dermatitis NICE CKS Seborrhoeic Dermatitis. Last Revised Nov 2024.
#### Chickenpox NICE CKS Chickenpox. Last revised Nov 2023.
#### Suspected Head and Neck Cancers Pathways NICE guideline [NG12] Suspected cancer: recognition and referral. Last updated: Apr 2025.
#### Acute Otitis Media (AOM) NICE CKS Otitis media - acute. Last revised Aug 2024.NICE guideline [NG91] Otitis media (acute): antimicrobial prescribing. Last updated Mar 2022.
#### Low Back Pain (LBP) and Sciatica NICE guideline [NG59] Low back pain and sciatica in over 16s: assessment and management. Last updated: Dec 2022.
#### Shingles NICE CKS Shingles. Last revised Nov 2024.NICE CKS Post-herpetic neuralgia. Last revised: Aug 2024.
#### Lower Urinary Tract Symptoms (LUTS) in Men NICE Clinical guideline [CG97] Lower urinary tract symptoms in men: management. Last updated: Jun 2015.
#### Urinary Incontinence in Women NICE guideline [NG123] Urinary incontinence and pelvic organ prolapse in women: management. Last updated: Jun 2019.
#### Acute Rhinosinusitis NICE guideline [NG79] Sinusitis (acute): antimicrobial prescribing. Published: Oct 2017.TO BE REVIEWED!
#### Hormone Replacement Therapy (HRT) NICE CKS Menopause - Hormone replacement therapy (HRT). Last revised: Nov 2024.STATUS: TO BE UPDATED
#### Parvovirus B19 in Pregnancy NICE CKS Parvovirus B19 infection. Last revised Feb 2022.
#### Intrahepatic Cholestasis of Pregnancy (ICP) RCOG Intrahepatic cholestasis of pregnancy (Green-top Guideline No. 43). Published Aug 2022.
#### Pregnancy of Unknown Location NICE Guideline [NG126] Ectopic pregnancy and miscarriage: diagnosis and initial management. Last updated: Aug 2023.
#### Ectopic Pregnancy NICE Guideline [NG126] Ectopic pregnancy and miscarriage: diagnosis and initial management. Last updated: Aug 2023.
#### Human Immunodeficiency Virus (HIV) BASHH/BHIVA guidelines (see below for individual references).
#### Pelvic Inflammatory Disease (PID) NICE CKS Pelvic inflammatory disease. Last revised Jun 2024.BASHH PID 2019. Last updated Jan 2019.
#### Oral Herpes Simplex NICE CKS Herpes simplex - oral. Last revised: May 2024.
#### Bacterial Vaginosis (BV) BASHH Bacterial Vaginosis. Last updated: Dec 2012.NICE CKS Bacterial vaginosis. Last revised: Jul 2023.
#### Gonorrhoea NICE CKS Gonorrhoea. Last revised Mar 2024.BASHH Gonorrhoea 2025: Updated Guideline. Last updated Apr 2025.
#### Chlamydia NICE CKS Chlamydia - uncomplicated genital. Last revised Feb 2025.BASHH Chlamydia 2015. Last updated Sep 2018.UK Health Security Agency Guidance NCSP: programme overview. Last updated Jun 2021.
#### Syphilis BASHH Syphilis 2024: Updated Guideline. Last updated Sep 2024.NICE CKS Syphilis. Last updated Mar 2025.
#### Psoriasis NICE Clinical guideline [CG153] Psoriasis: assessment and management. Last updated Sep 2017.NICE CKS Psoriasis. Last revised Dec 2024.
#### Allergic Rhinitis NICE CKS Allergic rhinitis. Last revised Jan 2024.
#### Suspected Lung and Pleural Cancers NICE guideline [NG12] Suspected cancer: recognition and referral. Last updated May 2025.This article only covers the recognition and referral recommendations of the named cancers. It does NOT cover further information or recommendations with regards to the names cancers.
#### Cluster Headache NICE Clinical guideline [CG150] Headaches in over 12s: diagnosis and management. Last updated Dec 2021.NICE CKS Headache - cluster. Last revised Apr 2022.
#### Bell's palsy NICE CKS Bell's palsy. Last revised Feb 2023.
#### Transient Ischaemic Attack (TIA) National Clinical Guideline for Stroke 2023 Edition.NICE Guideline [NG128] Stroke and transient ischaemic attack in over 16s: diagnosis and initial management. Last updated: Apr 2022TO BE REVIEWED!
#### Epistaxis (Nosebleeds) NICE CKS Epistaxis. Last revised Oct 2024.
#### Motion Sickness BNF Treatment summaries Nausea and labyrinth disorders.
#### Post-Operative Nausea and Vomiting BNF Treatment summaries Nausea and labyrinth disorders.
#### Neuropathic pain NICE Clinical guideline [CG173] Neuropathic pain in adults: pharmacological management in non-specialist settings. Last updated Sep 2020.
#### Diabetic Retinopathy NICE guideline [NG242] Diabetic retinopathy: management and monitoring. Published Aug 2024.
#### Chronic Open Angle Glaucoma (COAG) NICE guideline [NG81] Glaucoma: diagnosis and management. Last updated Jan 2022.This article only covers open angle glaucoma and ocular hypertension. Angle closure glaucoma is covered in a separate article.
#### Thromboprophylaxis in Orthopaedic Surgery NICE guideline [NG 89] Venous thromboembolism in over 16s: reducing the risk of hospital-acquired deep vein thrombosis or pulmonary embolism. Last updated Aug 2019.
#### Osteoarthritis (OA) NICE guideline [NG226] Osteoarthritis in over 16s: diagnosis and management. Published Oct 2022.NICE CKS Osteoarthritis. Last revised Dec 2023.
#### Kidney Transplantation BTS/RA Living Donor Kidney Transplantation Guidelines Fourth Edition Mar 2018.RA Clinical Practice Guidelines Post-Operative Care in the Kidney Transplant Recipient. Last revised Feb 2017.
#### Contraception (Non-Emergency) FSRH UK Medical Eligibility Criteria For Contraceptive Use UKMEC 2016 (Amended Sep 2019)FSRH CEU Guidance: Recommended Actions after incorrect Use of Combined Hormonal Contraception (e.g. late or missed pills, ring and patch) (Mar 2020, amended Jul 2021)
#### Miscarriage (Spontaneous Abortion) NICE guideline [NG126] Ectopic pregnancy and miscarriage: diagnosis and initial management. Last updated Aug 2023.
#### Palliative Care Prescribing BNF Medicines guidance Prescribing in palliative care
#### Allergic Conjunctivitis NICE CKS Conjunctivitis - allergic. Last revised: May 2022.
#### Gout NICE guideline [NG219] Gout: diagnosis and management. Published: Jun 2022.
#### Prostatitis (Acute) NICE guideline [NG110] Prostatitis (acute): antimicrobial prescribing. Published: Oct 2018.
#### Neonatal Hypoglycaemia BNFC Treatment summaries Hypoglycaemia - Neonatal hypoglycaemia in term babies.BAPM Identification and Management of Neonatal Hypoglycaemia in the Full-Term Infant (Birth – 72 hours) A BAPM Framework for Practice. Jan 2024
#### Candidiasis (Vulvovaginal) NICE CKS Candida - female genital. Last revised: Oct 2023.BASHH Vulvovaginal Candidiasis 2019. Last updated: Jan 2021.
#### Emergency Contraception (EC) FSRH Clinical Guideline: Emergency Contraception (March 2017, amended July 2023)
#### Neonatal Group B Streptococcal Disease Prevention RCOG Green-top Guideline No. 36 Prevention of Early-onset Group B Streptococcal Disease. Last reviewed Sep 2017.
#### Nausea and Vomiting and Hyperemesis Gravidarum in Pregnancy RCOG Green-top Guideline No. 69 The Management of Nausea and Vomiting of Pregnancy and Hyperemesis Gravidarum. Last revised: Jun 2025.NICE CKS Nausea / vomiting in pregnancy. Last revised: Apr 2025.
#### Hypertension in Pregnancy NICE guideline [NG133] Hypertension in pregnancy: diagnosis and management. Last updated: Apr 2023NICE CKS Hypertension in pregnancy. Last revised: Jan 2025.
#### Induction of Labour NICE guideline [NG207] Inducing labour. Published: Nov 2021.
#### Intrapartum Care NICE guideline [NG235] Intrapartum Care. Last updated: Jun 2025.
#### Postpartum Haemorrhage (PPH) RCOG Green Guideline No. 52 Prevention and Management of Postpartum HaemorrhageNICE [NG235] Intrapartum care - Third Stage of Labour - Postpartum haemorrhage. Last updated: Jun 2025.
#### Placental Abruption BMJ Best Practice Clinical Guideline Placental abruption. Last reviewed: Jun 2025.
#### Folic Acid and Pregnancy NICE guideline [NG247] Maternal and child nutrition: nutrition and weight management in pregnancy, and nutrition in children up to 5 years. Published: Jan 2025.NICE CKS Pre-conception - advice and management: Scenarior: Pre-conception advice for all women. Last revised: Jun 2025.
#### Thyroid Disorders in Pregnancy RCOG Green-top Guideline No. 76 Management of Thyroid Disorders in Pregnancy. Last reviewed: Apr 2025.
#### Cervical Screening Public Health England Guidance NHS cervical screening (CSP) programme. Last updated: Jul 2025.NICE CKS Cervical screening. Last revised: Sep 2022.
#### Endometriosis NICE guideline [NG73]. Endometriosis: diagnosis and management. Last updated: Nov 2024.NICE CKS Endometriosis. Last revised: Nov 2024.
#### Menopause and Premature Ovarian Insufficiency NICE guideline [NG23] Menopause: identification and management. Last updated: Nov 2024.NICE CKS Menopause. Last revised: Apr 2025.
#### Premenstrual Syndrome (PMS) RCOG Green-top Guideline No. 48 Management of Premenstrual Syndrome. Last reviewed: May 2023.NICE CKS Premenstrual syndrome. Last revised: Sep 2024.
#### Pelvic Organ Prolapse (POP) NICE guideline [NG123] Urinary incontinence and pelvic organ prolapse in women: management. Last updated: Jun 2019.NICE guideline [NG210] Pelvic floor dysfunction: prevention and non-surgical management. Published: Dec 2021.
#### Otitis Media With Effusion (OME) / Glue Ear NICE CKS Otitis media with effusion. Last revised Nov 2023.
#### Acute Otitis Externa NICE CKS Otitis externa. Last revised: May 2024.NICE BNF Treatment summaries Ear - Otitis externa
#### Meniere's Disease NICE CKS Meniere's disease. Last revised: Mar 2023.
#### Vestibular Neuronitis NICE CKS Vestibular neuronitis. Last revised: Jan 2023.
#### Benign Paroxysmal Positional Vertigo (BPPV) NICE CKS Benign paroxysmal positional vertigo. Last revised: Apr 2022.
#### Hearing Loss in Adults NICE guideline [NG98] Hearing loss in adults: assessment and management. Last updated: Oct 2023.NICE CKS Hearing loss in adults. Last revised: May 2024.NICE Technology appraisal guidance [TA566] Cochlear implants for children and adults with severe to profound deafness. Published: Mar 2019.NICE Interventional procedures guidance [IPG108] Auditory brain stem implants. Published: Jan 2005.
#### Ear Wax (Cerumen) NICE guideline [NG98] Hearing loss in adults: assessment and management. Last updated: Oct 2023.NICE CKS Earwax. Last revised: Mar 2025.
#### Eczema (Atopic Dermatitis) NICE guideline [CG57] Atopic eczema in under 12s: diagnosis and management. Last updated: Jun 2023.NICE CKS Eczema - atopic. Last revised: Mar 2025.
#### Cellulitis and Erysipelas NICE guideline [NG141] Cellulitis and erysipelas: antimicrobial prescribing. Published: Sep 2019.NICE CKS Cellulitis - acute. Last revised: Nov 2024.
#### Contact Dermatitis NICE CKS Dermatitis - contact. Last revised: Jan 2024.
#### Fungal Skin Infection NICE CKS Fungal skin infection - scalp. Last revised: Feb 2025.NICE CKS Fungal skin infection - body and groin. Last revised: Jul 2023.NICE CKS Fungal skin infection - foot. Last revised: Jun 2023.
#### Fungal Nail Infection (Onychomycosis) NICE CKS Fungal nail infection. Last revised: Aug 2023.
#### Molluscum Contagiosum NICE CKS Molluscum contagiosum. Last revised: Mar 2022.BASHH Molluscum Contagiosum 2021. Last updated: Mar 2022.
#### Rosacea NICE CKS Rosacea. Last revised: Dec 2024.
#### Scabies NICE CKS Scabies. Last revised: May 2025.
#### Skin Cancer - Recognition and Referral NICE CKS Skin cancers - recognition and referral. Last revised: Apr 2025.NICE guideline [NG12] Suspected cancer: recognition and referral. Last updated: May 2025.
#### Cutaneous Squamous Cell Carcinoma (cSCC) British Association of Dermatologists guidelines for the management of people with cutaneous squamous cell carcinoma 2020
#### Melanoma NICE guideline [NG14] Melanoma: assessment and management. Last updated: Jul 2022.
#### Epilepsy NICE guideline [NG217] Epilepsies in children, young people and adults. Last updated: Jan 2025.
#### Childhood Onset Epilepsy NICE guideline [NG217] Epilepsies in children, young people and adults - 6. Treating childhood-onset epilepsies. Last updated: Jan 2025.
#### Status Epilepticus NICE guideline [NG217] Epilepsies in children, young people and adults 7. Treating status epilepticus, repeated or cluster seizures, and prolonged seizures. Last updated: Jan 2025.
#### Meningitis NICE guideline [NG240] Meningitis (bacterial) and meningococcal disease: recognition, diagnosis and management. Published: Mar 2024.NICE BNF Treatment summaries. Central nervous system infections, antibacterial therapyNICE BNF Treatment summaries. Antibacterials, use for prophylaxis Meningococcal disease: prevention of secondary cases.
#### Migraine NICE clinical guideline [CG150] Headaches in over 12s: diagnosis and management. Last updated: Jun 2025.NICE CKS Migraine. Last revised: Feb 2024.
#### Tension-Type Headache NICE clinical guideline [CG150] Headaches in over 12s: diagnosis and management. Last updated: Jun 2025.NICE CKS Headache - tension-type. Last revised: Jul 2022.
#### Trigeminal Neuralgia NICE CKS Trigeminal neuralgia. Last revised: Jan 2024.NICE Clinical guideline [CG173] Neuropathic pain in adults: pharmacological management in non-specialist settings. Last updated: Sep 2020.
#### Multiple Sclerosis (MS) NICE guideline [NG220] Multiple sclerosis in adults: management. Published: Jun 2022.NICE CKS Multiple sclerosis. Last revised: May 2024.ECTRIMS International Advisory Committee on Clinical Trials in MS. McDonald Diagnostic Criteria
#### Parkinson's Disease (PD) NICE guideline [NG71] Parkinson's disease in adults. Published: Jul 2017.
#### Age-Related Macular Degeneration (AMD) NICE guideline [NG82] Age-related macular degeneration. Published: Jan 2018.NICE CKS Macular degeneration - age-related. Last revised: Aug 2022.
#### Infective Conjunctivitis NICE CKS Conjunctivitis - infective. Last revised: Oct 2022.
#### Compartment Syndrome BOAST Diagnosis and Management of Compartment Syndrome of the Limbs. Last reviewed: Dec 2016.
#### Hip Fracture NICE Clinical guideline [CG124] Hip fracture: management. Last updated: Jan 2023.
#### Carpal Tunnel Syndrome (CTS) NICE CKS Carpal tunnel syndrome. Last revised: Aug 2022.
#### Osteomyelitis NICE BNF Treatment summaries Musculoskeletal system infections, antibacterial therapy
#### Septic Arthritis NICE BNF Treatment summaries Musculoskeletal system infections, antibacterial therapy
#### Osteoporosis NICE clinical guideline [CG146] Osteoporosis: assessing the risk of fragility fracture. Last updated: Feb 2017.NOGG Clinical guideline for the prevention and treatment of osteoporosis. Last updated: Dec 2024.
#### Rheumatoid Arthritis (RA) NICE guideline [NG100] Rheumatoid arthritis in adults: management. Last updated: Oct 2020.
#### Systemic Lupus Erythematosus (SLE) The British Society for Rheumatology guideline for the management of systemic lupus erythematosus in adults. Published: Oct 2017.
#### Antiphospholipid Syndrome (APS) BSH Guidelines on the investigation and management of antiphospholipid syndrome. Published: Jul 2024.
#### Giant Cell Arteritis (GCA) NICE CKS Giant cell arteritis. Last revised: Jul 2024.British Society for Rheumatology guideline on diagnosis and treatment of giant cell arteritis. Published: Jan 2020.
#### Polymyalgia Rheumatica (PMR) NICE CKS Polymyalgia rheumatica. Feb 2024.
#### Mental Health Act (MHA) 1983 GOV.UK Statutory guidance Code of practice: Mental Health Act 1983. Last updated: Oct 2017.
#### Psychosis and Schizophrenia NICE clinical guideline [CG178] Psychosis and schizophrenia in adults: prevention and management. Last updated: Mar 2014.NICE CKS Psychosis and schizophrenia. Last revised: May 2025.
#### Generalised Anxiety Disorder (GAD) NICE clinical guideline Generalised anxiety disorder and panic disorder in adults: management. Last updated: Jun 2020.NICE CKS Generalized anxiety disorder. Last revised: Apr 2025.
#### Obsessive-Compulsive Disorder (OCD) NICE CKS Obsessive-compulsive disorder. Last revised: Feb 2024.
#### Depression NICE guideline Depression in adults: treatment and management. Published: Jun 2022.NICE CKS Depression. Last revised: Apr 2025.
#### Bipolar Disorder NICE clinical guideline [CG185] Bipolar disorder: assessment and management. Last updated: Dec 2023.NICE CKS Bipolar disorder. Last revised: May 2025.
#### Eating Disorder NICE guideline [NG69] Eating disorders: recognition and treatment. Last updated: Dec 2020.
#### Delirium NICE Clinical guideline [CG103] Delirium: prevention, diagnosis and management in hospital and long-term care. Last updated: Jan 2023.
#### Post-Traumatic Stress Disorder (PTSD) NICE guideline [NG116] Post-traumatic stress disorder. Published: Dec 2018.
#### Dementia NICE guideline [NG97] Dementia: assessment, management and support for people living with dementia and their carers. Published: Jun 2018.
#### Atrial Fibrillation (AF) NICE guideline [NG196] Atrial fibrillation: diagnosis and management. Last updated: Jun 2021.
#### Asthma (Acute) Asthma Pathway (BTS, NICE, SIGN) [SIGN 244]
#### Bronchiectasis NICE guideline [NG117] Bronchiectasis (non-cystic fibrosis), acute exacerbation: antimicrobial prescribing. Published: Dec 2018.NICE CKS Bronchiectasis. Last revised: Mar 2025.BTS Guideline for Bronchiectasis in Adults. Published: Jan 2019.
#### Chronic Obstructive Pulmonary Disease (COPD) NICE guideline [NG115] Chronic obstructive pulmonary disease in over 16s: diagnosis and management. Last updated: Jul 2019.
#### Cystic Fibrosis (CF) NICE guideline [NG78] Cystic fibrosis: diagnosis and management. Published: Oct 2017.
#### Venous Thromboembolism (DVT and PE) NICE guideline [NG158] Venous thromboembolic diseases: diagnosis, management and thrombophilia testing. Last updated: Aug 2023.
#### Obstructive Sleep Apnoea (OSA) and Obesity Hypoventilation Syndrome NICE guideline [NG202] Obstructive sleep apnoea/hypopnoea syndrome and obesity hypoventilation syndrome in over 16s. Published: Aug 2021.
#### Oxygen Therapy and Respiratory Failure BTS Guideline for oxygen use in adults in healthcare and emergency settings. Published: May 2017.
#### Pleural Effusion BTS Guideline for Pleural Disease. Published: Jul 2023.
#### Pneumonia NICE guideline [NG250] Pneumonia: diagnosis and management. Published: Sep 2025.
#### Smoking Cessation NICE guideline [NG209] Tobacco: preventing uptake, promoting quitting and treating dependence. Last updated: Feb 2025.
#### Tuberculosis (TB) NICE guideline [NG33] Tuberculosis. Last updated: Feb 2024.
#### Alcohol Use Disorders NICE Clinical guideline [CG115] Alcohol-use disorders: diagnosis, assessment and management of harmful drinking (high-risk drinking) and alcohol dependence. Last updated: Oct 2014.NICE Clinical guideline [CG100] Alcohol-use disorders: diagnosis and management of physical complications. Last updated: Apr 2017.NICE CKS Alcohol - problem drinking. Last revised May 2025.
#### Wernicke's Encephalopathy and Korsakoff Syndrome NICE Clinical guideline [CG100] Alcohol-use disorders: diagnosis and management of physical complications. Last updated: Apr 2017.
#### Crohn's Disease NICE guideline [NG129] Crohn's disease: management. Published: May 2019.NICE Clinical guideline [CG118] Colorectal cancer prevention: colonoscopic surveillance in adults with ulcerative colitis, Crohn's disease or adenomas. Last updated: Sep 2022.NICE CKS Crohn's disease. Last revised: May 2024.
#### Ulcerative Colitis (UC) NICE guideline [NG130] Ulcerative colitis: management. Published: May 2019.NICE CKS Ulcerative colitis. Last revised: Mar 2024.
#### Clostridioides Difficile Infection NICE guideline [NG199] Clostridioides difficile infection: antimicrobial prescribing. Published: Jul 2021.
#### Dyspepsia NICE Clinical guideline [CG184] Gastro-oesophageal reflux disease and dyspepsia in adults: investigation and management. Last updated: Oct 2019.
#### Helicobacter Pylori Infection NICE Clinical guideline [CG184] Gastro-oesophageal reflux disease and dyspepsia in adults: investigation and management. Last updated: Oct 2019.BNF Treatment summaries. Helicobacter pylori infection.
#### Gastro-Oesophageal Reflux Disease (GORD) NICE CKS Dyspepsia - proven GORD. Last revised: Jul 2023.NICE Clinical guideline [CG184] Gastro-oesophageal reflux disease and dyspepsia in adults: investigation and management. Last updated: Oct 2019.
#### Upper Gastrointestinal Bleeding NICE Clinical guideline [CG141] Acute upper gastrointestinal bleeding in over 16s: management. Last updated: Aug 2016.NICE guideline [NG24] Blood transfusion. Published: Nov 2015.MHRA Direct-acting oral anticoagulants (DOACs): reminder of bleeding risk, including availability of reversal agents. Published: Jun 2020.BSG Best Practice Guidance: outpatient management of cirrhosis – part 2: decompensated cirrhosis.
#### Lower Gastrointestinal Bleeding Diagnosis and management of acute lower gastrointestinal bleeding: guidelines from the British Society of Gastroenterology. Published: Jan 2019
#### Paracetamol Overdose NICE BNF Treatment summaries. Poisoning, emergency treatment. Paracetamol poisoning.
#### Overdose and Poisoning Treatment NICE BNF Treatment summaries. Poisoning, emergency treatment.
#### Coeliac Disease NICE guideline [NG20] Coeliac disease: recognition, assessment and management. Published: Sep 2015.NICE CKS Coeliac disease. Last revised: Jun 2025.
#### Alcohol-Related Liver Disease NICE Clinical guideline [CG100] Alcohol-use disorders: diagnosis and management of physical complications. Last updated: Apr 2017.
#### Cirrhosis NICE guideline [NG50] Cirrhosis in over 16s: assessment and management. Last updated: Sep 2023.BSC Guidelines on the Management of Ascites in CirrhosisBSG Best Practice Guidance: outpatient management of cirrhosis - part 1: compensated cirrhosisBSG Best Practice Guidance: outpatient management of cirrhosis - part 2: decompensated cirrhosis
#### Non-Alcoholic Fatty Liver Disease (NAFLD) NICE guideline [NG49] Non-alcoholic fatty liver disease (NAFLD): assessment and management. Published: Jul 2016.
#### Anal Fissure NICE CKS Anal Fissure. Last revised: Apr 2021.NICE BNF Treatment summaries. Anal fissure.
#### Haemorrhoids NICE CKS Haemorrhoids. Last revised: Jul 2021.NICE BNF Treatment summaries. HaemorrhoidsEuropean Society of ColoProctology: guideline for haemorrhoidal disease. Published: Feb 2020.
#### Constipation (Adults) NICE CKS Constipation. Last revised: Jun 2025.
#### Irritable Bowel Syndrome (IBS) NICE Clinical guideline [CG61] Irritable bowel syndrome in adults: diagnosis and management. Last updated: Apr 2017.NICE CKS Irritable bowel syndrome. Last revised: Aug 2023.
#### Polycystic Ovary Syndrome (PCOS) NICE CKS Polycystic ovary syndrome. Last revised: Mar 2025.NICE Clinical guideline [CG156] Fertility problems: assessment and treatment. Last updated: Sep 2017.
#### Type 1 Diabetes Mellitus (T1DM) NICE guideline [NG17] Type 1 diabetes in adults: diagnosis and management. Last updated: Aug 2022.NICE CKS Diabetes - type 1. Last revised: Dec 2024.
#### Type 2 Diabetes (T2DM) NICE guideline [NG28] Type 2 diabetes in adults: management. Last updated: Jun 2022.NICE CKS Diabetes - type 2. Last revised: Jul 2025.
#### Diabetic Ketoacidosis (DKA) JBDS-IP The Management of Diabetic Ketoacidosis in Adults. Last revised: Mar 2023.
#### Hyperosmolar Hyperglycaemia State (HHS) JBDS 06 The management of the hyperosmolar hyperglycaemic state (HHS) in adults with diabetes
#### Adrenal Insufficiency (Hypoadrenalism) NICE guideline [NG243] Adrenal insufficiency: identification and management. Published: Aug 2024.
#### Cushing's Syndrome NICE CKS Cushing's syndrome. Last revised: Dec 2024.NICE BNF Treatment summaries Cushing's syndrome.
#### Overweight and Obesity NICE guideline [NG246] Overweight and Obesity Management. Published: Jan 2025.
#### Hypothyroidism NICE CKS Hypothyroidism. Last revised: Nov 2024.NICE guideline [NG145] Thyroid disease: assessment and management. Last updated: Oct 2023.
#### Hyperthyroidism and Thyrotoxicosis NICE CKS Hyperthyroidism. Last revised: Jan 2025.NICE guideline [NG145] Thyroid disease: assessment and management. Last updated: Oct 2023.
#### Thyroid Cancer NICE guideline [NG230] Thyroid cancer: assessment and management. Published: Dec 2022.NICE guideline [NG145] Thyroid disease: assessment and management. Last updated: Oct 2023.
#### Hyperparathyroidism NICE guideline [NG132] Hyperparathyroidism (primary): diagnosis, assessment and initial management. Published: May 2019.
#### Hyperkalaemia UK Kidney Association Clinical Practice Guidelines Treatment of Acute Hyperkalaemia in Adults. Published: Oct 2023.
#### Renal and Ureteric Stones NICE guideline [NG118] Renal and ureteric stones: assessment and management. Published: Jan 2019.
#### Acute Kidney Injury (AKI) NICE guideline [NG148] Acute kidney injury: prevention, detection and management. Last updated: Oct 2024.
#### Chronic Kidney Disease (CKD) NICE guideline [NG203] Chronic kidney disease: assessment and management. Last updated: Nov 2021.
#### Diabetic Foot Problems NICE guideline [NG19] Diabetic foot problems: prevention and management. Last updated: Oct 2019.
#### Urinary Tract Infection (UTI) in Adults NICE guideline [NG109] Urinary tract infection (lower): antimicrobial prescribing. Published: Oct 2018.NICE CKS Urinary tract infection (lower) - men. Last revised: Apr 2025.NICE CKS Urinary tract infection (lower) - women. Last revised: Feb 2025.
#### Paediatric IV Fluid Therapy NICE guideline [NG29] Intravenous fluid therapy in children and young people in hospital. Last updated: Jun 2020.
#### Neonatal Jaundice NICE Clinical guideline [CG98] Jaundice in newborn babies under 28 days. Last updated: Oct 2023.
#### Newborn Life Support (NLS) Resuscitation Council UK Newborn resuscitation and support of transition of infants at birth Guidelines
#### Bronchiolitis NICE guideline [NG9] Bronchiolitis in children: diagnosis and management. Last updated: Aug 2021.
#### Croup NICE CKS Croup. Last revised: May 2022.NICE BNF Treatment summaries. Croup
#### Gastro-Oesophageal Reflux Disease (GORD) in Children NICE guideline [NG1] Gastro-oesophageal reflux disease in children and young people: diagnosis and management. Last updated: Oct 2019.
#### Febrile Seizure NICE CKS Febrile seizure. Last revised: Jan 2024.
#### Fever in Under 5s NICE guideline [NG143] Fever in under 5s: assessment and initial management. Last updated: Nov 2021.
#### Kawasaki Disease (KD) NICE guideline [NG143] Fever in under 5s: assessment and initial management. Kawasaki Disease. Last updated: Nov 2021.Update on Diagnosis and Management of Kawasaki Disease: A Scientific Statement From the American Heart Association
#### Heart Valve Disease NICE guideline [NG208] Heart valve disease presenting in adults: investigation and management. Published: Nov 2021.
#### Idiopathic Pulmonary Fibrosis (IPF) NICE Clinical guideline [CG163] Idiopathic pulmonary fibrosis in adults: diagnosis and management. Last updated: May 2017.
#### Lung Cancer NICE guideline [NG122] Lung cancer: diagnosis and management. Last updated: Mar 2024.NHS England Standard protocol and quality assurance standards for the Lung Cancer Screening Programme. Last updated: Apr 2025.
#### Acute Bronchitis NICE CKS Chest infections - adult. Last revised: Jan 2025.
#### Hyposplenism Prevention and treatment of infection in patients with an absent or hypofunctional spleen: A British Society for Haematology guideline. Published: Apr 2024.
#### Idiopathic Nephrotic Syndrome in Children NHS Scotland GGC Clinical guidelines 201 Idiopathic Nephrotic Syndrome Management in Children. Last revised: Sep 2022.
#### Bladder Cancer NICE guideline [NG2] Bladder cancer: diagnosis and management. Published: Feb 2015.NICE guideline [NG12] Suspected cancer: recognition and referral 1.6 Urological cancers. Last updated: May 2025.
#### Hypoglycaemia NICE BNF Treatment summaries. Hypoglycaemia.JBDS 01 The Hospital Management of Hypoglycaemia in Adults with Diabetes Mellitus. Last revised: Jan 2023.
#### Acid-Base Abnormalities
#### Acute Heart Failure NICE guidelines [CG187] Acute heart failure: diagnosis and management. Last updated: Nov 2021.
#### Diabetes Medications
#### Hyperprolactinaemia
#### Acromegaly
#### Hypoparathyroidism
#### Osteomalacia and Rickets
Home
Category
Gastrointestinal including Liver
Alcohol Use Disorders
Alcohol Use Disorders
NICE Clinical guideline [CG115] Alcohol-use disorders: diagnosis, assessment and management of harmful drinking (high-risk drinking) and alcohol dependence. Last updated: Oct 2014. NICE Clinical guideline [CG100] Alcohol-use disorders: diagnosis and management of physical complications. Last updated: Apr 2017. NICE CKS Alcohol - problem drinking. Last revised May 2025.
Acute Alcohol Intoxication Guidelines
Management
Management issupportive:
Maintaina clear airway
Measures to reduce risk of aspiration
Measure blood glucoseand treat if indicated
Alcohol Misuse Guidelines
Identification and Assessment
NICE recommends the following assessment tools:
| Assessment tool | Purpose |
--- |
| AUDIT | Identification of alcohol misuse and routine outcome measure 0-7: low risk 8-15: increasing risk 16-19: higher risk ≥20: possible dependence AUDIT PC and AUDIT-C questionnaires are shorter and can be used in primary care or where time is limited |
| SADQ / LDQ | Assessseverity of alcohol dependence |
Alcohol Cessation / Reduction
Offermotivational interventionat initial assessment
Consider residential rehabilitation (maximum 3 months) in those who are homeless
For harmful drinkers A pattern of alcohol consumption that is causing mental or physical damage Drinking ≥35 units a week (women) or ≥50 units a week (men) with mild alcohol dependence
1st line:psychological intervention (e.g.cognitive behavioural therapies, behavioural therapies or social network and environment-based therapies)
2nd line: psychological intervention+ drug (acamprosate / naltrexone)
NICE noted that evidence for acamprosate is less robust than naltrexone
Nalmefene is an option if there is no need for immediate detoxification + alcohol consumption >60g per day (men)>40g per day (women)+ without withdrawal symptoms
Alcohol Withdrawal Guidelines
Assisted Alcohol Withdrawal (No Active Withdrawal)
Indications and Choice of Setting
Medically assisted withdrawalis indicated in the appropriate setting if the following are met, otherwise patient could be managed with active observation without medication.
>15 units per day and/or AUDIT ≥20 (possible dependence):
Community-based assisted withdrawal or
Specialist alcohol service (if there are safety concerns)
Inpatient or residential assisted withdrawalis indicated if any of the following:
>30 units per day
≥30 on SADQ
History of epilepsy / experience of withdrawal-related seizures or delirium tremens during previous assisted withdrawal programmes
Need concurrent withdrawal from alcohol and benzodiazepines
Regularly drink between 15-30 units per day and significant psychiatric or physical comorbidities (e.g.chronic severe depression, psychosis, malnutrition, congestive cardiac failure, unstable angina, chronic liver disease) or significant learning disability or cognitive impairment
Choice of Regimen
Choice of regimen:
Community settings:fixed dose regimen Dose reduced to 0 over 7-10 days according to a standard protocol. That is NOT determined by the level of alcohol withdrawal
Inpatient / residential settings: fixed dose regimen Dose reduced to 0 over 7-10 days according to a standard protocol. That is NOT determined by the level of alcohol withdrawal orsymptom-triggered regimen Involves only giving the medication when the patient demonstrate features of alcohol withdrawal Rather than on a fixed schedule Lasts 2-3 weeks longer than fixed dose regimen
Choice of Drug
Benzodiazepineis the class of choice for medically assisted withdrawal:
1st line:chloridazepoxide /diazepam
If patient has liver impairment: lorazepampreferred
Family member or carer should preferably oversee the administration of medication
Monitor the patient every other day during assisted withdrawal
Use fixed-dose regimen or symptom-triggered medication regimens for inpatient or residential settings
Last 2-3 weeks or longer
After Successful Withdrawal
1st line: psychological intervention+ acamprosate / naltrexone
2nd line:psychological intervention + disulfiram
Management of Active (Symptomatic) Withdrawal
CIWA-Aris used to assess the severity of alcohol withdrawal and to decide where medication is needed or not:
CIWA-Ar ≥10is a threshold to consider medications to treat withdrawal symptoms
0-9: mild withdrawal symptoms
10-15: moderate withdrawal symptoms
15: severe withdrawal symptoms
Use a symptom-triggered regimen Involves only giving the medication when the patient demonstrate features of alcohol withdrawal Rather than on a fixed schedule Lasts 2-3 weeks longer than fixed dose regimen in hospital or where 24-hour assessment and monitoring are available
1st line: benzodiazepine(chloridazepoxide / diazepam, use lorazepam in liver impairment)or carbamazepine
2nd line: clomethiazole
Disclaimer: NICE does not explicitly recommend a CIWA-Ar score cutoff to guide when medically assisted withdrawal is indicated., NICE recommends usingclinical judgmentcombined withCIWA-Ar.
However, many clinical protocols and other guidelines use approximately CIWA-Ar score of 10 as a cut-off to consider medically assisted withdrawal to prevent progression.
Delirium Tremens
1st line:oral lorazepam
2nd line: parenteral lorazepam or haloperidol
Alcohol Withdrawal Seizures
1st line: IVlorazepam
Do not offer phenytoin to treat alcohol withdrawal seizures
References
Original Guideline
NICE guideline Alcohol-use disorders: diagnosis, assessment and management of harmful drinking (high-risk drinking) and alcohol dependence
NICE guideline Alcohol-use disorders: diagnosis and management of physical complications
NICE CKS Alcohol - problem drinking
Background Information
Definitions and Classifications
hover here (click if on phone/tablet)You’ve hovered here! This shows helpful medical details, definitions, and tips without leaving the page.
click here You clicked here, explore important medical info and insights.
Guidelines
Red Flags
Investigation and Diagnosis
References
Original Guideline
Author: Adams Lau
Reviewer:
Last edited: 03/08/25
UK medical guidelines made easy. From guidelines to genius in minutes!
Quick Links
Home
Categories
Subscribe
Contact
Address: 128 City Road, London, United Kingdom, EC1V 2NX
Email: guidelinegenius@gmail.com
Social Media
© 2025 GUIDELINE GENIUS LTD
Terms & Conditions and Medical Disclaimer
By accessing or using this website, you agree to comply with and be bound by these terms and conditions. If you do not agree with any part of these terms and conditions, you must not use this website.
Use of the website: This website is for informational and educational purposes only and is intended for medical professionals. You agree to use the website in a lawful manner and to refrain from activities that may disrupt its functionality or compromise its security.
Intellectual property: All content on this website, including but not limited to text, graphics, logos, images, and code, is the property of Guideline Genius LTD unless otherwise stated. Unauthorised reproduction, distribution, or use of the website's content is strictly prohibited without prior written consent.
Subscriptions and payments: Certain sections of the website may require a subscription for access. By subscribing, you agree to pay any fees specified during the subscription process. All fees are non-refundable unless otherwise specified.
Accuracy and updates: While we strive to ensure the accuracy of the information on this website, we do not guarantee that the content is error-free or up-to-date. Guideline Genius LTD reserves the right to modify or update the content of this website at any time without prior notice.
Third-party links: This website may contain links to third-party websites for your convenience. Guideline Genius LTD is not responsible for the content, accuracy, or availability of these external sites.
Disclaimer of liability: Guideline Genius LTD disclaims any liability for damages resulting from the use of or reliance on this website or its content. Your use of the website is at your own risk, and we shall not be held liable for any direct, indirect, or consequential damages.
Termination: We reserve the right to terminate your access to the website at our discretion, without notice, for any breach of these terms.
Governing law: These terms and conditions shall be governed by and construed in accordance with the laws of England and Wales.
Medical disclaimer
No advocacy or endorsement: Guideline Genius LTD does not advocate or endorse the use of any specific drug, therapy, medical intervention, or any other content mentioned on this website, including but not limited to diagnostic tools, procedures, or treatment recommendations. All content is for informational and educational purposes only and does not constitute medical advice or diagnosis.
Professional responsibility: As a medical professional, you retain full responsibility for the care and treatment of your patients. You should use your own clinical judgment, expertise, and knowledge of local regulations when interpreting and applying the information provided on this website.
Scope of information: This content does not aim to cover all possible diagnostic methods, treatments, follow-up protocols, drugs, contraindications, or side effects. Medical standards and practices evolve as new data become available, and we strongly recommend consulting multiple reputable sources to validate the information provided.
Verification of information: You are advised to independently verify all diagnoses, treatments, and follow-up plans to ensure they are appropriate for your patient and compliant with the regulations in your region. When dealing with prescription medication, always consult the product information sheet and the British National Formulary to verify conditions of use, dosage schedules, indications, forms of administration, adverse effects, contraindications, cautions, and any circumstances under which the medication should be avoided.
Accuracy and updates: Information on this website is provided on an "as is" basis without representations, warranties, or conditions regarding its accuracy, completeness, or timeliness. While we strive to provide up-to-date content, Guideline Genius LTD and its contributors do not guarantee the information will always reflect the latest medical standards or practices.
Liability disclaimer: To the fullest extent permitted by law, Guideline Genius LTD disclaims all liability for any loss, injury, or damage arising from the use of this website's content. All implied warranties and conditions, including those of satisfactory quality, fitness for a particular purpose, and use of reasonable care and skill, are excluded.
Local guidelines: Where regional or local guidelines are applicable, users should refer to those resources for contextually relevant information. Guideline Genius LTD disclaims all liability for reliance on content not specifically aligned with your local medical guidelines or practices.
Emergency situations: In case of a medical emergency, contact your local emergency services or healthcare provider immediately.
[x] I have read and agree to all the terms & conditions and medical disclaimer.
I Agree
Renew Subscription |
13003 | https://www.teacherspayteachers.com/Product/FREEBIE-7RPA2A-Identify-Proportional-Relationships-Classwork-or-Homework-8551921 | FREEBIE 7.RP.A.2.A Identify Proportional Relationships Classwork or Homework
Log InSign Up
Cart is empty
Total:
$0.00
View Wish ListView Cart
Grade
Elementary
Preschool
Kindergarten
1st grade
2nd grade
3rd grade
4th grade
5th grade
Middle school
6th grade
7th grade
8th grade
High school
9th grade
10th grade
11th grade
12th grade
Adult education
Resource type
Student practice
Independent work packet
Worksheets
Assessment
Graphic organizers
Task cards
Flash cards
Teacher tools
Classroom management
Teacher manuals
Outlines
Rubrics
Syllabi
Unit plans
Lessons
Activities
Games
Centers
Projects
Laboratory
Songs
Clip art
Classroom decor
Bulletin board ideas
Posters
Word walls
Printables
Seasonal
Holiday
Black History Month
Christmas-Chanukah-Kwanzaa
Earth Day
Easter
Halloween
Hispanic Heritage Month
Martin Luther King Day
Presidents' Day
St. Patrick's Day
Thanksgiving
New Year
Valentine's Day
Women's History Month
Seasonal
Autumn
Winter
Spring
Summer
Back to school
End of year
ELA
ELA by grade
PreK ELA
Kindergarten ELA
1st grade ELA
2nd grade ELA
3rd grade ELA
4th grade ELA
5th grade ELA
6th grade ELA
7th grade ELA
8th grade ELA
High school ELA
Elementary ELA
Reading
Writing
Phonics
Vocabulary
Grammar
Spelling
Poetry
ELA test prep
Middle school ELA
Literature
Informational text
Writing
Creative writing
Writing-essays
ELA test prep
High school ELA
Literature
Informational text
Writing
Creative writing
Writing-essays
ELA test prep
Math
Math by grade
PreK math
Kindergarten math
1st grade math
2nd grade math
3rd grade math
4th grade math
5th grade math
6th grade math
7th grade math
8th grade math
High school math
Elementary math
Basic operations
Numbers
Geometry
Measurement
Mental math
Place value
Arithmetic
Fractions
Decimals
Math test prep
Middle school math
Algebra
Basic operations
Decimals
Fractions
Geometry
Math test prep
High school math
Algebra
Algebra 2
Geometry
Math test prep
Statistics
Precalculus
Calculus
Science
Science by grade
PreK science
Kindergarten science
1st grade science
2nd grade science
3rd grade science
4th grade science
5th grade science
6th grade science
7th grade science
8th grade science
High school science
By topic
Astronomy
Biology
Chemistry
Earth sciences
Physics
Physical science
Social studies
Social studies by grade
PreK social studies
Kindergarten social studies
1st grade social studies
2nd grade social studies
3rd grade social studies
4th grade social studies
5th grade social studies
6th grade social studies
7th grade social studies
8th grade social studies
High school social studies
Social studies by topic
Ancient history
Economics
European history
Government
Geography
Native Americans
Middle ages
Psychology
U.S. History
World history
Languages
Languages
American sign language
Arabic
Chinese
French
German
Italian
Japanese
Latin
Portuguese
Spanish
Arts
Arts
Art history
Graphic arts
Visual arts
Other (arts)
Performing arts
Dance
Drama
Instrumental music
Music
Music composition
Vocal music
Special education
Speech therapy
Social emotional
Social emotional
Character education
Classroom community
School counseling
School psychology
Social emotional learning
Specialty
Specialty
Career and technical education
Child care
Coaching
Cooking
Health
Life skills
Occupational therapy
Physical education
Physical therapy
Professional development
Service learning
Vocational education
Other (specialty)
FREEBIE 7.RP.A.2.A Identify Proportional Relationships Classwork or Homework
Rated 4.29 out of 5, based on 7 reviews
4.3(7 ratings)
FREE
Log in to Download
Wish List
DescriptionReviews 7Q&AStandards 9 More from Catherine's Class
Thumbnail 1 Thumbnail 2 Thumbnail 3 Thumbnail 4
Share
Description
2 double-sided worksheets covering 7.RP.A.2.A of the CCSS grade 7 ratios and proportional relationships domain including: determining whether two quantities are in a proportional relationship and looking for equivalent ratios in tables, all working with real-world problems.An answer key is included for each worksheet.
The worksheets are intended to be assigned as homework or as classwork for independent practice or group work. They are also great resources to use for sub plans. The worksheets are meant to be given after a skill has been introduced in class.
If you enjoy this freebie, please follow my store for more freebies, resources, and updates.
You can check out the complete set of 5 double-sided worksheets for this standard by clicking here.
I have created worksheet sets for each of the grade 7 standards, which you can find here!
Other Resources You Might Like:
7th Grade Math Statistics Classwork or Homework BUNDLE
7th Grade Math Probability Classwork or Homework BUNDLE
7th Grade Math Ratios & Proportional Relationships Worksheet BUNDLE
7th Grade Math The Number System Worksheet BUNDLE
7th Grade Math Expressions & Equations Worksheet BUNDLE
7th Grade Math Review Worksheets BUNDLE
FREEBIE! 7.G.A.2 & 7.G.A.3 Classwork or Homework
Ideas for using this resource:
Assign one worksheet a day or a packet of worksheets per week for homework.
Assign worksheets soon after teaching this standard, or give them to students later to reinforce previously-taught skills.
Assign these as independent practice or group work in class.
Give these as classwork or homework to review for quizzes, unit tests, or standardized assessments.
Give homework or classwork packets containing just one standard, or mix-and-match topics in one packet! (Click here to find worksheet sets focused on each of the 7th grade standards.)
Use these worksheets for your sub plans! Just make copies of worksheets covering skills you’ve already taught students and leave the answer keys for the sub. :)
Use as a tool for intervention.
Give these worksheets to students for summer homework after completing grade 7.
The student worksheets do not have “grade 7” written on them; instead, the standard (7.RP.A.2.A) appears on the bottom left corner of each worksheet. This gives you the flexibility to use these worksheets with any grade level.
© Catherine Howland 2022
Report this resource to TPT
Reported resources will be reviewed by our team. Report this resource to let us know if this resource violates TPT's content guidelines.
FREEBIE 7.RP.A.2.A Identify Proportional Relationships Classwork or Homework
Rated 4.29 out of 5, based on 7 reviews
4.3(7 ratings)
Catherine's Class
Follow
72 Followers
FREE
Log in to Download
Wish List
Specs
What's Included
Grade
6 th - 8 th
Subject
Math, Math Test Prep, Other (Math)
Standards
CCSS 7.RP.A.2
CCSS 7.RP.A.2a
CCSS MP1
show more
Tags
Homeschool, Homework, Independent Work Packet, Worksheets
Reviews
4.3
Rated 4.29 out of 5, based on 7 reviews
7
ratings
This product (7)
All products (22)
All verified TPT purchases
All ratings
5 stars
4 stars
3 stars
All grades
7th grade
8th grade
Population
Learning difficulties
Sort by: Most recent
Most relevant
Most recent
Highest rating
Lowest rating
Identify Proportional Relationships
Rated 5 out of 5
August 26, 2025
Wonderful Resource! I was able to scaffold to what my students needed
Adriana B.
25 reviews
Grades taught:7th
Response from Catherine's Class(TPT Seller)
Aug 26, 2025
Wonderful! I'm glad that you were able to scaffold it for your students. Thanks for the review!
Rated 5 out of 5
January 10, 2025
I was very pleased how the resource went over with the students.
Kimberly G.
8 reviews
Grades taught:7th
Response from Catherine's Class(TPT Seller)
Aug 26, 2025
I'm so glad that this worked for your students!
Rated 4 out of 5
November 3, 2024
My students loved using this resource. They were engaged in the resource and had no trouble getting started. It also didn't take much time to prepare this resource. I did modify it a little for my emerging multilingual students. I highly recommend using the resource as either a practice or review assignment.
SALOME W.
308 reviews
Grades taught:8th
Student populations:Learning difficulties
Response from Catherine's Class(TPT Seller)
Nov 3, 2024
I'm so glad to hear that your students loved this resource! Thank you for the review.
Rated 5 out of 5
September 5, 2024
This resource is absolutely perfect for the standard. My students really love this resource. It was very useful for breaking down the concept. They were engaged and had no trouble getting started.
Candace S.
413 reviews
Grades taught:7th
Response from Catherine's Class(TPT Seller)
Sep 6, 2024
Thank you so much! I'm thrilled to hear that it worked out well for your students. I hope you all have a great year!
Rated 3 out of 5
April 2, 2024
Was a good review for students to do with the sub. We did this lesson a while ago. Good review.
KRYSTAL W.
43 reviews
Response from Catherine's Class(TPT Seller)
Apr 17, 2024
I'm glad that this was a good review for your students!
Rated 5 out of 5
September 10, 2023
I utilized this as a homework assignment for my students, and we went over in class the next day. Students did well and liked the simplicity of the worksheet.
Jeffrey C.
1 review
Grades taught:7th
Response from Catherine's Class(TPT Seller)
Sep 10, 2023
I'm so glad to hear that the assignment went well for your students. Thank you for your review!
Rated 3 out of 5
February 14, 2023
I had to modify due to students abilities but the premis was fantastic!
Math to the Seventh Power
(TPT Seller)
319 reviews
Response from Catherine's Class(TPT Seller)
Feb 14, 2023
Thanks so much for your feedback! I'm glad you were able to modify it for your students. :)
Questions & Answers
Please log into post a question.
Be the first to ask Catherine's Class a question about this product.
Standards
Log in
to see state-specific standards (only available in the US).
CCSS 7.RP.A.2
Recognize and represent proportional relationships between quantities.
CCSS 7.RP.A.2a
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
CCSS MP1
Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
Show more
Meet the Teacher-Author
### Catherine's Class
Follow
With a decade of experience teaching middle school math in both urban and rural public schools, I am passionate about creating standards-aligned resources. I've developed over 165 double-sided worksheets aimed at helping 7th grade students build their math skills and confidence.
Illinois, United States
4.68
Store rating after 22 reviews
72
Followers
You may also like
previous
7.SP.A.1 & 7.SP.A.2 Representative Samples Classwork or Homework
$3.00 Original Price $3.00
7.NS.A.1.D Properties of Addition & Subtraction Classwork or Homework
$3.00 Original Price $3.00 Rated 5 out of 5, based on 1 reviews 5.0 (1)
7.RP.A.2.B Constant of Proportionality Classwork or Homework
$3.00 Original Price $3.00 Rated 5 out of 5, based on 1 reviews 5.0 (1)
7.RP.A.2.D Graphs of Proportional Relationships Classwork or Homework
$3.00 Original Price $3.00
next
0
1
More from this Teacher-Author
BUNDLESRatios & Proportional RelationshipsThe Number SystemExpressions & Equations
TPT is the largest marketplace for PreK-12 resources, powered by a community of educators.
Facebook
Instagram
Pinterest
Twitter
About
Who we are
We're hiring
Press
Blog
Gift Cards
Support
Help & FAQ
Security
Privacy policy
Student privacy
Terms of service
Tell us what you think
Updates
Get our weekly newsletter with free resources, updates, and special offers.
Get newsletter
IXL family of brands
IXL Comprehensive K-12 personalized learning
Rosetta Stone Immersive learning for 25 languages
Wyzant Trusted tutors for 300 subjects
Education.com 35,000 worksheets, games, and lesson plans
Vocabulary.com Adaptive learning for English vocabulary
Emmersion Fast and accurate language certification
Thesaurus.com Essential reference for synonyms and antonyms
Dictionary.com Comprehensive resource for word definitions and usage
SpanishDictionary.com Spanish-English dictionary, translator, and learning
FrenchDictionary.com French-English dictionary, translator, and learning
Ingles.com Diccionario inglés-español, traductor y sitio de aprendizaje
ABCya Fun educational games for kids
© 2025 by IXL Learning
|Protected by reCAPTCHAPrivacy•Terms |
13004 | https://erikdemaine.org/foldcut/ | Erik Demaine's Folding and Unfolding:
The Fold-and-Cut Problem
Problem
What shapes can result from the following fold-and-cut process?
Take a piece of paper.
Fold it flat.
Make one complete straight cut.
Unfold the pieces.
Are all shapes possible? Refering back to the original sheet of
paper, what patterns of cuts can be achieved by this process?
The answer turns out to be any pattern of straight cuts.
Try it out yourself by printing one of our several examples.
History
Wakoku Chiyekurabe by Kan Chu Sen, 1721.
The first published reference to folding and cutting of which we are aware is a
Japanese book, Wakoku Chiyekurabe
(Mathematical Contests), by Kan Chu Sen, published in 1721. This book contains
a variety of problems for testing mathematical intelligence. One of the
problems asks to fold a rectangular piece of paper flat and make one complete
straight cut, so as to make a typical Japanese crest called sangaibisi, which
translates to “three folded rhombics”. The author gives a solution
consisting of a sequence of simple folds, each of which folds along a line.
Scanned images of the relevant pages in the book
are available.
Another early reference to folding and cutting is a July 1873 article
“National Standards and Emblems” in Harper's New Monthly Magazine,
volume 47, number 278. This article tells the story about Betsy Ross and her
relation to the American flag. It claims that in 1777, George Washington and a
committee of the Congress showed Betsy Ross plans for a flag with thirteen
six-pointed stars, and asked whether she could make such a flag. She said that
she would be willing to try, but suggested that the stars should have five
points. To support her argument, she showed how easily such a star could be
made, by folding a sheet of paper and making one cut with scissors. The
committee decided to accept her changes, and George Washington made a new
drawing, which Betsy Ross followed to make the first American flag.
One method for
folding and cutting a five-pointed star is described on ushistory.org's
Betsy Ross page.
The truth of this story is unclear,
but it still makes for an early reference.
Fold and cut was the basis for several early magic tricks. The earliest such
reference we know is 1833, in a book called “The Girl's Own Book”
by Lydia Marie Child. See
The Public Paperfolding History Project
for images from this and many more references.
In particular,
before Houdini was a
famous escape artist, he was a general magician.
His 1922 book Paper Magic
(E. P. Dutton & Company, pages 176–177)
describes a method for making a five-pointed star.
According to David Lister,
this book appears to have been ghostwritten by another magician, Walter Gibson,
but Houdini still may have performed the trick.
Another magician, Gerald Loe, studied the fold-and-cut idea in some
detail. His 1955 book Paper Capers (Magic, Inc., Chicago)
describes how to cut out arrangements of various geometric objects,
such as a circular chain of stars.
Martin Gardner wrote about the fold-and-cut problem in his famous series in
Scientific American (“Paper cutting”, chapter 5 of New
Mathematical Diversions (Revised Edition), Mathematical Association of
America, Washington, D.C., 1995.). Gardner was particularly impressed with
Loe's ability to cut out any desired letter of the alphabet. He was also the
first to state cutting out complex polygons as an open problem. What are the
limits of this fold-and-cut process? What polygonal shapes can be cut out?
Theorem
The theorem is that every pattern (plane graph) of straight-line cuts
can be made by folding and one complete straight cut. Thus it is possible
to make single polygons (possibly nonconvex), multiple disjoint polygons,
nested polygons, adjoining polygons, and even floating line segments and
points.
There are two methods for solving this problem, by different sets of people.
I call them the straight-skeleton and
disk-packing methods.
You can read descriptions of them below, or watch the
free
online video lecture describing them:
6.849 Lecture 7
Straight-Skeleton Method
Analyzing the straight-skeleton method, from
the book.
Martin Demaine, Anna Lubiw, and I solved the fold-and-cut problem using the
“straight skeleton”. This structure is defined (roughly) as follows. For
each face of the desired cut pattern (region between cuts), shrink the face so
that the edges stay parallel and move a constant speed in a perpendicular
direction. Stop whenever the boundary becomes nonsimple (intersects itself),
and continue shrinking each piece. The straight skeleton is the
trajectories of the vertices of the desired cut pattern during this shrinking
process. Not described here are the details for vertices of degree zero and
one, which require special care.
For more about the straight skeleton, see
Jeff Erickson's
straight-skeleton page
or
Aichholzer et al.'s
original paper about the straight skeleton.
The straight skeleton consists of the majority of the creases and achieve the
desired “lining up” of cuts. In addition, there are perpendicular
creases. Basically, from each vertex of the straight skeleton, we shoot a ray
perpendicular to each reachable cut edge, and the ray “bounces” (reflects)
through any skeleton edges it meets.
The difficult part is proving that the straight-skeleton creases together with
a subset of the perpendicular creases (and a few more auxiliary creases) can be
folded into a flat origami. This is done by exhibiting a folded state,
that is, by demonstrating how the piece of paper looks when folded. This
folded state must satisfy the properties that each face is isometrically
preserved and that paper does not intersect itself.
The most detailed description is the book
Geometric Folding Algorithms:
Linkages, Origami, Polyhedra.
The full paper (which includes all proof details) is still in progress. The
longest
description includes many of the proofs and was published in the
Proceedings of the Japan Conference on Discrete and Computational
Geometry (JCDCG'98),
to appear as a volume in Lecture Notes in Computer Science.
A 2-page
description of the method (without proofs) appears in the
Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms
(SODA'99).
This work is related to Robert Lang's
TreeMaker,
although it was initially developed in isolation. The intersection is that
they both solve the fold-and-cut problem when the faces (shapes to be cut out)
are all convex. Both TreeMaker and our work extend far beyond this common
base, in different directions. Essentially,
TreeMaker tries to collapse a polygon to a specified folded shape but only
addresses convex polygons, whereas the straight-skeleton method focuses on the
unfolded shape and addresses nonconvex and disconnected polygons.
Disk-Packing Method
An example of the disk-packing method, from
the paper.
Inspired by early versions of the straight-skeleton
method, Marshall Bern, David Eppstein, Barry Hayes, and I solved
the fold-and-cut problem using disk packing. Specifically, we place
disks on the piece of paper, such that
the disks do not properly overlap (but may touch),
the gaps between disks have either three or four sides,
there is a disk centered at each vertex of the desired cut pattern, and
the edges of the desired cut pattern (i.e., desired cuts) are the union
of radii of disks.
As a result, we can decompose the desired cut pattern by adding edges between
centers of touching disks, and this results in a collection of triangles and
quadrangles. We then fold each of these triangles and quadrangles using
molecules that line up the boundaries of the triangles and quadrangles.
This is the basic idea--a couple of tricks are needed to show that the
molecules can be joined together, and a further trick is needed so that only
the desired lines are the ones that come together onto a common line.
The most detailed description is the book
Geometric Folding Algorithms:
Linkages, Origami, Polyhedra.
The paper “A
Disk-Packing Algorithm for an Origami Magic Trick”
also describes the method in detail, though it has some small issues.
The most recent version
includes some simplifications, and appears in the
Proceedings of the 3rd International Meeting of Origami Science, Math, and
Education (OSME
2001).
The original version
was published in the Proceedings of the International Conference
on Fun with Algorithms (FUN'98).
Our use of disk packing is quite different from Robert Lang's
TreeMaker,
although it is intriguing that the same method comes up in two different
contexts in origami mathematics.
Related
Try out our new simple fold and cut font!
A 2010
paper with Martin Demaine, Andrea Hawksley, Hiro Ito, Po-Ru Loh, Shelly Manber,
and Omari Stephens considers what happens when we restrict the folds to
a sequence of simple folds, folding along a single line at a time.
Check out our new font based on simple
fold and cut!
Ivars Peterson briefly describes the fold-and-cut results in his article,
“Fold-and-Cut
Magic”,
which appears in
Science News,
volume 162, number 22, November 30, 2002.
Joseph O'Rourke briefly describes the two fold-and-cut methods in his
“Computational
Geometry Column 36”, which appears in
International Journal of Computational Geometry and Applications,
volume 9, number 6, pages 615-618, 1999; and SIGACT News, volume 30,
number 3, issue 112, September 1999, pages 35-38.
One application of the fold-and-cut theorem is
the design of the logo for CCCG 2001.
Some of our earlier papers on the fold-and-cut problem are the following:
The technical report
“Computing
Extreme Origami Bases” studies the special case of cutting out
a convex polygon.
In this case, we can not only compute a folded state exists,
but we also provide the continuous motion of the paper through time
that reaches this folded state. In fact, the motion can be achieved
while keeping rigid the regions between creases rigid, called
rigid origami.
“Planar Drawings of Origami Polyhedra” studies some properties of the
foldings of convex polygons using graph drawing.
A 2-page
summary was published in the Proceedings of the 6th
Symposium on Graph Drawing
(GD'98).
The full version
is a technical report.
Last updated March 3, 2025 by
Erik Demaine.
— Accessibility |
13005 | https://macaulay2.com/doc/Macaulay2/share/doc/Macaulay2/SpechtModule/html/_permutation__Sign.html | permutationSign -- the sign of a permutation
Macaulay2 » Documentation
Packages » SpechtModule :: permutationSign
next | previous | forward | backward | up | index | toc
permutationSign -- the sign of a permutation
Usage:
macaulay2
permutationSign(perm)
Inputs:
perm, a list, a permutation of the numbers from 0 to n-1
p, an instance of the type Partition, a partition that represents the conjugacy class of the permutation
Outputs:
an integer, 1 or -1, the sign of the permutation
Description
Every permutation can be decompose as a product of transpositions. This decomposition is not unique, however the parity of the number of transpositions that appears in the decomposition is always the same. Thus the sign is defined as (−1)l(-1)^l(−1)l where l l l is the number of transposition.
The sign can be calculated if the cycle decomposition if known because the sign is multiplicative and the sign of a k k k-cycle is (−1)(k+1)(-1)^(k+1)(−1)(k+1). This is the way the method permutationSign calculates the sign.
The sign permutation is used to calculate polytabloids and higher Specht polynomials.
```macaulay2
i1 : perm = {2,1,4,3,0}
o1 = {2, 1, 4, 3, 0}
o1 : List
macaulay2
i2 : c = cycleDecomposition perm
o2 = {{0, 2, 4}, {1}, {3}}
o2 : List
macaulay2
i3 : permutationSign perm
o3 = 1
macaulay2
i4 : perm2 = {4,2,1,0,3}
o4 = {4, 2, 1, 0, 3}
o4 : List
macaulay2
i5 : c2 = cycleDecomposition perm2
o5 = {{0, 4, 3}, {1, 2}}
o5 : List
macaulay2
i6 : permutationSign perm2
o6 = -1
```
Ways to use permutationSign:
permutationSign(List)
permutationSign(Partition)
For the programmer
The object permutationSign is a method function.
The source of this document is in SpechtModule.m2:3305:0. |
13006 | https://www.proedinc.com/Downloads/20306A_MM_TE_LF_SamplePgs.pdf | 1 UNIT Multiplying and Dividing Whole Numbers1 multiplication Objectives By the end of this lesson: • Students will understand that multiplication represents putting together groups of equal size to get a total.
• Students will be able to represent factors and products using multiplication, repeated addition, and arrays.
Vocabulary Review the term multiplication. Ask students to describe the way they think about multiplication. Some students may say that multiplication is repeated addition. Others may describe it as a fast way of counting equal groups or amounts. Help students connect the visual model (known as an array) with the multiplication sentence (4 # 5 = 20).
Problem of the Day Invite students to try this problem; assess their responses.
Ask students how they solved the problem. Emphasize that all of the strategies below will lead to the same total: • Count all eight markers one at a time.
• Add 2 + 2 + 2 + 2 = 8.
• Multiply 4 # 2 = 8.
Introduce the lesson question: “What does multiplication represent?” Connect the Problem of the Day to the lesson by explaining that it shows that counting, addition, and multiplication are all related. Then explain that students will learn more about the relationships between counting, addition, and multiplication in this lesson.
1 A UNIT Multiplying and Dividing Whole Numbers1 multiplication Today’s Destination What does multiplication represent? Vocabulary Array A grid of squares or other objects arranged in equal rows Multiplication An operation that represents putting equal groups together to get a total 5 4 For example: 4 # 5 = 20 factors the numbers you multiply product the total in a multiplication sentence 4 rows of 5 equals 20 Problem of the Day How many markers does Matt have in total? 8 markers From Momentum Math Level F Teacher's Edition, 2011, Austin, TX: PRO-ED, Copyright 2011, by PRO-ED, Inc.
2 Momentum Math Teacher’s Edition Level F 2 A Momentum Math Level F Rules Of the ROaD Addition means putting groups together to get a total. The total is called the sum. + + = 4 + 2 + 1 = 7 Repeated addition means putting together equal groups. + + = 2 + 2 + 2 = 6 If the groups are equal, you can use multiplication as a shortcut to repeated addition. Multiplication means putting equal groups together to get a total. The total is called the product. “Three groups of two equal six.” 3 # 2 = 6 Circle all the situations that show multiplication. Rules of the Road Instruction approx. 20 min.
Delivery Define multiplication as the repeated addition of equal groups.
• Read the introductory text and examples with students.
• Contrast the two examples. The first illustration shows addition of unequal groups of magazines, while the second illustration shows three equal groups.
• Emphasize that addition involves putting together groups, but multiplication always involves putting together equal groups.
Monitor students while they complete the Try It Out exercise.
Ask students to explain what helped them decide whether each situation represented addition or multiplication.
• The first comic represents multiplication (5 # 3) because it involves repetition of an equal number of words several times.
• The middle comic represents addition (8 + 4) because 8 and 4 are unequal amounts.
• The final comic represents multiplication (2 # 3) because there are two equal groups of three pieces of gum each.
Strategies for Differentiation Support for English Language Learners and Students with Special Needs The word product has everyday meanings that students may feel are dissimilar from its mathematical meaning. However, there is a connection: the equal groups produce a total, or product. Be sure to use product instead of answer when students multiply in this lesson and in other lessons. ELL students may not be comfortable with phrases such as “three groups of two.” Put these phrases in context by saying “three groups of two magazines.” Use magazines or books to act out the introductory examples. Make sure that you not only represent the three equal groups of two magazines, but also represent the total of six books by combining and counting them as one group.
From Momentum Math Level F Teacher's Edition, 2011, Austin, TX: PRO-ED, Copyright 2011, by PRO-ED, Inc.
3 UNIT Multiplying and Dividing Whole Numbers1 3 A UNIT Multiplying and Dividing Whole Numbers1 CHECKPOINT Kenya has 3 brothers. Each brother gave her 2 presents for her birthday. How many presents did Kenya’s brothers give her in all?
Draw a diagram to show the situation above. Write an addition sentence and a multiplication sentence to show the situation above. Nadia has 4 packs of pens. Each pack has 3 pens in it. How many pens does Nadia have in all?
THIS PROBLEM IS ABOUT PUTTING EQUAL GROUPS TOGETHER TO GET A TOTAL. I SEE FOUR GROUPS OF 3. addition sentence 3 + 3 + 3 + 3 = 12 multiplication sentence 4 # 3 = 12 In Kami’s classroom, there are 5 tables. Each table seats 6 students. How many seats are there in all? addition sentence multiplication sentence 6 + 6 + 6 + 6 + 6 = 30 5 # 6 = 30 2 + 2 + 2 =6 3 # 2 = 6 Delivery Guide students through Marta’s example problem.
• Ask, “Why do both of Marta’s number sentences have the same total?” • Emphasize that even though one sentence uses addition and the other uses multiplication, they all represent the same total number of pens.
• Encourage students to conceptualize the multiplication sentences as either “four groups of three pens” or “three pens, counted four times.” Monitor students while they complete the Try It Out exercise.
Allow students to write the product as either 5 # 6 = 30 or 6 # 5 = 30. However, make sure that students do not add 5 + 5 + 5 + 5 + 5 + 5 = 30 because this problem does not involve six equal groups of five. Encourage students to draw a diagram to represent this problem if needed.
Checkpoint Assessment and Reteaching Have students complete the Checkpoint independently.
Redirect struggling students with effective questions, such as: • How can you draw each present in a simple way? Draw a square.
• How can you show that each brother gave two presents? Draw the presents in pairs.
• What is the total number of presents? 6 • How big is each equal group of presents? 2 If students write 3 + 3 = 6 as the sum… Students may already have a strong sense of the commutativity of multiplication. Reinforce this sense while emphasizing that 3 + 3 represents two groups of three. In this problem, however, there are three groups of two.
If students draw a diagram with five squares… Students may have added 2 + 3. It may be helpful to make the problem more concrete by having students act out the situation with actual objects.
If students write 4 + 2 = 6 or some other addition sentence… Students may not have thought of using repeated addition to represent this problem. Ask students, “How can you represent equal groups using addition?” From Momentum Math Level F Teacher's Edition, 2011, Austin, TX: PRO-ED, Copyright 2011, by PRO-ED, Inc. |
13007 | https://courses.lumenlearning.com/calculus2/chapter/summary-of-the-definite-integral/ | Summary of the Definite Integral | Calculus II
Skip to main content
Calculus II
Module 1: Integration
Search for:
Summary of the Definite Integral
Essential Concepts
The definite integral can be used to calculate net signed area, which is the area above the x x-axis minus the area below the x x-axis. Net signed area can be positive, negative, or zero.
The component parts of the definite integral are the integrand, the variable of integration, and the limits of integration.
Continuous functions on a closed interval are integrable. Functions that are not continuous may still be integrable, depending on the nature of the discontinuities.
The properties of definite integrals can be used to evaluate integrals.
The area under the curve of many functions can be calculated using geometric formulas.
The average value of a function can be calculated using definite integrals.
Key Equations
Definite Integral
∫b a f(x)d x=lim n→∞n Σ i=1 f(x∗i)Δ x∫a b f(x)d x=lim n→∞Σ n i=1 f(x i∗)Δ x
Properties of the Definite Integral
∫a a f(x)d x=0∫a a f(x)d x=0
∫a b f(x)d x=−∫b a f(x)d x∫b a f(x)d x=−∫a b f(x)d x
∫b a[f(x)+g(x)]d x=∫b a f(x)d x+∫b a g(x)d x∫a b[f(x)+g(x)]d x=∫a b f(x)d x+∫a b g(x)d x
∫b a[f(x)−g(x)]d x=∫b a f(x)d x−∫b a g(x)d x∫a b[f(x)−g(x)]d x=∫a b f(x)d x−∫a b g(x)d x
∫b a c f(x)d x=c∫b a f(x)d x∫a b c f(x)d x=c∫a b f(x)d x for constant c c
∫b a f(x)d x=∫c a f(x)d x+∫b c f(x)d x∫a b f(x)d x=∫a c f(x)d x+∫c b f(x)d x
Glossary
average value of a function(or f ave f ave) the average value of a function on an interval can be found by calculating the definite integral of the function and dividing that value by the length of the interval definite integral a primary operation of calculus; the area between the curve and the x x-axis over a given interval is a definite integral integrable function a function is integrable if the limit defining the integral exists; in other words, if the limit of the Riemann sums as n n goes to infinity exists integrand the function to the right of the integration symbol; the integrand includes the function being integrated limits of integration these values appear near the top and bottom of the integral sign and define the interval over which the function should be integrated net signed area the area between a function and the x x-axis such that the area below the x x-axis is subtracted from the area above the x x-axis; the result is the same as the definite integral of the function total area total area between a function and the x x-axis is calculated by adding the area above the x x-axis and the area below the x x-axis; the result is the same as the definite integral of the absolute value of the function variable of integration indicates which variable you are integrating with respect to; if it is x x, then the function in the integrand is followed by d x d x
Candela Citations
CC licensed content, Shared previously
Calculus Volume 2. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at
Licenses and Attributions
CC licensed content, Shared previously
Calculus Volume 2. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at
PreviousNext
Privacy Policy |
13008 | https://edu.rsc.org/resources/practical | Skip to main content
Skip to navigation
Resources
Practical
Close menu
Home
I am a …
Back to parent navigation item
I am a …
Primary teacher
Secondary/FE teacher
Early career or student teacher
Technician
HE teacher
Student
Resources
Back to parent navigation item
Resources
Primary
Secondary
Higher education
Curriculum support
Practical
Analysis
Literacy in science teaching
Periodic table
Back to parent navigation item
Periodic table
Interactive periodic table
Climate change and sustainability
Careers
Resources shop
Collections
Back to parent navigation item
Collections
Remote teaching support
Starters for ten
Screen experiments
Assessment for learning
Microscale chemistry
Faces of chemistry
Classic chemistry experiments
Nuffield practical collection
Anecdotes for chemistry teachers
Literacy in science teaching
More …
Climate change and sustainability
Alchemy
On this day in chemistry
Global experiments
PhET interactive simulations
Chemistry vignettes
Context and problem based learning
Journal of the month
Chemistry and art
Back to parent navigation item
Chemistry and art
Techniques
Art analysis
Pigments and colours
Ancient art: today's technology
Psychology and art theory
Art and archaeology
Artists as chemists
The physics of restoration and conservation
Cave art
Ancient Egyptian art
Ancient Greek art
Ancient Roman art
Classic chemistry demonstrations
In search of solutions
In search of more solutions
Creative problem-solving in chemistry
Solar spark
Chemistry for non-specialists
Health and safety in higher education
Analytical chemistry introductions
Exhibition chemistry
Introductory maths for higher education
Commercial skills for chemists
Kitchen chemistry
Journals how to guides
Chemistry in health
Chemistry in sport
Chemistry in your cupboard
Chocolate chemistry
Adnoddau addysgu cemeg Cymraeg
The chemistry of fireworks
Festive chemistry
Education in Chemistry
Teach Chemistry
Events
Teacher PD
Back to parent navigation item
Teacher PD
Courses
Back to parent navigation item
Courses
On-demand online
Live online
Resources
Selected PD articles
PD for primary teachers
PD for secondary teachers
What we offer
Chartered Science Teacher (CSciTeach)
Teacher mentoring
Enrichment
Back to parent navigation item
Enrichment
UK Chemistry Olympiad
Back to parent navigation item
UK Chemistry Olympiad
Who can enter?
How does it work?
Resources and past papers
FAQs
Top of the Bench
Schools' Analyst
Our work
Back to parent navigation item
Our work
Regional support
Back to parent navigation item
Regional support
Education coordinators
RSC Yusuf Hamied Inspirational Science Programme
Policy
Back to parent navigation item
Policy
Science Education Policy Alliance
RSC Education News
Supporting teacher training
Interest groups
Home
I am a …
Resources
Education in Chemistry
Teach Chemistry
Events
Teacher PD
Enrichment
Our work
More navigation items
Practical videos
Practical videos | 14–16 years
Videos of core practical experiments for flipped learning, remote teaching or revision
Practical videos | 16–18 years
Videos of key practical techniques and apparatus for revision, flipped learning or remote teaching
Simulations
Support practical chemistry with new ways to explore experimental work
Aspirin screen experiment
Get students aged 16–18 running their own organic synthesis on a computer or tablet before taking part in the real thing. This resource also features an in-depth practical guide and a set of worksheets covering the theory.
Titration screen experiment
Give students the opportunity to conduct their own titration experiment on a computer or tablet. This resource also includes a redox titration experiment.
Exhibition chemistry
Spectacular demonstrations
Boiling without heating
Overcome common misconceptions about changes of state and vaporisation with water, a syringe and reduced pressure
Evaporation, entropy and the Marangoni effect
Demonstrate thermodynamics to your 16–18 learners with this simple experiment
Demonstrations
Diffusion of colour in water | Johnstone’s triangle worksheets | 11–14 years
Use this worksheet to develop learners’ understanding of diffusion as they observe coloured dyes diffusing in water
Investigating the sugar content and pH of fizzy drinks | 11–14 years
Compare fizzy drinks’ ingredients, acidities and health implications
Salt (for cooking) activities | 11–14 years
Uncover the properties and health implications of salt in our diets
Custard experiment | 11–14 years
Introduce your learners to Newtonian and non-Newtonian fluids as well as the explosive nature of powders
Investigating cold reactions | 11–14 years
Explore what is meant by ‘cold’ and the applications of endothermic reactions
Extracting DNA experiment | 11–14 years
Remove and compare genetic material in different plant cells
Emulsifiers in cleaning | 11–14 years
Investigate and explain what emulsions are and their applications in cleaning
View more demonstrations
Global experiments
Engaging hands-on group activities, adaptable for all ages and fully resourced
Mission starlight
Measuring vitamin C in food
The art of crystallisation
Water and hydrogels
Chemistry in sport
Mission starlight
Measuring vitamin C in food
Measuring vitamin C in food
The art of crystallisation
Water and hydrogels
Chemistry in sport
Mission starlight
Previous
Next
View more global experiments
Class practicals
A practical investigation of cis-trans isomerism in transition metal complexes
Investigate isomerism in transition metal complexes with this experiment, including pre- and post-lab questions
Chromatography challenge | 16–18 years
Explore analytical techniques and their applications with a chromatography investigation and research activity
Paracetamol book | Using thin-layer chromatography to investigate the reactions
In this activity you investigate the purity and identity of your laboratory prepared samples of nitrophenol or paracetamol using thin-layer chromatography
Paracetamol book | The reduction of a nitro group to an amine
Nitro group reduction into an amine is covered in this practical class experiment.
Paracetamol book | The quantitative analysis of various formulations of paracetamol
An experiment to show learners to analyse several different formulations of paracetamol and create and quantitative judgment of them.
Paracetamol book | The preparation of paracetamol
The first of three steps, in practical experiments, that show learners how to prepare paracetamol
Changing the properties of polymers and plastics
Learners consider how changes to polymers, for example adding plasticisers or cross-linking, affects the properties of the objects made from the polymers.
Polylactic acid
Explore an example of condensation polymerisation with this practical activity where lactic acid is reacted to make short strands of polylactic acid.
View more class practicals
Practical collections
Previous
Next
Microscale chemistry
Classic chemistry experiments
Nuffield practical collection
Global experiments
Classic chemistry demonstrations
In search of solutions
In search of more solutions
Chemistry for non-specialists
Exhibition chemistry
Contact us
Topics
Issues
Contributors
Email alerts
FAQs
Safety
Site powered by Webvision Cloud
This site uses cookies
This website uses cookies and similar technologies to store and retrieve information about your use of this website. This information helps us to provide, analyse and improve our services, which may include personalised content or advertising. We may share this information with Google and other third parties. Some cookies are necessary for our website to work properly and others are optional. You can review and configure your cookie settings below.
Privacy preference center
When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalised web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Manage consent preferences
Strictly necessary cookies
Always active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Analytical cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Functional cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Targeting cookies
These cookies may be set through our site by our advertising partners. They may be used by those companies to build a profile of your interests and show you relevant adverts on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device. If you do not allow these cookies, you will experience less targeted advertising.
Social media cookies
These cookies are set by a range of social media services that we have added to the site to enable you to share our content with your friends and networks. They are capable of tracking your browser across other sites and building up a profile of your interests. This may impact the content and messages you see on other websites you visit. If you do not allow these cookies you may not be able to use or see these sharing tools. |
13009 | http://mrsgalgebra.pbworks.com/Zero-Product-Property | Mrs. Grieser's Algebra Wiki: WikiGrieser / Zero Product Property
Mrs. Grieser's Algebra Wiki: WikiGrieser
log inhelp
Get a free wiki|Try our free business product
Wiki
Pages & Files
View Edit To edit this page, request access to the workspace. Already have an account? Log in! Zero Product Property ===================== Page history last edited by Andrea Grieser;)15 years, 2 months ago The zero product property states that for real numbers a and b, where a/= b, if ab = 0 then either a = 0 or b = 0 What this means is that if we multiply a set of factors and the quotient is 0, then at least one of the factors must be 0. Zero Product Property ===================== #### Page Tools ### Insert links Insert links to other pages or uploaded files. PagesImages and files Insert a link to a new page 1. Loading... 1. No images or files uploaded yet. Insert image from URL Tip: To turn text into a link, highlight the text, then click on a page or file from the list above. ### Comments (0) You don't have permission to comment on this page. Printable version PBworks / Help Terms of use / Privacy policy / GDPR About this workspace Contact the owner / RSS feed / This workspace is public### Join this workspace To join this workspace, request access. Already have an account? Log in! ### Navigator Algebra Relations Right Triangles Scale Drawings Sets Sigma Notation (Summation Notation) Simplifying Radicals Slide and Divide Method for Factoring Polynomials Slope Slope-Intercept Form of a Line Solving Equations Solving Equations for a Variable Solving Inequalities Solving Linear Systems of Equations Solving Linear Systems of Inequalities Solving Quadratic Equations Solving Quadratic Equations by Completing the Square Spheres Standard Deviation Standard Form of a Line Statistical Data Representation Statistical Deviation Statistical Variation Statistics Subtracting Polynomials Surface Area Synthetic Division Systems of Equations and Inequalities Three-Dimensional Figures Transformations Triangles Trinomials Volume Wiki rules X-intercept Y-intercept Zero Product Property optionsPagesFiles ### SideBar Mrs. G's Algebra Wiki. See rules for more information. Virginia Standards of Learning (SOLs) 2009 Index ### Recent Activity Show 0 new item s Logicedited by Andrea Grieser Deductive Reasoningedited by Andrea Grieser Inductive Reasoningedited by Andrea Grieser Logicedited by Andrea Grieser Deductive Reasoningedited by Andrea Grieser Deductive Reasoningedited by Andrea Grieser Deductive Reasoningadded by Andrea Grieser More activity... |
13010 | http://xahlee.info/tiling/tiling.html | Reading Notes on Tilings and Patterns
This is my notes on reading
Tilings and Patterns Book, by Branko Grunbaüm, G C Shaphard
it's randomish and incomplete. mostly written in 1997.
Some definitions
A Plane Tiling PT is a countable family of closed sets PT = {T1, T2,…} which cover the plane without gaps or overlaps.
to narrow down our study, we require that each tile is a (closed) topological disk. By topological disk, we mean any set whose boundary is a single simple closed curve. (this way, we eliminates some possibly useful but weird situations where "tiles" that are disconnected, has a hole, connected by just a point, connected by a line, or has lines, or unbounded.)
A vertice of a tiling is a point in the plane that are intersection of two tiles. A edge of a tiling is a curve that is the intersection of two tiles. The vertices, edges, tiles of a tiling are called its elements.
A vertice is connected to a number of edges. This number is called the valence of the vertice.
in the context of tilings by polygons, there is the concept of edge-to-edge tilings. That is, if every side of a polygon is also a edge of the tiling, than the tiling is edge-to-edge (as opposed to a edge a tiling consisting several sides of a tile). More to the point, all edges in the tiling are straight lines.
1.2 Tilings with tiles of a few shapes
A tiling is call monohedral if every tile in the tiling PT is congruent (directly or reflectively) to one fixed set T. The set T is called the prototile of PT.
1.3 Symmetry, transitivity, and regularity
An isometry or congruence transformation is any mapping of the Euclidean plane onto itself which preserves all distances.
If a tiling admits any symmetry in addition to the identity symmetry then it will be called symmetric. If its symmetry group contains at least two translations in nonparallel directions then the tiling will be called periodic. The linear combinations of any two nonparallel translations generates a lattice. Thus with every periodic tiling PT is associated a lattice, and the points of the lattice can be regarded (in many ways) as the vertices of a parallelogram tiling PT; the tiles of PT are known as period parallelograms.
Two tiles T1 and T2 of a tiling PT are said to be equivalent if the symmetry group S(PT) contains a transformation that maps T1 onto T2; the collection of all tiles of PT that are equivalent to T1 is called the transitivity class of T1. If all tiles of PT form one transitive class we say that PT is tile-transitive or isohedral.
If PT is a tiling with precisely k transitivity classes then PT is called k-isohedral.
If the symmetry group S(PT) of PT contains operations that map every vertex of PT onto any other vertex, then we say that the vertices form one transitivity class, or that the tiling is isogonal. A tiling is k-isogonal if its vertices form k transitivity classes, where k ≥ 1 is any integer.
1-gonal is called monogonal. It's the number of vertex types.
1-hedral is called monohedral. It's the number of tiles types.
1-toxal is called monotoxal. It's the number of edge types.
With symmetry of the tiling considered, we have k-isogonal, k-isohedral, and k-isotoxal corresponding to the number of classes with vertexes, tiles, and edges respectively.
By a flag in a tiling we mean a triple (V,E,T) consisting of a vertex V, an edge E and a tile T which are mutually incident.
A tiling PT is called regular if its symmetry group S(PT) is transitive on the flags of PT.
1.4 Symmetry groups of tilings
1.5 Monomorphism and k-morphism
A tile is called k-morphic if it can be used to tile the plane in exactly n ways. is the prototile of a unique monohedral tiling of the plane. Similarly, a tile is n-morphic if it is the prototile of exactly n tilings of the plane.
open questions
For every positive integer r, is it possible to find a tile which is r-morphic?
Is there a tile that admits a countable infinity of distinct tilings?
Xah: I thought so, but don't know. For example, figure 1.5.8, just change the neghboring row placements, one should be able to come up with infinity distinct tilings, where most of them are not symmetric. Form p.59, the discussion seems to indicate that my method above allows an “uncountable infinity”, so is not a solution.
2. Tilings by regular polygons and star polygons
Edge-to-edge regular polygon tilings
k-hedral, k-gonal, k-toxal and k-isohedral, k-isogonal, k-isotoxal
if the tiling has k types of tiles (that is, uses k prototiles), then it is called k-hedral.
if the tiling has k types of vertexes, then it is called k-gonal.
if the tiling has k types edges, then it is called k-toxal.
If the tiles in a tiling forms h transitive classes with respect to the tiling's symmetry, then the tiling is called h-isohedral (or, having isohedral index of h). (two tiles are of the same transitive class if one can be mapped to the other by the symmetries of the tiling. Obviously, the isoheral index is always less or equal to the hedral index.) A tiling's isogonal and isotoxal indexes are similarly defined with respect to vertexes and edges.
A edge-to-edge regular polygon tiling has the following type of properties:
isohedral index (k-isohedral) and number of tile types (k-hedral). (if the two are equal, the tiling is called equitransitive.)
isogonal index (k-isogonal) and number of vertex types (k-gonal).
isotoxal index (k-isotoxal) and number of edge types (k-toxal). (the latter is always 1.)
its symmetry group.
k-uniform is the name for edge-to-edge regular polygon tilings that's k-isohedral. Archimedean tiling are 1-gonal edge-to-edge regular polygon tilings (by definition). Archimedean tilings are also 1-isogonal(i.e. happens to be), therefore also uniform.
2.8 Extension of Patches
2.9 Archimedean and UniformColorings
4. The topology of tilings
4.1 Homeomorphisms and topologicalequivalence
A mapping φ:E²→E²
of the plane onto itself is called a homeomorphism or a topological transformation if it is one-to-one and bicontinuous. (bicontinuous means both pre-image and image are continuous.)
Two tilings are said to be of the same topological type (or topologically equivalent) if there is a homeomorphism which maps one onto the other. The fact that the composition of two homeomorphisms, and the inverse of a homeomorphism, are also homeomorphisms shows that topological equivalence is an equivalence relation (inthat it is reflexive, symmetric, and transitive). It therefore partitions the set of all tilings into “topological types”.
A normal tiling after a homeomorphism may no longer be normal. (see text.)
Two tilings are said to be combinatorial equivalent if one is the image of an inclusion-preserving map of the other. (inclusion meaning subset.) Basically meaning that elements (vertexes, edges, tiles) remain neighbors. For normal tilings the concepts of topological equivalence and combinatorial equivalence coincide.
Two tilings are said to be isoptic if there is a series of continuous mappings that transforms one to the other. i.e. Their topological equivalence can be demonstrated visually. (the book notes that this is not easy to prove that isotopy and topoligical equivalence are equivalent.)
4.2 Duality
Two tilings PT and PT are said to be dual to each other if there exists a one-to-one inclusion-reversing map phi; from a set E[PT] onto the set E[PT]. (we recall that E[PT] is the set of elements -- vertices, edges, and tiles — of the tiling PT.) By inclusion-reversing we mean that whenever e1 and e2 are elements of T, then phi[e1] includes phi[e2] iff e2 includes e1.
Two dual tilings are dually-situated if one can be superimposed on the other to make their dual relation visually apparant.
4.3 Homeohedral Tilings
A tiling PT is called homeohedral or topologically tiletransitive if it is a normal tiling and is such that for any two tiles T1, T2 of PT there exists a homeomorphism of the plane that maps PT onto PT and T1 onto T2.
If T is a tile of the tiling PT then we say that T is of valence-type {j1,j2,…,jk} provided T has k vertices which, if considered in a suitable cyclic order, have valences {j1,j2,…,jk}. If all tiles of a normal tiling PT have the same valence-type {j1,j2,…,jk}, we say that PT is homogeneous of type {j1,j2,…,jk}. It is clear that veery homeohedral tiling is homogeneous. We do not distinguish between types if their symbols can be obtained from each other by a cyclic permutation or reversal of order. We will choose the standard order to be the one ordered lexicographically.
Theorem: If PT is a homogeneous tiling then it is of one of the element types corresponding to the vertice types of the 11 Lave tilings. Each homogeneous tiling is homeohedral, all homogeneous tiling of the same type are topologically equivalent to each other.
Personal notes
To do
Investigate tilings by regular square and triangle.
Find m-isogonal tiling with n vertice types, where m > n.
possible errors in the book
Suggestion:
p.37, section 1.4.
Considering introducting the orbifold notation for symmetry groups.
Suggestion:
p.44, Table 1.4.2.
Is there a reason why the transitive class are considered within the type of symmetry?
Error?
p.56, upper right.
According to Symmetries of Islamic Geometrical Patterns, by Syed Jan Abas, Amer Shaker Salman (World Scientific. 1995), all types of wallpaper groups are found in Islamic area. They also provide photographs.
Suggestion:
p.61, Table 2.1.1
The middle section of the table is annoying. The info is explicit in the tilings' symbol.
Suggestion:
p.95, section 2.7
While writing computer program to generate Archimedean tilings and their duals, I found that the centroids of Archimedean tiles are the vertexes of the Lave tilings, but the centroid of Lave tiles are not vertexes of Archimedean tilings. This is suprising, because I think intuitive we'd think that the centroids mutually forms the other's vertexes. Perhaps a good exercise.
Suggestion:
p.221, lower left
“We find it convenient to consider …”
Q: How exactly is group diagram defined? Does it contain the pattern themselves?
probably not a good comment.
Book wishlist
A chapter that intorduce Bill Thurston's idea of viewing wallpaper as an orbifold, and the orbifold notation for wallpaper groups.
A chapter on layered patterns and isonemal fabric.
Possibly a chapter on group theory, and Cayley diagram.
tilings and patterns topic
Reading Notes on Tilings and Patterns
Tilings and Patterns Book, by Branko Grunbaüm, G C Shaphard
Math Tiling and Patterns Gallery
Wolfram: Plane Tiling Package 📦
Discontinuous Groups of Rotation and Translation in the Plane
Great Math Software: Tilings and Patterns |
13011 | https://www.sgo.org/wp-content/uploads/2012/09/posttreatment-surveillance-paper_RESIZE.pdf | ONCOLOGY Posttreatment surveillance and diagnosis of recurrence in women with gynecologic malignancies: Society of Gynecologic Oncologists recommendations Ritu Salani, MD, MBA; Floor J. Backes, MD; Michael Fung Kee Fung, MB, BS; Christine H. Holschneider, MD; Lynn P. Parker, MD; Robert E. Bristow, MD, MBA; Barbara A. Goff, MD I n 2010, gynecologic malignancies were expected to afflict approxi-mately 80,000 women within the United States.1 Advances within the field of gynecologic oncology have resulted in long-term survivals and a high rate of survivors. Because long-term survival is becoming more common in this patient population, insights into cancer surveil-lance and detection of recurrence and addressing side-effects from treatment are of utmost importance.
Currently, posttreatment guidelines call for frequent visits immediately after treatment, followed by increasing inter-vals over time. Typically, after the first 2-3 years, patients are transitioned back to their primary care providers. How-ever, primary care physicians may not be comfortable with guidelines or surveil-lance for each specific cancer type.2 This is in part due to a lack of training and in part to unclear expectations for the pri-mary care provider by the oncologist.2-4 As survivorship continues to grow, coor-dination of care between gynecologic oncologists, primary care providers, other healthcare providers (such as radi-ation oncologists), and patients ideally will allow for compliance with cancer follow-up care and routine health main-tenance. The provision of a clear under-standing of recommendations and re-sponsibilities of appropriate surveillance will reduce unnecessary tests and ulti-mately result in cost savings.
The role of surveillance is to provide clinicalandcost-effectivepracticesthatde-tect recurrence and impact survival out-comes. Acceptance of surveillance should beconsideredifthereisutilityoftreatment for recurrence and decreased morbidity from both monitoring for disease recur-renceandtreatment.Oneshouldalsocon-sider the costs and the use of resources for conducting these tests. Last, patients should be counseled on the benefits and pitfalls of disease monitoring, which should include the psychologic impact of surveillance programs.5 Unfortunately, most studies across all cancer sites are based predominantly on retrospective studies and provide limited insight into the true benefit of recommended guide-lines for posttreatment surveillance.
There is a real need for prospective stud-ies to establish the most cost-effective methods for the detection of recurrent disease. In addition, surveillance tests should be directed at detecting recur-rences that are amenable to curative or significant palliative treatment. There-fore, the primary objective of this review is to provide the most recent data on surveillance for cancer recurrence in women who have had a complete re-sponse to primary cancer therapy for gynecologic malignancies. Additionally, we have included routine health screen-Although gynecologic cancers account for only 10% of all new cancer cases in women, these cancers account for 20% of all female cancer survivors. Improvements in cancer care have resulted in almost 10 million cancer survivors, and this number is expected to grow. Therefore, determining the most cost-effective clinical surveillance for detection of recurrence is critical. Unfortunately, there has been a paucity of research in what are the most cost-effective strategies for surveillance once patients have achieved a complete response. Currently, most recommendations are based on retrospective studies and ex-pert opinion. Taking a thorough history, performing a thorough examination, and educating cancer survivors about concerning symptoms is the most effective method for the detec-tion of most gynecologic cancer recurrences. There is very little evidence that routine cytologic procedures or imaging improves the ability to detect gynecologic cancer recur-rence at a stage that will impact cure or response rates to salvage therapy. This article will review the most recent data on surveillance for gynecologic cancer recurrence in women who have had a complete response to primary cancer therapy.
Key words: cervical cancer, cytology, endometrial cancer, gynecologic cancer, imaging, ovarian cancer, surveillance From The Ohio State University, Columbus, OH (Drs Salani and Backes); the University of Ottawa, Ontario, Canada (Dr Fung Kee Fung); the David Geffen School of Medicine at UCLA, Los Angeles, CA (Dr Holschneider); the University of Louisville Louisville, KY (Dr Parker); the University of California Irvine Medical Center Orange, CA (Dr Bristow); and the University of Washington School of Medicine Seattle, WA (Dr Goff).
Received Feb. 19, 2011; accepted March 8, 2011.
Reprints: Ritu Salani, MD, MBA, The Ohio State University, 320 W 10th Ave., M210 Starling Loving, Columbus, OH 43210. ritu.salani@osumc.edu.
Authorship and contribution to the article is limited to the 7 authors indicated. There was no outside funding or technical assistance with the production of this article.
0002-9378/$36.00 • © 2011 Mosby, Inc. All rights reserved. • doi: 10.1016/j.ajog.2011.03.008 Reviews www.AJOG.org 466 American Journal of Obstetrics & Gynecology JUNE 2011 ing guidelines to allow for enhanced communication between oncologists and primary care providers.
Endometrial cancer Endometrial cancer is the most common gynecologic cancer and the fourth most common cancer in women. Yearly, there are approximately 44,000 new endome-trial cancer diagnoses and 8000 deaths in the United States.1 Commonly, patients experience symptoms such as abnormal or postmenopausal bleeding, which war-rant further investigation with ultrasound scanning and/or endometrial sampling.
The combination of symptoms and diag-nostic testing results in 83% of patients being diagnosed in the early stages of the disease.6 As a result of localized disease, 5-year survival rates exceed 95% for stage I and approach 83% overall. How-ever, recurrence rates for patients with early-stage disease range from 2–15% and reach as high as 50% in advanced stages or in patients with aggressive his-tologic condition.7-10 Many local recur-rences from endometrial cancer are cur-able; therefore, the determination of the ideal time interval and diagnostic tools for surveillance of recurrent endometrial cancer that can impact survival out-comes is critical.
Typically, surveillance guidelines are more intensive the first few years after diagnosis because many studies have shown that most (70-100%) recurrences occur within 3 years after primary treat-ment.11-14 Current guidelines of the Na-tional Comprehensive Cancer Network (NCCN) and the American Congress of Obstetricians and Gynecologists recom-mend physical examination every 3-6 months for 2 years, then every 6 months or annually.15,16 Further evaluation with vaginal cytologic evidence is recom-mended every 6 months for 2 years and annually thereafter.16 To date, there are no prospective studies that have evalu-ated the role of surveillance in endome-trial cancer follow-up evaluation. Based on recommended guidelines and institu-tional practices, retrospective research and literature reviews comprise the best evidence that is available.
The most consistently used method for surveillance is the physical examina-tion. This alone accounts for a high rate of detection that ranges from 35-68% of cases.11,13,17-19 Even more striking is that the combination of physical examination and symptoms has resulted in rates of de-tectionthatexceed80%.18,19Inarecentlit-erature review, Sartori et al20 report that only physical examination has shown util-ity in the detection of endometrial cancer recurrence. Therefore, physical examina-tion,whichincludesathoroughspeculum, pelvic, and rectovaginal examination, should be conducted during each fol-low-up assessment.
The role of surveillance is based on the concept that detection of recurrences in the asymptomatic stage results in better therapeutic options and out-comes. Interestingly, even in spite of in-tensive surveillance, many recurrences are detected based on the presence of symptoms, which occurs in 41-83% of patients.11-13,18,19,21-24 A common symptom, vaginal bleeding, is indicative of a local recurrence that is often curable if it is an isolated site of disease.13,18,19 However, other common symptoms in-clude abdominal and/or pelvic pain, lethargy, and weight loss.13,25,26 Even in the face of monitoring for recurrence, patients who experience a distant recur-rence are symptomatic in 70% of cases, such as coughing or headaches.13,21 Therefore, patient education about the signs and symptoms is a critical compo-nent of posttreatment care and may lead to the detection of recurrent disease.
Survival outcomes have been evalu-ated on the basis of the presence or ab-sence of symptoms at the time of recur-rence. In a report by Sartori et al,11 52% of patients were diagnosed with recur-rence after they had symptoms; these pa-tients had a median postrecurrence sur-vival of 7 months. This was significantly less than the 20-month survival that pa-tients experienced if they were diagnosed with recurrence in an asymptomatic state that was based on examination or imaging. Several other series have evalu-ated the role of routine surveillance for the follow up evaluation of patients with stage I endometrial cancer and re-ported no difference in survival based on the presence or absence of symp-toms.13,21-23,26,27 Of note, even pa-tients who had symptoms were under-going the recommended follow-up evaluations, which provided an argu-ment against the use of routine surveil-lance. Although all of these studies were retrospective, they reiterate the impor-tance of prospective trials to determine the true role and regimen for surveillance.
Because most recurrences occur at the vaginal cuff, the use of cytologic evalua-tionhasbeenadvocated.However,many gynecologic oncologists challenge this recommendation. Rates of recurrence detection on vaginal cytologic evidence range from 0–6.8%, even in asymptom-atic patients.11,17-25,28 Although Ber-chuck et al19 and Owen and Duncan28 report that cytologic evaluation detected 25% of all recurrences and that cytologic evaluationalonedetectedonly3ofthe44 (7%) recurrences. Furthermore, in addi-tion to a low yield of detection, Agboola et al13 reported that the use of vaginal cytologic evaluation at each visit resulted in a cost of $27,000 per case detected. Be-cause most recurrences at the vaginal cuff can be found on examination, vagi-nal cytologic evaluation adds only signif-icant healthcare costs without added benefit.
Similarly to ovarian cancer, the use of cancer antigen 125 (CA125) level has been investigated as a marker for recur-rence. In asymptomatic patients with en-dometrial cancer, the use of CA125 levels accounted for 15% of detections.12 Rose et al29 reported that CA125 levels were elevated in more than one-half of the pa-tients with advanced stage and/or high-grade histologic evidence and that of these patients most had an elevated pre-treatment level. However, one must be aware of elevated CA125 levels because of other conditions or even previous ra-diotherapy. In addition, the role of CA125 levels for the detection of recur-rence was negligible in patients with low-risk disease.26,29 At present, the use of CA125 levels should not be used rou-tinely in patients with endometrial can-cer but may be appropriate in select pa-tients with advanced disease, serous histologic condition, or a CA125 level that is elevated before treatment.
The use of radiographic imaging has been suggested for the detection of re-www.AJOG.org Oncology Reviews JUNE 2011 American Journal of Obstetrics & Gynecology 467 current disease. Because of low costs, chest radiographs have been advocated for the detection of asymptomatic recur-rences, often on a semiannual or annual basis. The rate of detection for asymp-tomatic chest recurrences that are found on chest radiographs ranges from 0– 20%.14,19 In another series, chest radio-graph detected 7 asymptomatic pulmo-nary recurrences and accounted for 0.34% of all chest radiographs that were performed for surveillance, which indi-cates low utility for this tool.13 Although reports of isolated pulmonary recur-rences, albeit rare, may be amenable to therapies that allow for long-term sur-vival outcomes, the routine use of chest radiographs is not recommended.25,30 In further evaluation of radiographic imaging for endometrial cancer surveil-lance, Fung Kee Fung et al14 conducted a review of the literature and found that only 5-21% of asymptomatic recur-rences were found by computed tomog-raphy (CT) scans. Other studies have agreed that the role of CT scanning for asymptomatic patients is not warranted, because survival of patients with disease that is detected on CT scan, compared with clinical examination, did not differ significantly.25,27 To increase the detec-tion of local recurrence, the use of pelvic ultrasound scans has also been reported.
Although detection rates for local recur-rence range from 4–31%, many of these recurrences were also detected on other diagnostic methods, which included physical examination.11,14,21,25,26 There-fore, the use of routine pelvic ultrasound and CT scanning is not advocated; how-ever, these modalities may play a role in the evaluation of patients with symp-toms, because the rates of detection ap-proach 50% of cases.27 More recently, attention has been fo-cused on positron emission test (PET) CT scans for endometrial cancer recur-rence. Park et al31 reported 100% sensi-tivity and 83% specificity when PET-CT scanning was used for suspected recur-rence and 100% diagnostic accuracy in 64 asymptomatic patients. However, its use for routine screening has not been well studied, and larger prospec-tive studies will determine whether PET/CT will have a role in endometrial cancer surveillance. In addition, the high cost of PET/CT may limit its use in routine surveillance (Table 1).
In conclusion, most patients with en-dometrial cancer will be a low risk for recurrence, and more than one-half of all recurrences will be detected through symptoms alone. With the exception of local disease, recurrent endometrial can-cer is associated with a poor prognosis, regardless of the time of detection. On the basis of the data, we recommend a surveillance regimen to include a thor-ough history and physical examination, which would include a speculum and pelvic examination, at scheduled inter-vals with further testing indicated to evaluate symptoms and abnormalities that are detected on examination. This approach may save valuable healthcare dollars. Cytologic evaluation and chest radiographs in asymptomatic women are not clearly beneficial. If patients do have a suspected recurrence, generally a CT scan of chest, abdomen, and pelvis or PET/CT scans may be performed to assess the extent of the disease (Table 2).32,33 Ovarian cancer Ovarian cancer affects almost 22,000 women each year in the United States and results in 13,000 deaths yearly.1 Although responsible for 30% of all gynecologic malignancies, ovarian can-cer accounts for 50% of deaths. These results stem from a lack of accurate screening tools and symptoms that are vagueandoftennotspecific,whichresult in approximately 75% of patients being diagnosed with advanced disease.6 Since the 1970s, the median overall survival of patients with advanced ovarian cancer has increased from 20 months up to 65 months because of advances in surgery and chemotherapy.34,35 Despite the achievement of a complete clinical response, recurrence rates remain high, occurring in 25% of patients with early-stage disease and 80% of patients with advanced disease.35,36 Although pa-tients with recurrent ovarian cancer rarely are cured, patients can have significant re-sponses to salvage treatments.
To detect recurrences, the NCCN guidelines for epithelial ovarian cancer, fallopian tube cancer, and primary peri-toneal cancer recommend follow-up vis-its every 2-4 months for the first 2 years, followed by 6-month intervals for the next 3 years. At each visit, physical exam-ination and identification of the CA125 level or corresponding tumor marker are recommended.37 Additionally, these guidelines advocate the use of radio-graphic imaging and laboratory testing, as clinically indicated. However, the im-pact of surveillance and guidelines are based predominantly on retrospective studies and expert opinions.
Because 26-50% of recurrences occur within the pelvis, a thorough physical ex-amination is an important part of a pa-tient’s follow-up care and should include a bimanual pelvic and rectovaginal ex-TABLE 1 Sensitivity/detection rate of the methods that were used to detect recurrence in patients at routine visits after treatment Method of detection Type of cancer, % Endometrial Ovarian Cervical Symptoms 41-83 — 46-95 ..............................................................................................................................................................................................................................................
Physical examination 35-68 15-78 29-75 ..............................................................................................................................................................................................................................................
Cytologic evidence 0-7 — 0-17 ..............................................................................................................................................................................................................................................
Chest radiograph 0-20 — 20-47 ..............................................................................................................................................................................................................................................
Cancer antigen 125 level 15 62-74 — ..............................................................................................................................................................................................................................................
Computed tomography scan 0-20 40-93 0-45a ..............................................................................................................................................................................................................................................
Positron emission test–computed tomography scan 100a 45-100 86 ..............................................................................................................................................................................................................................................
a Limited data.
Salani. Surveillance for gynecologic cancers. Am J Obstet Gynecol 2011.
Reviews Oncology www.AJOG.org 468 American Journal of Obstetrics & Gynecology JUNE 2011 amination.36,38 However, the rates of de-tection by physical examination vary sig-nificantly from 15-78%.39,40 Although physical examination is one of the most commonly used tools and is associated with low cost, the reproducibility is low and may not detect other common sites of disease recurrence, such as the retro-peritoneal lymph nodes, upper abdomi-nal organs, or lungs.41,42 Thus, in a pa-tient with symptoms or tests that are concerning for recurrence, physical ex-amination alone may not be sufficient.
Historically, second-look surgeries have been used to assess disease response to primary treatment. Despite negative findings, recurrence rates that range from 35-50% have been reported, and no benefit in overall survival was noted.
Thus, this procedure fell out of favor and is used rarely today.42,43 Since its discovery in 1981, the use of CA125 level for tumor recurrence has been evaluated extensively. Approxi-mately 80% of epithelial tumors will have an elevated CA125 level at the time of diagnosis. Studies have shown that CA125levelcorrelateswithdiseasestatus in most cases and is often elevated 2-5 months before clinical detection of re-lapse.38 Generally, the sensitivity and specificityforCA125levelanddiseasere-currence ranges from 62–94% and 91–100%, respectively.41,42,44,45 In 255 patients who had completed primary therapy, a CA125 level twice the upper limits of normal was consistent with dis-ease progression in almost all patients who were evaluated.46 Santillan et al46 reported that CA125 levels with a persis-tently low level of increase, even within normal values of the test, were often con-sistent with tumor recurrence. However, other reports found that the detection of recurrent disease by CA125 level alone yielded no prognostic benefit and advo-cate the use of CA125 level for surveil-lance only after a discussion that would explain the interpretation of the test.47 Furthermore, in a recently completed prospective randomized trial, the Euro-pean Organization for Research and Treatment of Cancer assessed the out-come of 527 patients who were treated for recurrent ovarian cancer based on CA125 level alone vs clinically evident recurrence. The overall survival out-come did not differ for either group, and the investigators concluded that routine measurement of CA125 level is not war-ranted for disease surveillance.48 To improve early detection of recur-rent disease, the role of radiographic im-aging modalities has been investigated.
In a retrospective analysis, surveillance with CT scans every 6 months for the first 2 years, followed by yearly intervals, demonstrated the ability to detect asymptomatic disease. The authors re-ported a higher rate of optimal second-ary cytoreductive surgery and an im-proved overall survival in the group with recurrence detected asymptomatically, compared with the symptomatic recur-rence.49 Other studies that have evalu-ated methods of surveillance for ovarian cancer have reported the sensitivity of CT scans to be 40-93% and the specific-ity to be 50-98% for recurrent disease.
On the contrary, in a study of 412 pa-tients, the use of surveillance techniques detected recurrence in 80% of patients with the following evaluations: examina-tion (15%), imaging (27%), CA125 level (23%), and CA125 level and imaging in (35%). However, the authors did not find a difference in survival, regardless of TABLE 2 Endometrial cancer surveillance recommendations Variable Months Years 0-12 12-24 24-36 3-5 >5 Review of symptoms and physical examination .......................................................................................................................................................................................................................................................................................................................................................................
Low risk (stage IA grade 1 or 2) Every 6 mo Yearly Yearlya Yearlya Yearlya .......................................................................................................................................................................................................................................................................................................................................................................
Intermediate risk (stage IB-II) Every 3 mo Every 6 mo Every 6 mob Every 6 mob Yearlya .......................................................................................................................................................................................................................................................................................................................................................................
High risk (stage III/IV, serous or clear cell) Every 3 mo Every 3 mo Every 6 mo Every 6 mo Yearlya ................................................................................................................................................................................................................................................................................................................................................................................
Papanicolaou test/cytologic evidence Not indicated Not indicated Not indicated Not indicated Not indicated ................................................................................................................................................................................................................................................................................................................................................................................
Cancer antigen 125 Insufficient data to support routine use Insufficient data to support routine use Insufficient data to support routine use Insufficient data to support routine use Insufficient data to support routine use ................................................................................................................................................................................................................................................................................................................................................................................
Radiographic imaging (chest x-ray, positron emission tomography/ computed tomography, magnetic resonance imaging) Insufficient data to support routine use Insufficient data to support routine use Insufficient data to support routine use Insufficient data to support routine use Insufficient data to support routine use ................................................................................................................................................................................................................................................................................................................................................................................
Recurrence suspected Computed tomography and/or positron emission tomography scan cancer antigen 125 Computed tomography and/or positron emission tomography scan cancer antigen 125 Computed tomography and/or positron emission tomography scan cancer antigen 125 Computed tomography and/or positron emission tomography scan cancer antigen 125 Computed tomography and/or positron emission tomography scan cancer antigen 125 ................................................................................................................................................................................................................................................................................................................................................................................
a May be followed by a generalist or gynecologic oncologist; b Consider alternating visits with a generalist and gynecologic oncologist.
Salani. Surveillance for gynecologic cancers. Am J Obstet Gynecol 2011.
www.AJOG.org Oncology Reviews JUNE 2011 American Journal of Obstetrics & Gynecology 469 the modality that was used.50 Ideally, prospective studies will help to deter-mine the true role of interval CT scans in ovarian cancer surveillance.
Because CT scans may lack the ability to detect a small volume of disease, other imaging modalities have been reviewed.
The use of magnetic resonance imaging has also been evaluated for its role in ovarian cancer surveillance. Although sensitivity ranges from 62-91% and specificity ranges from 40-100%, com-parable detection rates to CT scans and increased costs have limited its general-ized acceptance.41 Ultrasound scanning has also been investigated for ovarian cancer surveillance. Studies have shown sensitivity that ranged from 45-85% and specificity that ranged from 60-100%.41 However, because of user variability and limited visibility, this modality typically is not used for the evaluation of recur-rent disease.
More recently, the use of PET-CT scans has been reported. Sensitivity var-ies from 45-100% and specificity ranges from 40-100%, although diagnostic accuracy rates approach as high as 95%.41,42,51 In patients with normal CA125 levels and clinical suspicion of disease (based on symptoms or surveil-lance CT scans), PET-CT was slightly more sensitive than CT scans for the de-tection of recurrent disease.52 Studies have shown that PET-CT will alter treat-ment in approximately 60% of patients with recurrent disease and many recom-mend PET-CT before secondary cytore-duction.53 However, the potential use of this modality for surveillance is limited, and currently the role of radiographic imaging is best reserved as a supplement to abnormalities in physical examina-tion, CA125 levels, or symptoms.
Although improvements in primary treatmentofovariancancerhaveoccurred, outcomes after recurrence remain disap-pointing. Many physicians hypothesize that the detection of recurrence early po-tentially may improve the benefit of avail-able treatments, especially surgery. Sec-ond-line therapies are rarely curative and often result in short-term progression-free survival. However, some patients, espe-cially those who are good candidates for secondary surgical cytoreduction and/or thosewhoremainplatinumsensitivewill have high response rates to salvage treat-ments. Until the ideal surveillance is de-termined, individualized patient plans that consist of a thorough assessment of symptoms and physical examination, which includes a pelvic examination, should be undertaken. The role for CA125 level monitoring should be dis-cussed with patients. The pros and cons of imaging should be discussed with the patients who do not have an elevated CA125 level at the time of diagnosis.
When a recurrence is suspected based on symptoms, examination, or CA125 level, a CT scan of the chest, abdomen, and pelvis should be obtained to determine the extent of the disease. PET scans are a useful adjunct when CT scans are inde-terminate (Table 3).54 Lowmalignantpotential(LMP)tumors Tumors of LMP, also called borderline tumors, account for 10-20% of epithelial ovarian tumors; approximately 4000 cases are diagnosed annually.55 The av-erage age of a woman at the time of diag-nosis is 40-60 years, but a significant pro-portion of these tumors occur in women in their child-bearing years.55 In general, the prognosis for women with LMP tu-mors is quite good, and most women (especially those with stage I disease) are at a very low risk of recurrence.2,3 Recur-rences tend to occur late, and, even in advanced stages, 70% of recurrences will be after 5 years, and 30% will be after 10 TABLE 3 Ovarian cancer surveillance recommendations Variable Months Years 0-12 12-24 24-36 3-5 >5 Review of symptoms and physical examination Every 3 mo Every 3 mo Every 4-6 mo Every 6 mo Yearlya ................................................................................................................................................................................................................................................................................................................................................................................
Papanicolaou test/ cytologic evidence Not indicated Not indicated Not indicated Not indicated Not indicated ................................................................................................................................................................................................................................................................................................................................................................................
Cancer antigen 125 Optional Optional Optional Optional Optional ................................................................................................................................................................................................................................................................................................................................................................................
Radiographic imaging (chest x-ray, positron emission tomography/ computed tomography, magnetic resonance imaging) Insufficient data to support routine use Insufficient data to support routine use Insufficient data to support routine use Insufficient data to support routine use Insufficient data to support routine use ................................................................................................................................................................................................................................................................................................................................................................................
Recurrence suspected Computed tomography and/or positron emission tomography scan Computed tomography and/or positron emission tomography scan Computed tomography and/or positron emission tomography scan Computed tomography and/or positron emission tomography scan Computed tomography and/or positron emission tomography scan ................................................................................................................................................................................................................................................................................................................................................................................
Cancer antigen 125 Cancer antigen 125 Cancer antigen 125 Cancer antigen 125 Cancer antigen 125 ................................................................................................................................................................................................................................................................................................................................................................................
a May be followed by a generalist or gynecologic oncologist.
Salani. Surveillance for gynecologic cancers. Am J Obstet Gynecol 2011.
Reviews Oncology www.AJOG.org 470 American Journal of Obstetrics & Gynecology JUNE 2011 years.3 Many patients with recurrent LMP tumors can be salvaged with additional surgery, and 5% eventually progress to invasive cancers.55-57 Current NCCN guidelines recom-mend physical examination, including pelvic examinations, CA125 level (if ini-tially elevated), every 3-6 months and pelvic ultrasound scans for those women with fertility-sparing surgery. Complete hysterectomy with bilateral salpingo-oophorectomy is recommended once fertility is completed.37 However, there are no studies that suggest that this ag-gressive surveillance improves prognosis for women with LMP tumors.
Retrospective studies suggest that, in women who have undergone a complete hysterectomy with bilateral salpingo-oophorectomy and resection of all gross disease, surveillance should be similar that used for those women with invasive ovarian cancer. For patients who have undergone fertility-sparing surgery, ei-ther a unilateral salpingo-oophorectomy or a cystectomy, the risk of recurrence ranges from 7–30%.58 Current surveil-lance recommendations for women who have undergone fertility-preserving sur-gery are to undergo serial pelvic sonog-raphy because this is the most sensitive method of detection of recurrent disease in residual ovary.59 Ultrasound scanning with or without tumor markers is rec-ommended on an every 6-month basis.
When recurrent disease is suspected, a CT scan of abdomen and pelvis is recom-mended to assess the extent of the dis-ease. Because most women with LMP tu-mors can be salvaged with additional surgery,56,57 prompt attention to symp-tomsorphysicalexaminationabnormal-ities is important; however, there is no evidence that routine radiographic sur-veillancewithCTscansisatallbeneficial.
Germ cell and sex-cord stromal tumors of the ovary Malignant germ cell tumors of the ovary are rare and account for 2.6% of all ovar-ian cancers.60 Most patients have ab-dominal pain and a palpable mass. Ma-lignant germ cell tumors can produce serum tumor markers that can prove helpful in the diagnosis and posttreat-ment surveillance if they are elevated at the time of diagnosis. Alpha-fetoprotein can be produced by yolk sac tumors, em-bryonal carcinomas, polyembryomas, and immature teratomas. Human chori-onic gonadotropin can be produced by choriocarcinomas, embryonal carcino-mas, polyembryomas, and, in low levels, in some dysgerminomas. Lactate dehy-drogenase can be a marker for dysgermi-noma.61-63 Because these tumors tend to occur in young women and most are unilateral, fertility-sparing surgery has been used to include pelvic washings, unilateral salpingo-oophorectomy, peri-toneal biopsies, omentectomy, and pel-vic and paraaortic lymph node dissec-tion.
NCCN guidelines recommend observation for low-risk tumors such as stage I dysgerminomas and stage IA, grade 1 immature teratomas.37,61-63 All other malignant ovarian germ cell tu-mors in this country receive postopera-tive chemotherapy with bleomycin, eto-poside, and platinum with excellent survival rates. However, in Europe some healthcare providers advocate observa-tion of all stage I germ cell tumors.
Sex cord stromal tumors are rare and account for 1.2% of ovarian cancers.60 Sex cord stromal tumors of the ovary can also produce serum tumor markers such as estradiol, inhibin, Müllerian inhibi-tory substance, and testosterone.63,64 Granulosa cell tumors also have the pos-sibility of late recurrence of disease, with a reported median time to recurrence of 4-6 years.62 Pelvic recurrence accounts for 30-45% of cases.64,65 Surveillance should include a thorough physical ex-amination and serum tumor markers for an extended period of time because of reports of recurrence even 20 years after the initial diagnosis. The utility of imaging in sex-cord stromal tumors has not been proven, so imaging should be limited to patients with symptoms or concerning findings on physical examination.64,65 Studies that have evaluated surveil-lance strategies for ovarian germ cell tu-mors and sex cord stromal tumors have not been performed; therefore, recom-mendations are based on expert opinion.
NCCN guidelines for surveillance rec-ommend tumor markers every 2-4 months for 2 years if the markers were elevated originally. Physical examina-tion that includes bimanual examination may be less helpful than serum tumor markers, especially in adolescent pa-tients. Although recurrences are rare and data about them in the gynecologic on-cology literature are small in number, they typically occur in the first 2 years after treatment. Although prognosis for recurrent germ cell tumors is usually poor, there are potentially curative treat-ment options that are available with multiagent chemotherapy regimens and high-dose chemotherapy with autolo-gous stem cell support. Recently, the American Society of Clinical Oncology issued guidelines for surveillance using serum tumor markers for men with tes-ticular cancer.66 The recommendations were similar to the current NCCN guide-lines for germ cell tumors of the ovary in the first 2 years, with the exception that the surveillance continues for 10 years after treatment because of a reported in-cidence of 50% of the recurrences occur-ring 5 years after treatment in men.66 The timing of surveillance imaging in ovarian germ cell tumors is less well-characterized.
NCCN guidelines for germ cell testicular tumors recommend CT scans every 3-6 months for the first 2 years then every 6-12 months until 6 years after treatment for those who re-ceived chemotherapy alone.67 Because germ cell tumors of the ovary occur in young women, because serum tumor markers are very sensitive for the pres-ence of disease, and because repeated CT scans can lead to significant radiation ex-posure over time, the argument could be made that imaging is not indicated with-out evidence of the elevation of serum tumor markers, clinical symptoms, or concerning findings on physical exami-nation. In addition, in those patients without elevated tumor markers, radio-logic assessment in the first 2 years can be helpful (Table 4).
Cervical cancer More than 12,000 women are diagnosed with cervical cancer each year in the United States.1 Patients are diagnosed with stage I disease in 50% of cases, and the 5-year survival rate for this group ex-www.AJOG.org Oncology Reviews JUNE 2011 American Journal of Obstetrics & Gynecology 471 ceeds 90%.6 However, recurrence rates for this group of patients are high, rang-ing from 10–20%.68 The treatment of re-current cervical cancer depends greatly on the primary therapy that is used and the location of recurrence. Patients with locally recurrent disease can be offered salvage treatments with the potential for cure. Distant metastases are rarely sal-vageable. In efforts to detect disease at curable states, surveillance has been ad-vocated in patients who have success-fully completed primary treatment.
Typically, more than three-fourths of recurrences will occur within the first 2-3 years after the initial treatment, which suggests a role for increased surveillance during this time frame.68-72 Thus, the NCCN guidelines recommend fol-low-up evaluation every 3-6 months for the first 2 years, followed by every 6 months for the next 3 years. These rec-ommendations include cytologic evalu-ation at each visit and recommend annual chest radiographs, although op-tional.71 Use of other imaging is advo-cated on the basis of clinical indications.
Similarly to most cancer surveillance, these recommendations are based on retrospective studies.
Although patients are often observed every 3-4 months during the first 2 years, recurrence is diagnosed during routine follow-up examination in few cases, ranging from 26–36% of cases.69,72 De-spitesurveillance,presentationwithsymp-tomsiscommon,rangingfrom46–95%of patients.73,74-81 These symptoms often in-clude abdominal and pelvic pain, leg symptoms such as pain or lymphedema, vaginal bleeding or discharge, urinary symptoms, cough, and weight loss.68,73 Additionally, the presence of symptoms or suspicion of recurrence prompted un-scheduled evaluation in approximately 40% of patients.77,78 Thus, counseling pa-tients about signs and symptoms remains an important part of survivorship care.
The use of physical examination for cervical cancer surveillance has been well accepted.Inareview,thissimplemethod accounted for the highest rate of asymp-tomatic disease, ranging from 29– 75%.11,68 Physical examination ac-counted for the highest detection rate when compared with cytologic evalua-tion and imaging modalities.11,68,77 The evaluation should include a complete as-sessment of areas that are susceptible to the human papilloma virus and a thor-ough speculum, bimanual, and recto-vaginal examination. Although there is insufficient evidence in cancer surveil-lance, cytologic evaluation may have value in the detection of other lower gen-ital tract neoplasia. Along with symp-toms, physical examination will detect most cases of recurrent cervical cancer.76 In efforts to detect patients with a vag-inal/local recurrence, surveillance with cytologic evaluation has been recom-mended.64,70-73 Unfortunately, retro-spective studies have shown cytologic evaluation to be consistently low yield, with detection rates that range from 0–17%.68 In addition, other studies have found that rarely was cytologic evidence theonlyabnormalityandthatclinicalev-TABLE 4 Nonepithelial ovarian cancer (germ cell and sex-cord stromal tumors) surveillance recommendations Variable Months Years 0-12 12-24 24-36 3-5 >5 Review of symptoms and physical examination .......................................................................................................................................................................................................................................................................................................................................................................
Germ cell tumors Every 2-4 mo Every 2-4 mo Yearly Yearly Yearly .......................................................................................................................................................................................................................................................................................................................................................................
Sex-cord stromal tumors Every 2-4 mo Every 2-4 mo Every 6 mo Every 6 mo Every 6 mo ................................................................................................................................................................................................................................................................................................................................................................................
Serum tumor markers .......................................................................................................................................................................................................................................................................................................................................................................
Germ cell tumors Every 2-4 mo Every 2-4 mo Not indicated Not indicated Not indicated .......................................................................................................................................................................................................................................................................................................................................................................
Sex-cord stromal tumors Every 2-4 mo Every 2-4 mo Every 6 mo Every 6 mo Every 6 mo ................................................................................................................................................................................................................................................................................................................................................................................
Radiographic imaging (chest x-ray, computed tomography, magnetic resonance imaging) .......................................................................................................................................................................................................................................................................................................................................................................
Germ cell tumors Not indicated unless tumor marker normal at initial presentation Not indicated unless tumor marker normal at initial presentation Not indicated Not indicated Not indicated .......................................................................................................................................................................................................................................................................................................................................................................
Sex-cord stromal tumors Insufficient data to support routine use Insufficient data to support routine use Insufficient data to support routine use Insufficient data to support routine use Insufficient data to support routine use ................................................................................................................................................................................................................................................................................................................................................................................
Recurrence suspected Computed tomography scan Computed tomography scan Computed tomography scan Computed tomography scan Computed tomography scan ................................................................................................................................................................................................................................................................................................................................................................................
Tumor markers Tumor markers Tumor markers Tumor markers Tumor markers ................................................................................................................................................................................................................................................................................................................................................................................
Salani. Surveillance for gynecologic cancers. Am J Obstet Gynecol 2011.
Reviews Oncology www.AJOG.org 472 American Journal of Obstetrics & Gynecology JUNE 2011 idenceofdiseasewasoftenorsoonthere-after apparent. These low rates of detec-tion have led to the recommendations by authors to eliminate the use of cytologic evaluation or to limit its use to once a year.68,74,75 Furthermore, the role of cy-tologic evaluation in patients who have undergone pelvic radiation therapy may be limited, and the elimination of its use from routine surveillance may be accept-able. Thus, the reduction of unnecessary cytologic evaluation may provide an op-portunity for significant cost-savings whilemaintainingqualityofcareinthese patients.
Imaging has also been suggested for surveillance in the asymptomatic pa-tient. In regards to chest radiographs, rates of detection range from 20– 47%.68,69,74 Because of a higher distant failure rate, Salmal et al69 advocated its use, particularly in patients who had re-ceived radiotherapy. However, because many of these cases are not salvageable, others have questioned its use.75,76 Al-though some studies have reported suc-cessful treatments for patients with iso-lated pulmonary recurrence, there is little evidence to support its use at this time.68,69,74 Other studies have evaluated the use of radiographic imaging modali-ties (such as CT scan and magnetic reso-nance imaging), pelvic ultrasound scans, and intravenous pyelograms.11,69 Unfor-tunately, the rates of detection are low, and these tests have not proven useful for routine surveillance. However, these tests may be indicated based on patient symptoms or findings on examination, and their use should be individualized.
PET CT scans have also been used for the evaluation of recurrent cervical cancer. In patients with clinical suspi-cion of recurrence, PET scans detected disease with high sensitivity (86%) and specificity (87%).77 More recently, its use as a surveillance tool has been stud-ied with promising results. In this series, PET-CT showed locoregional disease in 8 of the 9 asymptomatic patients, com-pared with 4 of the 21 with symptoms that were being evaluated.78 Because pelvic recurrences may be amenable to salvage therapy, with radiation or ex-enteration, this modality may have po-tential benefit; further investigations are ongoing.
One of the major components of sur-veillance is its ability to impact survival.
Survival for women with recurrent cer-vical cancer has been assessed only in retrospective analyses, which compare those women with or without symptoms at the time recurrence is diagnosed. Me-dian survival rates in asymptomatic and symptomatic patients ranged from 8–53 months and 8–38 months, respectively.68 Several studies have reported improved median survival in patients who were de-tected with asymptomatic recurrence, re-gardless of the method of diagnosis, and advocate the need for surveillance pro-grams.69,74,77-80 Other reports have noted similar survival regardless of symptoms and have questioned the effectiveness of routine surveillance.72,76 Surveillance should be focused on re-current disease that is amenable to treat-ment and that will result in cure or long-term survival. Unfortunately, in regards to cervical cancer, this is limited pre-dominantly to locoregional recurrence.
The potential of newer modalities, which includes PET/CT scanning, must be in-vestigated further in prospective studies, especially given the high cost. Although only retrospective data are available, his-tory and physical examination are the only consistent methods that have been reported for the detection of recurrence; and specific follow-up plans should be discussed with patients. If recurrent dis-ease is suspected based on symptoms or examination, a CT scan of the chest, ab-domen, and pelvis is recommended to evaluate the extent of disease, and a bi-opsy should be obtained to confirm re-currence. PET/CT scanning usually is performed before definitive radiation or exenterative surgery to identify distant disease that would alter management (Table 5).81 Vulvar cancer With 3900 new cases and 920 deaths an-nually in the United States, vulvar cancer is uncommon and represents approxi-mately 4% of malignancies of the female genital tract and 0.6% of all cancers in women.1 Radical local excision of the vulva and inguinofemoral lymphade-nectomy has been the standard surgical therapy for nearly 8 decades. More re-cent advances have included the intro-duction of preoperative chemoradiation for large primary tumors that involve the urethra, vagina, or anus and the investi-gation of the sentinel lymph node tech-nique. Survival of patients with vulvar cancer correlates with International Fed-eration of Gynecology and Obstetrics stage. The prognosis for patients with early-stage disease is generally good.
Lymph node status is the single most im-portant prognostic factor. Patients with negative lymph nodes have a 5–year sur-vival rate of 80%, which falls to 50% for patients with positive lymph nodes and to as low as 13% for those with 4 positive nodes.82 Although patients with local recurrences may be salvageable, groin or distant recurrences generally are fatal.
There is no direct evidence to inform surveillance strategies for patients with vulvar cancer after definitive treatment.
There are no NCCN practice guidelines to address this issue. Thus, surveillance strategies for patients with definitively treated vulvar cancer are extrapolated from other disease sites, mainly cervix cancer. A report from the Mayo Clinic on 330 patients with primary squamous cell carcinoma of the vulva, 95% of whom underwent bilateral inguinofemo-ral lymphadenectomy, underscores the significant correlation between lymph node status and the risk of treatment fail-ure in the first 2 years after initial ther-apy: 44.2% overall recurrence rate with positive vs 17.5% with negative lymph nodes. After 2 years, patients with posi-tive and patients with negative nodes had similar recurrence rates. Importantly, more than one-third of relapses oc-curred 5 years after the initial therapy.
In other words, nearly 1 in 10 patients hadalate(5years)reoccurrenceofdis-ease (same site recurrence or second pri-mary vulvar site), which demonstrates the need for long-term surveillance.
More than 95% of those late relapses had local reoccurrences; 13% of the relapses also demonstrated evidence of distant disease.83 This pattern of predominantly local recurrence is confirmed by another study of 399 patients with node-negative www.AJOG.org Oncology Reviews JUNE 2011 American Journal of Obstetrics & Gynecology 473 squamous cell carcinoma of the vulva, 23%ofwhichrecurredwith90%ofthe recurrences in the vulva.84 Because of the propensity for local re-currence (regular and long-term), care-ful examinations of the vulva and groin constitute the cornerstone of posttreat-ment surveillance for these patients. This should include careful visual inspection of the vulva, skin bridge, and inguinal lymph nodes. Because a significant num-ber of vulvar cancers are human papillo-mavirus associated, such examination should survey not only for vulvar reoc-currence or multifocal vulvar cancer but also for cervical, vaginal, and perianal neoplasia. Whether asymptomatic pa-tients with positive groin nodes benefit from additional imaging for the assess-ment of distant sites of failure is un-proven and generally not recommended because salvage therapies are relative in-effective. Patients whose symptoms or review of systems suggests the possibility for distant failure should undergo addi-tional imaging and may be evaluated similarly as with patients with cervical cancer. If exenterative surgery is con-sidered for local recurrence, PET-CT should be performed to rule out distant disease that would alter management (Table 5).84 Vaginal cancer Primary cancer of the vagina is an un-common malignancy. With approxi-mately 2300 cases diagnosed annually in the United States, vaginal cancer com-prises approximately 3% of all malignant neoplasms of the female genital tract.1 Given the rarity of the disease, there is a paucity of information to guide post-treatment surveillance for patients with vaginal cancer. There are no data to sup-port the routine use of follow-up vaginal cytologic evaluation or imaging in the asymptomatic patient.
Posttreatment surveillance relies primarily on the care-ful assessment of symptoms and physical examination, which should survey not only for vaginal recurrence or multifocal vaginal cancer but also for cervical, vul-var, and perianal neoplasia. Patients with a suspicion of recurrent disease should undergo additional imaging for the eval-uation of disease extent that may help guide treatment options (Table 5).85 Comment Although gynecologic cancers account for only 10% of all new cancer cases in women, the number of survivors from these malignancies approaches 20%.1,86 Improvements in cancer care have re-sulted in almost 10 million cancer survi-vors, and this number is expected to grow at an even faster rate than ever be-fore.86 Thus, the determination of the most clinically and cost-effective surveil-lance for the detection of recurrence is critical.
As survivorship increases, transition-ing patients from oncology care to the primary care setting is becoming a com-mon practice. However, this shift results in the burden of care falling on primary care providers who may not be comfort-able or trained to deal with follow-up TABLE 5 Cervical, vulvar, and vaginal cancer surveillance recommendations Variable Months Years 0-12 12-24 24-36 3-5 >5 Review of symptoms and physical examination .......................................................................................................................................................................................................................................................................................................................................................................
Low risk (early stage, treated with surgery alone, no adjuvant therapy) Every 6 mo Every 6 mo Yearlya Yearlya Yearlya .......................................................................................................................................................................................................................................................................................................................................................................
High risk (advanced stage, treated with primary chemotherapy/ radiation therapy or surgery plus adjuvant therapy) Every 3 mo Every 3 mo Every 6 mo Every 6 mo Yearlya ................................................................................................................................................................................................................................................................................................................................................................................
Papanicolaou test/cytologic evidence Yearlyb Yearlyb Yearlyb Yearlyb Yearlyb ................................................................................................................................................................................................................................................................................................................................................................................
Routine radiographic imaging (chest x-ray, positron emission tomography/computed tomography, magnetic resonance imaging) Insufficient data to support routine use Insufficient data to support routine use Insufficient data to support routine use Insufficient data to support routine use Insufficient data to support routine use ................................................................................................................................................................................................................................................................................................................................................................................
Recurrence suspected Computed tomography and/or positron emission tomography scan Computed tomography and/or positron emission tomography scan Computed tomography and/or positron emission tomography scan Computed tomography and/or positron emission tomography scan Computed tomography and/or positron emission tomography scan ................................................................................................................................................................................................................................................................................................................................................................................
a May be followed by a generalist or gynecologic oncologist; b Insufficient evidence for cancer recurrence but may have value in the detection of other lower genital tract neoplasia.
Salani. Surveillance for gynecologic cancers. Am J Obstet Gynecol 2011.
Reviews Oncology www.AJOG.org 474 American Journal of Obstetrics & Gynecology JUNE 2011 needs or practice standards for patients with cancer. Although the Institute of Medicine’s report advocates for open communication between oncologists and primary care providers, almost 50% of primary care physicians did not feel comfortable with cancer surveillance and standard guidelines for cancer re-currence.86 However, primary care pro-viders generally are willing to assume cancer follow-up care, typically after 2 years from treatment. In a survey, pri-mary care providers believed the transi-tion of oncology patients could be im-proved with an individualized treatment summary, guidelines for surveillance, and expedited routes of rereferral for suspected recurrence.2-4,86-88 Thus, the provision of up-to-date information and the education of both patients and phy-sicians are mandatory.
However,arecentevaluationofcancer survivorship care demonstrated a signif-icant discordance among primary care providers, oncologists, and patients.
This discrepancy was seen with primary cancer surveillance and with the recom-mendation of cancer screening and preventative healthcare management.3 Therefore, it is important not only to specify routine cancer surveillance but alsotocontinueroutinescreeningguide-lines in cancer survivors and to promote healthy behaviors. As rates of the devel-opment of a second cancer approach 10% within 30 years,4 communication between providers and with patients will improve adherence to guidelines and re-duce repetitive testing.3 Despite its asso-ciation with cancer and comorbidities, almost one-quarter of cancer survivors continue to use tobacco after the first year of diagnosis; rates, which exceed 37%, are highest in patients with a history of gynecologic malignancies.89 Thus, both oncologists and primary care providers should advocate for smoking cessation in these patients. In addition, the promotion of exercise and weight re-duction (if indicated), of the monitoring of bone density, and of breast and colo-rectal screening is important.
If not previously done, the surveil-lance period may provide an opportu-nity to assess patients who are at a higher risk for cancer than the general popula-FIGURE Checklist for surveillance of gynecologic malignancies Patient name___ Visit date____ Disease site and stage_____ Date of diagnosis/surgery___ Date treatment completed__ Symptoms review and treatment side-effects •Pain (abdominal or pelvic, hip or back) •Abdominal bloating •Vaginal bleeding (also rectum, bladder) •Weight loss •Nausea and/or vomiting •Cough or shortness of breath •Lethargy/fatigue •Swelling of abdomen or leg(s) •Depression •Sexual dysfunction •Neuropathy •Fatigue Physical examination •General physical examination targeted to symptoms •Lymph node assessment (axillary, supraclavicular, and inguinal) •Pelvic examination (evaluation of lower genital tract, speculum, bimanual, rectovaginal examination) Laboratory •Tumor markers_____ Disease status •No evidence of disease •Suspect recurrence ••Radiographic imaging ___ ••Biopsy______ ••Refer to gynecologic oncologist Routine health maintenance Breast cancer screening •Yearly clinical breast examination __ •Mammogram _____ Every 1-2 years starting with ages 40-49 years, then yearly Colon cancer screening •Colonoscopy or flexible sigmoidoscopy ___ Every 5-10 years beginning at age 50 years Genetic screening •Not indicated •Recommended/completed _______ Consider if patient is diagnosed at a young age, strong family history, multiple primaries (see specific surveillance guidelines) Menopausal assessment Osteoporosis prevention Calcium (1200-1500 mg) and vitamin D (800 IU) Bone mineral density testing Begin at age 65 years (sooner if high risk factors) Smoking cessation Weight maintenance (exercise, diet) Salani. Surveillance for gynecologic cancers. Am J Obstet Gynecol 2011.
www.AJOG.org Oncology Reviews JUNE 2011 American Journal of Obstetrics & Gynecology 475 tion. Obtaining a thorough personal and family history, which would include can-cer type and age at diagnosis, may help to identify patients who are at risk and re-sult in a referral to genetic counseling for additional assessment. Furthermore, pa-tients and family members with a known or suspected genetic predisposition may require a more intensive screening pro-gram. Improving one’s awareness of risk will enhance compliance with these rec-ommendations and ultimately decrease preventable cancers.90 The goal of follow-up evaluation for the detection of recurrent disease re-quires both clinical and cost-effective-ness. Failure to adhere to recommended guidelines results in unnecessary tests, and efforts should be made to provide effective surveillance, which will result in cost-savings.86-91 Currently, the ideal tests and schedule for gynecologic cancer surveillance have not yet been estab-lished; however, a detailed review of symptoms and physical examination at each visit results in the detection of most recurrences (Figure). The use of addi-tional modalities has not been well-sup-ported; and individualized treatment plans should be made with each patient.
The lack of evidence-based guidelines for surveillance can be addressed only with prospective studies; the incorpora-tion of cost-effective follow-up plans into the design of clinical trials will help to establish the ideal regimens.
f ACKNOWLEDGMENT The Society of Gynecologic Oncologists’ (SGO) Clinical Practice Committee has developed a series of Clinical Documents that are designed in part to improve the overall quality of women’s cancer care, to reduce the use of unnecessary, ineffective, or harmful interventions, and to fa-cilitate the treatment of patients with a goal to maximum the chance of benefit with a minimum risk of harm and at an acceptable cost.
SGO Clinical Documents remain strictly confi-dential and are not to be disclosed or dissemi-nated by any participant in the process before the Document’s publication.
SGO Clinical Documents may have direct im-pact on the practice of treating women with gynecologic malignancies.
Clinical Documents are intended to be educa-tional devices that provide information that may assist healthcare providers in caring for pa-tients. This Clinical Document is not a rule and should not be construed as establishing a legal standard of care or as encouraging, advocat-ing, requiring, or discouraging any particular treatment. Clinical Documents are not intended to supplant the judgment of the healthcare pro-vider with respect to particular patients or spe-cial clinical situations. Clinical decisions in any particular case involve a complex analysis of a patient’s condition and available courses of ac-tion with the ultimate determination to be made by the healthcare provider in light of each indi-vidual patient’s circumstances. Therefore, clin-ical considerations may lead a healthcare pro-vider to take a course of action appropriately that varies from this Document.
This Clinical Document has met SGO’s criteria of an Expert Clinical Opinion Document.
REFERENCES 1. Jemal A, Siegel R, Xu J, Ward E. Cancer sta-tistics, 2010. CA Can J Clin 2010;60:277-300.
2. Nissen MJ, Beran MS, Lee MW, et al. Views of primary care providers on follow up care of cancer patients. Fam Med 2007;39:477-82.
3. Cheung WY, Neville BA, Cameron DB, Cook EF, Earle CC. Comparison of patient and phy-sician expectations for cancer survivorship care. J Clin Oncol 2009;17:2489-95.
4. Jacobs LA, Palmer SC, Schwartz LA, et al.
Adult cancer survivorship: evolution, research, and planning care. CA Cancer J Clin 2009; 59:391-410.
5. Markman M. Follow-up of the asymptomatic patient with ovarian cancer. Gynecol Oncol 1994;55:S134-7.
6. Altekruse SF, Krapcho M, Neyman N, et al.
SEER cancer statistics review. Bethesda, MD: National Cancer Institute; 1975-2007.
7. Randall ME, Filliaci VL, Muss H, et al. Ran-domized phase III trial of whole abdominal irra-diation versus doxorubicin and cisplatin chemo-therapy in advanced endometrial carcinoma: a Gynecologic Oncology Group study. J Clin On-col 2006;24:36-44.
8. Greer BE, Goff BA, Koh WJ. Endometrial car-cinoma. In: Johnson FR, Virgo KS, eds. Cancer patient follow-up. St. Louis: Mosby; 1997:357-77.
9. Keys HM, Roberts JA, Bruneto VL, et al. A phase III trial of surgery with or without adjunctive external pelvic radiation therapy in intermediate risk endometrial adenocarcinoma: a Gynecologic Oncology Group study. Gynecol Oncol 2004; 92:744-51.
10. Creutzberg CL, van Putten WL, Koper PC, et al (PORTEC Study Group). Survival after re-lapse in patients with endometrial cancer: re-sults from a randomized trial. Gynecol Oncol 2003;89:201-9.
11. Sartori E, Pasinett B, Carrara L, Gambino A, Odicino F, Pecorelli S. Pattern of failure and value of follow-up procedures in endometrial and cervical cancer patients. Gynecol Oncol 2007;107:S241-7.
12. Reddoch JM, Burke TW, Morris M, Tornos C, Levenback C, Gershenson DM. Surveillance for recurrent endometrial carcinoma: develop-ment of a follow-up scheme. Gynecol Oncol 1995;59:221-5.
13. Agboola OO, Grunfeld E, Coyle D, Perry GA. Costs and benefits of routine follow-up af-ter curative treatment for endometrial cancer.
Can Med Assoc 1997;157:879-86.
14. Fung Kee Fung M, Dodge J, Elit L, Lukka H, Chambers A, Oliver T. Follow-up after primary therapy for endometrial cancer: a systematic review. Gynecol Oncol 2006;101:520-9.
15. American College of Obstetricians and Gy-necologists. ACOG practice bulletin no. 65: clinical management guidelines for obstetrician-gynecologists, August 2005: management of endometrial cancer. Obstet Gynecol 2005; 106:413-25.
16. Teng N, Greer B, Kapp D, Kavanagh J, Koh WJ. NCCN practice guidelines for endometrial carcinoma. Oncology 1999;13:45.
17. Bristow RE, Purinton SC, Santillan A, Diaz-Montes TP, Gardner GJ, Giuntoli RL 2nd. Cost-effectiveness of routine vaginal cytology for en-dometrial cancer surveillance. Gynecol Oncol 2006;103:709-13.
18. Tjalma WAA, Van Dam PA, Makar AP, Cruickshank DJ. The clinical value and the cost-effectiveness of follow-up in endometrial cancer patients. Int J Gynecol Cancer 2004;14:931-7.
19. Berchuck A, Anspach C, Evans AC, et al.
Postsurgical surveillance of patients with FIGO stage I/II endometrial adenocarcinoma. Gyne-col Oncol 1995;59:20-4.
20. Sartori E, Pasinetti B, Chiudinelli F, et al.
Surveillance procedures for patients treated for endometrial cancer: a review of the literature. Int J Gynecol Cancer 2010;20:985-92.
21. Gadducci A, Cosio S. Fanucchi A, Cristo-fani R, Genazzani AR. An intensive follow-up does not change survival of patients with clinical stage I endometrial cancer. Anticancer Res 2000;20:1977-84.
22. Shumsky AG, Stuart GC, Brasher PM, Na-tion JG, Robertson DI, Sangkarat S. An evalu-ation of routine follow-up of patients treatment for endometrial carcinoma. Gynecol Oncol 1994;55:229-33.
23. Smith CJ, Heeren M, Nicklin JL, et al. Effi-cacy of routine follow-up in patients with recur-rent uterine cancer. Gynecol Oncol 2007; 107:124-9.
24. Pastner B, Orr JW Jr, Mann WJ Jr. Use of serum CA125 measurement in post treatment surveillance of early-stage endometrial carci-noma. Am J Obstet Gynecol 1990;162:427-9.
25. Morice P, Levy-Piedbois C, Ajaj S, et al.
Value and cost evaluation of routine follow-up for patients with clinical stage I/II endometrial cancer. Eur J Cancer 2001;37:985-90.
26. Sartori E, Lafare B, Gadducci A, et al. Fac-tors influencing survival in endometrial cancer relapsing patients: a Cooperation Task Force (CTF) study. Int J Gynecol Cancer 2003;13: 458-65.
27. Connor JP, Andrews JI, Anderson B, Buller RE. Computed tomography in endometrial car-cinoma. Obstet Gynecol 2000;95:692-6.
Reviews Oncology www.AJOG.org 476 American Journal of Obstetrics & Gynecology JUNE 2011 28. Owen P, Duncan ID. Is there any value in long term follow up of women treated for endo-metrial cancer? BJOG 1996;103:710-3.
29. Rose PG, Sommers RM, Reale FR, Hunter RE, Fournier L, Nelson BE. Serial serum CA 125 measurements for evaluation of recurrence in patients with endometrial carcinoma. Obstet Gynecol 1994;84:12-6.
30. Dowdy SC, Mariani A, Bakkum JN, et al.
Treatment of pulmonary recurrences in patients with endometrial cancer. Gynecol Oncol 2007; 107:242-7.
31. Park JY, Kim EN, Kim DY, et al. Clinical impact of positron emission tomography or positron emission tomography/computed to-mography in the posttherapy surveillance of en-dometrial carcinoma: evaluation of 88 patients.
Int J Gynecol Cancer 2008;18:1332-8.
32. Kitajima K, Murakami K, Yamasake E. Per-formance of FDG-PET/CT in the diagnosis of recurrent endometrial cancer. Ann Nucl Med 2008;22:103-9.
33. Chung HH, Kang WJ, Kim JW, et al. The clinical impact of [(18)F]FDG PET/CT for the management of recurrent endometrial cancer: correlation with clinical and histological findings.
Eur J Nucl Med Mol Imaging 2008;36:1081-8.
34. Harries M, Gore M. Part I: chemotherapy for epithelial ovarian cancer: treatment at first diag-nosis. Lancet Oncol 2002;3:529-36.
35. Armstrong DK, Bundy B, Wenzel L, et al.
Intraperitoneal cisplatin and paclitaxel in ovarian cancer. N Engl J Med 2006;354:34-43.
36. Trimbos JB, Parmar M, Vergote I, et al. In-ternational Collaborative Ovarian Neoplasm trial 1 and Adjuvant Chemo Therapy in Ovarian Neo-plasm trial: two parallel randomized phase III trials of adjuvant chemotherapy in patients with early stage ovarian carcinoma. J Natl Cancer Inst 2003;95:105-12.
37. Morgan RJ Jr, Alvarez RD, Armstrong DL, et al. The NCCN ovarian cancer clinical practice guidelines in oncology. J Natl Compr Cancer Network 2008;6:766-94.
38. Gadducci A, Cosio S, Zola P, Landoni F, Maggino T, Sartori E. Surveillance procedures for patients treated for epithelial ovarian cancer: a review of the literature. Int J Gynecol Cancer 2007;17:21-31.
39. von Georgi R, Schubert K, Grant P, Munst-edt K. Post-therapy surveillance and after-care in ovarian cancer. Eur J Obstet Gynecol Reprod Biol 2004;114:228-33.
40. Fehm T, Heller F, Krämer S, Jäger W, Ge-bauer G. Evaluation of CA125, physical and ra-diological findings in follow-up of ovarian cancer patients. Anticancer Res 2005;25:1551-4.
41. Gadducci A, Cosio S. Surveillance of pa-tients after initial treatment of ovarian cancer.
Crit Rev Oncol Hematol 2009;71:43-52.
42. Vaidya AP, Curtin JP. The follow-up of ovar-ian cancer. Semin Oncol 2003;30:401-12.
43. Greer BE, Bundy BN, Ozols RF, et al. Impli-cations of second-look laparotomy in the con-text of optimally resected stage II ovarian can-cer: a non-randomized comparison using an explanatory analysis: a Gynecologic Oncology Group study. Gynecol Oncol 2005;99:71-9.
44. Prat A, Parera M, Adamo B, et al. risk of re-currenceduringfollow-upforoptimallytreatedad-vanced epithelial ovarian cancer (EOC) with a low-level increase of serum CA-125 levels. Ann Oncol 2009;20:294-7.
45. Rustin GJS, Nelstrop AE, Tuxen MK, Lam-bert HE. Defining progression of ovarian carci-noma during follow-up according to CA 125: a North Thames Ovary Group Study. Ann Oncol 1996;7:361-4.
46. Santillan A, Gary R, Zahurak ML, et al. Risk of epithelial ovarian cancer recurrence in patient with rising serum CA-125 levels within the nor-mal range. J Clin Oncol 2005;23:9338-43.
47. von Georgi, Hopkins ML, Coyle D, Le T, Fung Kee Fung M, Wells G. Cancer antigen 125 in ovarian cancer surveillance: a decision anal-ysis model. Curr Oncol 2007;14:167-72.
48. Rustin GJ, van der Burg ME; on behalf of MRC and EORTC collaborators. A randomized trial in ovarian cancer (OC) of early treatment of relapse based on CA125 level alone versus de-layed treatment based on conventional clinical indicators (MRC OV05/EORTC 55955 trials).
J Clin Oncol 2009;27:18s.
49. Tanner EJ, Chi DS, Eisenhauer EL, Diaz-Montes TP, Santillan A, Bristow RE. Surveil-lance for recurrent ovarian cancer: survival im-pact or lead-time bias? Gynecol Oncol 2010; 117:336-40.
50. Gadducci A, Fuso L, Cosio S, et al. Are surveillance procedures of clinical benefit for patients treated for ovarian cancer? A retro-spective Italian multicentric study. Int J Gynecol Cancer 2009;19:367-74.
51. Gu P, Pan L-L, Wu S-Q, Huang G. CA125, PET alone, PET-CT, CT and MRI in diagnosing recurrent ovarian carcinoma: a systematic re-view of meta-analysis. Eur J Rad 2009;71: 164-74.
52. Bhosale P, Peungjesada S, Wei W, et al.
Clinical utility of positron emission tomography/ computer tomography in the evaluation of sus-pected recurrent ovarian cancer in the setting of normal CA-125 levels. Int J Gynecol Cancer 2010;20:936-44.
53. Fulham MJ, Carter J, Baldey A, Hicks RJ, Ramshaw JE, Gibson M. The impact of PET-CT in suspected recurrent ovarian cancer: a pro-spective multi-centre study as part of the Aus-tralian PET Data Collection Project. Gynecol Oncol 2009;112:462-8.
54. Basu S, Rubello D. PET imaging in the man-agement of tumors of testis and ovary: current thinking and future directions. Minerva Endocri-nol 2008;33:229-56.
55. Trimble CL, Trimble EL. Ovarian tumors of low malignant potential. Oncology (Williston Park) 2003;17:1563-7.
56. Kane A, Uzan C, Rey A, et al. Prognostic factors in patients with ovarian serous low ma-lignant potential (borderline) tumors with perito-neal implants. Oncologist 2009;14:591-600.
57. Silva EG, Gershenson DM, Malpica A, Deavers M. The recurrence and the overall sur-vival rates of ovarian serous borderline neoplasms with noninvasive implants is time dependent.
Am J Surg Pathol 2006;30:1367-71.
58. Boran N, Cil AP, Tulunay G, et al. Fertility and recurrence results of conservative surgery for borderline ovarian tumors. Gynecol Oncol 2005;97:845-51.
59. Zanetta G, Rota S, Lissoni A, et al. Ultra-sound, physical examination, and CA125 mea-surement for the detection of recurrence after conservative surgery for early borderline ovarian tumors. Gynecol Oncol 2001;81:63-6.
60. Quirk JT, Natarajan N. Ovarian cancer inci-dence in the United States, 1992–1999. Gyne-col Oncol 2005;97:519-23.
61. Gershenson DM. Management of ovarian germ cell tumors.
J Clin Oncol 2007;25: 2938-43.
62. Bridgewater JA, Rustin GJS. Management of non-epithelial ovarian tumors. Oncology 1999;57:89-98.
63. Koulouris CR, Penson RT. Ovarian stromal and germ cell tumors. Semin Oncol 2009; 36:126-36.
64. Gilligan TM, Seidenfeld J, Basch EM, et al.
American Society of Clinical Oncology Clinical Practice Guidelines on uses of serum tumor markers in adult males with germ cell tumors.
J Clin Oncol 2010;28:3388-404.
65. Motzer RJ, Agarwal N, Beard C, et al.
NCCN clinical practice guidelines in Oncology: testicular cancer. J Natl Comp Canc Netw 2009;7:672-93.
66. Schumer ST, Cannistra SA. Granulosa cell tumor of the ovary. J Clin Oncol 2003;21: 1180-9.
67. Pectasides D, Pectasides E, Psyrri A. Gran-ulosa cell tumor of the ovary. Cancer Treat Rev 2008;34:1-12.
68. Elit L, Fyles AW, Oliver TK, Devrie-Aboud MC, Fung-Kee-Fung M. Follow-up for women after treatment for cervical cancer. Curr Oncol 2010;17:65-9.
69. Samlal RAK, Van Der Velden J, Eerden TV, Schilthuis MS, Gonzalez D, Lammes FB. Recur-rent cervical carcinoma after radical hysterec-tomy: an analysis of clinical aspects and prog-nosis. Int J Gynecol Cancer 1998;8:78-84.
70. Larson DM, Copeland LJ, Stringer CA, Ger-shenson DM, Malone JM Jr, Edwards CL. Re-current cervical carcinoma after radical hyster-ectomy. Gynecol Oncol 1988;30:381-7.
71. Greer BE, Abu-Rustin NR, Campos SM, et al. Clinical practice guidelines in oncology: cer-vical cancer. J Natl Compr Cancer Network 2010;8:1388-416.
72. Duyn A, Elikeren MV, Kenter G, Zinderman K, Ansink A. Recurrent cervical cancer: detec-tion and prognosis. Acta Obstet Gynecol Scand 2008;81:759-63.
73. Soisson AP, Geszler G, Soper JT, et al. A comparison of symptomatology, physical ex-amination, and vaginal cytology in the detection of recurrent cervical carcinoma after radical hysterectomy. Obstet Gynecol 1990;76:106-9.
74. Bodurka-Bevers D, Morris M, Eifel PJ, et al.
Posttherapy surveillance of women with cervi-www.AJOG.org Oncology Reviews JUNE 2011 American Journal of Obstetrics & Gynecology 477 cal cancer: an outcomes analysis. Gynecol Oncol 2000;78:187-93.
75. Zanagnolo V, Minig LA, Gadducci A, et al.
Surveillance procedures for patients with cervi-cal carcinoma: a review of the literature. Int J Gynecol Cancer 2009;19:306-13.
76. Morice P, Deyrolled C, Rey A, et al. Value of routine follow-up procedures for patients with stage I/II cervical cancer treated with combined surgery-radiation therapy. Ann Oncol 2004; 15:218-23.
77. Havrilesky LJ, Wong TZ, Secord AA, Ber-chuck A, Clarke-Pearson DL, Jones EL. The role of PET scanning in the detection of recur-rent cervical cancer. Gynecol Oncol 2003;90: 186-90.
78. Brooks RA, Rader JS, Dehdashti F, et al.
Surveillance FDG-PET detection of asymptom-atic recurrences in patients with cervical cancer.
Gynecol Oncol 2009;112:104-9.
79. Ansink A, de Barros Lopes A, Naik R, Mon-aghan JM. Recurrent stage IB cervical carci-noma: evaluation of the effectiveness of rou-tine follow-up surveillance. BJOG 1996;103: 1156-8.
80. Zola P, Fuso L, Mazzola S, et al. Could fol-low-up different modalities play a role in asymp-tomatic cervical cancer relapses diagnosis? An Italian multicenter retrospective analysis.
Gynecol Oncol 2007;107:S150-4.
81. Husain A, Akhurst T, Larson S, Alekitar K, Barakat RR, Chi DS. A prospective study of the accuracy of 18Fluorodeoxyglucose positron emission tomography (18FDG PET) in identify-ing sites of metastasis prior to pelvic exentera-tion. Gynecol Oncol 2007;106:177-80.
82. Beller U, Quinn MA, Benedet JL, et al. 26th Annual report on the results of treatment in gy-necological cancer: carcinoma of the vulva. Int J Gynaecol Obstet 2006;95:S7-27.
83. Gonzalez Bosquet J, Magrina JF, Gaffey TA, et al. Long-term survival and disease recur-rence in patients with primary squamous cell carcinoma of the vulva. Gynecol Oncol 2005; 97:828-33.
84. Tantipalakorn C, Robertson G, Marsden DE, et al. Outcome and patterns of recurrence for International Federation of Gynecology and Obstetrics (FIG(O) stages I and II squa-mous cell vulvar cancer. Obstet Gynecol 2009;113:895-901.
85. Basu S, Li G, Alavi A. PET and PET-CT im-aging of gynecological malignancies: present role and future promise. Expert Rev Anticancer Ther 2009;9:75-96.
86. Hewitt M, Greenfield S, Stovall E, et al. From cancer patient to cancer survivor: lost in transi-tion. Washington, DC: National Academies Press: 2006.
87. Del Giudice ME, Grunfeld E, Harvey BJ, Bili-otis E, Verma S. Primary care physicians’ views of routine follow up care of cancer survivors.
J Clin Oncol 2009;27:3338-45.
88. Hewitt M, Bamundo A, Day R, Harvey C.
Perspectives on post treatment cancer care: qualitative research with survivors, nurses, and physicians. J Clin Oncol 2007;24:2270-3.
89. Belizzi KM, Rowland JH, Jeffrey DD, Mc-Neel T. Health behaviors of cancer survivors: examining opportunities for cancer control in-tervention. J Clin Oncol 2005;23:8884-93.
90. American College of Obstetricians and Gy-necologists. Guidelines for Women’s Health Care: a resource manual, 3rd ed. Bethesda, MD: The College: 2007.
91. Keating NL, Landrum MB, Guadagnoli E, Winer EP, Ayanian JZ. Surveillance testing among survivors of early stage breast cancer.
J Clin Oncol 2007;25:1074-81.
Reviews Oncology www.AJOG.org 478 American Journal of Obstetrics & Gynecology JUNE 2011 |
13012 | https://mathoverflow.net/questions/158575/more-than-n-approximately-orthonormal-vectors-in-rn | linear algebra - More than $n$ approximately orthonormal vectors in $R^n$ - MathOverflow
Join MathOverflow
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
MathOverflow helpchat
MathOverflow Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
More than n n approximately orthonormal vectors in R n R n
Ask Question
Asked 11 years, 7 months ago
Modified4 years ago
Viewed 2k times
This question shows research effort; it is useful and clear
16
Save this question.
Show activity on this post.
This question was asked at math.stackexchange, where it got several upvotes but no answers.
It is impossible to find n+1 n+1 mutually orthonormal vectors in R n R n.
However, it is well established that the central angle between legs of a regular simplex with n n-dimensional volume goes as θ=a r c c o s(−1/n)θ=a r c c o s(−1/n). This approaches 90 90 degrees as n→∞n→∞, so since there are n+1 n+1 vertices of a simplex with n n-dimensional volume, we can conclude
Given ϵ>0 ϵ>0, there exists a n n such that we can find n+1 n+1 approximately mutually orthogonal vectors in R n R n, up to tolerance ϵ ϵ. (Unit vectors u u and v v are said to be approximately orthogonal to tolerance ϵ ϵ if their inner product satisfies ⟨u,v⟩<ϵ⟨u,v⟩<ϵ)
My question is a natural generalization of this - if we can squeeze n+1 n+1 approximately mutually orthonormal vectors into R n R n for n n sufficiently large, how many more vectors can we squeeze in? n+2 n+2? n+m n+m for any m m? 2 n 2 n? n 2 n 2? e n e n?
Actually the n+m n+m case is easy to construct from the n+1 n+1 case. Given ϵ ϵ, one finds the k k such that you can have k+1 k+1 ϵ ϵ-approximate mutually orthogonal unit vectors in R k R k. Call these vectors v 1,v 2,...,v k v 1,v 2,...,v k. Then you could squeeze m k+m m k+m vectors in R m k R m k, by using the vectors
⎡⎣⎢⎢⎢⎢v 1 0⋮0⎤⎦⎥⎥⎥⎥,⎡⎣⎢⎢⎢⎢v 2 0⋮0⎤⎦⎥⎥⎥⎥,⎡⎣⎢⎢⎢⎢v k+1 0⋮0⎤⎦⎥⎥⎥⎥,⎡⎣⎢⎢⎢⎢0 v 1⋮0⎤⎦⎥⎥⎥⎥,⎡⎣⎢⎢⎢⎢0 v 2⋮0⎤⎦⎥⎥⎥⎥,⎡⎣⎢⎢⎢⎢0 v k+1⋮0⎤⎦⎥⎥⎥⎥,…⎡⎣⎢⎢⎢⎢0 0⋮v 1⎤⎦⎥⎥⎥⎥,⎡⎣⎢⎢⎢⎢0 0⋮v 2⎤⎦⎥⎥⎥⎥,⎡⎣⎢⎢⎢⎢0 0⋮v k+1⎤⎦⎥⎥⎥⎥.[v 1 0⋮0],[v 2 0⋮0],[v k+1 0⋮0],[0 v 1⋮0],[0 v 2⋮0],[0 v k+1⋮0],…[0 0⋮v 1],[0 0⋮v 2],[0 0⋮v k+1].
So, setting n=m k n=m k, we have found an n n such that we can fit n+m n+m ϵ ϵ-orthogonal unit vectors in R n R n.
I haven't been able to construct anything stronger than n+m n+m, but I also haven't been able to show that this is the upper bound.
linear-algebra
mg.metric-geometry
measure-concentration
Share
Share a link to this question
Copy linkCC BY-SA 4.0
Cite
Improve this question
Follow
Follow this question to receive notifications
edited Sep 14, 2021 at 1:16
Tony Huynh
32.5k 11 11 gold badges 114 114 silver badges 189 189 bronze badges
asked Feb 25, 2014 at 1:00
Nick AlgerNick Alger
1,180 7 7 silver badges 15 15 bronze badges
2
5 The tag "measure-concentration" is a good hint: in large dimension, two uniform random unit vectors are close to orthogonal with very high probability, so you get (exponentially) many pairwise almost orthogonal vectors by simply drawing them at random.Benoît Kloeckner –Benoît Kloeckner 2014-02-25 09:53:34 +00:00 Commented Feb 25, 2014 at 9:53
7 Getting ϵ−2 exp n ϵ−2 expn vectors is a special case of a very general result that is easy to prove. Google "Johnson-Lindenstrauss Lemma".Bill Johnson –Bill Johnson 2014-02-25 14:53:44 +00:00 Commented Feb 25, 2014 at 14:53
Add a comment|
4 Answers 4
Sorted by: Reset to default
This answer is useful
20
Save this answer.
Show activity on this post.
A set of points on the unit sphere in R n R n with ⟨x,y⟩≤cos θ⟨x,y⟩≤cosθ for all distinct x x and y y is called a spherical code with minimum angle θ θ. For 0<θ<π/2 0<θ<π/2, Kabatiansky and Levenshtein gave an exponential upper bound (of the form exp(C(θ)n)exp(C(θ)n)) for the maximum number of points in such a spherical code. There is also an exponential lower bound. This is related to sphere packings. See for example the recent paper by Cohn and Zhao, which will have more references:
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Improve this answer
Follow
Follow this answer to receive notifications
answered Feb 25, 2014 at 3:33
LuciaLucia
44.2k 6 6 gold badges 198 198 silver badges 222 222 bronze badges
1
1 Here is the paper by Kabatiansky & Levenshtein. The original Russian-language version is available for free. mathnet.ru/php/… I can't find the English-translation online; Springer only has "Problems of Information Transmission" back to volume 37 (January 2001). Cohn & Zhao claim simplify and strengthen their argument, so unlikely to be worth tracking down.Jess Riedel –Jess Riedel 2016-12-28 20:44:28 +00:00 Commented Dec 28, 2016 at 20:44
Add a comment|
This answer is useful
8
Save this answer.
Show activity on this post.
From the area estimates you get that that for fixed ε>0 ε>0 this number, say M ε(n)M ε(n), grows quite fast.
Direct calculations show that the total area of the locus of unit vectors in R n R n which are not ε ε-perependicular to the given vector u u is about
2⋅e−(n−2)⋅ε 2/2⋅area S n−1.2⋅e−(n−2)⋅ε 2/2⋅areaS n−1.
In particular,
M ε(n)>1 2⋅e(n−2)⋅ε 2/2>1 100⋅λ n M ε(n)>1 2⋅e(n−2)⋅ε 2/2>1 100⋅λ n
where λ=e ε 2/2>1 λ=e ε 2/2>1
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Improve this answer
Follow
Follow this answer to receive notifications
edited Feb 25, 2014 at 3:39
answered Feb 25, 2014 at 3:18
Anton PetruninAnton Petrunin
46.7k 17 17 gold badges 141 141 silver badges 311 311 bronze badges
4
Question: this 1/2 e^((n-2)𝜀^2/2) lower bound on the maximum number of nearly orthogonal vectors in R^n seems to be tighter than this other answer which states a lower bound of e^(n𝜀^2/8). How did you arrive at your lower bound, or do you have a source for it? I'm curious to know what is at the root of this discrepancy (or if I'm just misunderstanding something).Hans Brende –Hans Brende 2025-06-16 00:41:55 +00:00 Commented Jun 16 at 0:41
@HansBrende It was a while ago; if I made a mistake, then feel free to correct it. Anyway, if p≈0 p≈0 denotes the probability that two unit vectors ar not ε ε-orthogonal, then p⋅(N 2)p⋅(N 2) is the expected number of pairs of vectors among N N that are not ε ε-orthogonal. Therefore M⩾N−2⋅p⋅(N 2)M⩾N−2⋅p⋅(N 2), and you can optimize for N N.Anton Petrunin –Anton Petrunin 2025-06-16 00:52:55 +00:00 Commented Jun 16 at 0:52
Full disclosure: I only started learning about this topic today, so I would have no idea if you made a mistake! But I am trying to determine what the tightest known deterministic lower bound on that maximum number is. Many answers I've found only show the upper bound on the maximum number, and a lot seem to be based on probabilistic arguments (e.g. your comment above)... but that confuses me: wouldn't a probabilistic argument only demonstrate a probable lower bound rather than an actual provable, deterministic lower bound? I feel like I am missing something here.Hans Brende –Hans Brende 2025-06-16 01:03:31 +00:00 Commented Jun 16 at 1:03
1 @HansBrende Probabolistic method gives actual provable boud, it is just a convinient way to count. Indeed if a configuration appears with positive probability then it exists! (Start with wikipdia en.wikipedia.org/wiki/Probabilistic_method)Anton Petrunin –Anton Petrunin 2025-06-16 04:13:10 +00:00 Commented Jun 16 at 4:13
Add a comment|
This answer is useful
5
Save this answer.
Show activity on this post.
Terry Tao has a nice blog post on a 'cheap version' of the Kabatjanskii-Levenstein bound mentioned in Lucia's answer, using the so-called 'tensor product trick'.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Improve this answer
Follow
Follow this answer to receive notifications
answered Feb 26, 2014 at 0:07
Tony HuynhTony Huynh
32.5k 11 11 gold badges 114 114 silver badges 189 189 bronze badges
Add a comment|
This answer is useful
3
Save this answer.
Show activity on this post.
This is also related to equiangular tight frames (ETFs). A frame is a kind of "overcomplete basis" of an inner product space; more precisely a family (f 1,…,f n)(f 1,…,f n) of vectors of an inner product space V V is a frame if there exist constants A,B>0 A,B>0 such that for all f∈V f∈V it holds that
A∥f∥2≤∑|⟨f,f k⟩|2≤B∥f∥2.A‖f‖2≤∑|⟨f,f k⟩|2≤B‖f‖2.
The frame is called tight, if A=B A=B and it is called equiangular if there is some c c such that for all k,l k,l: |⟨f k,f l⟩|=c|⟨f k,f l⟩|=c.
"Equiangular tight frame" usually also means that the vector f k f k are normalized.
If you have an equiangular tight frame consisting of n n vectors in R d R d (or C d C d) then it holds that
|⟨f k,f l⟩|=n−d d(n−1)−−−−−−−−√.|⟨f k,f l⟩|=n−d d(n−1).
Note that for n=d+m n=d+m and also for n=d m n=d m this number always goes to zero for d→∞d→∞ saying that the vectors become asymptotically orthogonal.
The only catch is, that equiangular tight frames do not always exist…
There are several construction, such as the Mecedes-Benz-frame (with n=d+1 n=d+1), see here . A pretty recent preprint on the existence of ETFs is "On the existence of equiangular tight frames" by Sustika, Tropp, Dhillon and Heath. In that paper you find the result that "maximal ETFs" (i.e. ones with the largest number of vectors) exist (not in all dimensions) but have the size N=d(d+1)/2 N=d(d+1)/2 in the real case and N=d 2 N=d 2 in the complex case.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Improve this answer
Follow
Follow this answer to receive notifications
answered Feb 25, 2014 at 16:34
DirkDirk
12.9k 7 7 gold badges 57 57 silver badges 100 100 bronze badges
1
The complex case with N=d 2 N=d 2 is also known as a SIC-POVM. They appear to have an interesting structure as the orbit of a certain representation of the Heisenberg group: mathoverflow.net/questions/2897/…Yoav Kallus –Yoav Kallus 2014-02-26 02:30:08 +00:00 Commented Feb 26, 2014 at 2:30
Add a comment|
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
linear-algebra
mg.metric-geometry
measure-concentration
See similar questions with these tags.
Featured on Meta
Spevacus has joined us as a Community Manager
Introducing a new proactive anti-spam measure
Linked
12A group action of the Heisenberg group with special symmetries
4Do kissing numbers with distance d d always grow polynomially or exponentially in dimension?
Related
5fast merging of orthogonal bases
14How far is a set of vectors from being orthogonal?
4How to efficiently compute the generalized cross product?
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
MathOverflow
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings |
13013 | https://chem.libretexts.org/Courses/Southwestern_College/Atoms_First_-_Introductory_Chemistry_for_Science_and_Engineering/07%3A_Aqueous_Solutions/7.04%3A_Quantifying_Solutions_-_Concentration/7.4.01%3A_Measuring_Concentration_with_Mass_and_Volume_Percents | Skip to main content
7.4.1: Measuring Concentration with Mass and Volume Percents
Last updated
: Jun 13, 2025
Save as PDF
7.4: Quantifying Solutions - Concentration
7.4.2: Measuring Concentration with Molarity
Page ID
: 523788
Anonymous
LibreTexts
( \newcommand{\kernel}{\mathrm{null}\,})
There are human cultures that do not recognize numbers above three. Anything greater than that is simply referred to as "much" or "many". Although this form of calculation may seem very limited, American culture does the same thing to a certain degree. For example, there are several ways to express the amount of solute in a solution in a quantitative manner. The concentration of a solution is a measure of the amount of solute that has been dissolved in a given amount of solvent or solution. A concentrated solution is one that has a relatively large amount of dissolved solute. A dilute solution is one that has a relatively small amount of dissolved solute. However, those terms are vague, and it is often necessary to express concentration with numbers.
Percent Solutions
One way to describe the concentration of a solution is by the percent of a solute in the solvent. The percent can further be determined in one of two ways: (1) the ratio of the mass of the solute divided by the mass of the solution or (2) the ratio of the volume of the solute divided by the volume of the solution.
Mass Percent
When the solute in a solution is a solid, a convenient way to express the concentration is by mass percent , which is the grams of solute per of solution.
Suppose that a solution was prepared by dissolving of sugar into of water. The percent by mass would be calculated by:
Sometimes you may want to make up a particular mass of solution of a given percent by mass, and need to calculate what mass of the solvent to use. For example, you need to make of a solution of sodium chloride. You can rearrange and solve for the mass of solute:
You would need to weigh out of and add it to of water. Notice that it was necessary to subtract the mass of the from the mass of solution to calculate the mass of the water that would need to be added.
Volume Percent
The percentage of solute in a solution can be determined by volume when the solute and solvent are both liquids. The volume of the solute divided by the volume of the solution, expressed as a percent, yields the percent by volume of the solution. If a solution is made by adding of ethanol to enough water to make of the resulting solution, the percent by volume is:
Frequently, ingredient labels on food products and medicines have amounts listed as percentages (see figure below).
Summary
The concentration of a solution is a measure of the amount of solute that has been dissolved in a given amount of solvent or solution.
A concentrated solution has a relatively large amount of dissolved solute.
A dilute solution has a relatively small amount of dissolved solute.
Techniques for calculation of percent mass and percent volume solution concentrations are described.
7.4: Quantifying Solutions - Concentration
7.4.2: Measuring Concentration with Molarity |
13014 | https://www.sciencefacts.net/acceleration-due-to-gravity.html | Physics
Why Does Water Expand When It Freezes
Gold Foil Experiment
Faraday Cage
Oil Drop Experiment
Magnetic Monopole
Life Science
Why Do Fireflies Light Up
Types of Blood Cells With Their Structure, and Functions
The Main Parts of a Plant With Their Functions
Parts of a Flower With Their Structure and Functions
Parts of a Leaf With Their Structure and Functions
Chemistry
Why Does Ice Float on Water
Why Does Water Expand When It Freezes
Why Does Oil Float on Water
Geology
How Do Clouds Form
What Causes Lightning
How are Diamonds Made
Types of Meteorites
Types of Volcanoes
Types of Rocks
Worksheets
Home / Physics / Acceleration due to Gravity
Acceleration due to Gravity
The acceleration due to gravity is the net acceleration that an object close to Earth’s surface experiences due to the combined effect of the gravitational force and the centrifugal force. It is denoted by the letter ‘g’. The numerical value of g is 9.81 m/s2. There are slight variations of this value depending on a few factors.
Formula
We shall use Newton’s laws to determine the formula for acceleration due to gravity. The gravitational force on an object close to the Earth’s surface is given by
[ F_g = \frac{GMm}{R^2} ]
Fg: Gravitational force
M: Mass of the Earth
m: Mass of the object
R: Radius of the Earth
G: Universal gravitational constant
From Newton’s second law, the force on an object due to Earth’s gravitation is
[ F_g = mg ]
Comparing the above two equations:
[ mg = \frac{GMm}{R^2} \ g = \frac{GM}{R^2} ]
Units of g
SI unit: meters per second squared or m/s2
cgs unit: centimeters per second squared or cm/s2
Imperial unit: feet per second squared or ft/s2
We can make the following conclusions from the above expression:
All bodies experience the same acceleration due to gravity, irrespective of their masses.
The acceleration value depends on Earth’s mass, not the object’s mass.
Gravitational Acceleration Value
From the above equation, it is clear that the acceleration due to gravity takes a constant value. Let us calculate this value using the following constants:
| | |
--- |
| Quantity | Value |
| G | 6.67 x 10-11 Nm2/kg2 |
| M | 6 x 1024 kg |
| R | 6.4 x 106 m |
Plugging in the values in the expression for g, we get
g = 6.67 x 10-11 Nm2/kg2 x (6 x 1024 kg/6.4 x 106 m)2
=> g = 9.8 m/s2
The above value is calculated for the metric system of units. In the imperial unit, 1 m = 3.28 ft. Therefore, g = 32.14 ft/s2.
Factors Affecting Acceleration due to Gravity
1. Height Above the Surface of the Earth or Altitude
To understand the effect of altitude, let us revisit the gravitational force equation
[ F_g = \frac{GMm}{R^2} ]
At a height h from Earth’s surface, this equation modifies to
[ F_g = \frac{GM}{(R+h)^2} ]
Therefore, by repeating the same steps as above, we get
[ g_h = \frac{GM}{(R+h)^2} \ \Rightarrow g_h = \frac{GM}{R^2\left[ 1+ \frac{h}{R}\right]^2 }\ \Rightarrow g_h = g\left[ 1+ \frac{h}{R}\right]^{-2} ]
Since GM/R2 is the acceleration g on the Earth’s surface
The above expression is the acceleration due to gravity at a height above Earth’s surface. By examining the formula, we can say that acceleration decreases with an increase in the height of an object and becomes zero at an infinite distance from the Earth.
Note that when h << R, the expression reduces to
[ \Rightarrow g_h = g \left[ 1- \frac{2h}{R} \right] ]
2. Depth Below the Surface of the Earth
Recall the expression for acceleration due to gravity at the surface of the Earth
[ g = \frac{GM}{R^2} ]
Earth’s mass M can be written in terms of density ρ and volume (\frac{4}{3}\pi R^3 ).
[ M = \rho \times \frac{4}{3} \pi R^3 ]
Therefore,
[ g = \frac{4}{3}G\pi \rho R ]
At a distance d below the Earth’s surface, the acceleration due to gravity is given by
[ g_d = \frac{4}{3} G\pi \rho (R – d) ]
Diving one equation by the other
[ g_d = g\left[ 1-\frac{d}{R} \right] ]
Note that at the Earth’s surface, d = 0 and gd = g
3. Latitude
Suppose an object with mass m lies on the Earth’s surface at a latitude θ. Let ω bet the angular velocity of the Earth. For the non-inertial frame, there exists a centrifugal force on the object given by
[ F_C = m \omega ^2 r ]
Where r is the shortest distance of the object from the axis of Earth’s rotation. It is also the radius of the circular path that the object traces. The gravitational force is
[ F_g = mg ]
The two co-planer forces will add vectorially to give the resultant force on the object. When two co-planer vectors form two sides of a parallelogram, the resultant vector is given by its diagonal. According to the parallelogram law of vector addition, the magnitude of the apparent gravitational force mg’ at a latitude θ is
(mg’)2 = (mg)2 + (mrω2)2 + 2 (mg) (mrω2) cos (180 – θ) …
The relationship between r and Earth’s radius R is
r = R cos θ
Substituting for r, we get
g’ = g – Rω2cos2θ
Here, g′ is the apparent value of acceleration due to gravity at a latitude θ due to Earth’s rotation, and g is the actual value of gravity at the same latitude without considering Earth’s rotation.
At the equator, θ = 0. Therefore, g’ = g – Rω2
At the poles, θ = ± 90. Therefore, g’ = g
We can conclude that the acceleration due to gravity at the poles is more than at the equator.
References
The Acceleration of Gravity – Physicsclassroom.com
Acceleration Due to Gravity – Studysmarter.us
Acceleration Due to Gravity – Energyeducation.ca
Acceleration – Uh.edu
Article was last reviewed on Tuesday, February 14, 2023
Related articles
Uniform Motion
Periodic Motion
Magnetic Vector Potential
Helmholtz Coil
Leave a comment Cancel reply
Popular Articles
Contact and Non-contact Forces
Applied Force
Join our Newsletter
Fill your E-mail Address
Related Worksheets
None Found
About Us
Contact Us
Privacy Policy
© 2025 (Science Facts). All rights reserved. Reproduction in whole or in part without permission is prohibited. |
13015 | https://physics.nist.gov/cgi-bin/ASD/ie.pl?spectra=barium&units=1&e_out=0&unc_out=1&at_num_out=1&el_name_out=1&ion_charge_out=1&biblio=1 | NIST Atomic Ionization Energies Output
Magnetic dipole transition occurs only between states of the same parity.
Electric quadrupole transition occurs only between states of the same parity.
Magnetic quadrupole transition occurs only between states of different parities.
Magnetic octupole transition occurs only between states of the same parity.
Two-photon transition occurs only between states of the same parity.
Hyperfine-induced transition may occur only in isotopes having non-zero nuclear spin.
Undefined transition type.
M1+E2 is a mix of a magnetic-dipole and an electric-quadrupole transition, both of which occur only between states of the same parity.
E1+M2 is a mix of an electric-dipole and a magnetic-quadrupole transition, both of which occur only between states of different parities.
Oscillator strength is a dimensionless quantity. For strong lines (both in atoms and in ions), it is of the order of unity.
This level/line may not be real.
This level was determined by interpolation or extrapolation of known experimental values or by semiempirical calculation; its absolute accuracy is reflected in the number of significant figures assigned to it.
This level was determined by interpolation or extrapolation of known experimental values or by semiempirical calculation; its absolute accuracy is reflected in the number of significant figures assigned to it.
Theoretical value.
Observed in absorption
Line or feature having large width due to autoionization broadening
Beam-foil measurement
Intensity is shared by several lines
Double line
Broad due to overexposure in the quoted reference
Line position estimated
Very hazy line
Superposed with neighbor line
See the original reference on other issues
Observed wavelength given is actually a rounded Ritz value (no wavelength measurement is available)
The relative positions of the levels within such a system are accurate within experimental uncertainties, but no experimental connection between this system and the other levels of the spectrum has been made.
This level may have substantial autoionization rate.
Band head
Blended with another line that may affect the wavelength and intensity
Complex line
Diffuse line
Forbidden line
Transition involving a level of the ground term
Hazy line
Line has hyperfine structure
Identification uncertain
Wavelength smoothed along isoelectronic sequence
Shaded to longer wavelengths
Masked by another line (no wavelength measurement)
Term assignment of the level is questionable
Perturbed by a close line
Asymmetric line
Easily reversed
Shaded to shorter wavelengths
Tentatively classified line.
Unresolved from close line
Wide line
Extrapolated wavelength
This wavelength is calculated on-line. The number of significant digits was obtained assuming the trailing zeros in the energies are insignificant, and thus the real accuracy may be higher.
This wavelength is calculated on-line with a proper account of significant digits
This is a Ritz wavelength calculated on-line from the stored values of the lower and upper energy levels. Its numerical precision is very approximate. It is determined by estimated uncertainty of the wave number, which is calculated as a combination in quadrature of the level uncertainties. If the latter are unknown, they are estimated as 10 units of the last significant digit of the level value. Even if they are known, presence of correlations or unknown systematic errors in level values may cause the number of significant digits in the Ritz wavelength to be wrong by up to ±2.
Unknown line type
Somewhat less intensity than the value given
A multiplet in the original compilation has been separated into its component lines and the transition probability was derived from the compiled value assuming spin-orbit coupling. This may decrease the listed accuracy, especially for weaker transitions.
Relative intensities provide a qualitative description of what the emission spectrum of a particular element in a particular (low-density) source looks like. More help.
NIST Atomic Spectra Database
Ionization Energies Data
Ba (all spectra)
56 Data Rows FoundExample of how to reference these results:
Kramida,A., Ralchenko,Yu., Reader,J., and NIST ASD Team (2024). NIST Atomic Spectra Database (ver.5.12), [Online]. Available: [2025, September 28]. National Institute of Standards and Technology, Gaithersburg, MD. DOI:
BibTex Citation (new window)
| At. Num. | Ion Charge | El. name | Ionization Energy (eV) | Uncertainty (eV) | References |
--- --- --- |
| 56 | 0 | Barium | 5.2116646 | 0.0000012 | L6347,L9224 |
| 56 | +1 | Barium | 10.003826 | 0.000012 | L9224 |
| 56 | +2 | Barium | 35.8438 | 0.0025 | L3667 |
| 56 | +3 | Barium | ))47.0[])) | 0.3 | L11023 |
| 56 | +4 | Barium | ())58.0))) | 1.9 | L8672c99 |
| 56 | +5 | Barium | ())71.0))) | 2.0 | L8672c99 |
| 56 | +6 | Barium | ())86.0))) | 2.0 | L8672c99 |
| 56 | +7 | Barium | ())101.0))) | 2.1 | L8672c99 |
| 56 | +8 | Barium | ))130.5[])) | 0.6 | L6980 |
| 56 | +9 | Barium | ))146.52[])) | 0.12 | L5438 |
| 56 | +10 | Barium | ())241.0))) | 2.4 | L8672c99 |
| 56 | +11 | Barium | ())267.1))) | 2.5 | L8672c99 |
| 56 | +12 | Barium | ())296))) | 3 | L8672c99 |
| 56 | +13 | Barium | ())325))) | 3 | L8672c99 |
| 56 | +14 | Barium | ())354))) | 3 | L8672c99 |
| 56 | +15 | Barium | ())390))) | 3 | L8672c99 |
| 56 | +16 | Barium | ())422))) | 3 | L8672c99 |
| 56 | +17 | Barium | ())455))) | 3 | L8672c99 |
| 56 | +18 | Barium | ())488))) | 3 | L8672c99 |
| 56 | +19 | Barium | ())520))) | 4 | L8672c99 |
| 56 | +20 | Barium | ())646))) | 4 | L8672c99 |
| 56 | +21 | Barium | ())679))) | 4 | L8672c99 |
| 56 | +22 | Barium | ())717))) | 4 | L8672c99 |
| 56 | +23 | Barium | ())752))) | 4 | L8672c99 |
| 56 | +24 | Barium | ())809))) | 4 | L8672c99 |
| 56 | +25 | Barium | ())846))) | 4 | L8672c99 |
| 56 | +26 | Barium | ())935))) | 4 | L8672c99 |
| 56 | +27 | Barium | ))976.62[])) | 0.11 | L5473 |
| 56 | +28 | Barium | ())1 695))) | 4 | L8672c99 |
| 56 | +29 | Barium | ())1 776))) | 4 | L8672c99 |
| 56 | +30 | Barium | ())1 864))) | 4 | L8672c99 |
| 56 | +31 | Barium | ())1 958))) | 4 | L8672c99 |
| 56 | +32 | Barium | ())2 047))) | 4 | L8672c99 |
| 56 | +33 | Barium | ())2 142))) | 5 | L8672c99 |
| 56 | +34 | Barium | ())2 256))) | 5 | L8672c99 |
| 56 | +35 | Barium | ())2 349))) | 5 | L8672c99 |
| 56 | +36 | Barium | ())2 452))) | 5 | L8672c99 |
| 56 | +37 | Barium | ())2 547))) | 4 | L8672c99 |
| 56 | +38 | Barium | ())2 814))) | 4 | L8672c99 |
| 56 | +39 | Barium | ())2 901))) | 4 | L8672c99 |
| 56 | +40 | Barium | ())2 994))) | 4 | L8672c99 |
| 56 | +41 | Barium | ())3 081))) | 4 | L8672c99 |
| 56 | +42 | Barium | ())3 266))) | 4 | L8672c99 |
| 56 | +43 | Barium | ())3 363))) | 4 | L8672c99 |
| 56 | +44 | Barium | ())3 546))) | 4 | L8672c99 |
| 56 | +45 | Barium | ())3 640))) | 4 | L8672c99 |
| 56 | +46 | Barium | ())8 326))) | 4 | L8672c99 |
| 56 | +47 | Barium | ())8 565))) | 4 | L8672c99 |
| 56 | +48 | Barium | ())8 831))) | 4 | L8672c99 |
| 56 | +49 | Barium | ())9 077))) | 4 | L8672c99 |
| 56 | +50 | Barium | ())9 739))) | 4 | L8672c99 |
| 56 | +51 | Barium | ())10 023))) | 4 | L8672c99 |
| 56 | +52 | Barium | ())10 376))) | 4 | L8672c99 |
| 56 | +53 | Barium | ())10 616.42))) | 0.05 | L16264c99 |
| 56 | +54 | Barium | ())43 485.37))) | 0.04 | L21139 |
| 56 | +55 | Barium | ())44 561.633))) | 0.024 | L19200 |
If you did not find the data you need, please inform the ASD Team. |
13016 | http://www.algebra.com/algebra/homework/Geometry-proofs/Geometry_proofs.faq.question.996174.html | SOLUTION: Given: AB=CD Prove: AC=BD
SOLUTION: Given: AB=CD Prove: AC=BD
Algebra->Geometry-proofs -> SOLUTION: Given: AB=CD Prove: AC=BD Log On
Geometry: Proofs in Geometry GeometrySolvers SolversLessons LessonsAnswers archive Answers Discover more CD Compact disc math Mathematics Math Algebra Inc Click here to see ALL problems on Geometry proofs Question 996174: Given: AB=CD Prove: AC=BD Answer byfarohw(175) (Show Source): You can put this solution on YOUR website! Suppose A, B, C, D are points on a line in this order. AB = CD given BC = BC reflexive property of segments AC = AB + BC, BD = CD + BC segment addition postulate AB + BC = BC + CD addition property of equality Therefore, AC = BD by substitution
Discover more
Compact disc
CD
Mathematics
Math
math
Algebra Inc |
13017 | https://www.ahajournals.org/doi/10.1161/hypertensionaha.106.076083 | Arterial Chemoreceptors and Sympathetic Nerve Activity | Hypertension
Skip to main content
Advertisement
Become a member
Volunteer
Donate
Journals
BrowseCollectionsSubjectsAHA Journal PodcastsTrend Watch
ResourcesCMEAHA Journals @ MeetingsJournal MetricsEarly Career Resources
InformationFor AuthorsFor ReviewersFor SubscribersFor International Users
Alerts
0 Cart
Search
Sign inREGISTER
Quick Search in Journals
Enter search term
Quick Search anywhere
Enter search term
Quick search in Citations
Journal Year Volume Issue Page
Searching:
This Journal
This JournalAnywhereCitation
Advanced SearchSearch
navigate the sidebar menu
Sign inREGISTER
Quick Search anywhere
Enter search term
Publications
Arteriosclerosis, Thrombosis, and Vascular Biology
Circulation
Circulation Research
Hypertension
Current Issue
Archive
Journal Information
About Hypertension
Author Instructions
Editorial Board
Meet the Editors
Information for Advertisers
Features
Best of Hypertension
Early Career Program
Education & Webinars
Statements & Guidelines
Stroke
Journal of the American Heart Association
Circulation: Arrhythmia and Electrophysiology
Circulation: Cardiovascular Imaging
Circulation: Cardiovascular Interventions
Circulation: Cardiovascular Quality & Outcomes
Circulation: Genomic and Precision Medicine
Circulation: Heart Failure
Stroke: Vascular and Interventional Neurology
Annals of Internal Medicine: Clinical Cases
Information
For Authors
For Reviewers
For Subscribers
For International Users
Submit & Publish
Submit to the AHA
Editorial Policies
Open Science
Value of Many Voices
Publishing with the AHA
Open Access Information
Resources
AHA Journals CME
AHA Journals @ Meetings
Metrics
AHA Journals Podcast Network
Early Career Resources
Trend Watch
Professional Heart Daily
AHA Newsroom
Current Issue
Archive
Journal Information
About Hypertension
Author Instructions
Editorial Board
Meet the Editors
Information for Advertisers
Features
Best of Hypertension
Early Career Program
Education & Webinars
Statements & Guidelines
Submit
Reference #1
Review Article
Originally Published 14 May 2007
Free Access
Arterial Chemoreceptors and Sympathetic Nerve Activity: Implications for Hypertension and Heart Failure
Harold D.Schultz, Yu L.Li, and Yanfeng DingAuthor Info & Affiliations
Hypertension
Volume 50, Number 1
37,111 124
Metrics
Total Downloads 37,111
Last 12 Months 2,568
Total Citations 124
Last 12 Months 5
View all metrics
Track CitationsAdd to favorites
PDF/EPUB
Contents
The Sympathetic Response to Arterial Chemoreceptor Activation
Arterial Chemoreflex in Heart Failure
Mechanisms of Enhanced Arterial Chemoreflex in Heart Failure
Blood Flow Effects on CB Function in Heart Failure
Arterial Chemoreflex in Hypertension
Mechanisms of Enhanced Arterial Chemoreflex in Hypertension
Vascular Effects on CB Function in Hypertension
Acknowledgments
Supplemental Material
References
eLetters
Information & Authors
Metrics & Citations
View Options
References
Figures
Tables
Media
Share
Chronic elevation in sympathetic nerve activity (SNA) is associated with the development and maintenance of certain types of hypertension1 and contributes to the progression of chronic heart failure (CHF).2 The mechanisms involved in sympathetic dysfunction in these disorders appear to be complex and multifactorial. A unified hypothesis is likely to encompass alterations in multiple autonomic reflex pathways, central integratory sites, and chemical mediators that control sympathetic outflow. For example, tonic restraint of sympathetic outflow by arterial and cardiopulmonary baroreflexes is depressed in CHF2 and depressed or reset in hypertension.3 Moreover, maladaptive changes also occur in the central nervous system at integrative sites for autonomic control in both disease processes.4,5 It is also clear that sympathoexcitatory cardiac,6 somatic,7 and central/peripheral chemoreceptor reflexes8 are enhanced in CHF and hypertension.
Arterial chemoreceptors serve an important regulatory role in the control of alveolar ventilation, but they also exert a powerful influence on cardiovascular function.9 Activation of arterial chemoreceptors by hypoxemia increases sympathetic outflow to systemic vascular beds to compensate for the direct vasodilating effects of hypoxia on these vessels and to redistribute blood flow to essential organs. In this review, we highlight relevant information that implicates the arterial chemoreflex as a contributory mechanism for the sympathetic hyperactivity in CHF and hypertension and illustrate proposed mechanisms for this altered function.
The Sympathetic Response to Arterial Chemoreceptor Activation
Arterial chemoreceptors located in the aortic and carotid bodies (CBs) respond to hypoxemia and hypercapnia. Because central chemoreceptors also respond to hypercapnia, hypoxia is typically used as a specific stimulus to arterial chemoreceptors. In some mammals, such as rats and rabbits, reflex responses to hypoxemia arise solely from the CB, whereas in other species, the aortic chemoreceptor contribution can be significant. However, it is not possible to experimentally separate the relative contribution of the aortic and CBs to reflex responses in conscious animals or humans. For these reasons, discussions on the arterial chemoreflex generally relate functionality to that of the CB, which has been more extensively studied.
Glomus cells in the CB depolarize in response to hypoxia and release multiple putative neurotransmitters (including acetylcholine, serotonin, ATP, and substance P) that activate impulses in afferent fibers traveling to the medulla via the carotid sinus nerve.10 Arterial chemoreceptor stimulation in freely breathing humans and conscious animals increases sympathetic vasoconstrictor outflow to muscle, splanchnic, and renal beds to elevate arterial pressure, and, in humans, increases cardiac sympathetic activity to increase heart rate and contractility.9 There is preferential activation of sympathetic fibers going to the adrenal gland to increase norepinephrine but not epinephrine release.11 However, sympathetic activity to brown adipose tissue is decreased12 to facilitate a lowering of body temperature, an adaptive response to reduce oxygen consumption during hypoxia. Parasympathetic and sympathetic fibers to the heart and cerebral vessels are concomitantly activated.9 This dual response likely serves to limit sympathetic vasoconstriction of coronary and cerebral vessels during hypoxemia and to limit or modify cardiac chronotropic and inotropic responses, depending on the conditions.9
Sympathetic efferent responses to arterial chemoreflex activation are tightly coupled to the increase in rate and magnitude of phrenic nerve activity (PNA) to increase ventilation.13 Significant interaction of central respiratory areas with presympathetic neurons in the medulla dictate that SNA is often clustered in bursts in phase with ventilation, peaking during early expiration.13 In addition, enhanced SNA during chemoreceptor activation is progressively blunted as tidal volume increases because of inhibitory feedback from pulmonary volume receptors.14 As a result, when chemoreceptor activation occurs during apnea, where pulmonary volume feedback is abrogated, sympathetic activation is maximized. The sympathetic response is also influenced by the rate of ventilation. Rapid breathing rate enhances the sympathetic response to chemoreflex activation.15 Conversely, slow breathing reduces arterial chemoreflex responsiveness.16
The dynamics of the ventilatory and sympathetic responses to arterial chemoreceptor stimulation are complex. PNA and SNA increase within seconds of exposure of the CB to hypoxemia, and as hypoxia ensues, the magnitudes of SNA and PNA increase asymptotically and are maintained, whereas the ventilatory rate increases transiently and then progressively declines.13 When hypoxic stimulation is removed, PNA and SNA gradually return to baseline rather than decrease abruptly (short-term potentiation). In humans, a sustained period of hypoxemia for ≈20 minutes can elevate muscle SNA for as long as 1 hour after blood gases and ventilation return to control.17 After repeated episodes of short hypoxic exposures in rats (acute intermittent hypoxia), both PNA and SNA remain elevated for as long as ≥1 hour, a phenomenon called long-term facilitation.18
The extent to which hypertension and CHF influence this complex and highly differentiated integration of sympathorespiratory responses to arterial chemoreflex activation is far from being fully realized. However, there is good evidence to indicate that the sensitivity of the arterial chemoreflex is enhanced in certain types of neurogenic hypertension and in CHF and that the enhanced CB function contributes to tonic elevation in sympathetic outflow to resistance vessels and kidney.
Arterial Chemoreflex in Heart Failure
The role of arterial chemoreflex mechanisms in heart failure has received considerable attention19 but not without controversy. An exaggerated ventilatory response to hyperoxic hypercapnia, indicative of an enhanced central chemoreflex, has been consistently observed in humans with CHF,20 but the ventilatory response to isocapnic hypoxemia, an index of arterial chemoreflex function, is not altered in many patients with CHF.20 Similarly, acute inhibition of arterial chemoreceptors with hyperoxia (Dejour’s effect) has no effect on resting muscle SNA in some studies on CHF patients.21 By contrast, other groups22,23 have found an enhanced ventilatory response to hypoxia in CHF patients, particularly those in more severe stages of the disease.
Although results from clinical studies have been variable, we have documented an enhanced arterial chemoreflex function in a rabbit model of pacing-induced CHF.24,25 Although the cardiac pathology of tachycardia-induced ventricular failure differs somewhat from ischemic heart disease in patients, it induces a similar systemic hemodynamic impairment, with sympathohumoral activation and altered autonomic reflexes, as observed in patients and animal models with ischemic heart failure.2
With monitoring of renal SNA (RSNA) in conscious animals as an index of sympathetic outflow, we have documented that resting (normoxic) and hypoxic activation of RSNA (Figure 1) and ventilation progressively increase in rabbits as they are paced over 3 to 4 weeks. During this interval, unpaced left ventricular contractility and ejection fraction decline to levels equivalent to moderate compensated CHF in patients.24,26,27 Single-unit discharge activity of CB chemoreceptors, both at rest and in response to hypoxia, is similarly enhanced over the same time course (Figure 2A).25 This increased central input from the CB provides a tonic excitatory influence on sympathetic outflow, because hyperoxia reduces resting RSNA in CHF but not control animals (Figure 2B).24 Moreover, resting RSNA and plasma norepinephrine levels are attenuated in CHF rabbits with CB denervation.19,28 These studies confirm clinical evidence suggesting that arterial chemoreflex function is enhanced in CHF and further demonstrate that enhanced CB function is an important contributory mechanism for sympathetic activation in CHF.
Open in Viewer
Figure 1.A, Effect of chronic cardiac pacing on resting (21% O 2) and hypoxia-activated RSNA in a conscious rabbit. RSNA obtained by chronic telemetric recording. Left ventricular fractional shortening, 77% of control after 1 week and 50% of control after 3 weeks of pacing. B, Carotid body CB chemoreflex function curves for RSNA before and during cardiac pacing. RSNA normalized among animals as percentage of maximal activity (evoked by the nasopharyngeal reflex). Left ventricular function for the group is listed in Table S1. n=4; P<0.05 vs prepacing.
Open in Viewer
Figure 2.Effects of hypoxia (10% O 2) and hyperoxia (100% O 2, Dejour’s effect) on resting CB chemoreceptor single unit activity (A) and RSNA (B) in normal and CHF (3 to 4 weeks pacing) rabbits. Resting and hypoxia-activated CB activity and RSNA are elevated in CHF rabbits compared with normal. Hyperoxia inhibited CB activity and normalized RSNA in CHF rabbits. P=0.05 vs room air; #P<0.05 vs normal.
Mechanisms of Enhanced Arterial Chemoreflex in Heart Failure
Because sleep apnea (SA) is highly correlated with CHF,29 and SA enhances arterial chemoreflex function (see discussion below), it is possible that SA may contribute to arterial chemoreflex activation in patients with CHF with breathing disturbances. However, chronic recordings of breathing in CHF rabbits discount this as an essential mechanism in this animal model.28 Enhancement of arterial chemoreflex function begins to occur as early as the first week of pacing, with no evidence of arterial hypoxemia, hypercapnia, or altered breathing. Similarly, the incidence of SA in patients with CHF correlates with the severity of the disease30 and, thus, does not seem to be a predisposing factor for altered chemoreflex function in early heart failure.
Although neither chronic nor intermittent hypoxia can be implicated in the alteration in CB function in CHF animals, there are numerous neurohumoral factors that are altered in the CHF, such as angiotensin II (Ang II), NO, catecholamines, and endothelin, among others, that are known to affect CB chemoreceptor function.10 The role of some of these factors in altered CB function in CHF remains largely unexplored, but there is good evidence that Ang II and NO play major roles.
Systemic and tissue Ang II levels are increased in patients with CHF, and Ang II enhances chemoreceptor activity via the angiotensin II receptor subtype 1 receptor (AT 1 R) in the CB.31 Furthermore, a local Ang II system is operational in the rabbit CB.27,32 CHF upregulates the Ang II level and the expression of the AT 1 R in the rabbit CB.27
Systemic administration of Ang II in conscious rabbits (to levels equivalent to the endogenous plasma Ang II level in CHF rabbits) enhances hypoxia-induced chemoreflex activation of sympathetic outflow.27 Furthermore, blockade of AT 1 R in CHF rabbits attenuates (ie, normalizes) the exaggerated hypoxia-induced chemoreflex responses observed in CHF rabbits.27 Afferent recordings from the isolated CB confirm that elevation of Ang II and AT 1 R in the CB enhances chemoreceptor sensitivity to hypoxemia in CHF rabbits.27
The mechanism by which Ang II enhances the hypoxic sensitivity of the CB chemoreceptors involves an interaction with oxygen sensitive, voltage-gated potassium channels (K vo 2) in CB glomus cells.33 Hypoxic inhibition of K vo 2 channels is enhanced in isolated CB glomus cells from CHF rabbits, and blockade of AT 1 R alone is capable of reversing this enhanced hypoxic sensitivity.33 In addition, exposing normal rabbit CB glomus cells to Ang II mimics this effect of CHF on K vo 2 channel function.33
AT 1 R promotes activation of reduced nicotinamide-adenine dinucleotide phosphate (NADPH) oxidase with superoxide anion production.34 In the CB, expression of NADPH oxidase subunits and superoxide anion production are enhanced in CHF rabbits.35 The elevated superoxide anion level can be normalized by AT 1 R antagonists, NADPH oxidase inhibitors, and the superoxide anion scavenger Tempol.35 Similarly, NADPH inhibitors and Tempol normalize the enhanced CB chemoreceptor discharge to hypoxemia and normalize the enhanced sensitivity of K vo 2 channels to hypoxia in CB glomus cells from CHF rabbits.35 These results suggest that the NADPH oxidase-superoxide anion pathway mediates the effects of Ang II in the CB to enhance hypoxic sensitivity in the CHF state.
Despite the potent effects of Ang II-AT 1 R on the sensitivity of the CB to hypoxemia, AT 1 R blockade does not alter chemoreceptor discharge or K vo 2 currents in glomus cells at normoxia in CHF animals.27,33 Thus, Ang II cannot account for the enhanced chemoreceptor activity under normal resting conditions observed in CHF rabbits.25–27 Other evidence suggests that downregulation of an NO mechanism is likely to participate in the enhanced basal activity in the CB in CHF.25,26
Both neuronal NO synthase (nNOS; NOS) and endothelial NOS are present in the CB, and NO is inhibitory to CB activity.10 Because O 2 is essential for biosynthesis of NO, during normal normoxic conditions, basal production of NO acts as an amplifier of O 2 to keep CB chemoreceptor discharge suppressed.10 However, basal NO production and nNOS expression within the CB are depressed in CHF.25,26 Thus, the tonic inhibitory effect of NO on the activity of CB chemoreceptors, demonstrated in the CB of normal rabbits, is virtually absent in CHF rabbits.25
The marked down regulation of endogenous nNOS in the CB plays a major role in the enhanced CB chemoreceptor activity in CHF rabbits.26 Adenoviral gene transfer of nNOS to the CB in CHF rabbits enhances protein expression and NO production in the CB and reverses the enhanced CB chemoreceptor activity seen in the CHF state.26 Furthermore, gene transfer of nNOS to the CB in CHF rabbits reduces resting RSNA.26 These results provide important evidence of a measurable contribution of enhanced CB chemoreceptor input to elevated sympathetic outflow in CHF and the contribution of nNOS downregulation in the CB to this effect.
We have shown further that NO inhibits CB chemoreceptor discharge by activation of calcium-dependent voltage-gated K (K Ca) channels via a cGMP-dependent pathway in CB glomus cells in rabbits.36 K Ca channel current is markedly suppressed in glomus cells from CHF rabbits under normoxic conditions because of NO depletion,36 and the attenuated K Ca channel current contributes to the depolarized resting membrane potential observed in the CHF glomus cells.36 However, an inwardly rectifying K+ current carried by the HERG-like K+ channel (named for the human ether-a-go-go related gene that encodes the channel protein) has been reported in rabbit glomus cells, which could also influence the resting membrane potential.37
The concurrent downregulation of the nNOS/NO pathway and upregulation of the Ang II/superoxide anion pathway in the CB of CHF rabbits raises the question of whether these pathways interact to enhance CB excitability via oxidative stress. Ang II may contribute to depressed bioavailable NO in the CB by suppressing nNOS gene expression38 and/or increased scavenging of NO through superoxide anion production. Conversely, the downregulation of NO production in the CB in CHF is likely to enhance the effects of Ang II by reduced scavenging of superoxide anion by NO. The relationship among nNOS, NO, Ang II, and superoxide anion on CB chemoreceptor function is not yet clear and deserves further study.
Existing evidence of the role of endothelial NOS versus nNOS in the CB is somewhat controversial. Studies have shown that specific nNOS inhibitors are ineffective in altering CB ventilatory responses in rats.39 On the other hand, transgenic mice lacking nNOS show greater ventilatory responses to hypoxia than wild-type controls, whereas responses to hypoxia are blunted in mutant mice lacking endothelial NOS.40 We have observed a downregulation endothelial NOS in the CB of CHF rabbits (YL Li and HD Schultz, unpublished data) but have not yet investigated the functional consequence of this change.
Blood Flow Effects on CB Function in Heart Failure
The stimulus for upregulation of local Ang II/AT 1 R signaling and downregulation of NOS/NO in CB glomus cells during CHF is not clear. However, the changes in potassium voltage channel function described in glomus cells in CHF rabbits33,36 are similar in most respects to those described in glomus cells from rabbits exposed to chronic hypoxia.41 Thus, a decreased cardiac output from the failing heart may decrease blood flow and O 2 delivery to the CB sufficiently to mimic effects of chronic hypoxia. Although the CB is known to acutely autoregulate its blood flow such that moderate acute changes in systemic hemodynamics have little effect on chemoreceptor activity,9 the effects of chronic reductions in cardiac output on CB afferent function are not clear.
In CHF rabbits, a progressive decline in carotid flow occurs as cardiac function deteriorates over the course of 3 weeks of pacing (please see the data supplement, available online at A similar profile of progressive reduction in carotid artery blood flow imposed with bilateral adjustable cuff occluders enhances CB chemoreflex activation of RSNA in response to hypoxia (Figure 3A and 3B) and enhances CB chemoreceptor discharge (Figure 3C), similar to changes observed in CHF rabbits.42 It is unlikely that these responses are driven simply by an acute reduction in blood flow to the CB, because chemoreflex and afferent responses are still normal after only 1 day of occlusion. Immunohistochemical examination of the CB after chronic carotid occlusion revealed an increase in AT 1 R and decrease in nNOS expression.42 These results support the concept that a chronic reduction in oxygen delivery to the CB precipitates altered CB function in CHF.
Open in Viewer
Figure 3.Effect of chronic carotid artery occlusion (CAO) for 3 weeks on resting and hypoxia activated RSNA (A and B) and on resting and hypoxia activated CB activity (C) in rabbits (CAO mimicked carotid flow during cardiac pacing: see online supplemental data). P<0.05 vs normal.
The link between reduced carotid flow and altered signaling in the CB has not yet been explored. Relevant evidence indicates that angiotensin-converting enzyme in the CB is markedly increased in the CB of rats within 1 week of exposure to chronic hypoxia.43 A similar phenomenon may occur in response to chronically reduced blood flow to the CB. Ultimately, such changes in protein expression must be related to effects on gene transcription. Possibilities include activation of hypoxia-inducible factor-1 and/or activator protein-1, gene transcription factors that are known to be activated in the CB in response to hypoxia.44
Although enhanced Ang II/AT 1 R and reduced NOS/NO in the CB were shown to have direct effects on K+ channel function in CB glomus cells,33,36 it is not known whether these altered signaling pathways also act to further reduce blood flow to the CB and enhance CB chemoreceptor sensitivity by way of their vasoactive properties. In addition, it is known that sympathetic efferent feedback to the CB can reduce CB blood flow and activate CB chemoreceptors.9,10 The contribution of these various factors to CB vascular function in CHF need to be elucidated.
Arterial Chemoreflex in Hypertension
Functional studies in humans with established and borderline essential hypertension45,46 and in genetic spontaneously hypertensive rats (SHRs)47 have shown a hyperventilation under resting conditions and enhanced respiratory and SNA reactions to hypoxia. In addition, CB chemoreceptor neural discharge to hypoxemia is exaggerated in SHRs.48
One study, however, suggests that the exaggerated CB chemoreflex in SHRs is not linked to hypertension but to differences in rat strain.49 In addition, not all forms of hypertension are correlated with enhanced chemoreflex drive. In renal hypertensive rabbits, CB function appears to be unaltered.50 Despite these inconsistencies, studies on humans with borderline and established essential hypertension consistently show enhanced arterial chemoreflex drive.46 Furthermore, acute inhibition of arterial chemoreceptors with hyperoxia evokes ventilatory and SNA depression to a much greater extent in patients with essential hypertension than in control subjects.51
A link between neurogenic hypertension and enhanced arterial chemoreflex function has been more extensively studied in patients and in animal models of SA. Patients with SA are at high risk for developing sustained arterial hypertension and exhibit an elevated level of resting muscle SNA that persists throughout the day when periods of apnea are nonexistent.52 Patients with SA exhibit markedly enhanced chemoreflex responses to hypoxemia, whereas SNA responses to hyperoxic hypercapnia (central chemoreflex) and cold pressor tests are normal.53 Similar to that seen in patients with essential hypertension, hyperoxia markedly blunts muscle SNA and reduces arterial pressure in patients with SA.53
A rat model of chronic intermittent hypoxia (CIH) exposure that mimics a pattern of hypoxic episodes experienced by patients with SA has been instrumental in the exploration of this issue.54 Rats exposed to extended periods of nighttime-only CIH develop hypertension and persistent sympathetic activation similar to that seen in patients with SA.54 Chronic CB denervation prevents the development of hypertension and SNA activation.55 Furthermore, CIH enhances CB chemoreceptor discharge to hypoxemia but not to hypercapnia.56 In fact, even a single bout of intermittent hypoxia evokes sustained activation of both CB chemoreceptor activity and SNA for extended periods after the hypoxic challenge is removed (sensory and functional long-term facilitation of the CB chemoreflex).57 These studies provide compelling evidence that intermittent periods of hypoxemia can tonically enhance arterial chemoreceptor drive to increase sympathetic outflow and precipitate hypertension.
There is a high correlation of obesity and metabolic syndrome with elevated SNA and hypertension as well.58 The fact that obstructive SA is often associated with these conditions may explain why arterial chemoreflex function is enhanced in these patients. Although the concept has been proposed,59 no one has evaluated whether chemoreflex function is enhanced in metabolic syndrome independent of SA as a risk factor or whether chemoreflex function contributes to the sympathetic hyperactivity.
Mechanisms of Enhanced Arterial Chemoreflex in Hypertension
To our knowledge, the mechanism by which arterial chemoreflex function is enhanced in SHRs has not been explored. Fukuda et al48 demonstrated that multiunit chemoreceptor activity is enhanced from the CB in SHRs in response to hypoxia but not hypercapnia, but mechanisms for the enhanced CB activity in SHRs are not known. Other studies have shown that inhibition or lesions of commissural subnuclei of the nucleus tractus solitarii in the medulla, where CB chemoreceptors project, reduce blood pressure in SHRs.60,61 Although such lesions may not be specific for the arterial chemoreflex, this evidence raises the possibility that enhanced CB chemoreceptor function contributes to elevated sympathetic activity in genetic hypertensive rats.
Essential hypertensive patients and SHRs exhibit hypertrophied CBs,45 resembling that which occurs in chronic exposure to high altitude or chronic hypoxia. The mechanisms responsible for this effect and its relation to enhanced CB function are still poorly understood. However, CB hypertrophy has not been consistently observed in patients or animal models of CHF and SA, despite enhanced CB function in these disease states.
Peng et al56 demonstrated that CIH in rats enhances normoxic and hypoxic CB chemoreceptor discharge. Chronic intermittent hypercapnia was without effect. It was proposed that the effects of CIH on the CB are comparable to that of ischemia-reperfusion in other tissue, which is known to enhance superoxide anion production and oxidative stress. Indeed, the facilitatory effect of CIH on CB afferent activity is completely abolished in rats treated with a superoxide scavenger.62 Congruently, superoxide anion levels are elevated in CBs from CIH-exposed animals.44,62
The cellular mechanisms of CIH-induced superoxide anion production in the CB suggest alterations in both mitochondrial and membrane redox pathways. Downregulation of mitochondrial complex I but not complex III occurs in CBs from CIH rats.63 In addition, inhibition of complex I increases reactive oxygen species in mitochondria from CB of normal rats but not from that of CIH rats, whereas inhibition of complex III increases oxygen radicals in both groups.63 On the other hand, evidence also suggests that 5-hydroxytryptamine in the CB may be involved in the response, because 5-hydroxytryptamine can induce long-term facilitation in the CB by way of activation of NADPH oxidase and superoxide anion production.64 Still unresolved is the mechanism by which CIH-induced superoxide anion production increases the sensitivity of CB glomus cells and afferent chemoreceptor activity to hypoxia. Based on our work described above,35 an inhibitory effect of superoxide anion on K vo 2 channels in glomus cells is possible.
Other investigators have demonstrated that endothelin-1 (ET-1) may also be involved in the enhanced CB function with CIH. ET-1 is expressed in the CB,65 induces CB chemosensory excitation,66 and potentiates the chemosensory response to acute hypoxia.67 ET-1 immunostaining in the CB vasculature and glomus cells is increased in cats exposed to CIH, whereas ET-1 plasma levels are unchanged.68 A nonselective ET receptor antagonist, bosentan, reduces the elevated normoxic and hypoxic CB discharge induced by CIH.67
Upregulation of both ET-1 expression and enhanced superoxide anion signaling in the CB in response to CIH appears to be related to activation of hypoxia-inducible factor-163,69 and/or activator protein-1.63 These transcription factors, induced by hypoxic challenges to the CB by either periodic bouts of hypoxemia during apneas or chronically reduced blood flow, may be the common denominator for altered CB function in hypertensive and CHF states.
Vascular Effects on CB Function in Hypertension
Indirect evidence suggests that the excitatory effect of ET-1 on CB chemoreceptor activity in CIH is because of its vasoconstrictive effect to reduce CB blood flow and O 2 delivery,66 at least during normoxic conditions. ET-1 itself has little effect on chemoreceptor activity under normoxic conditions in the superfused CB preparation, which is devoid of vascular effects.67 Nevertheless, ET-1 can enhance CB responses to hypoxia in superfused CB preparations, possibly by enhancing voltage-gated (L-type) calcium current in CB glomus cells in response to hypoxia via a cAMP/inositol triphosphate pathway.67 However, because CB chemoreceptor activity is elevated under normoxic conditions in CIH animals, the vascular effects of the peptide are likely to play a major role in this tonic activation of afferent input.
The sympathetic innervation of the CB and/or circulating catecholamines could conceivably play a role in enhanced CB excitability in certain hypertensive states by way of α 1 adrenergic vasoconstriction with reduction of CB blood flow.9,10 However, α 2 adrenergic receptors in the CB inhibit chemoreceptor discharge by inhibiting norepinephrine release70; thus, the net effects of sympathetic feedback on the CB are difficult to predict. Although it is generally thought that sympathetic activation in response to hypoxemia has an inhibitory effect on CB responsiveness,10 the effects of a chronic elevation in sympathetic feedback to the CB under normoxic conditions in hypertensive states has not been addressed.
Based on evidence that Ang II and downregulation of NO are important contributory mechanisms to the development of neurogenic hypertension, these factors may exert important roles on CB function in hypertension, as well as in CHF. Alternatively, in a hypothesis article in 1981, Przybylski71 proposed that atherosclerosis or vascular hypertrophy may act to reduce CB blood flow and enhance chemoreceptor activity in hypertensive states. This concept also seems worthy of pursuit based on our results with chronic reduction in carotid flow.
Perspectives
To what extent does the arterial chemoreflex contribute to a chronic sympathetic activation in either CHF or hypertension? Neurohumoral control of sympathetic activity is a highly complex network of peripheral and central neural pathways, neurotransmitters, and neuromodulators. Most if not all of these integrative inputs are altered to variable degrees in these disease states and are likely to play a collective role in the sympathetic dysfunction. In this regard, there is indisputable evidence that arterial chemoreceptor activity is elevated in CHF and in essential and SA-related hypertension, and this input can contribute to sympathetic activation. Indeed, interruption of afferent input from CB chemoreceptors, by way of either hyperoxia or denervation, reduces sympathetic outflow in many patients and in animal models of these diseases. The relative importance of this chemoreflex effect, however, must be viewed in context with other changes that occur in the sympathetic system as a whole.
Summary
Chemoreceptor input is exaggerated in CHF and hypertension by chemical and hemodynamic factors in the CB. As we have summarized here, a predisposing factor involved in the enhanced chemoreceptor activity appears to be oxidative stress because of impaired oxygen delivery to the CB, either by way of intermittent hypoxic exposure from periodic apneas or reduced blood flow from impaired cardiac or vascular function (Figure 4). In CHF, upregulation of Ang II/NADPH oxidase and downregulation of NOS in the CB secondary to reduced blood flow contribute to this effect. In SA-related hypertension, upregulation of 5-hydroxytryptamine/NADPH oxidase and downregulation of mitochondrial complex I in the CB secondary to periodic bouts of hypoxemia contribute to the effect. In essential hypertension, mechanisms for enhanced CB function remain unexplored but may be related to altered vascular function in the CB. Regardless of the initial stimulus, the resultant chronic activation of sympathetic outflow is likely to amplify and perpetuate the CB disturbance, in a feed-forward manner, by exacerbating deterioration of cardiac function and cardiac output with reduced flow to the CB in CHF and by vasoconstriction with reduced flow to the CB in hypertension. Future inroads in this area will rely on further identification of chemical stimuli, signaling pathways, altered gene expression, and transcription factors responsible for this cascade of events on neural and vascular function within the CB.
Open in Viewer
Figure 4.The vicious cycle of CB chemoreflex activation in CHF and hypertension. Left, Chronic reduction in CB blood flow (BF) because of left ventricular dysfunction (LVD) evokes altered signaling pathways and oxidative stress in the CBs that enhance CB chemoreceptor activity and reflex activation of SNA (see text). The enhanced SNA exacerbates progression of cardiac deterioration, further reduces BF, and enhances CB excitability in CHF. Right, CIH evokes altered signaling pathways and oxidative stress in the CBs that enhance chemoreceptor activity and reflex activation of SNA (see text). The enhanced SNA exacerbates progression of hypertension and further increases CB vascular resistance (VR) to reduce BF and enhance CB excitability. Mechanisms of CB activation in essential hypertension (EH) are unknown but may follow similar pathways.
Acknowledgments
We thank Lisa Rasmussen for technical assistance and Dr Irving Zucker for editorial comments.
Source of Funding
This work was supported by National Institutes of Health grant PO-1 HL62222.
Disclosures
None.
Supplemental Material
File(sup_zhy076083-s2.doc)
Download
534.00 KB
References
1.
Esler M, Rumantir M, Kaye D, Jennings G, Hastings J, Socratous F, Lambert G. Sympathetic nerve biology in essential hypertension. Clin Exp Pharmacol Physiol . 2001; 28: 986–989.
Go to Citation
Crossref
PubMed
Google Scholar
2.
Zucker IH, Schultz HD, Li YF, Wang Y, Wang W, Patel KP. The origin of sympathetic outflow in heart failure: the roles of angiotensin II and nitric oxide. Prog Biophys Mol Biol . 2004; 84: 217–232.
Crossref
PubMed
Google Scholar
a [...] progression of chronic heart failure (CHF).
b [...] baroreflexes is depressed in CHF
c [...] animal models with ischemic heart failure.
3.
Korner PI. Baroreceptor resetting and other determinants of baroreflex properties in hypertension. Clin Exp Pharmacol Physiol . 1989; 15 (suppl): 45–64.
Go to Citation
Google Scholar
4.
Zucker IH. Novel mechanisms of sympathetic regulation in chronic heart failure. Hypertension . 2006; 48: 1005–1011.
Go to Citation
Crossref
PubMed
Google Scholar
5.
Sved AF, Ito S, Sved JC. Brainstem mechanisms of hypertension: role of the rostral ventrolateral medulla. Curr Hypertens Rep . 2003; 5: 262–268.
Crossref
PubMed
Google Scholar
6.
Malliani A, Montano N. Emerging excitatory role of cardiovascular sympathetic afferents in pathophysiological conditions. Hypertension . 2002; 39: 63–68.
Go to Citation
Crossref
PubMed
Google Scholar
7.
Sinoway LI, Li J. A perspective on the muscle reflex: implications for congestive heart failure. J Appl Physiol . 2005; 99: 5–22.
Go to Citation
Crossref
PubMed
Google Scholar
8.
Schultz HD, Sun SY. Chemoreflex function in heart failure. Heart Fail Rev . 2000; 5: 45–56.
Go to Citation
Crossref
PubMed
Google Scholar
9.
Marshall JM. Peripheral chemoreceptors and cardiovascular regulation. Physiol Rev . 1994; 74: 543–594.
Crossref
PubMed
Google Scholar
a [...] influence on cardiovascular function.
b [...] to increase heart rate and contractility.
c [...] vessels are concomitantly activated.
d [...] responses, depending on the conditions.
e [...] little effect on chemoreceptor activity,
f [...] blood flow and activate CB chemoreceptors.
g [...] with reduction of CB blood flow.
10.
Prabhakar NR. Neurotransmitters in the carotid body. In: O’Regan RG, Nolan P, McQueen DS, Paterson DJ, eds. Arterial Chemoreceptors: Cell to System. New York, NY: Plenum Press; 1994: 57–69.
Google Scholar
a [...] to the medulla via the carotid sinus nerve.
b [...] known to affect CB chemoreceptor function.
c [...] CB, and NO is inhibitory to CB activity.
d [...] keep CB chemoreceptor discharge suppressed.
e [...] an inhibitory effect on CB responsiveness,
11.
Cao WH, Morrison SF. Differential chemoreceptor reflex responses of adrenal preganglionic neurons. Am J Physiol Regul Integr Comp Physiol . 2001; 281: R1825–R1832.
Go to Citation
Crossref
PubMed
Google Scholar
12.
Madden CJ, Morrison SF. Hypoxic activation of arterial chemoreceptors inhibits sympathetic outflow to brown adipose tissue in rats. J Physiol . 2005; 566: 559–573.
Go to Citation
Crossref
PubMed
Google Scholar
13.
Taylor EW, Jordan D, Coote JH. Central control of the cardiovascular and respiratory systems and their interactions in vertebrates. Physiol Rev . 1999; 79: 855–916.
Crossref
PubMed
Google Scholar
a [...] activity (PNA) to increase ventilation.
b [...] peaking during early expiration.
c [...] and then progressively declines.
14.
Somers VK, Mark AL, Zavala DC, Abboud FM. Influence of ventilation and hypocapnia on sympathetic nerve responses to hypoxia in normal humans. J Appl Physiol . 1989; 67: 2095–2100.
Go to Citation
Crossref
PubMed
Google Scholar
15.
Narkiewicz K, van de Borne P, Montano N, Hering D, Kara T, Somers VK. Sympathetic neural outflow and chemoreflex sensitivity are related to spontaneous breathing rate in normal men. Hypertension . 2006; 47: 51–55.
Go to Citation
Crossref
PubMed
Google Scholar
16.
Bernardi L, Porta C, Spicuzza L, Sleight P. Cardiorespiratory interactions to external stimuli. Arch Ital Biol . 2005; 143: 215–221.
Go to Citation
PubMed
Google Scholar
17.
Xie A, Skatrud JB, Puleo DS, Morgan BJ. Exposure to hypoxia produces long-lasting sympathetic activation in humans. J Appl Physiol . 2001; 91: 1555–1562.
Go to Citation
Crossref
PubMed
Google Scholar
18.
Dick TE, Hsieh YH, Wang N, Prabhakar NR. Acute intermittent hypoxia increases both phrenic and sympathetic nerve. Activities in the rat. Exp Physiol . 2007; 92: 87–97.
Go to Citation
Crossref
PubMed
Google Scholar
19.
Schultz HD, Li YL. Carotid body function in heart failure. Respir Physiol Neurobiol . 2007; 157: 171–185.
Crossref
PubMed
Google Scholar
a [...] failure has received considerable attention
b [...] in CHF rabbits with CB denervation.
20.
Narkiewicz K, Pesek CA, van de Borne PJ, Kato M, Somers VK. Enhanced sympathetic and ventilatory responses to central chemoreflex activation in heart failure. Circulation . 1999; 100: 262–267.
Crossref
PubMed
Google Scholar
a [...] consistently observed in humans with CHF,
b [...] is not altered in many patients with CHF.
21.
Haque WA, Boehmer J, Clemson BS, Leuenberger UA, Silber DH, Sinoway LI. Hemodynamic effects of supplemental oxygen administration in congestive heart failure. J Am Coll Cardiol . 1996; 27: 353–357.
Go to Citation
Crossref
PubMed
Google Scholar
22.
Chua T, Clark AL, Amadi AA, Coats AJ. Relation between chemosensitivity and the ventilatory response to exercise in chronic heart failure. J Am Col Cardiol . 1996; 27: 650–657.
Go to Citation
Crossref
PubMed
Google Scholar
23.
Ponikowski P, Banasiak W. Chemosensitivity in chronic heart failure. Heart Fail Monit . 2001; 1: 126–131.
PubMed
Google Scholar
24.
Sun SY, Wang W, Zucker IH, Schultz HD. Enhanced peripheral chemoreflex function in conscious rabbits with pacing-induced heart failure. J Appl Physiol . 1999; 86: 1264–1272.
Crossref
PubMed
Google Scholar
a [...] in a rabbit model of pacing-induced CHF.
b [...] to moderate compensated CHF in patients.
c [...] resting RSNA in CHF but not control animals
25.
Sun SY, Wang W, Zucker IH, Schultz HD. Enhanced activity of carotid body chemoreceptors in rabbits with heart failure: role of nitric oxide. J Appl Physiol . 1999; 86: 1273–1282.
Crossref
PubMed
Google Scholar
a [...] enhanced over the same time course
b [...] resting conditions observed in CHF rabbits.
c [...] enhanced basal activity in the CB in CHF.
d [...] within the CB are depressed in CHF.
e [...] is virtually absent in CHF rabbits.
26.
Li YL, Li YF, Liu D, Cornish KG, Patel KP, Zucker IH, Channon KM, Schultz HD. Gene transfer of neuronal nitric oxide synthase to carotid body reverses enhanced chemoreceptor function in heart failure rabbits. Circ Res . 2005; 97: 260–267.
Crossref
PubMed
Google Scholar
a [...] CB chemoreceptor activity in CHF rabbits.
b [...] activity seen in the CHF state.
c [...] the CB in CHF rabbits reduces resting RSNA.
27.
Li YL, Xia XH, Zheng H, Gao L, Li YF, Liu D, Patel KP, Wang W, Schultz HD. Angiotensin II enhances carotid body chemoreflex control of sympathetic outflow in chronic heart failure rabbits. Cardiovasc Res . 2006; 71: 129–138.
Crossref
PubMed
Google Scholar
a [...] II system is operational in the rabbit CB.
b [...] R in the rabbit CB.
c [...] activation of sympathetic outflow.
d [...] responses observed in CHF rabbits.
e [...] sensitivity to hypoxemia in CHF rabbits.
f [...] in glomus cells at normoxia in CHF animals.
28.
Xia XH, Sun SY, Cornish KG, Zeng YC, Wang W, Schultz HD. Effect of carotid body denervation on Cheyne-Stokes respiration and baroreflex function in heart failure. Circulation . 2000; 102: 700–701.Abstract.
Go to Citation
Google Scholar
29.
Coccagna G, Pollini A, Provini F. Cardiovascular disorders and obstructive sleep apnea syndrome. Clin Exp Hypertens . 2006; 28: 217–224.
Go to Citation
Crossref
PubMed
Google Scholar
30.
Caples SM, Wolk R, Somers VK. Influence of cardiac function and failure on sleep-disordered breathing: evidence for a causative role. J Appl Physiol . 2005; 99: 2433–2439.
Go to Citation
Crossref
PubMed
Google Scholar
31.
Allen AM. Angiotensin AT 1 receptor-mediated excitation of rat carotid body chemoreceptor afferent activity. J Physiol . 1998; 510: 773–781.
Go to Citation
Crossref
PubMed
Google Scholar
32.
Lam SY, Leung PS. A locally generated angiotensin system in rat carotid body. Regul Pept . 2002; 107: 97–103.
Crossref
PubMed
Google Scholar
33.
Li YL, Schultz HD. Enhanced sensitivity of Kv channels to hypoxia in the rabbit carotid body in heart failure: role of angiotensin II. J Physiol . 2006; 575: 215–227.
Crossref
PubMed
Google Scholar
a [...] ) in CB glomus cells.
b [...] this enhanced hypoxic sensitivity.
c [...] channel function.
d [...] described in glomus cells in CHF rabbits
e [...] channel function in CB glomus cells,
34.
Griendling KK, Minieri CA, Ollerenshaw JD, Alexander RW. Angiotensin II stimulates NADH and NADPH oxidase activity in cultured vascular smooth muscle cells. Circ Res . 1994; 74: 1141–1148.
Go to Citation
Crossref
PubMed
Google Scholar
35.
Li YL, Gao L, Zucker IH, Schultz HD. NADPH oxidase-derived superoxide anion mediates angiotensin II-enhanced peripheral chemoreceptor sensitivity in heart failure rabbits. Cardiovas Res. In press.
Google Scholar
a [...] production are enhanced in CHF rabbits.
b [...] and the superoxide anion scavenger Tempol.
c [...] in CB glomus cells from CHF rabbits.
d [...] hypoxia. Based on our work described above,
36.
Li YL, Sun SY, Overholt JL, Prabhakar NR, Rozanski GJ, Zucker IH, Schultz HD. Attenuated outward potassium currents in carotid body glomus cells of heart failure rabbit: involvement of nitric oxide. J Physiol . 2004; 555: 219–229.
Crossref
PubMed
Google Scholar
a [...] pathway in CB glomus cells in rabbits.
b [...] conditions because of NO depletion,
c [...] potential observed in the CHF glomus cells.
37.
Overholt JL, Ficker E, Yang T, Shams H, Bright GR, Prabhakar NR. HERG-like potassium current regulates the resting membrane potential in glomus cells of the rabbit carotid body. J Neurophysiol . 2000; 83: 1150–1157.
Go to Citation
Crossref
PubMed
Google Scholar
38.
Kihara M, Umemura S, Kadota T, Yabana M, Tamura K, Nyuui N, Ogawa N, Murakami K, Fukamizu A, Ishii M. The neuronal isoform of constitutive nitric oxide synthase is up-regulated in the macula densa of angiotensinogen gene-knockout mice. Lab Invest . 1997; 76: 285–294.
Go to Citation
PubMed
Google Scholar
39.
Gozal D, Gozal E, Gozal YM, Torres JE. Nitric oxide synthase isoforms and peripheral chemoreceptor stimulation in conscious rats. Neuroreport . 1996; 7: 1145–1148.
Go to Citation
Crossref
PubMed
Google Scholar
40.
Kline DD, Prabhakar NR. Peripheral chemosensitivity in mutant mice deficient in nitric oxide synthase. Adv Exp Med Biol . 2000; 475: 571–579.
Go to Citation
PubMed
Google Scholar
41.
Kääb S, Miguel-Velado E, López-López JR, Pérez-García MT. Downregulaion of Kv3.4 channels by chronic hypoxia increases acute oxygen sensitivity in rabbit carotid body. J Physiol . 2005; 566: 395–408.
Go to Citation
Crossref
PubMed
Google Scholar
42.
Ding YF, Li YL, Cornish KG, Schultz HD. Reduced blood flow in carotid arteries is a trigger contributing to peripheral chemoreflex hypersensitivity in chronic heart failure rabbits. FASEB J . 2007; 21: A1268. Abstract.
Google Scholar
a [...] online at
b [...] similar to changes observed in CHF rabbits.
c [...] R and decrease in nNOS expression.
43.
Lam SY, Fung ML, Leung PS. Regulation of the angiotensin-converting enzyme activity by a time-course hypoxia in the carotid body. J Appl Physiol . 2004; 96: 809–813.
Go to Citation
Crossref
PubMed
Google Scholar
44.
Prabhakar NR. Oxygen sensing during intermittent hypoxia: cellular and molecular mechanisms. J Appl Physiol . 2001; 90: 1986–1994.
Crossref
PubMed
Google Scholar
a [...] activated in the CB in response to hypoxia.
b [...] elevated in CBs from CIH-exposed animals.
45.
Habeck JO. Peripheral arterial chemoreceptors and hypertension. J Auton Nerv Syst . 1991; 34: 1–7.
Crossref
PubMed
Google Scholar
a [...] and borderline essential hypertension
b [...] and SHRs exhibit hypertrophied CBs,
46.
Somers VK, Mark AL, Abboud FM. Potentiation of sympathetic nerve responses to hypoxia in borderline hypertensive subjects. Hypertension . 1988; 11: 608–612.
Go to Citation
Crossref
PubMed
Google Scholar
47.
Ohta H, Talman WT. Alteration of baroreceptor and chemoreceptor reflexes in spontaneously hypertensive rats. Clin Exp Pharmacol Physiol . 1995; 22 (suppl): S60–S61.
Go to Citation
Crossref
Google Scholar
48.
Fukuda Y, Sato A, Trzebski A. Carotid chemoreceptor discharge responses to hypoxia and hypercapnia in normotensive and spontaneously hypertensive rats. J Auton Nerv Syst . 1987; 19: 1–11.
Crossref
PubMed
Google Scholar
a [...] to hypoxemia is exaggerated in SHRs.
b [...] in SHRs has not been explored. Fukuda et al
49.
Hayward LF, Johnson AK, Felder RB. Arterial chemoreflex in conscious normotensive and hypertensive adult rats. Am J Physiol . 1999; 276: H1215–H1222.
Go to Citation
Crossref
PubMed
Google Scholar
50.
Angell-James JE, Clarke JA, Daly MD, Taton A. Respiratory and cardiovascular responses to hyperoxia, hypoxia and hypercapnia in the renal hypertensive rabbit: role of carotid body chemoreceptors. J Hypertens . 1985; 3: 213–223.
Go to Citation
Crossref
PubMed
Google Scholar
51.
Izdebska E, Cybulska I, Sawicki M, Izdebski J, Trzebski A. Postexercise decrease in arterial blood pressure, total peripheral resistance and in circulatory responses to brief hyperoxia in subjects with mild essential hypertension. J Hum Hypertens . 1998; 12: 855–860.
Go to Citation
Crossref
PubMed
Google Scholar
52.
Smith ML, Pacchia CF. Sleep apnoea and hypertension. Role of chemoreflexes in humans. Exp Physiol . 2007; 92: 45–50.
Go to Citation
Crossref
PubMed
Google Scholar
53.
Narkiewicz K, van de Borne PJ, Pesek CA, Dyken ME, Montano N, Somers VK. Selective potentiation of peripheral chemoreflex sensitivity in obstructive sleep apnea. Circulation . 1999; 99: 1183–1189.
Crossref
PubMed
Google Scholar
a [...] and cold pressor tests are normal.
b [...] arterial pressure in patients with SA.
54.
Fletcher EC. Invited review: physiological consequences of intermittent hypoxia: systemic blood pressure. J Appl Physiol . 2001; 90: 1600–1605.
Crossref
PubMed
Google Scholar
a [...] in the exploration of this issue.
b [...] similar to that seen in patients with SA.
55.
Lesske J, Fletcher EC, Bao G, Unger T. Hypertension caused by chronic intermittent hypoxia-influence ofchemoreceptors and sympathetic nervous system. J Hypertens . 1997; 15: 1593–1603.
Go to Citation
Crossref
PubMed
Google Scholar
56.
Peng Y, Kline DD, Dick TE, Prabhakar NR. Chronic intermittent hypoxia enhances carotid body chemoreceptor response to low oxygen. Adv Exp Med Biol . 2001; 499: 33–38.
Crossref
PubMed
Google Scholar
a [...] to hypoxemia but not to hypercapnia.
b [...] Peng et al
57.
Peng YJ, Overholt JL, Kline D, Kumar GK, Prabhakar NR. Induction of sensory long-term facilitation in the carotid body by intermittent hypoxia: implications for recurrent apneas. Proc Natl Acad Sci U S A . 2003; 100: 10073–10078.
Go to Citation
Crossref
PubMed
Google Scholar
58.
Esler M, Straznicky N, Eikelis N, Masuo K, Lambert G, Lambert E. Mechanisms of sympathetic activation in obesity-related hypertension. Hypertension . 2006; 48: 787–796.
Go to Citation
Crossref
PubMed
Google Scholar
59.
Nimbkar NV, Lateef F. Carotid body dysfunction: the possible etiology of non-insulin dependent diabetes mellitus and essential hypertension. Med Hypotheses . 2005; 65: 1067–1075.
Go to Citation
Crossref
PubMed
Google Scholar
60.
Sato MA, Colombari E, Morrison SF. Inhibition of neurons in commissural nucleus of solitary tract reduces sympathetic nerve activity in SHR. Am J Physiol Heart Circ Physiol . 2002; 282: H1679–H1684.
Go to Citation
Crossref
PubMed
Google Scholar
61.
Sato MA, Menani JV, Lopes OU, Colombari E. Lesions of the commissural nucleus of the solitary tract reduce arterial pressure in spontaneously hypertensive rats. Hypertension . 2001; 38: 560–564.
Crossref
PubMed
Google Scholar
62.
Peng YJ, Prabhakar NR. Reactive oxygen species in the plasticity of respiratory behavior elicited by chronic intermittent hypoxia. J Appl Physiol . 2003; 94: 2342–2349.
Go to Citation
Crossref
PubMed
Google Scholar
63.
Prabhakar NR, Kumar GK. Oxidative stress in the systemic and cellular responses to intermittent hypoxia. Biol Chem . 2004; 385: 217–221.
PubMed
Google Scholar
a [...] complex III occurs in CBs from CIH rats.
b [...] increases oxygen radicals in both groups.
c [...] to activation of hypoxia-inducible factor-1
d [...] and/or activator protein-1.
64.
Peng YJ, Yuan G, Jacono FJ, Kumar GK, Prabhakar NR. 5-HT evokes sensory long-term facilitation of rodent carotid body via activation of NADPH oxidase. J Physiol . 2006; 576: 289–295.
Go to Citation
Crossref
PubMed
Google Scholar
65.
Chen J, He L, Dinger B, Stensaas L, Fidone S. Role of endothelin and endothelin A-type receptor in adaptation of the carotid body to chronic hypoxia. Am J Physiol Lung Cell Mol Physiol . 2002; 282: L1314–L1323.
Go to Citation
Crossref
PubMed
Google Scholar
66.
Rey S, Del Rio R, Iturriaga R. Contribution of endothelin-1 to the enhanced carotid body chemosensory responses induced by chronic intermittent hypoxia. Brain Res . 2006; 1086: 152–159.
Crossref
PubMed
Google Scholar
a [...] induces CB chemosensory excitation,
b [...] delivery,
67.
Chen J, He L, Dinger B, Fidone S. Cellular mechanisms involved in rabbit carotid body excitation elicited by endothelin peptides. Respir Physiol . 2000; 121: 13–23.
Crossref
PubMed
Google Scholar
a [...] the chemosensory response to acute hypoxia.
b [...] and hypoxic CB discharge induced by CIH.
c [...] which is devoid of vascular effects.
d [...] via a cAMP/inositol triphosphate pathway.
68.
Rey S, Corthorn J, Chacon C, Iturriaga R. Expression and immunolocalization of endothelin peptides and its receptors, ET A and ET B, in the carotid body exposed to chronic intermittent hypoxia. J Histochem Cytochem . 2007; 55: 167–174.
Go to Citation
Crossref
PubMed
Google Scholar
69.
Lam SY, Tipoe GL, Liong EC, Fung ML. Hypoxia-inducible factor (HIF)-1alpha and endothelin-1 expression in the rat carotid body during intermittent hypoxia. Adv Exp Med Biol . 2006; 580: 21–27;discussion 351–359.
Crossref
PubMed
Google Scholar
70.
Almaraz L, Perez-Garcia MT, Gomez-Nino A, Gonzalez C. Mechanisms of alpha2-adrenoceptor-mediated inhibition in rabbit carotid body. Am J Physiol . 1997; 272: C628–C637.
Go to Citation
Crossref
PubMed
Google Scholar
71.
Przybylski J. Do arterial chemoreceptors play a role in the pathogenesis of hypertension? Med Hypotheses . 1981; 7: 127–131.
Go to Citation
Crossref
PubMed
Google Scholar
Show all references
eLetters
eLetters should relate to an article recently published in the journal and are not a forum for providing unpublished data. Comments are reviewed for appropriate use of tone and language. Comments are not peer-reviewed. Acceptable comments are posted to the journal website only. Comments are not published in an issue and are not indexed in PubMed. Comments should be no longer than 500 words and will only be posted online. References are limited to 10. Authors of the article cited in the comment will be invited to reply, as appropriate.
Comments and feedback on AHA/ASA Scientific Statements and Guidelines should be directed to the AHA/ASA Manuscript Oversight Committee via its Correspondence page.
Sign In to Submit a Response to This Article
Information & Authors
Information Authors
Information
Published In
Hypertension
Volume 50 • Number 1 • 1 July 2007
Pages: 6 - 13
PubMed: 17502495
Copyright
© 2007.
Versions
You are viewing the most recent version of this article.
14 May 2007: Previous PDF (Version 1)
History
Received: 31 January 2007
Revision received: 19 February 2007
Accepted: 19 April 2007
Published online: 14 May 2007
Published in print: 1 July 2007
Permissions
Request permissions for this article.
Request permissions
Subjects
Animal Models of Human Disease
Chronic Ischemic Heart Disease
Clinical Studies
Congenital Heart Disease
Etiology
Heart Failure
Hypertension
Authors
Affiliations Expand All
Harold D.Schultz
From the Department of Cellular and Integrative Physiology, University of Nebraska Medical Center, Omaha.
View all articles by this author
Yu L.Li
From the Department of Cellular and Integrative Physiology, University of Nebraska Medical Center, Omaha.
View all articles by this author
Yanfeng Ding
From the Department of Cellular and Integrative Physiology, University of Nebraska Medical Center, Omaha.
View all articles by this author
Notes
Correspondence to Harold D. Schultz, Department of Cellular and Integrative Physiology, University of Nebraska Medical Center, 985850 Nebraska Medical Center, Omaha, NE 68198-5850. E-mail hschultz@unmc.edu
Metrics & Citations
Metrics Citations 124
Metrics
Article Metrics
View all metrics
Downloads
Citations
No data available.
37,111
124
Total
6 Months
12 Months
Total number of downloads and citations
See more details
Referenced in 1 patents
100 readers on Mendeley
Citations
Download Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Select your manager software from the list below and click Download.
Please select your download format:
[x] Direct Import
Xiang-Qun Hu,
Chiranjib Dasgupta,
Yu Gao,
Isaac Kremsky,
Eugenia Mata-Greenwood,
Lubo Zhang,
The uteroplacental circulation at high altitude: adaptation and maladaptation, Philosophical Transactions of the Royal Society B: Biological Sciences, 380, 1933, (2025).
Crossref
Ana Santamaría-Santiago,
Verónica Sobrino,
Luis Luna-Ramírez,
José López-Barneo,
Ricardo Pardal,
Genetic map of the carotid body stem cell niche with focus on the O2-sensing chemoreceptor cell lineage, Scientific Reports, 15, 1, (2025).
Crossref
Franziska Koppe-Schmeißer,
Karl Fengler,
Karl-Patrik Kresoja,
Philipp Lurz,
Karl-Philipp Rommel,
Renal denervation in the setting of heart failure, Heart Failure Reviews, (2025).
Crossref
Anastasiia Vasileva,
Laura D. M. Flores,
Mikhail Vasilyev,
Michelle A. Buckman,
Lara R. DeRuisseau,
Michael H. Tomasson,
Melissa L. Bates,
Cardiovascular function and autonomic regulation in urethane-anesthetized and conscious mice, American Journal of Physiology-Regulatory, Integrative and Comparative Physiology, 328, 1, (R133-R144), (2025).
Crossref
LUIS ANDRES DULCEY SARMIENTO,
Jhon Fredy Castillo Blanco,
Juan Sebastian Theran Leon,
Raimondo Caltagirone Miceli,
Eficacia de la Denervación simpática renal como ultimo escalafón terapéutico en hipertensión refractaria, revisión de la literatura., Revista Cuarzo, 30, 1, (26-31), (2024).
Crossref
Mariana Reis Guedes,
Sylvana Izaura Salyba Rendeiro de Noronha,
Máira Tereza Talma Chírico,
Gabriela Dias Carvalho da Costa,
Thalles de Freitas Castro,
Rory Cristiane Fortes de Brito,
Lucas Gabriel Vieira,
Thayane Oliveira Reis,
Marcelo Carlos Ribeiro,
Alexandre Barbosa Reis,
Cláudia Martins Carneiro,
Frank Silva Bezerra,
Nicola Montano,
Valdo José Dias da Silva,
Rodrigo Cunha Alvim de Menezes,
Deoclécio Alves Chianca-Jr,
Fernanda Cacilda dos Santos Silva,
Ivabradine restores tonic cardiovascular autonomic control and reduces tachycardia, hypertension and left ventricular inflammation in post-weaning protein malnourished rats, Life Sciences, 346, (122636), (2024).
Crossref
Sheng-Feng Tsai,
Yu-Min Kuo,
The Role of Central Oxytocin in Autonomic Regulation, Journal of Physiological Investigation, 67, 1, (3-14), (2024).
Crossref
R.M. Lataro,
F. Brognara,
R. Iturriaga,
J.F.R. Paton,
Inflammation of some visceral sensory systems and autonomic dysfunction in cardiovascular disease, Autonomic Neuroscience, 251, (103137), (2024).
Crossref
Ana I. Afonso,
Ângela Amaro-Leal,
Filipa Machado,
Isabel Rocha,
Vera Geraldes,
Doxorubicin Dose-Dependent Impact on Physiological Balance—A Holistic Approach in a Rat Model, Biology, 12, 7, (1031), (2023).
Crossref
Alberto Giannoni,
Chiara Borrelli,
Francesco Gentile,
Paolo Sciarrone,
Jens Spießhöfer,
Massimo Piepoli,
George B. Richerson,
John S. Floras,
Andrew J.S. Coats,
Shahrokh Javaheri,
Michele Emdin,
Claudio Passino,
Autonomic and respiratory consequences of altered chemoreflex function: clinical and therapeutic implications in cardiovascular diseases, European Journal of Heart Failure, 25, 5, (642-656), (2023).
Crossref
See more
Loading...
View Options
View options
PDF and All Supplements
Download PDF and All Supplements
Download is in progress
PDF/EPUB
View PDF/EPUB
Figures
Open all in viewer
Figure 1.A, Effect of chronic cardiac pacing on resting (21% O 2) and hypoxia-activated RSNA in a conscious rabbit. RSNA obtained by chronic telemetric recording. Left ventricular fractional shortening, 77% of control after 1 week and 50% of control after 3 weeks of pacing. B, Carotid body CB chemoreflex function curves for RSNA before and during cardiac pacing. RSNA normalized among animals as percentage of maximal activity (evoked by the nasopharyngeal reflex). Left ventricular function for the group is listed in Table S1. n=4; P<0.05 vs prepacing.
Go to FigureOpen in Viewer
Figure 2.Effects of hypoxia (10% O 2) and hyperoxia (100% O 2, Dejour’s effect) on resting CB chemoreceptor single unit activity (A) and RSNA (B) in normal and CHF (3 to 4 weeks pacing) rabbits. Resting and hypoxia-activated CB activity and RSNA are elevated in CHF rabbits compared with normal. Hyperoxia inhibited CB activity and normalized RSNA in CHF rabbits. P=0.05 vs room air; #P<0.05 vs normal.
Go to FigureOpen in Viewer
Figure 3.Effect of chronic carotid artery occlusion (CAO) for 3 weeks on resting and hypoxia activated RSNA (A and B) and on resting and hypoxia activated CB activity (C) in rabbits (CAO mimicked carotid flow during cardiac pacing: see online supplemental data). P<0.05 vs normal.
Go to FigureOpen in Viewer
Figure 4.The vicious cycle of CB chemoreflex activation in CHF and hypertension. Left, Chronic reduction in CB blood flow (BF) because of left ventricular dysfunction (LVD) evokes altered signaling pathways and oxidative stress in the CBs that enhance CB chemoreceptor activity and reflex activation of SNA (see text). The enhanced SNA exacerbates progression of cardiac deterioration, further reduces BF, and enhances CB excitability in CHF. Right, CIH evokes altered signaling pathways and oxidative stress in the CBs that enhance chemoreceptor activity and reflex activation of SNA (see text). The enhanced SNA exacerbates progression of hypertension and further increases CB vascular resistance (VR) to reduce BF and enhance CB excitability. Mechanisms of CB activation in essential hypertension (EH) are unknown but may follow similar pathways.
Go to FigureOpen in Viewer
Tables
Media
Share
Share
Share article link
Copy Link
Copied!
Copying failed.
Share
FacebookLinkedInX (formerly Twitter)emailWeChatBluesky
References
References
1.
Esler M, Rumantir M, Kaye D, Jennings G, Hastings J, Socratous F, Lambert G. Sympathetic nerve biology in essential hypertension. Clin Exp Pharmacol Physiol . 2001; 28: 986–989.
Go to Citation
Crossref
PubMed
Google Scholar
2.
Zucker IH, Schultz HD, Li YF, Wang Y, Wang W, Patel KP. The origin of sympathetic outflow in heart failure: the roles of angiotensin II and nitric oxide. Prog Biophys Mol Biol . 2004; 84: 217–232.
Crossref
PubMed
Google Scholar
a [...] progression of chronic heart failure (CHF).
b [...] baroreflexes is depressed in CHF
c [...] animal models with ischemic heart failure.
3.
Korner PI. Baroreceptor resetting and other determinants of baroreflex properties in hypertension. Clin Exp Pharmacol Physiol . 1989; 15 (suppl): 45–64.
Go to Citation
Google Scholar
4.
Zucker IH. Novel mechanisms of sympathetic regulation in chronic heart failure. Hypertension . 2006; 48: 1005–1011.
Go to Citation
Crossref
PubMed
Google Scholar
5.
Sved AF, Ito S, Sved JC. Brainstem mechanisms of hypertension: role of the rostral ventrolateral medulla. Curr Hypertens Rep . 2003; 5: 262–268.
Crossref
PubMed
Google Scholar
6.
Malliani A, Montano N. Emerging excitatory role of cardiovascular sympathetic afferents in pathophysiological conditions. Hypertension . 2002; 39: 63–68.
Go to Citation
Crossref
PubMed
Google Scholar
7.
Sinoway LI, Li J. A perspective on the muscle reflex: implications for congestive heart failure. J Appl Physiol . 2005; 99: 5–22.
Go to Citation
Crossref
PubMed
Google Scholar
8.
Schultz HD, Sun SY. Chemoreflex function in heart failure. Heart Fail Rev . 2000; 5: 45–56.
Go to Citation
Crossref
PubMed
Google Scholar
9.
Marshall JM. Peripheral chemoreceptors and cardiovascular regulation. Physiol Rev . 1994; 74: 543–594.
Crossref
PubMed
Google Scholar
a [...] influence on cardiovascular function.
b [...] to increase heart rate and contractility.
c [...] vessels are concomitantly activated.
d [...] responses, depending on the conditions.
e [...] little effect on chemoreceptor activity,
f [...] blood flow and activate CB chemoreceptors.
g [...] with reduction of CB blood flow.
10.
Prabhakar NR. Neurotransmitters in the carotid body. In: O’Regan RG, Nolan P, McQueen DS, Paterson DJ, eds. Arterial Chemoreceptors: Cell to System. New York, NY: Plenum Press; 1994: 57–69.
Google Scholar
a [...] to the medulla via the carotid sinus nerve.
b [...] known to affect CB chemoreceptor function.
c [...] CB, and NO is inhibitory to CB activity.
d [...] keep CB chemoreceptor discharge suppressed.
e [...] an inhibitory effect on CB responsiveness,
11.
Cao WH, Morrison SF. Differential chemoreceptor reflex responses of adrenal preganglionic neurons. Am J Physiol Regul Integr Comp Physiol . 2001; 281: R1825–R1832.
Go to Citation
Crossref
PubMed
Google Scholar
12.
Madden CJ, Morrison SF. Hypoxic activation of arterial chemoreceptors inhibits sympathetic outflow to brown adipose tissue in rats. J Physiol . 2005; 566: 559–573.
Go to Citation
Crossref
PubMed
Google Scholar
13.
Taylor EW, Jordan D, Coote JH. Central control of the cardiovascular and respiratory systems and their interactions in vertebrates. Physiol Rev . 1999; 79: 855–916.
Crossref
PubMed
Google Scholar
a [...] activity (PNA) to increase ventilation.
b [...] peaking during early expiration.
c [...] and then progressively declines.
14.
Somers VK, Mark AL, Zavala DC, Abboud FM. Influence of ventilation and hypocapnia on sympathetic nerve responses to hypoxia in normal humans. J Appl Physiol . 1989; 67: 2095–2100.
Go to Citation
Crossref
PubMed
Google Scholar
15.
Narkiewicz K, van de Borne P, Montano N, Hering D, Kara T, Somers VK. Sympathetic neural outflow and chemoreflex sensitivity are related to spontaneous breathing rate in normal men. Hypertension . 2006; 47: 51–55.
Go to Citation
Crossref
PubMed
Google Scholar
16.
Bernardi L, Porta C, Spicuzza L, Sleight P. Cardiorespiratory interactions to external stimuli. Arch Ital Biol . 2005; 143: 215–221.
Go to Citation
PubMed
Google Scholar
17.
Xie A, Skatrud JB, Puleo DS, Morgan BJ. Exposure to hypoxia produces long-lasting sympathetic activation in humans. J Appl Physiol . 2001; 91: 1555–1562.
Go to Citation
Crossref
PubMed
Google Scholar
18.
Dick TE, Hsieh YH, Wang N, Prabhakar NR. Acute intermittent hypoxia increases both phrenic and sympathetic nerve. Activities in the rat. Exp Physiol . 2007; 92: 87–97.
Go to Citation
Crossref
PubMed
Google Scholar
19.
Schultz HD, Li YL. Carotid body function in heart failure. Respir Physiol Neurobiol . 2007; 157: 171–185.
Crossref
PubMed
Google Scholar
a [...] failure has received considerable attention
b [...] in CHF rabbits with CB denervation.
20.
Narkiewicz K, Pesek CA, van de Borne PJ, Kato M, Somers VK. Enhanced sympathetic and ventilatory responses to central chemoreflex activation in heart failure. Circulation . 1999; 100: 262–267.
Crossref
PubMed
Google Scholar
a [...] consistently observed in humans with CHF,
b [...] is not altered in many patients with CHF.
21.
Haque WA, Boehmer J, Clemson BS, Leuenberger UA, Silber DH, Sinoway LI. Hemodynamic effects of supplemental oxygen administration in congestive heart failure. J Am Coll Cardiol . 1996; 27: 353–357.
Go to Citation
Crossref
PubMed
Google Scholar
22.
Chua T, Clark AL, Amadi AA, Coats AJ. Relation between chemosensitivity and the ventilatory response to exercise in chronic heart failure. J Am Col Cardiol . 1996; 27: 650–657.
Go to Citation
Crossref
PubMed
Google Scholar
23.
Ponikowski P, Banasiak W. Chemosensitivity in chronic heart failure. Heart Fail Monit . 2001; 1: 126–131.
PubMed
Google Scholar
24.
Sun SY, Wang W, Zucker IH, Schultz HD. Enhanced peripheral chemoreflex function in conscious rabbits with pacing-induced heart failure. J Appl Physiol . 1999; 86: 1264–1272.
Crossref
PubMed
Google Scholar
a [...] in a rabbit model of pacing-induced CHF.
b [...] to moderate compensated CHF in patients.
c [...] resting RSNA in CHF but not control animals
25.
Sun SY, Wang W, Zucker IH, Schultz HD. Enhanced activity of carotid body chemoreceptors in rabbits with heart failure: role of nitric oxide. J Appl Physiol . 1999; 86: 1273–1282.
Crossref
PubMed
Google Scholar
a [...] enhanced over the same time course
b [...] resting conditions observed in CHF rabbits.
c [...] enhanced basal activity in the CB in CHF.
d [...] within the CB are depressed in CHF.
e [...] is virtually absent in CHF rabbits.
26.
Li YL, Li YF, Liu D, Cornish KG, Patel KP, Zucker IH, Channon KM, Schultz HD. Gene transfer of neuronal nitric oxide synthase to carotid body reverses enhanced chemoreceptor function in heart failure rabbits. Circ Res . 2005; 97: 260–267.
Crossref
PubMed
Google Scholar
a [...] CB chemoreceptor activity in CHF rabbits.
b [...] activity seen in the CHF state.
c [...] the CB in CHF rabbits reduces resting RSNA.
27.
Li YL, Xia XH, Zheng H, Gao L, Li YF, Liu D, Patel KP, Wang W, Schultz HD. Angiotensin II enhances carotid body chemoreflex control of sympathetic outflow in chronic heart failure rabbits. Cardiovasc Res . 2006; 71: 129–138.
Crossref
PubMed
Google Scholar
a [...] II system is operational in the rabbit CB.
b [...] R in the rabbit CB.
c [...] activation of sympathetic outflow.
d [...] responses observed in CHF rabbits.
e [...] sensitivity to hypoxemia in CHF rabbits.
f [...] in glomus cells at normoxia in CHF animals.
28.
Xia XH, Sun SY, Cornish KG, Zeng YC, Wang W, Schultz HD. Effect of carotid body denervation on Cheyne-Stokes respiration and baroreflex function in heart failure. Circulation . 2000; 102: 700–701.Abstract.
Go to Citation
Google Scholar
29.
Coccagna G, Pollini A, Provini F. Cardiovascular disorders and obstructive sleep apnea syndrome. Clin Exp Hypertens . 2006; 28: 217–224.
Go to Citation
Crossref
PubMed
Google Scholar
30.
Caples SM, Wolk R, Somers VK. Influence of cardiac function and failure on sleep-disordered breathing: evidence for a causative role. J Appl Physiol . 2005; 99: 2433–2439.
Go to Citation
Crossref
PubMed
Google Scholar
31.
Allen AM. Angiotensin AT 1 receptor-mediated excitation of rat carotid body chemoreceptor afferent activity. J Physiol . 1998; 510: 773–781.
Go to Citation
Crossref
PubMed
Google Scholar
32.
Lam SY, Leung PS. A locally generated angiotensin system in rat carotid body. Regul Pept . 2002; 107: 97–103.
Crossref
PubMed
Google Scholar
33.
Li YL, Schultz HD. Enhanced sensitivity of Kv channels to hypoxia in the rabbit carotid body in heart failure: role of angiotensin II. J Physiol . 2006; 575: 215–227.
Crossref
PubMed
Google Scholar
a [...] ) in CB glomus cells.
b [...] this enhanced hypoxic sensitivity.
c [...] channel function.
d [...] described in glomus cells in CHF rabbits
e [...] channel function in CB glomus cells,
34.
Griendling KK, Minieri CA, Ollerenshaw JD, Alexander RW. Angiotensin II stimulates NADH and NADPH oxidase activity in cultured vascular smooth muscle cells. Circ Res . 1994; 74: 1141–1148.
Go to Citation
Crossref
PubMed
Google Scholar
35.
Li YL, Gao L, Zucker IH, Schultz HD. NADPH oxidase-derived superoxide anion mediates angiotensin II-enhanced peripheral chemoreceptor sensitivity in heart failure rabbits. Cardiovas Res. In press.
Google Scholar
a [...] production are enhanced in CHF rabbits.
b [...] and the superoxide anion scavenger Tempol.
c [...] in CB glomus cells from CHF rabbits.
d [...] hypoxia. Based on our work described above,
36.
Li YL, Sun SY, Overholt JL, Prabhakar NR, Rozanski GJ, Zucker IH, Schultz HD. Attenuated outward potassium currents in carotid body glomus cells of heart failure rabbit: involvement of nitric oxide. J Physiol . 2004; 555: 219–229.
Crossref
PubMed
Google Scholar
a [...] pathway in CB glomus cells in rabbits.
b [...] conditions because of NO depletion,
c [...] potential observed in the CHF glomus cells.
37.
Overholt JL, Ficker E, Yang T, Shams H, Bright GR, Prabhakar NR. HERG-like potassium current regulates the resting membrane potential in glomus cells of the rabbit carotid body. J Neurophysiol . 2000; 83: 1150–1157.
Go to Citation
Crossref
PubMed
Google Scholar
38.
Kihara M, Umemura S, Kadota T, Yabana M, Tamura K, Nyuui N, Ogawa N, Murakami K, Fukamizu A, Ishii M. The neuronal isoform of constitutive nitric oxide synthase is up-regulated in the macula densa of angiotensinogen gene-knockout mice. Lab Invest . 1997; 76: 285–294.
Go to Citation
PubMed
Google Scholar
39.
Gozal D, Gozal E, Gozal YM, Torres JE. Nitric oxide synthase isoforms and peripheral chemoreceptor stimulation in conscious rats. Neuroreport . 1996; 7: 1145–1148.
Go to Citation
Crossref
PubMed
Google Scholar
40.
Kline DD, Prabhakar NR. Peripheral chemosensitivity in mutant mice deficient in nitric oxide synthase. Adv Exp Med Biol . 2000; 475: 571–579.
Go to Citation
PubMed
Google Scholar
41.
Kääb S, Miguel-Velado E, López-López JR, Pérez-García MT. Downregulaion of Kv3.4 channels by chronic hypoxia increases acute oxygen sensitivity in rabbit carotid body. J Physiol . 2005; 566: 395–408.
Go to Citation
Crossref
PubMed
Google Scholar
42.
Ding YF, Li YL, Cornish KG, Schultz HD. Reduced blood flow in carotid arteries is a trigger contributing to peripheral chemoreflex hypersensitivity in chronic heart failure rabbits. FASEB J . 2007; 21: A1268. Abstract.
Google Scholar
a [...] online at
b [...] similar to changes observed in CHF rabbits.
c [...] R and decrease in nNOS expression.
43.
Lam SY, Fung ML, Leung PS. Regulation of the angiotensin-converting enzyme activity by a time-course hypoxia in the carotid body. J Appl Physiol . 2004; 96: 809–813.
Go to Citation
Crossref
PubMed
Google Scholar
44.
Prabhakar NR. Oxygen sensing during intermittent hypoxia: cellular and molecular mechanisms. J Appl Physiol . 2001; 90: 1986–1994.
Crossref
PubMed
Google Scholar
a [...] activated in the CB in response to hypoxia.
b [...] elevated in CBs from CIH-exposed animals.
45.
Habeck JO. Peripheral arterial chemoreceptors and hypertension. J Auton Nerv Syst . 1991; 34: 1–7.
Crossref
PubMed
Google Scholar
a [...] and borderline essential hypertension
b [...] and SHRs exhibit hypertrophied CBs,
46.
Somers VK, Mark AL, Abboud FM. Potentiation of sympathetic nerve responses to hypoxia in borderline hypertensive subjects. Hypertension . 1988; 11: 608–612.
Go to Citation
Crossref
PubMed
Google Scholar
47.
Ohta H, Talman WT. Alteration of baroreceptor and chemoreceptor reflexes in spontaneously hypertensive rats. Clin Exp Pharmacol Physiol . 1995; 22 (suppl): S60–S61.
Go to Citation
Crossref
Google Scholar
48.
Fukuda Y, Sato A, Trzebski A. Carotid chemoreceptor discharge responses to hypoxia and hypercapnia in normotensive and spontaneously hypertensive rats. J Auton Nerv Syst . 1987; 19: 1–11.
Crossref
PubMed
Google Scholar
a [...] to hypoxemia is exaggerated in SHRs.
b [...] in SHRs has not been explored. Fukuda et al
49.
Hayward LF, Johnson AK, Felder RB. Arterial chemoreflex in conscious normotensive and hypertensive adult rats. Am J Physiol . 1999; 276: H1215–H1222.
Go to Citation
Crossref
PubMed
Google Scholar
50.
Angell-James JE, Clarke JA, Daly MD, Taton A. Respiratory and cardiovascular responses to hyperoxia, hypoxia and hypercapnia in the renal hypertensive rabbit: role of carotid body chemoreceptors. J Hypertens . 1985; 3: 213–223.
Go to Citation
Crossref
PubMed
Google Scholar
51.
Izdebska E, Cybulska I, Sawicki M, Izdebski J, Trzebski A. Postexercise decrease in arterial blood pressure, total peripheral resistance and in circulatory responses to brief hyperoxia in subjects with mild essential hypertension. J Hum Hypertens . 1998; 12: 855–860.
Go to Citation
Crossref
PubMed
Google Scholar
52.
Smith ML, Pacchia CF. Sleep apnoea and hypertension. Role of chemoreflexes in humans. Exp Physiol . 2007; 92: 45–50.
Go to Citation
Crossref
PubMed
Google Scholar
53.
Narkiewicz K, van de Borne PJ, Pesek CA, Dyken ME, Montano N, Somers VK. Selective potentiation of peripheral chemoreflex sensitivity in obstructive sleep apnea. Circulation . 1999; 99: 1183–1189.
Crossref
PubMed
Google Scholar
a [...] and cold pressor tests are normal.
b [...] arterial pressure in patients with SA.
54.
Fletcher EC. Invited review: physiological consequences of intermittent hypoxia: systemic blood pressure. J Appl Physiol . 2001; 90: 1600–1605.
Crossref
PubMed
Google Scholar
a [...] in the exploration of this issue.
b [...] similar to that seen in patients with SA.
55.
Lesske J, Fletcher EC, Bao G, Unger T. Hypertension caused by chronic intermittent hypoxia-influence ofchemoreceptors and sympathetic nervous system. J Hypertens . 1997; 15: 1593–1603.
Go to Citation
Crossref
PubMed
Google Scholar
56.
Peng Y, Kline DD, Dick TE, Prabhakar NR. Chronic intermittent hypoxia enhances carotid body chemoreceptor response to low oxygen. Adv Exp Med Biol . 2001; 499: 33–38.
Crossref
PubMed
Google Scholar
a [...] to hypoxemia but not to hypercapnia.
b [...] Peng et al
57.
Peng YJ, Overholt JL, Kline D, Kumar GK, Prabhakar NR. Induction of sensory long-term facilitation in the carotid body by intermittent hypoxia: implications for recurrent apneas. Proc Natl Acad Sci U S A . 2003; 100: 10073–10078.
Go to Citation
Crossref
PubMed
Google Scholar
58.
Esler M, Straznicky N, Eikelis N, Masuo K, Lambert G, Lambert E. Mechanisms of sympathetic activation in obesity-related hypertension. Hypertension . 2006; 48: 787–796.
Go to Citation
Crossref
PubMed
Google Scholar
59.
Nimbkar NV, Lateef F. Carotid body dysfunction: the possible etiology of non-insulin dependent diabetes mellitus and essential hypertension. Med Hypotheses . 2005; 65: 1067–1075.
Go to Citation
Crossref
PubMed
Google Scholar
60.
Sato MA, Colombari E, Morrison SF. Inhibition of neurons in commissural nucleus of solitary tract reduces sympathetic nerve activity in SHR. Am J Physiol Heart Circ Physiol . 2002; 282: H1679–H1684.
Go to Citation
Crossref
PubMed
Google Scholar
61.
Sato MA, Menani JV, Lopes OU, Colombari E. Lesions of the commissural nucleus of the solitary tract reduce arterial pressure in spontaneously hypertensive rats. Hypertension . 2001; 38: 560–564.
Crossref
PubMed
Google Scholar
62.
Peng YJ, Prabhakar NR. Reactive oxygen species in the plasticity of respiratory behavior elicited by chronic intermittent hypoxia. J Appl Physiol . 2003; 94: 2342–2349.
Go to Citation
Crossref
PubMed
Google Scholar
63.
Prabhakar NR, Kumar GK. Oxidative stress in the systemic and cellular responses to intermittent hypoxia. Biol Chem . 2004; 385: 217–221.
PubMed
Google Scholar
a [...] complex III occurs in CBs from CIH rats.
b [...] increases oxygen radicals in both groups.
c [...] to activation of hypoxia-inducible factor-1
d [...] and/or activator protein-1.
64.
Peng YJ, Yuan G, Jacono FJ, Kumar GK, Prabhakar NR. 5-HT evokes sensory long-term facilitation of rodent carotid body via activation of NADPH oxidase. J Physiol . 2006; 576: 289–295.
Go to Citation
Crossref
PubMed
Google Scholar
65.
Chen J, He L, Dinger B, Stensaas L, Fidone S. Role of endothelin and endothelin A-type receptor in adaptation of the carotid body to chronic hypoxia. Am J Physiol Lung Cell Mol Physiol . 2002; 282: L1314–L1323.
Go to Citation
Crossref
PubMed
Google Scholar
66.
Rey S, Del Rio R, Iturriaga R. Contribution of endothelin-1 to the enhanced carotid body chemosensory responses induced by chronic intermittent hypoxia. Brain Res . 2006; 1086: 152–159.
Crossref
PubMed
Google Scholar
a [...] induces CB chemosensory excitation,
b [...] delivery,
67.
Chen J, He L, Dinger B, Fidone S. Cellular mechanisms involved in rabbit carotid body excitation elicited by endothelin peptides. Respir Physiol . 2000; 121: 13–23.
Crossref
PubMed
Google Scholar
a [...] the chemosensory response to acute hypoxia.
b [...] and hypoxic CB discharge induced by CIH.
c [...] which is devoid of vascular effects.
d [...] via a cAMP/inositol triphosphate pathway.
68.
Rey S, Corthorn J, Chacon C, Iturriaga R. Expression and immunolocalization of endothelin peptides and its receptors, ET A and ET B, in the carotid body exposed to chronic intermittent hypoxia. J Histochem Cytochem . 2007; 55: 167–174.
Go to Citation
Crossref
PubMed
Google Scholar
69.
Lam SY, Tipoe GL, Liong EC, Fung ML. Hypoxia-inducible factor (HIF)-1alpha and endothelin-1 expression in the rat carotid body during intermittent hypoxia. Adv Exp Med Biol . 2006; 580: 21–27;discussion 351–359.
Crossref
PubMed
Google Scholar
70.
Almaraz L, Perez-Garcia MT, Gomez-Nino A, Gonzalez C. Mechanisms of alpha2-adrenoceptor-mediated inhibition in rabbit carotid body. Am J Physiol . 1997; 272: C628–C637.
Go to Citation
Crossref
PubMed
Google Scholar
71.
Przybylski J. Do arterial chemoreceptors play a role in the pathogenesis of hypertension? Med Hypotheses . 1981; 7: 127–131.
Go to Citation
Crossref
PubMed
Google Scholar
Advertisement
Recommended
October 2015
Hypothalamic Paraventricular and Arcuate Nuclei Contribute to Elevated Sympathetic Nerve Activity in Pregnant Rats
Zhigang Shi,
Priscila A. Cassaglia,
Laura C. Gotthardt, and
[...]
Virginia L. Brooks
+0 authors
September 2015
Cerebrospinal Fluid Hypernatremia Elevates Sympathetic Nerve Activity and Blood Pressure via the Rostral Ventrolateral Medulla
Sean D. Stocker,
Susan M. Lang,
Sarah S. Simmonds,
Megan M. Wenner, and
[...]
William B. Farquhar
+1 authors
April 2018
Abstract 437: Inhibition of Endoplasmic Reticulum Function in Pre-sympathetic PVN Neurons by Thapsigargin Increased Neuronal Excitability and Sympathetic Nerve Activity
Qing Hui Chen,
Le Gui,
Andrew Chapp,
Robert Larson,
Michael Huber, and
[...]
Jianhua Zhu
+2 authors
Advertisement
Submit a Response to This Article
Close
Compose eLetter
Title:
Comment text:
Contributors
(all fields are required)
Remove Contributor
First Name:
Last Name:
Email:
Affiliation:
Add Another Contributor
Statement of Competing Interests
Competing Interests?
YES
NO
Please describe the competing interests
Cancel Submit
View full text|Download PDF
Figures Tables
Close figure viewer
Back to article
Figure title goes here
Change zoom level
Go to figure location within the article
Toggle download panel
Toggle download panel
Download figure
Toggle share panel
Toggle share panel
Share
Toggle information panel
Toggle information panel
All figures All tables
View all material
View all material
xrefBack.goTo
xrefBack.goTo
Request permissions Expand All
Collapse
Expand Table
Show all references
SHOW ALL BOOKS
Authors Info & Affiliations
Comment Response
Now Reading:
Arterial Chemoreceptors and Sympathetic Nerve Activity: Implications for Hypertension and Heart Failure
Track Citations
Add to favorites
Share
PDF/EPUB
###### PREVIOUS ARTICLE Personalized Medicine for High Blood Pressure Previous###### NEXT ARTICLE Cerebrovascular Function in Pregnancy and Eclampsia Next
Hypertension
Submit
BrowseBrowse
Collections
Subject Terms
AHA Journal Podcasts
Trend Watch
ResourcesResources
CME
Journal Metrics
Early Career Resources
AHA Journals @ Meetings
InformationInformation
For Authors
For Reviewers
For Subscribers
For International Users
Arteriosclerosis, Thrombosis, and Vascular Biology
Circulation
Circulation Research
Hypertension
Stroke
Journal of the American Heart Association
Circulation: Arrhythmia and Electrophysiology
Circulation: Cardiovascular Imaging
Circulation: Cardiovascular Interventions
Circulation: Cardiovascular Quality & Outcomes
Circulation: Genomic and Precision Medicine
Circulation: Heart Failure
Stroke: Vascular and Interventional Neurology
Annals of Internal Medicine: Clinical Cases
This page is managed by Wolters Kluwer Health, Inc. and/or its affiliates or subsidiaries.Wolters Kluwer Privacy Policy
Your California Privacy Choices
Manage Cookie Preferences
Back to top
National Center
7272 Greenville Ave.Dallas, TX 75231
Customer Service
1-800-AHA-USA-1
1-800-242-8721
Hours
Monday - Friday: 7 a.m. – 7 p.m. CT
Saturday: 9 a.m. - 5 p.m. CT
Closed on Sundays
Tax Identification Number
13-5613797
ABOUT US
About the AHA/ASA
Annual report
AHA Financial Information
Careers
International Programs
Latest Heart and Stroke News
AHA/ASA Media Newsroom
GET INVOLVED
Donate
Advocate
Volunteer
ShopHeart
ShopCPR
OUR SITES
American Heart Association
American Stroke Association
CPR & ECC
Go Red For Women
More Sites
AHA Careers
AHA Privacy Policy
Medical Advice Disclaimer
Copyright Policy
Accessibility Statement
Ethics Policy
Conflict of Interes Policy
Linking Policy
Whistleblower Policy
Content Editorial Guidelines
Diversity
Suppliers & Providers
State Fundraising Notices
©2025 American Heart Association, Inc. All rights reserved. Unauthorized use prohibited.
The American Heart Association is a qualified 501(c)(3) tax-exempt organization.
Red Dress ™ DHHS, Go Red ™ AHA ; National Wear Red Day® is a registered trademark.
✓
Thanks for sharing!
AddToAny
More…
__("articleCrossmark.closePopup")
Your Privacy
To give you the best possible experience we use cookies and similar technologies. We use data collected through these technologies for various purposes, including to enhance website functionality, remember your preferences, show the most relevant content, and show the most useful ads. You can select your preferences by clicking the link. For more information, please review ourPrivacy & Cookie Notice
Manage Cookie Preferences Accept All Cookies
Privacy Preference Center
When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device. Because we respect your right to privacy, you can choose not to allow certain types of cookies on our website. Click on the different category headings to find out more and manage your cookie preferences. However, blocking some types of cookies may impact your experience on the site and the services we are able to offer.
Privacy & Cookie Notice
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function. They are usually set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, this may have an effect on the proper functioning of (parts of) the site.
View Vendor Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality, user experience and personalization, and may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies, then some or all of these services may not function properly.
View Vendor Details
Performance Cookies
[x] Performance Cookies
These cookies support analytic services that measure and improve the performance of our site. They help us know which pages are the most and least popular and see how visitors move around the site.
View Vendor Details
Advertising Cookies
[x] Advertising Cookies
These cookies may collect insights to issue personalized content and advertising on our own and other websites, and may be set through our site by third party providers. If you do not allow these cookies, you may still see basic advertising on your browser that is generic and not based on your interests.
View Vendor Details
Vendors List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Allow All Reject All Confirm My Choices |
13018 | https://www.quora.com/How-many-5-digit-combinations-can-be-made-with-4-digits-Must-each-digit-be-used-at-least-once | How many 5-digit combinations can be made with 4 digits? Must each digit be used at least once? - Quora
Something went wrong. Wait a moment and try again.
Try again
Skip to content
Skip to search
Sign In
How many 5-digit combinations can be made with 4 digits? Must each digit be used at least once?
All related (32)
Sort
Recommended
Mario Skrtic
IT at Public Libraries (2003–present) · Author has 4.4K answers and 1.1M answer views
·2y
Since you have 5 digit combination and only 4 digits, some digit must be used more than once.
If you have 4 distinct digits and there is no rule that all 4 digits must be used, the answer is 44444=4^5.
Your second question is case where all digits must be used at least once:
You have one pair and 3 more distinct digits. 11234,11243,…,43121,43211,12234,…,43221,12334,…,43321,12344,…,44321
You can use computer or you can use some math.
Hint: permutations where one digit is repeated.
4 (5!/2!) =…
Upvote ·
9 3
Sponsored by Adobe Photoshop
Select details, fast.
Quickly select tricky details for faster edits.
Learn More
9 3
Related questions
More answers below
How many 4 digit combinations can be made with 5?
How many possible combinations are possible in a 5 digit number?
How many 4-digit numbers are there whose digits add to 5?
What is the 4-digit number with the digits 5, 8, 7, and 4?
How many 4-digit numbers can be formed from 1,2,3,4, and 5 if no repetition is allowed, and if repetition is allowed?
David Vanderschel
PhD in Mathematics&Physics, Rice (Houston neighborhood) (Graduated 1970) · Author has 37.1K answers and 49.3M answer views
·Apr 5
Assuming each digit must be used , you can make 4!=24 4!=24 distinct 4 4-digit sequences. Now there are 4 4 ways to choose a digit to repeat and 4 4 distinguishable positions to place it. So that would make 16⋅24=384 16⋅24=384 distinct 5 5-digit sequences. (It can’t be a mathematical combination because those don’t admit repeats.)
If there is no requirement to use all 4 4 digits, the answer is 4 5=1024.4 5=1024.
Upvote ·
9 1
Jayanta Mukherjee
B Tech IEE in Instrumentation Engineering, Jadavpur University (Graduated 1990) · Author has 41K answers and 10.9M answer views
·7y
Related
How many 6 digit numbers can be formed from 1, 2, 3, and 4 such that each digit is used at least once?
So, at least four of the six places are to be occupied by the digits ‘1’, ‘2’, ‘3’ and ‘4’. Hence, we may call them as ‘reserved’ places. There are two patterns available for us to form such six-digit numbers.
Pattern-1: Now, if the first place (10^5’s place) is a reserved one:
We have 4 options for this place (any one among ‘1’, ‘2’, ‘3’ or ‘4’).
Now, among the five remaining places; three have to be reserved. These three can be selected in (5C3) = 10 ways.
For these three reserved places we have three digits for using once each (since we have used up one digit at the first place). So, these thre
Continue Reading
So, at least four of the six places are to be occupied by the digits ‘1’, ‘2’, ‘3’ and ‘4’. Hence, we may call them as ‘reserved’ places. There are two patterns available for us to form such six-digit numbers.
Pattern-1: Now, if the first place (10^5’s place) is a reserved one:
We have 4 options for this place (any one among ‘1’, ‘2’, ‘3’ or ‘4’).
Now, among the five remaining places; three have to be reserved. These three can be selected in (5C3) = 10 ways.
For these three reserved places we have three digits for using once each (since we have used up one digit at the first place). So, these three digits can be permuted in (3!) = 6 ways.
Now, for each of the remaining two places, we can use any of the ten digits between ‘0’ and ‘9’. So, these two places can be filled in (1010) = 100 ways.
Therefore, in Pattern-1; (4106100) = 24000 possible six-digit numbers can be generated.
Pattern-2: Now, if the first place (10^5’s place) is a non-reserved one:
We have 9 options for this place (since digit ‘0’ can’t be considered for this place).
Now, among the five remaining places; four have to be reserved. These four can be selected in (5C4) = 5 ways.
For these four reserved places we have four digits for using once each. So, these three digits can be permuted in (4!) = 24 ways.
Now, for the remaining one place, we can use any of the ten digits between ‘0’ and ‘9’. So, this place can be filled in 10 ways.
Therefore, in Pattern-2; (952410) = 10800 possible six-digit numbers can be generated.
So, formation of total (24000 + 10800) = 34800 such six-digit numbers are possible.
Upvote ·
9 1
Mergers Milers
Studied at The Open University
·1y
Related
How many four-digit numbers can be formed using all the digits 0-9 with at least one repeated digit? How many of these numbers will have exactly two repeated digits?
How many with at least one repeated digit?
I’m going to exclude numbers with a leading zero such as 0100, because to my mind, they are not really four digit numbers
There are 9000 numbers from 1000 to 9999
The numbers with no repeated digits are of the form abcd where a is in the range [1,9] and b,c,d are in the range [0,9] and they are different to a. So the number of numbers with no repeated digits is 9987, which is 4536
Therefore there are 9000–4536=4464 numbers with at least one repeated digit
How many have exactly two repeated digits?
Consider all the three digit numbers with no repeated dig
Continue Reading
How many with at least one repeated digit?
I’m going to exclude numbers with a leading zero such as 0100, because to my mind, they are not really four digit numbers
There are 9000 numbers from 1000 to 9999
The numbers with no repeated digits are of the form abcd where a is in the range [1,9] and b,c,d are in the range [0,9] and they are different to a. So the number of numbers with no repeated digits is 9987, which is 4536
Therefore there are 9000–4536=4464 numbers with at least one repeated digit
How many have exactly two repeated digits?
Consider all the three digit numbers with no repeated digits, of the form abc. There are 998=648 of these
From each abc, you can make aabc, abac, abca, abbc, abcb&abcc, and these are all different numbers. (You don’t want to count babc, for example, because then you start counting numbers twice.)
So there are 6486=3888 with exactly two repeated digits
(14:31 - corrected a typo in the final answer. Have also checked this answer with Haskell code:
import Data.List
twoRepeatedDigits n = length . nub . show $ n == 3
length $ filter twoRepeatedDigits [1000..9999]
(14:55 - link to a relevant table from OEIS: A337127 - OEIS)
Upvote ·
9 3
Promoted by Grammarly
Grammarly
Great Writing, Simplified
·Updated 3y
What makes a cover letter stand out?
When you’re writing a cover letter, the little things can make a big difference. The better your letter is, the better your chances are of getting a job interview. Follow these tips for composing a winning cover letter, and you will increase your chances of landing the job you want.
Research and brainstorm
Before you start writing your cover letter, familiarize yourself with the role requirements. Read the job listing carefully and pull out the most important information. Then, spend time on the company’s website to get a strong sense of the company’s culture, values, and mission. Brainstorm the
Continue Reading
When you’re writing a cover letter, the little things can make a big difference. The better your letter is, the better your chances are of getting a job interview. Follow these tips for composing a winning cover letter, and you will increase your chances of landing the job you want.
Research and brainstorm
Before you start writing your cover letter, familiarize yourself with the role requirements. Read the job listing carefully and pull out the most important information. Then, spend time on the company’s website to get a strong sense of the company’s culture, values, and mission. Brainstorm the most effective way to communicate your suitability for the role in your cover letter. Brainstorming is a key part of the writing process.
Personalize your greeting
The first thing the recruiter or hiring manager will notice in your cover letter is whether you addressed it to them personally. It’s not always easy to find the recruiter’s or hiring manager’s name, but it’s always worth your time to directly address the person who will be making the hiring decision.
Grab the reader’s attention
Your cover letter needs to grab attention within the first sentence or two. Remember, the recruiter is going to be reading lots of cover letters that will contain pretty similar content. If your cover letter doesn’t captivate them from the get-go, you could get overlooked.
Show your enthusiasm about the role
Throughout your cover letter, use language that communicates your passion for the kind of work you do. Your word choice plays a big role in shaping how recruiters perceive your attitude toward your work experience and your enthusiasm for the role.
Sign up for a free Grammarly account and ensure your cover letter is readable, clear, and concise with real-time suggestions for stronger, more precise language.
Upvote ·
999 814
999 101
99 41
Related questions
More answers below
How many 4-digit number combinations using 1 and 5 are there?
How many 5 digit combinations can be made using 1-15?
How many 5 digit combinations are there from 00000-99999?
How many 6 digit combinations can you make with the numbers 4,5,6?
How many ways can the digits 1, 2, 3, 4, and 5 be arranged?
John Chadwick
Former Head of Electrical, Mechanical & MV Engineering at South Cheshire College (1972–2007) · Author has 1K answers and 581.1K answer views
·Updated 1y
Related
Using 1,2,3,4,5,6 how many 4 digit combinations are there?
Assuming REPETITION there are 6666 =6^4 = 1296 four digit possible permutations if REPETITION of the digits is employed.
Assuming NO repetitions there are 6543 = 360 FOUR digit possible permutations.
These arise from 6C4 = 15 FOUR digit COMBINATIONS of DIFFERENT digits 1 ==> 6 and these are shown below:-
(1, 2, 3, 4), (1, 2, 3, 5), 1, 2, 3, 6), (1, 2, 4, 5), (1, 2, 4, 6), (1, 2, 5, 6), (1, 3, 4, 5), (1, 3, 4, 6), (1, 3, 5,6), (1, 4, 5, 6),
(2, 3, 4, 5), (2, 3, 4, 6), (2, 3, 5, 6), (2, 4, 5, 6), (3, 4, 5, 6).
Total: 15.
EACH of these 15 combinations can be PERMUTED 4321 = 24 ways which in tu
Continue Reading
Assuming REPETITION there are 6666 =6^4 = 1296 four digit possible permutations if REPETITION of the digits is employed.
Assuming NO repetitions there are 6543 = 360 FOUR digit possible permutations.
These arise from 6C4 = 15 FOUR digit COMBINATIONS of DIFFERENT digits 1 ==> 6 and these are shown below:-
(1, 2, 3, 4), (1, 2, 3, 5), 1, 2, 3, 6), (1, 2, 4, 5), (1, 2, 4, 6), (1, 2, 5, 6), (1, 3, 4, 5), (1, 3, 4, 6), (1, 3, 5,6), (1, 4, 5, 6),
(2, 3, 4, 5), (2, 3, 4, 6), (2, 3, 5, 6), (2, 4, 5, 6), (3, 4, 5, 6).
Total: 15.
EACH of these 15 combinations can be PERMUTED 4321 = 24 ways which in turn will give 1524 = 360 different permutations WITHOUT digit repetition.
Upvote ·
9 7
Calvin Campbell
B.Sc. in Computer Science&Mathematics, University of the West Indies (Graduated 1984) · Author has 3.8K answers and 4M answer views
·5y
Related
How many 4-digit numbers are there whose digits add to 5?
A simple count from 1004 to 5000, but only those numbers whose digits add up to 5 will do the trick!
1004,1013,1022,1031,1040,1103,1112,1121,1130,1202,1211,1220,1301,1310,1400,2003,2012,2021,2030,2102,2111,2120,2201,2210,2300,3002,3011,3020,3101,3110,3200,4001,4010,4100,5000.
There is 1 number starting with 5; 3 numbers starting with 4; 6 numbers starting with 3; 10 numbers starting with 2; and 15 numbers starting with 1.
The total is 35.
It can be seen as the familiar sum of sums of the first n natural numbers series:
1+3+6+10+15+… +1/2(n)(n+1), which has a sum formula to n terms of:
1/6(n)(n+1)(n+
Continue Reading
A simple count from 1004 to 5000, but only those numbers whose digits add up to 5 will do the trick!
1004,1013,1022,1031,1040,1103,1112,1121,1130,1202,1211,1220,1301,1310,1400,2003,2012,2021,2030,2102,2111,2120,2201,2210,2300,3002,3011,3020,3101,3110,3200,4001,4010,4100,5000.
There is 1 number starting with 5; 3 numbers starting with 4; 6 numbers starting with 3; 10 numbers starting with 2; and 15 numbers starting with 1.
The total is 35.
It can be seen as the familiar sum of sums of the first n natural numbers series:
1+3+6+10+15+… +1/2(n)(n+1), which has a sum formula to n terms of:
1/6(n)(n+1)(n+2).
In our case, n is 5. So the formula gives our answer:
1/6(5)(6)(7)
= 57
=35
Good luck!
Upvote ·
9 2
Sponsored by Book Geists
If you're a Kiwi, this could be the best day of your life!
Available to Kiwis only. Read today.
Learn More
1.2K 1.2K
Jahnvee Gajjar
M.Tech (Computer Science) from Ahmedabad University (Graduated 2020)
·6y
Related
How many 4 digit numbers can you make using 0,1,3,and 5?
Without repetition of numbers
With repetition of numbers
Here we want 4 digit number so at 1st position, there are 3 possibility (1,3,5) we can not put ‘0’ at 1st position in both the cases.
Without Repetition of numbers :
3321 = 18
At 2nd position we have either 0 or any two from (1,3,5)
At 3rd position we have remaining two
At 4th position we have remaining one
So here we can make 18 - 4 digit numbers without repetition.
With Repetition of numbers :
3444 = 192
At any position except 1st position we have either 0 or any two from (1,3,5), that means 4 possibilities.
Here we can make 192 - 4 digit nu
Continue Reading
Without repetition of numbers
With repetition of numbers
Here we want 4 digit number so at 1st position, there are 3 possibility (1,3,5) we can not put ‘0’ at 1st position in both the cases.
Without Repetition of numbers :
3321 = 18
At 2nd position we have either 0 or any two from (1,3,5)
At 3rd position we have remaining two
At 4th position we have remaining one
So here we can make 18 - 4 digit numbers without repetition.
With Repetition of numbers :
3444 = 192
At any position except 1st position we have either 0 or any two from (1,3,5), that means 4 possibilities.
Here we can make 192 - 4 digit numbers with repetition.
Upvote ·
9 2
Rahul Vaidya
Always learning!
·Updated 8y
Related
How many 4 digit numbers can be formed from 1,2,4,5,7 and 8?
Two answers are possible.
One is with the repetition of digits.
The other is without repetition of digits.
ANSWER 1: With Repetition
Numbers: 1,2,4,5,7,8
There are 4 digits __ __ __ __
1st digit can be filled by any of the 6 numbers.
Similarly 2nd, 3rd and 4th digits.
Hence each digit will have 6 possibilities.
Therefore no.of 4 digit numbers that can be formed are 6 x 6 x 6 x 6 = 1296
ANSWER 2: Without Repetition
4 digits __ __ __ __
Now the first digit can be filled by any of the six numbers. Therefore there are 6 possibilities
The second digit will have only 5 possibilities as one of the nu
Continue Reading
Two answers are possible.
One is with the repetition of digits.
The other is without repetition of digits.
ANSWER 1: With Repetition
Numbers: 1,2,4,5,7,8
There are 4 digits __ __ __ __
1st digit can be filled by any of the 6 numbers.
Similarly 2nd, 3rd and 4th digits.
Hence each digit will have 6 possibilities.
Therefore no.of 4 digit numbers that can be formed are 6 x 6 x 6 x 6 = 1296
ANSWER 2: Without Repetition
4 digits __ __ __ __
Now the first digit can be filled by any of the six numbers. Therefore there are 6 possibilities
The second digit will have only 5 possibilities as one of the numbers gets used up by the 1st digit.
Similarly the 3rd and 4th digits will have 4 and 3 possibilities respectively.
Therefore no.of 4 digit numbers that can be formed are 6 x 5 x 4 x 3 = 360
Upvote ·
99 87
99 10
9 1
Sponsored by Adsy
Content Marketing Platform.
Explore over 100K websites ready to publish your articles. Sign up now.
Sign Up
99 37
Aayush Mittal
Product & BD @ MBAtrek | IIM Kashipur | Intrapreneur | Music · Upvoted by
Sin Keong Tong
, M.Sc. Mathematics, University of New South Wales · Author has 89 answers and 691.9K answer views
·9y
Related
How many combinations of the digits from 1 to 10 are there that sum to 40, using each digit at most once? Order does not matter.
20.
The sum of first 10 digits is 55. But you wanted the sum to be 40. So you have to subtract 15 from it. You can do that in 20 ways as follows.
(5,10) (6,9) (7,8) (1,4,10) (1,5,9) (1,6,8) (2,3,10) (2,4,9) (2,5,8) (2,6,7) (3,4,8) (3,5,7) (4,5,6) (1,2,3,9) (1,2,4,8) (1,2,5,7) (1,3,4,7) (1,3,5,6) (2,3,4,6) (1,2,3,4,5).
The sum of first 9 digits is 45. But you want the sum to be 40. So you have to subtract 5 from it. You can do that in 2 ways as follows.
(1,4) (2,3)
But if you look carefully here, (1,4) is already present in the pair (1,4,10) and (2,3) is already present in (2,3,10) and you did no
Continue Reading
20.
The sum of first 10 digits is 55. But you wanted the sum to be 40. So you have to subtract 15 from it. You can do that in 20 ways as follows.
(5,10) (6,9) (7,8) (1,4,10) (1,5,9) (1,6,8) (2,3,10) (2,4,9) (2,5,8) (2,6,7) (3,4,8) (3,5,7) (4,5,6) (1,2,3,9) (1,2,4,8) (1,2,5,7) (1,3,4,7) (1,3,5,6) (2,3,4,6) (1,2,3,4,5).
The sum of first 9 digits is 45. But you want the sum to be 40. So you have to subtract 5 from it. You can do that in 2 ways as follows.
(1,4) (2,3)
But if you look carefully here, (1,4) is already present in the pair (1,4,10) and (2,3) is already present in (2,3,10) and you did not take 10 here. So these 2 cases are similar to the former ones mentioned. (Thanks for edit Paul )
The sum of first 8 digits is 36. But you wanted 40. So you need to add 4 to this. But you cannot repeat the digits. Hence, the end.
So, overall, the number of ways in which we can choose digits from 1 to 10 such that sum of the chosen digits is equal to 40 and each digit is used only once is 20.
:)
Upvote ·
99 12
9 6
Jayanta Mukherjee
B Tech IEE in Instrumentation Engineering, Jadavpur University (Graduated 1990) · Author has 41K answers and 10.9M answer views
·1y
Related
How many five-digit numbers can you make that end in 5? If the digits can be repeated.
Any five-digit number that ends with “5” be expressed as “abcde".
There is only 1 choice for e ………… because all the numbers have to end with “5”.
There are 10 choices for d ……………. {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
There are 10 choices for c ……………. {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
There are 10 choices for b ……………. {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
There are 9 choices for d ……………. {1, 2, 3, 4, 5, 6, 7, 8, 9}.
Therefore, there can be (9 10 10 10 1) = 9000 five-digit numbers that end with “5”.
Upvote ·
9 3
9 1
Sol Warda
Author has 5.8K answers and 2.4M answer views
·1y
Related
From the digits 1, 2, 3, 4, 5, how many arrangements can be found using 1 digit, 2 digits, 3 digits, 4 digits, and 5 digits?
[5 P 1] + [5 P 2] + [5 P 3] + [5 P 4] + [5 P 5]==5 + 20 + 60 + 120 + 120 ==325 permutations.
[(1,), (2,), (3,), (4,), (5,), (1, 2), (1, 3), (1, 4), (1, 5), (2, 1), (2, 3), (2, 4), (2, 5), (3, 1), (3, 2), (3, 4), (3, 5), (4, 1), (4, 2), (4, 3), (4, 5), (5, 1), (5, 2), (5, 3), (5, 4), (1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 3, 2), (1, 3, 4), (1, 3, 5), (1, 4, 2), (1, 4, 3), (1, 4, 5), (1, 5, 2), (1, 5, 3), (1, 5, 4), (2, 1, 3), (2, 1, 4), (2, 1, 5), (2, 3, 1), (2, 3, 4), (2, 3, 5), (2, 4, 1), (2, 4, 3), (2, 4, 5), (2, 5, 1), (2, 5, 3), (2, 5, 4), (3, 1, 2), (3, 1, 4), (3, 1, 5), (3, 2, 1), (3, 2, 4)
Continue Reading
[5 P 1] + [5 P 2] + [5 P 3] + [5 P 4] + [5 P 5]==5 + 20 + 60 + 120 + 120 ==325 permutations.
[(1,), (2,), (3,), (4,), (5,), (1, 2), (1, 3), (1, 4), (1, 5), (2, 1), (2, 3), (2, 4), (2, 5), (3, 1), (3, 2), (3, 4), (3, 5), (4, 1), (4, 2), (4, 3), (4, 5), (5, 1), (5, 2), (5, 3), (5, 4), (1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 3, 2), (1, 3, 4), (1, 3, 5), (1, 4, 2), (1, 4, 3), (1, 4, 5), (1, 5, 2), (1, 5, 3), (1, 5, 4), (2, 1, 3), (2, 1, 4), (2, 1, 5), (2, 3, 1), (2, 3, 4), (2, 3, 5), (2, 4, 1), (2, 4, 3), (2, 4, 5), (2, 5, 1), (2, 5, 3), (2, 5, 4), (3, 1, 2), (3, 1, 4), (3, 1, 5), (3, 2, 1), (3, 2, 4), (3, 2, 5), (3, 4, 1), (3, 4, 2), (3, 4, 5), (3, 5, 1), (3, 5, 2), (3, 5, 4), (4, 1, 2), (4, 1, 3), (4, 1, 5), (4, 2, 1), (4, 2, 3), (4, 2, 5), (4, 3, 1), (4, 3, 2), (4, 3, 5), (4, 5, 1), (4, 5, 2), (4, 5, 3), (5, 1, 2), (5, 1, 3), (5, 1, 4), (5, 2, 1), (5, 2, 3), (5, 2, 4), (5, 3, 1), (5, 3, 2), (5, 3, 4), (5, 4, 1), (5, 4, 2), (5, 4, 3), (1, 2, 3, 4), (1, 2, 3, 5), (1, 2, 4, 3), (1, 2, 4, 5), (1, 2, 5, 3), (1, 2, 5, 4), (1, 3, 2, 4), (1, 3, 2, 5), (1, 3, 4, 2), (1, 3, 4, 5), (1, 3, 5, 2), (1, 3, 5, 4), (1, 4, 2, 3), (1, 4, 2, 5), (1, 4, 3, 2), (1, 4, 3, 5), (1, 4, 5, 2), (1, 4, 5, 3), (1, 5, 2, 3), (1, 5, 2, 4), (1, 5, 3, 2), (1, 5, 3, 4), (1, 5, 4, 2), (1, 5, 4, 3), (2, 1, 3, 4), (2, 1, 3, 5), (2, 1, 4, 3), (2, 1, 4, 5), (2, 1, 5, 3), (2, 1, 5, 4), (2, 3, 1, 4), (2, 3, 1, 5), (2, 3, 4, 1), (2, 3, 4, 5), (2, 3, 5, 1), (2, 3, 5, 4), (2, 4, 1, 3), (2, 4, 1, 5), (2, 4, 3, 1), (2, 4, 3, 5), (2, 4, 5, 1), (2, 4, 5, 3), (2, 5, 1, 3), (2, 5, 1, 4), (2, 5, 3, 1), (2, 5, 3, 4), (2, 5, 4, 1), (2, 5, 4, 3), (3, 1, 2, 4), (3, 1, 2, 5), (3, 1, 4, 2), (3, 1, 4, 5), (3, 1, 5, 2), (3, 1, 5, 4), (3, 2, 1, 4), (3, 2, 1, 5), (3, 2, 4, 1), (3, 2, 4, 5), (3, 2, 5, 1), (3, 2, 5, 4), (3, 4, 1, 2), (3, 4, 1, 5), (3, 4, 2, 1), (3, 4, 2, 5), (3, 4, 5, 1), (3, 4, 5, 2), (3, 5, 1, 2), (3, 5, 1, 4), (3, 5, 2, 1), (3, 5, 2, 4), (3, 5, 4, 1), (3, 5, 4, 2), (4, 1, 2, 3), (4, 1, 2, 5), (4, 1, 3, 2), (4, 1, 3, 5), (4, 1, 5, 2), (4, 1, 5, 3), (4, 2, 1, 3), (4, 2, 1, 5), (4, 2, 3, 1), (4, 2, 3, 5), (4, 2, 5, 1), (4, 2, 5, 3), (4, 3, 1, 2), (4, 3, 1, 5), (4, 3, 2, 1), (4, 3, 2, 5), (4, 3, 5, 1), (4, 3, 5, 2), (4, 5, 1, 2), (4, 5, 1, 3), (4, 5, 2, 1), (4, 5, 2, 3), (4, 5, 3, 1), (4, 5, 3, 2), (5, 1, 2, 3), (5, 1, 2, 4), (5, 1, 3, 2), (5, 1, 3, 4), (5, 1, 4, 2), (5, 1, 4, 3), (5, 2, 1, 3), (5, 2, 1, 4), (5, 2, 3, 1), (5, 2, 3, 4), (5, 2, 4, 1), (5, 2, 4, 3), (5, 3, 1, 2), (5, 3, 1, 4), (5, 3, 2, 1), (5, 3, 2, 4), (5, 3, 4, 1), (5, 3, 4, 2), (5, 4, 1, 2), (5, 4, 1, 3), (5, 4, 2, 1), (5, 4, 2, 3), (5, 4, 3, 1), (5, 4, 3, 2), (1, 2, 3, 4, 5), (1, 2, 3, 5, 4), (1, 2, 4, 3, 5), (1, 2, 4, 5, 3), (1, 2, 5, 3, 4), (1, 2, 5, 4, 3), (1, 3, 2, 4, 5), (1, 3, 2, 5, 4), (1, 3, 4, 2, 5), (1, 3, 4, 5, 2), (1, 3, 5, 2, 4), (1, 3, 5, 4, 2), (1, 4, 2, 3, 5), (1, 4, 2, 5, 3), (1, 4, 3, 2, 5), (1, 4, 3, 5, 2), (1, 4, 5, 2, 3), (1, 4, 5, 3, 2), (1, 5, 2, 3, 4), (1, 5, 2, 4, 3), (1, 5, 3, 2, 4), (1, 5, 3, 4, 2), (1, 5, 4, 2, 3), (1, 5, 4, 3, 2), (2, 1, 3, 4, 5), (2, 1, 3, 5, 4), (2, 1, 4, 3, 5), (2, 1, 4, 5, 3), (2, 1, 5, 3, 4), (2, 1, 5, 4, 3), (2, 3, 1, 4, 5), (2, 3, 1, 5, 4), (2, 3, 4, 1, 5), (2, 3, 4, 5, 1), (2, 3, 5, 1, 4), (2, 3, 5, 4, 1), (2, 4, 1, 3, 5), (2, 4, 1, 5, 3), (2, 4, 3, 1, 5), (2, 4, 3, 5, 1), (2, 4, 5, 1, 3), (2, 4, 5, 3, 1), (2, 5, 1, 3, 4), (2, 5, 1, 4, 3), (2, 5, 3, 1, 4), (2, 5, 3, 4, 1), (2, 5, 4, 1, 3), (2, 5, 4, 3, 1), (3, 1, 2, 4, 5), (3, 1, 2, 5, 4), (3, 1, 4, 2, 5), (3, 1, 4, 5, 2), (3, 1, 5, 2, 4), (3, 1, 5, 4, 2), (3, 2, 1, 4, 5), (3, 2, 1, 5, 4), (3, 2, 4, 1, 5), (3, 2, 4, 5, 1), (3, 2, 5, 1, 4), (3, 2, 5, 4, 1), (3, 4, 1, 2, 5), (3, 4, 1, 5, 2), (3, 4, 2, 1, 5), (3, 4, 2, 5, 1), (3, 4, 5, 1, 2), (3, 4, 5, 2, 1), (3, 5, 1, 2, 4), (3, 5, 1, 4, 2), (3, 5, 2, 1, 4), (3, 5, 2, 4, 1), (3, 5, 4, 1, 2), (3, 5, 4, 2, 1), (4, 1, 2, 3, 5), (4, 1, 2, 5, 3), (4, 1, 3, 2, 5), (4, 1, 3, 5, 2), (4, 1, 5, 2, 3), (4, 1, 5, 3, 2), (4, 2, 1, 3, 5), (4, 2, 1, 5, 3), (4, 2, 3, 1, 5), (4, 2, 3, 5, 1), (4, 2, 5, 1, 3), (4, 2, 5, 3, 1), (4, 3, 1, 2, 5), (4, 3, 1, 5, 2), (4, 3, 2, 1, 5), (4, 3, 2, 5, 1), (4, 3, 5, 1, 2), (4, 3, 5, 2, 1), (4, 5, 1, 2, 3), (4, 5, 1, 3, 2), (4, 5, 2, 1, 3), (4, 5, 2, 3, 1), (4, 5, 3, 1, 2), (4, 5, 3, 2, 1), (5, 1, 2, 3, 4), (5, 1, 2, 4, 3), (5, 1, 3, 2, 4), (5, 1, 3, 4, 2), (5, 1, 4, 2, 3), (5, 1, 4, 3, 2), (5, 2, 1, 3, 4), (5, 2, 1, 4, 3), (5, 2, 3, 1, 4), (5, 2, 3, 4, 1), (5, 2, 4, 1, 3), (5, 2, 4, 3, 1), (5, 3, 1, 2, 4), (5, 3, 1, 4, 2), (5, 3, 2, 1, 4), (5, 3, 2, 4, 1), (5, 3, 4, 1, 2), (5, 3, 4, 2, 1), (5, 4, 1, 2, 3), (5, 4, 1, 3, 2), (5, 4, 2, 1, 3), (5, 4, 2, 3, 1), (5, 4, 3, 1, 2), (5, 4, 3, 2, 1)]>>Total distinct permutations ==325
Upvote ·
9 1
9 1
Shambhu Bhat
Retired professor in engineering ;Very fond of mathematics · Author has 6.6K answers and 4.9M answer views
·6y
Related
How many 6-digit numbers contain exactly 4 different digits?
How many 6-digit numbers contain exactly 4 different digits?
,………………………………………………………………………………………..
Awdhesh has done a nice job. But the second part seemed too complicated for me.Some of what i am writing is repetition. I have an easy way out.
Select 4 4 digits out of total possible 10 10
This can be done in 10 C 4 10 C 4 ways.
The number of arrangements will be
10 C 4(4 C 1 6!3!+4 C 2 6!2!⋅2!)10 C 4(4 C 1 6!3!+4 C 2 6!2!⋅2!)
Now since no discrimination has been made all 10 10 digits have equal chance of leading the numbers. So leading 0 0 will be found in 1 10 1 10 of the total numbers. rejecting t
Continue Reading
How many 6-digit numbers contain exactly 4 different digits?
,………………………………………………………………………………………..
Awdhesh has done a nice job. But the second part seemed too complicated for me.Some of what i am writing is repetition. I have an easy way out.
Select 4 4 digits out of total possible 10 10
This can be done in 10 C 4 10 C 4 ways.
The number of arrangements will be
10 C 4(4 C 1 6!3!+4 C 2 6!2!⋅2!)10 C 4(4 C 1 6!3!+4 C 2 6!2!⋅2!)
Now since no discrimination has been made all 10 10 digits have equal chance of leading the numbers. So leading 0 0 will be found in 1 10 1 10 of the total numbers. rejecting them we get
10 C 4(4 C 1 6!3!+4 C 2 6!2!⋅2!)⋅9 10 10 C 4(4 C 1 6!3!+4 C 2 6!2!⋅2!)⋅9 10
Upvote ·
99 15
9 1
9 1
Alankritha Mudiraj
love Math · Author has 77 answers and 293.9K answer views
·9y
Related
How many even three digit numbers can be formed from the digits 3,4,5,6,9 if each digit is used only once?
24
Let's take box with 3 columns( since its a 3 digit number)
For the number to be even, it's last digit has to be a even number I.e. 4,6 Can be used.
Using 4, write the digit 4 in the units place of the number I.e. In the box. Apart from the digit 4 we have four more numbers which can be filled in the hundreds place in 4 ways and tens place in 3 ways ( 3 Cz one of the four digit is used)
Number of ways =43=12
Similarly for 6.
Total number of ways =12+12=24.
Continue Reading
24
Let's take box with 3 columns( since its a 3 digit number)
For the number to be even, it's last digit has to be a even number I.e. 4,6 Can be used.
Using 4, write the digit 4 in the units place of the number I.e. In the box. Apart from the digit 4 we have four more numbers which can be filled in the hundreds place in 4 ways and tens place in 3 ways ( 3 Cz one of the four digit is used)
Number of ways =43=12
Similarly for 6.
Total number of ways =12+12=24.
Upvote ·
99 10
Related questions
How many 4 digit combinations can be made with 5?
How many possible combinations are possible in a 5 digit number?
How many 4-digit numbers are there whose digits add to 5?
What is the 4-digit number with the digits 5, 8, 7, and 4?
How many 4-digit numbers can be formed from 1,2,3,4, and 5 if no repetition is allowed, and if repetition is allowed?
How many 4-digit number combinations using 1 and 5 are there?
How many 5 digit combinations can be made using 1-15?
How many 5 digit combinations are there from 00000-99999?
How many 6 digit combinations can you make with the numbers 4,5,6?
How many ways can the digits 1, 2, 3, 4, and 5 be arranged?
How many two-digit combinations are possible between numbers 1 and 3?
How many four digit combinations can you make with 1123?
Using 1,2,3,4,5,6 how many 4 digit combinations are there?
How many 4 digit numbers can be formed from 0-9 without repetition?
How many combinations of 2 digits can be made from 6 digits?
Related questions
How many 4 digit combinations can be made with 5?
How many possible combinations are possible in a 5 digit number?
How many 4-digit numbers are there whose digits add to 5?
What is the 4-digit number with the digits 5, 8, 7, and 4?
How many 4-digit numbers can be formed from 1,2,3,4, and 5 if no repetition is allowed, and if repetition is allowed?
How many 4-digit number combinations using 1 and 5 are there?
Advertisement
About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press ·
© Quora, Inc. 2025 |
13019 | https://math.stackexchange.com/questions/2830891/digital-sum-invariant-problem-from-engels-book | Skip to main content
Digital sum Invariant problem from Engels Book
Ask Question
Asked
Modified
7 years, 2 months ago
Viewed
229 times
This question shows research effort; it is useful and clear
1
Save this question.
Show activity on this post.
I'm having a hard time with this proof. Perhaps I'm not understanding the question correctly. Is what they're saying to detatch the first digit then add it to the total, e.g 567→5+67=72 until we get a sum that is 10 digits long. But this is different from the digital sum so I don't see how the ideas from the digital sum apply to this operation. And beyond that, in the proof, they say 73≡1 mod 9 implies 71996≡71 mod 9. How does this follow. Multiplying in modular arithmetic allows us to get to 74≡7 mod 9 from 73≡1 mod 9, and I see that 4|1996 but I'm still unsure of how this implication is made.
Thanks
Solution
elementary-number-theory
contest-math
proof-explanation
Share
CC BY-SA 4.0
Follow this question to receive notifications
asked Jun 25, 2018 at 0:36
john fowlesjohn fowles
2,55511 gold badge2323 silver badges3939 bronze badges
Add a comment
|
1 Answer 1
Reset to default
This answer is useful
3
Save this answer.
Show activity on this post.
Claim:After each transition, the new number is congruent to the old one, mod 9.
Let say at a given stage, we have the number x, with leading digit a.
Then x has the form x=a10k+b, for some positive integers k,b.
The next transition yields the number y=b+a, hence
x=a10k+b≡a10k+b(mod9)≡a(1)k+b(mod9)≡a+b(mod9)≡y(mod9)
as claimed.
Thus, at the end, the final 10-digit number is congruent to the starting number, mod 9.
But any nonnegative integer is congruent, mod 9 to the sum of its digits.
If the final 10-digit number had 10 distinct digits, it would be congruent, mod 9, to
0+1+2+⋯+9=45
which is congruent to 0, mod 9, contradiction, since 71996 is not a multiple of 9.
Note:
There is a subtle point here which the author may have missed . . .
How do we know that the process ends with a 10-digit number (with nonzero leading digit)?
One way to show this is to compute 71996 mod 1010, and see that the result is an actual 10-digit number. Then argue that the number of steps is insufficient to change, at any stage, the digit which is tenth from the right. But computing 71996 mod 1010 is a lot of work.
Without that verification, the first transition which yields a final number less than 1010 might actually yield a number with less than 10 digits, in which case, it's conceivable that the final number has all distinct digits.
Share
CC BY-SA 4.0
Follow this answer to receive notifications
edited Jun 25, 2018 at 3:09
answered Jun 25, 2018 at 1:09
quasiquasi
61.2k33 gold badges4646 silver badges105105 bronze badges
Add a comment
|
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
elementary-number-theory
contest-math
proof-explanation
See similar questions with these tags.
Featured on Meta
Community help needed to clean up goo.gl links (by August 25)
Related
2
Prove that if the sum of digits of a number is divisible by 3, so is the number itself.
2
Prove the digital root of a square can be anything other than 2,3,5,6,8?
9
Can the magician guess the number?
2
18x≡1(mod25). Computing inverses modulo a square.
6
Find sum of the digits of 1395t+2015
Hot Network Questions
Error siunitx and tabularray
What’s the point of passing an ordinance that only recommends residents limit smartphone use to 2 hours/day if there are no penalties for exceeding it
How can I parse a string to a float in C in a way that isn't affected by the current locale?
Three Houses in Mile Lane
How do we deal with trainees/students using LLM for higher academic work?
Would weekly seasonal changes still allow a habitable planet?
table with diagbox
How to make Rsync newly write all file perms, ownership etc. only?
Fastest double check?
How to do Lucas' quest in Lil Gator Game?
How do I push back on an impossible scope?
Why does he have a phobia on these dates?
Worried travelling to America once more after long stays in the past
Can I seal a crack in acryl shower-pan by glueing a strip of EPDM rubber over? Any better suggestions?
A week after the Putin-Trump meeting at Anchorage, what's the quick elevator pitch on proposals currently publicly on the table?
Legit or scam? What's a good filter when asked for emergency help?
Why can we ignore the effect the electron has on the EM field in electron orbitals in quantum mechanics?
How to call Mathematica commands from Maple on same PC without leaving Maple?
What happens to the charges in metals?
Wearing a Tallis by Mincha
Shebang apparently calls wrong shell
What building(s) were used as the filming locations for the “Jump Program”?
Does using dispel magic on one victim of glitterdust dispel it for all?
Funny looking hydrogen spectrum
more hot questions
Question feed
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device. |
13020 | https://www.youtube.com/watch?v=dvEVYrSUYJM | Simple Machines
Khan Academy - Projects
36600 subscribers
328 likes
Description
51327 views
Posted: 13 Dec 2013
10 comments
Transcript:
Introduction okay so in this video we're going to show you how to uh calculate mechanical advantage uh for pulleys uh now here's a basic pulley a pulley is just a grooved wheel that a that a a chain or a strand or a rope or a cable can slide through and you can see here's a very simple pulley system this is called a fixed pulley right here and that's because this pulley even though it rotates around does not move up or or down so fixed pulleys rotate but they don't move up and down and that's going to be really important because we need to uh understand movable pulleys in order to do mechanical advantage calculations now fixed pulleys will allow us to transfer the force um so we can turn uh we can turn we can actually pull down it it allows us to turn the force down so we can pull down to lift something up so this is our effort force and this is our load force and so we can overcome our load Force by pulling down instead of instead of pulling up so this allows us to change the direction uh with which we pull but it doesn't give us a mechanical advantage to get a mechanical advantage we need to have a movable Movable Pulley pulley and we've got a movable pulley here uh in this in this setup so this is the movable pulley so the first thing we want to do is we want to define the pulley so this is a fixed pulley and this one's movable and so if we pull on this this strand string cable uh this pulley here will move up so it is a movable pulley and we calculate the mechanical advantage by adding the number of ropes supporting each movable pulley so we've got one two so then our mechanical advantage equals two and you may say well how does that help us well we can say remember we said that our effort force equal our resistance force or our load Force so this is our load force and this is our effort force over here our effort force equals our load force uh divided by the mechanical advantage which is two so so let's say let's take our load force and say it's uh 100 Newtons and now we can figure out what our effort force is so we can just put it plug it right in so we got 100 Newtons and we're going to divide that by two so that means that our effort force is 50 Newtons so this would be 50 Newtons of force um so the mechanical advantage of this system is two and that means that the uh amount of force we have to apply is half of what the weight um is that we're lifting now of course that that means we still have to uh that means we're going to have to pull more cable in order for that to uh in order for us to lift this weight uh and so remember uh work equals force times displacement and so uh in order for us to do the same amount of work we if we are decreasing the force we have to increase the displacement otherwise the amount of work will change and energy won't be conserved so we we that that's really important important concept there all right so let's go to the next one and see if we can do one with more pulleys a little bit more complexity Pulley System okay so wow this one's crazy it's got a lot of pulley all right so don't stress when you see a problem like this it's really simple all you need to do is figure out which ones are the movable pulley so these guys are the movable they're the ones attached to the load they're the ones that can move up and down they can move in this direction remember all the pulleys can move around like that so uh let's figure out the fixed pulley so this one's fixed this one's fixed this one's fixed this is movable one movable two movable three okay so we have three movable pulleys that means we have one I'll just go ahead and write that one two three four five six strands attached to movable pulleys so our mechanical advantage equals six all right so if if we want to do our calculations remember we can say that our effort force which is here has to equal our load force and this is our load divided by our mechanical advantage which is in this situation is six okay so let's say our load is 10 Newtons what is the effort force that we need well if our load force is 10 Newtons so we can s send we can then say 10 Newtons and uh then what we need to do is we need to to divide 10 Newtons by 6 so we can say 10 / 6 equals 1.66 repeating so our effort force and that's Newtons and our effort so our effort force equals 1.66 repeating Newtons in this system we can also flip it around we can calculate the the load if we know the effort so let's say um in this system we're going to put in 100 Newtons of effort so let's just plug that right into the same equation so we' have 100 Newtons equals LF which is our load Force divided by six well we can multiply by six on this side cancel that out multiply by six on this side and so that's going to be 6 newtons that we can lift with this system now of course that means that we are going to have to pull that that uh that uh the cord or cable or strand we're going to have to pull that six times as far so we're going to have to pull a whole lot of that but this is really handy because things like cranes for example um they could have huge Motors to lift the kind of loads that they uh that they lift or they could take advantage of um simple machines so they could have a smaller motor that can't lift quite as much uh uh let's say a a motor that can lift you know maybe 100 Newtons but that motor can wind up as much string as it wants on its spool so it can wind up all of that extra six times string and and lift loads that are way more heavy than it could normally lift without the simple machine and that's that's why we use these things okay so you may also see another pulley problem that looks like this it's a block and tackle pulley if you see this don't stress out I know it looks really complicated but it's the same thing so again we're going to count the strands for the movable pulley so the first thing we need to do is figure out which pulley are fixed and which are movable so this one's fixed fixed and fixed how do we know they're all connected to this beam up here at the top so they will rotate again they'll rotate but they're not not going to move up and down these pulleys because they're connected to the load they have to move up and down in order for us to raise the load so there's one two three movable pulleys so that means that we have one two three four five six strands six strands connected to our movable pulley pulley um and so that means that our mechanical advantage equals six now keep in mind that uh you can also just look at the number of pulleys and and and count up and multiply by two and count up the uh mechanical advantage you can just say well we have three pulleys and you know every pulley is going to need at least two strand or is going to need two strands so it's going to be 3 2 equals 6 and that's your mechanical advantage so let's do a problem with this one as well uh and remember we want to say our effort force uh equals our load Force divided by our mechanical advantage okay so let's say that our effort force is 500 Newtons how much can we lift with this block and tackle system how much can we lift we lift using this system I'll give you just a second to see if you can figure that out how much can we lift okay have you figured it out well we're going to take 500 and we're going to say 500 equals our LF so we don't know what the load force uh that we can lift is but we know what our mechanical advantage is it's six so what we can do is we can multiply so we're dividing by six so we can multiply by six cancel that out so that's going to be 500 6 so 500 6 equal 3,000 3,000 and that's 500 Newtons don't don't forget to include those those units there so 3,000 Newtons so this system will let us turn 500 uh Newtons of lifting power into 3,000 Newtons of lifting power power and again it does that by um allowing us to take up more of this more of the Strand or more the cable and so we can lift this it does slow the uh the lifting process down um you get a a quicker lift if you just have one pulley and you're lifting at the same rate you can lift it very quickly uh generally um but that depends on a number of things too you can also lift this slowly but if you have something turning at the same rate very slow SL winding up the string it's going to go uh up at the same rate here as it as it winds up here but in this system because it's going through all these different things the uh the load will lift more slowly uh if it's the if it's moving at the same rate as it was over here I hope that helps |
13021 | https://www.quora.com/What-is-the-definition-of-an-absolute-value-Why-does-it-equal-two-values-in-mathematics | Something went wrong. Wait a moment and try again.
Absolute Value
Algebra A
Number Theory
Math and Algebra
Mathematical Sciences
Basic Algebra
5
What is the definition of an absolute value? Why does it equal two values in mathematics?
Zulqarnain Janjua
M.SC PHYSICS GCUF PAK in Mathematics & Physics, Govt High School Maan Kot Khanewal Pakistan (Graduated 2018)
·
1y
The absolute value of a number is its distance from zero on the number line, regardless of its direction. In mathematical notation, it is often denoted by vertical bars, such as |x|, where x represents any real number.The absolute value function returns a non-negative value. For positive numbers, the absolute value remains the same (e.g., |5| = 5). For negative numbers, it changes the sign to positive (e.g., |-5| = 5). Therefore, it "equals two values" because it effectively removes the negative sign if present, resulting in a positive value.
Related questions
What is an absolute value? What are some examples?
What are absolute values in mathematics?
What is the absolute value of two?
What is absolute value of |2/3|?
What is the absolute value of two numbers?
Gerald Hawkins
Math Teacher
·
Author has 19.9K answers and 8.5M answer views
·
1y
An absolute value is the distance from zero on the number line.
It does not “equal two values.” That is the opposite of how it works.
Both the numbers -3 and 3 are exactly 3 units from 0. The absolute value of -3 is 3. The absolute value of 3 is also 3.
Sometimes people say (incorrectly) that absolute value “just takes away the minus sign.” That’s an oversimplification that will cause mistakes later.
It is correct to say that absolute value forces a result that is positive (or zero).
Some books state that the absolute value of (x) is the principal square root of (x²). That is also correct.
Garrett
Chief Technology Officer at ConcentricDAO (1989–present)
·
1y
The magnitude of the number. It doesn’t ’equal’ 2 values, it can be of a positive or a negative number. The absolute value, for example of 5, expressed as
| 5 |
is equal to 5, just as the absolute value of -5, expressed as
|-5|
is also equal to 5.
Both +5 & -5 have a magnitude of 5.
Senia Sheydvasser
PhD in Mathematics
·
Upvoted by
David Joyce
, Ph.D. Mathematics, University of Pennsylvania (1979) and
Erik Bergland
, PhD Mathematics, Brown University (2024) · Author has 2.5K answers and 39.9M answer views
·
5y
Related
What is an absolute value? What are some examples?
It is very possible that you meant the absolute value, defined as
|x|∞={xif x≥0−xotherwise
for all real numbers. But since you asked about an absolute value, I am going to assume that you are talking about the more general concept from abstract algebra (of which the usual absolute value is just one example).
Absolute values are special functions that we define on integral domains that measure the “sizes” of elements in nice ways. Now, what is an integral domain? Recall from my answer to What are fields, rings, an
It is very possible that you meant the absolute value, defined as
|x|∞={xif x≥0−xotherwise
for all real numbers. But since you asked about an absolute value, I am going to assume that you are talking about the more general concept from abstract algebra (of which the usual absolute value is just one example).
Absolute values are special functions that we define on integral domains that measure the “sizes” of elements in nice ways. Now, what is an integral domain? Recall from my answer to What are fields, rings, and groups? that a ring is a set R together with two operations + and ⋅, and two special elements 0 and 1, satisfying the following properties.
For all x,y,z, x⋅(y⋅z)=(x⋅y)⋅z and x+(y+z)=(x+y)+z—that is, + and ⋅ are associative.
For all x,y, x+y=y+x—that is, + is commutative.
For all x,y,z, x⋅(y+z)=(x⋅y)+(x⋅z)— that is, ⋅ distributes over + on the left.
For all x,y,z, (x+y)⋅z=(x⋅z)+(y⋅z)— that is, ⋅ distributes over + on the right.
For all x, x+0=0+x=x and x⋅1=1⋅x=x—that is, 0 is the additive identity, and 1 is the multiplicative identity.
For all x, there exists a y such that x+y=0—that is, every element has an additive inverse.
I gave many examples of rings in that answer, so I shall skip doing so again now. An absolute value on a ring R is a real-valued function |⋅|:R→R that satisfies the following properties:
For all x in R, |x|≥0, and |x|=0 if and only if x=0.
For all x,y in R, |xy|=|x||y|.
For all x,y in R, |x+y|≤|x|+|y|.
There are many examples of absolute values. The usual example on the real numbers obviously qualifies, but here are some other examples that might be a bit more esoteric (but not, as I will show, without use).
Let R=Z (i.e. the integers), and choose a prime number p. Define νp(n) to be the largest exponent k such that pk divides n. Then we can define
|n|p=p−νp(n)
and we can check that this is actually an absolute value. (It is, in fact, a very interesting and very useful absolute value—more on this later.)
Let R=Q[X] (i.e. the collection of polynomials in X with rational coefficients). For every such polynomial p, define νX(p) to be the largest exponent k such that we can write p(X)=Xkq(X) for some polynomial q(X). Then
|p|X=2−νX(p)
is in fact an absolute value.
Let R=H (i.e. the quaternions). If ¯¯¯z denotes the conjugate of z, then z↦(z¯¯¯z)1/2 is an absolute value.
Let R=F3 (i.e. the field of three elements). Then we can define
|x|d={0if x=01otherwise
and we can check that this is an absolute value.
If you are wondering what types of rings allow you to define an absolute value, the answer is very simple: a ring R permits an absolute value if and only if R has no zero divisors—that is, if xy=0 implies x=0 or y=0. Let’s give a proof.
Theorem: A ring R permits an absolute value if and only if R has no zero divisors.
Proof:
Suppose that R admits an absolute value. Suppose that xy=0. Then |xy|=|x||y|=|0|=0. But, since |x| and |y| are real numbers, |x||y|=0 implies that |x|=0 or |y|=0, ergo x=0 or y=0.
In the other direction, assume that R has no zero divisors, and consider the map
|⋅|d:R→Rr↦{0if x=01otherwise.
It is easy to check that |⋅|d is an absolute value (I leave this as an exercise to the reader), and therefore R admits an absolute value. □
Since |⋅|d is an absolute value that is always available (whenever there is any absolute value that is available), we refer to it as the trivial absolute value. As we have shown above, there are often many, many more choices for an absolute value. On the other hand, sometimes it really is the only choice. Here is an exercise for the reader: show that if R is a finite ring, then the trivial absolute value is the only absolute value on R.
There is a nice classification of absolute values defined on the integers due to Ostrowski's theorem—in essence, it says that the only options are either |⋅|∞ (i.e. the usual absolute value) or |⋅|p (the funky absolute value that we defined above in terms of a prime number p). This is technically a lie because something like |⋅|1/2p is also an absolute value, but it turns out that raising the aforementioned absolute values by powers is the only thing you can do. (Moreover, in every single way that matters, |⋅|1/2p behaves exactly the same way as |⋅|p.) This turns out to be directly relevant to algebraic number theory via the notion of localization, which I discussed in my answer to What are local-global principles? Why are they useful?, and which is, of course, my bread and butter as a mathematician.
Sponsored by Grammarly
Is your writing working as hard as your ideas?
Grammarly’s AI brings research, clarity, and structure—so your writing gets sharper with every step.
Related questions
How can the absolute value of any quantity be determined?
What is the absolute value of a difference?
What is the mathematical definition of absolute value?
What is absolute value in math?
What is the absolute difference between two values?
Danya Rose
I'm an Aussie feller, I've been one all my life. It may make me seem funny...
·
Upvoted by
Robby Goetschalckx
, Computer scientist for 11+ years and passionate about math since childhood. · Author has 2K answers and 7M answer views
·
6y
Related
What is the mathematical definition of absolute value?
There’s no one definition of the absolute value function. The precise form that you use best depends on what’s convenient in a given situation.
If we consider abs:R→R, then we can use
abs(x)={x,if x≥0,−x,if x<0,
which leaves x as it is if it’s already non-negative, or takes the additive inverse −x of x if x is negative (and thus returns a positive number).
Alternatively, there is the definition abs(x)=√x2, which works nicely, since √(a), for a≥0, only returns the n
There’s no one definition of the absolute value function. The precise form that you use best depends on what’s convenient in a given situation.
If we consider abs:R→R, then we can use
abs(x)={x,if x≥0,−x,if x<0,
which leaves x as it is if it’s already non-negative, or takes the additive inverse −x of x if x is negative (and thus returns a positive number).
Alternatively, there is the definition abs(x)=√x2, which works nicely, since √(a), for a≥0, only returns the non-negative number b that satisfies b2=a, and for all x∈R, x2≥0.
This second definition is closely related to the conventional definition for abs:C→C, which is abs(z)=√z¯z, where if z=a+ib, ¯z=a−ib is the complex conjugate of z. Unpacked a little, that’s equivalent to abs(z)=√a2+b2 (notice that if b=0 then z is equivalent to a member of R, and this definition amounts to the same as above). Not unrelated, if z∈C is represented in some polar form, such as z=reiθ, with r≥0, then abs(z)=r.
In words (though no less mathematical), the absolute value of some number is the non-negative real number which denotes some sense of the “size” of “magnitude” of that number. Such a definition, by conveying the quality of what we want the absolute value to be, leaves open the precise way in which it might be represented or computed, which is also useful in its own way.
Senia Sheydvasser
PhD in Mathematics
·
Upvoted by
Michael Jørgensen
, PhD in mathematics and
Alon Amit
, Lover of math. Also, Ph.D. · Author has 2.5K answers and 39.9M answer views
·
Updated Apr 10
Related
Does absolute value have any practical applications?
This is a question that is surprisingly difficult to answer—not because absolute values don’t show up in any practical applications, but because the conversation
A: What does the absolute value do?
B: It gives you the magnitude of a real number.
A: What practical applications does that have?
B: Well…
feels very similar to this one:
A: What do nails do?
B: They hold things together.
A: What practical applications does that have?
B: Well…
I’m not saying that it’s a bad question, but it is a question about an object so fundamentally basic yet nondescript that you start wracking your head thinking of some s
This is a question that is surprisingly difficult to answer—not because absolute values don’t show up in any practical applications, but because the conversation
A: What does the absolute value do?
B: It gives you the magnitude of a real number.
A: What practical applications does that have?
B: Well…
feels very similar to this one:
A: What do nails do?
B: They hold things together.
A: What practical applications does that have?
B: Well…
I’m not saying that it’s a bad question, but it is a question about an object so fundamentally basic yet nondescript that you start wracking your head thinking of some simple but clearly useful examples. Does that IKEA shelf over there use nails? No, it’s mostly held together by nuts and bolts. Think, Senia, think! Just so, if you look up tables of identities useful for engineering, you’ll quickly find examples using the absolute value.
But then you realize that you probably don’t want to talk about integrals and Fourier transforms to a question about the practical applications of the absolute value, as that is probably going to seem much too complicated. For similar reasons, the idea of using Weyl's inequality to prove that sufficiently small changes to a quantum mechanical system cannot radically change the allowed energies is thrown out the window. Discussing what the Bernstein inequalities from probability theory tell you about polling is also discarded.
So. With this preamble in mind, let me begin again.
The absolute value gives you the magnitude of a real number. This is applicable because there are many situations where you might have something that naturally comes with a sign, but you don’t actually really care about the sign at all: you just want to know how large it is.
The prototypical example is separation and distance. Roughly speaking, the separation between two points is the difference of their coordinates—i.e. if I am standing at a position of x=0 meters, Jack is standing at x=1 meters, and Kate is standing at x=3 meters, then if I want to find the separation between Jack and Kate I simply calculate it as 1−3=−2 meters. However, while there are many times in physics and engineering where we do want to keep track of the sign, there are almost as many when we don’t. In which case, the absolute value takes care of such issues automatically: the absolute value of the separation is the distance. E.g. |1−3|=|−2|=2 meters is the distance between Jack and Kate.
This seems so darn easy that one almost wonders why we would define a function to capture this idea. But the thing is, it’s easy here when we only have two data points to keep track of, and everything is given completely explicitly. What if we were given 1,000 data points and we were asked to compute the average distance between them? (Maddeningly, this is apparently a real thing that real biologists care about, but I was unable to find out why. If anyone in the comments knows the answer, please do tell me.) I wouldn’t want to do this by hand, which means giving it to a computer to crunch, which means telling it what to do if it does a subtraction and gets a negative answer—which, naturally, is precisely what the absolute value takes care of.
Of course, distance doesn’t have to mean actual physical distance. There are all sorts of times when we talk about various approximations getting progressively better, and it is very common to talk about their “distance” away from either each other or from the thing they are trying to approximate. As a result, absolute values show up in real analysis and anything that uses real analysis everywhere. If you look up the Wikipedia list of inequalities, you’ll find examples that use the absolute value very, very quickly. (Incidentally, this is how I found Weyl's inequality and the Bernstein inequalities above.) And as a result, yes, absolute values show up in practical applications quite often.
Here is an example which doesn’t require any calculus to understand. Suppose you are trying to find a mathematical model that describes some data set. The easiest possible case is that you are trying to fit a line of best fit—I previously described this in my answer to What are the benefits of understanding geometry in more than 3 dimensions?.
In that answer, I showed that if you try to minimize the sum of the squares of the errors between the model line and the actual data points, there is a nice geometric interpretation of the problem that allows you a very efficient solution. In the comments, Joachim Pense asked why we look at the square of the errors, rather than the absolute values, as that would seem to be easier. I answered that computing the sum of the absolute values of the errors is significantly harder, as it doesn’t have the same nice geometric interpretation. This wasn’t wrong, but it wasn’t the complete truth. Sure, if you care primarily about ease of computation, then the sum of squares is probably the right way to go. However, there are cases where you might want to try to minimize the sum of the absolute values instead.
Here’s a simple example: suppose that we are collecting data using some instrument, which generally gives a correct reading, but unfortunately has a 10% chance of giving a skewed reading. This creates a problem when you are trying to find a line of best fit: the usual method of the sum of squares is sensitive to outliers, so those skewed readings can lead to the sum being substantially shifted from what they should be, despite your data being mostly accurate.
Here is an illustration of the problem. I have plotted 200 data points: most of them would seem to be linearly distributed, but there seem to be some outliers that are pulling up the line of best fit, particularly at the origin. In principle, we could try and go through the data by hand and throw out apparent outliers—no doubt, there have been past generations of scientists who did precisely that. But apart from the obvious issue of possible human error being added as a result, this approach might be impossible if you have literally thousands of data points. So, can we do better and find a method of finding a line of best fit that is robust enough that it will mostly ignore outliers? The answer is ‘yes,’ and one comparatively simple solution is to replace the sum of square errors with the sum of absolute values of errors.
Why does the sum of absolute values of errors work better? There is more than one reason, but, for one thing, in the sum of squares, large deviations are weighed very heavily (since the error is squared), which is not the case for the sum of absolute values of errors. In any case, you can directly see this working.
The new purple line represents the best fit line computed by minimizing the sum of the absolute values of errors: you will notice that it better fits the bulk of the data, as desired. There are, of course, important trade-offs here: as previously mentioned, minimizing the sum of the absolute values is harder than minimizing the sum of squares. Minimizing the sum of absolute values also by no means guarantees that it will ignore the right things as outliers.
The point is, if you want to do minimization this way, I strongly recommend talking with someone who has actually studied it extensively (i.e. not me), but it is still a useful part of any data scientist’s toolkit.
Promoted by The Penny Hoarder
Lisa Dawson
Finance Writer at The Penny Hoarder
·
Updated Sep 16
What's some brutally honest advice that everyone should know?
Here’s the thing: I wish I had known these money secrets sooner. They’ve helped so many people save hundreds, secure their family’s future, and grow their bank accounts—myself included.
And honestly? Putting them to use was way easier than I expected. I bet you can knock out at least three or four of these right now—yes, even from your phone. Don’t wait like I did.
Cancel Your Car Insurance
You might not even realize it, but your car insurance company is probably overcharging you. In fact, they’re kind of counting on you not noticing. Luckily, this problem is easy to fix.
Don’t waste your time
Here’s the thing: I wish I had known these money secrets sooner. They’ve helped so many people save hundreds, secure their family’s future, and grow their bank accounts—myself included.
And honestly? Putting them to use was way easier than I expected. I bet you can knock out at least three or four of these right now—yes, even from your phone. Don’t wait like I did.
Cancel Your Car Insurance
You might not even realize it, but your car insurance company is probably overcharging you. In fact, they’re kind of counting on you not noticing. Luckily, this problem is easy to fix.
Don’t waste your time browsing insurance sites for a better deal. A company calledInsurify shows you all your options at once — people who do this save up to $996 per year.
If you tell them a bit about yourself and your vehicle, they’ll send you personalized quotes so you can compare them and find the best one for you.
Tired of overpaying for car insurance? It takes just five minutes to compare your options with Insurify andsee how much you could save on car insurance.
Ask This Company to Get a Big Chunk of Your Debt Forgiven
A company calledNational Debt Relief could convince your lenders to simply get rid of a big chunk of what you owe. No bankruptcy, no loans — you don’t even need to have good credit.
If you owe at least $10,000 in unsecured debt (credit card debt, personal loans, medical bills, etc.), National Debt Relief’s experts will build you a monthly payment plan. As your payments add up, they negotiate with your creditors to reduce the amount you owe. You then pay off the rest in a lump sum.
On average, you could become debt-free within 24 to 48 months. It takes less than a minute to sign up and see how much debt you could get rid of.
Set Up Direct Deposit — Pocket $300
When you set up direct deposit withSoFi Checking and Savings (Member FDIC), they’ll put up to $300 straight into your account. No… really. Just a nice little bonus for making a smart switch.
Why switch? With SoFi, you can earn up to 3.80% APY on savings and 0.50% on checking, plus a 0.20% APY boost for your first 6 months when you set up direct deposit or keep $5K in your account. That’s up to 4.00% APY total. Way better than letting your balance chill at 0.40% APY.
There’s no fees. No gotchas.Make the move to SoFi and get paid to upgrade your finances.
You Can Become a Real Estate Investor for as Little as $10
Take a look at some of the world’s wealthiest people. What do they have in common? Many invest in large private real estate deals. And here’s the thing: There’s no reason you can’t, too — for as little as $10.
An investment called the Fundrise Flagship Fund lets you get started in the world of real estate by giving you access to a low-cost, diversified portfolio of private real estate. The best part? You don’t have to be the landlord. The Flagship Fund does all the heavy lifting.
With an initial investment as low as $10, your money will be invested in the Fund, which already owns more than $1 billion worth of real estate around the country, from apartment complexes to the thriving housing rental market to larger last-mile e-commerce logistics centers.
Want to invest more? Many investors choose to invest $1,000 or more. This is a Fund that can fit any type of investor’s needs. Once invested, you can track your performance from your phone and watch as properties are acquired, improved, and operated. As properties generate cash flow, you could earn money through quarterly dividend payments. And over time, you could earn money off the potential appreciation of the properties.
So if you want to get started in the world of real-estate investing, it takes just a few minutes tosign up and create an account with the Fundrise Flagship Fund.
This is a paid advertisement. Carefully consider the investment objectives, risks, charges and expenses of the Fundrise Real Estate Fund before investing. This and other information can be found in the Fund’s prospectus. Read them carefully before investing.
Cut Your Phone Bill to $15/Month
Want a full year of doomscrolling, streaming, and “you still there?” texts, without the bloated price tag? Right now, Mint Mobile is offering unlimited talk, text, and data for just $15/month when you sign up for a 12-month plan.
Not ready for a whole year-long thing? Mint’s 3-month plans (including unlimited) are also just $15/month, so you can test the waters commitment-free.
It’s BYOE (bring your own everything), which means you keep your phone, your number, and your dignity. Plus, you’ll get perks like free mobile hotspot, scam call screening, and coverage on the nation’s largest 5G network.
Snag Mint Mobile’s $15 unlimited deal before it’s gone.
Get Up to $50,000 From This Company
Need a little extra cash to pay off credit card debt, remodel your house or to buy a big purchase?
We found a company willing to help.
Here’s how it works: If your credit score is at least 620, AmONE can help you borrow up to $50,000 (no collateral needed) with fixed rates starting at 6.40% and terms from 6 to 144 months.
AmONE won’t make you stand in line or call a bank. And if you’re worried you won’t qualify, it’s free tocheck online. It takes just two minutes, and it could save you thousands of dollars.
Totally worth it.
Get Paid $225/Month While Watching Movie Previews
If we told you that you could get paid while watching videos on your computer, you’d probably laugh.
It’s too good to be true, right?
But we’re serious. By signing up for a free account with InboxDollars, you could add up to $225 a month to your pocket. They’ll send you short surveys every day, which you can fill out while you watch someone bake brownies or catch up on the latest Kardashian drama.
No, InboxDollars won’t replace your full-time job, but it’s something easy you can do while you’re already on the couch tonight, wasting time on your phone.
Unlike other sites, InboxDollars pays you in cash — no points or gift cards. It’s already paid its users more than $56 million.
Signing up takes about one minute, and you’ll immediately receive a $5 bonus to get you started.
Earn $1000/Month by Reviewing Games and Products You Love
Okay, real talk—everything is crazy expensive right now, and let’s be honest, we could all use a little extra cash. But who has time for a second job?
Here’s the good news. You’re already playing games on your phone to kill time, relax, or just zone out. So why not make some extra cash while you’re at it?
WithKashKick, you can actually get paid to play. No weird surveys, no endless ads, just real money for playing games you’d probably be playing anyway. Some people are even making over $1,000 a month just doing this!
Oh, and here’s a little pro tip: If you wanna cash out even faster, spending $2 on an in-app purchase to skip levels can help you hit your first $50+ payout way quicker.
Once you’ve got $10, you can cash out instantly through PayPal—no waiting around, just straight-up money in your account.
Seriously, you’re already playing—might as well make some money while you’re at it.Sign up for KashKick and start earning now!
Work Find
1y
The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|.
Read more
What is the definition of an absolute value? Why does it equal two values in mathematics?
Absolute value
Devash Roy
1y
Absolute value (represented by the “||” signs) is how far a number is to zero on a number line in terms of distance.
|2| = 2, since 2 is 2 units away from 0 on a number line.
|-2| = 2, since -2 is also 2 units away from 0 on a number line.
Hope this helped!
Sponsored by CDW Corporation
What does your AI strategy look like?
CDW can help you define a clear strategy with actionable steps for AI adoption success.
David Joyce
Ph.D. in Mathematics, University of Pennsylvania (Graduated 1979)
·
Author has 9.9K answers and 68.4M answer views
·
6y
Related
What does absolute value mean?
The absolute value of a number is the distance it is from zero.
For example, the number 3 is three units from 0, so |3|=3. Likewise the number 5 is five units from 0, so |5|=5.
The negative number −3 is also three units from 0, so |−3|=3.
For real numbers, all that the absolute value does is strip the minus sign from negative numbers but it leaves positive numbers alone.
Note that the absolute value |a−b| of the difference of two numbers a and b is the distance between them. That’s one of the main uses of absolute value. It allows you to use algebra to work with distances, and therefore you can us
The absolute value of a number is the distance it is from zero.
For example, the number 3 is three units from 0, so |3|=3. Likewise the number 5 is five units from 0, so |5|=5.
The negative number −3 is also three units from 0, so |−3|=3.
For real numbers, all that the absolute value does is strip the minus sign from negative numbers but it leaves positive numbers alone.
Note that the absolute value |a−b| of the difference of two numbers a and b is the distance between them. That’s one of the main uses of absolute value. It allows you to use algebra to work with distances, and therefore you can use algebra to study a lot of geometry.
There’s also an absolute value for complex numbers z=x+yi. It still is the distance from zero. For example, the imaginary unit i is one unit from 0, so |i|=1. The absolute value of a complex number z=x+yi can be found algebraically by the formula
|z|=|x+yi|=√x2+y2.
Philip Lloyd
Specialist Calculus Teacher, Motivator and Baroque Trumpet Soloist.
·
Author has 6.8K answers and 52.8M answer views
·
6y
Related
What is absolute value in math?
The absolute value of something is its length!
Let me explain…
This is an Argand Plane and I have drawn t...
The absolute value of something is its length!
Let me explain…
——————————————————————————————————-
This is an Argand Plane and I have drawn t...
Nadav Bhonker
Upvoted by
Justin Rising
, PhD in statistics and
Jack Huizenga
, Math Professor, Penn State
·
10y
Related
Why do we need the absolute value?
If g(t) is a complex number then the absolute value makes a difference:
If g(t) is a complex number then the absolute value makes a difference:
Rory Barrett
M.Sc in Mathematics, University of Auckland
·
Author has 2.1K answers and 1.8M answer views
·
6y
Related
What is the mathematical definition of absolute value?
What follows only applies to real numbers. The absolute value of positive numbers is the actual value. If the number is negative then it is the product of -1 and the number.
The symbol for absolute value is ||.
So |3| = 3
|-5| = -1-5 = 5
|0| = 0.
Brett Schmidt
Upvoted by
David Joyce
, Ph.D. Mathematics, University of Pennsylvania (1979) · Author has 2K answers and 3.2M answer views
·
Updated 6y
Related
What is the absolute value?
The absolute value of a number n is the distance of the number from zero on the number line and/ or the complex plane.
The absolute value of 0 is 0.
If n is a positive real number, the absolute value of n is n.
If n is a negative real number, the absolute value of n is negative n.
If z = a+bi is any complex number, then the absolute value of z is the non-negative square root of a^2+b^2. (Rules 1 to 3 are special cases of this rule.)
Related questions
What is an absolute value? What are some examples?
What are absolute values in mathematics?
What is the absolute value of two?
What is absolute value of |2/3|?
What is the absolute value of two numbers?
How can the absolute value of any quantity be determined?
What is the absolute value of a difference?
What is the mathematical definition of absolute value?
What is absolute value in math?
What is the absolute difference between two values?
What is the definition of "absolute value"? What are its applications in real life and mathematics?
What is the concept of absolute value and its applications?
What is the definition of absolute value? What are some properties of absolute value? Why is it important to take the absolute value on both sides of an equation?
Does absolute value come before parenthesis, or does parenthesis come before absolute value?
How can you show that two numbers have the same absolute value (mathematically)?
About
·
Careers
·
Privacy
·
Terms
·
Contact
·
Languages
·
Your Ad Choices
·
Press
·
© Quora, Inc. 2025 |
13022 | https://www.euroflor.pro/system-pages/download-product-leaflet/flora-britannica-prod2544?LanguageID=LANG1&PDF=true&LeftRightMargin=-23&TopBottomMargin=1&Filename=FLORA%20BRITANNICA.pdf | A mix of 100% native and naturalised species found in the British Isles For creating natural flower rich meadows Good for ecological renovations Recommend to be sown along rural roadsides, nature reserves, ecological conservation schemes, riverside edges, on agricultural set-aside land and natural areas of parks.
Flora Britannica is a universal mixture meaning it’s plant base is common to many habitats in the UK. It can be sown alone or associated with companion grass species (be aware of the possible dominance of grasses). It also can be sown with other Euroflor flower mixes or with specialised habitat species such as wetland, acid soil, shade, chalk land flower species etc.
Spring and Autumn sowing Flowering height: 20-80 cm Including 25 species EUROFLOR - Fleurissement et biodiversité - contact@euroflor.pro FLORA BRITANNICA |
13023 | https://mathleaks.com/study/trigonometric_ratios_of_acute_angles | Unlocking the Secrets of Trigonometric Ratios of Acute Angles
{{ 'ml-label-loading-course' | message }}
{{ toc.signature }}
{{ tocHeader }}{{ 'ml-btn-view-details' | message }}
arrow_back{{ tocSubheader }}
arrow_right{{ child.name }}{{ child.label }} {{ child.sublabel }} {{ child.name }}{{ child.label }}
{{ child2.name }}{{ child2.label }}{{ child2.sublabel }}radio_button_unchecked radio_button_checked{{ child2.name }}{{ child2.label }}
{{ child2.name }}{{ child2.label }}{{ child2.sublabel }}{{ child2.name }}{{ child2.label }}
arrow_right{{ child3.label }} {{ child3.sublabel }} info_outline{{ child3.name }}{{ child3.label }}
arrow_right{{ child4.name }}{{ child4.label }}{{ child4.sublabel }}radio_button_unchecked radio_button_checked{{ child4.label }}
{{ child5.name }}{{ child5.label }}
{{ 'ml-toc-proceed-mlc' | message }}
{{ 'ml-toc-proceed-tbs' | message }}
LessonExercisesRecommendedTests
View as slides View as column
An error ocurred, try again later!
Try again
arrow_back
eCourses/
Geometry/
Trigonometric Ratios of Acute Angles expand_more
Chapter{{ article.chapter.number }}
{{ article.number }}.
{{ article.displayTitle }}
{{ article.intro.summary }}
Show less Show more expand_more
{{ ability.description }}
Lesson Settings & Tools
Show Calculators
Lock Screen
Set Timer
{{ 'ml-lesson-number-slides' | message : article.intro.bblockCount }}
{{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount }}
{{ 'ml-lesson-time-estimation' | message }}
Start
Image Credits expand_more
{{ item.file.title }}{{ presentation }}
No file copyrights entries found
In this lesson, the concept of similarity will be used to understand that side ratios in right triangles are properties of the angles. This will lead to the definition of trigonometric ratios for acute angles.
Catch-Up and Review
Here are a few recommended readings before getting started.
Similar triangles
Right angle
Right triangles
Pythagorean Theorem
Try your knowledge on these topics.
a Select all the triangles that are similar to △A B C.
{"type":"multichoice","form":{"alts":["Triangle <\/span>1<\/span><\/span><\/span><\/span>","Triangle <\/span>2<\/span><\/span><\/span><\/span>","Triangle <\/span>3<\/span><\/span><\/span><\/span>"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[0,2]}
b Which of the following angles is a right angle?
{"type":"choice","form":{"alts":["<\/span>\u2220<\/span>1<\/span><\/span><\/span><\/span>","<\/span>\u2220<\/span>2<\/span><\/span><\/span><\/span>","<\/span>\u2220<\/span>3<\/span><\/span><\/span><\/span>","<\/span>\u2220<\/span>4<\/span><\/span><\/span><\/span>","<\/span>\u2220<\/span>5<\/span><\/span><\/span><\/span>","<\/span>\u2220<\/span>6<\/span><\/span><\/span><\/span>"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":3}
c Which of the following are right triangles?
{"type":"multichoice","form":{"alts":["Triangle <\/span>1<\/span><\/span><\/span><\/span>","Triangle <\/span>2<\/span><\/span><\/span><\/span>","Triangle <\/span>3<\/span><\/span><\/span><\/span>","Triangle <\/span>4<\/span><\/span><\/span><\/span>","Triangle <\/span>5<\/span><\/span><\/span><\/span>"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[0,3]}
d Find the value of x by using the Pythagorean Theorem.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<\/span><\/span>"},"formTextBefore":"<\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["13"]}}
Challenge
Investigating a Side of a Right Triangle
The Leaning Tower of Pisa has a tilt of 4 degrees. Once, a worker maintaining it accidentally dropped a hammer from the top. The hammer landed 3 meters away from the base of the tower. Luckily, it did not hurt anyone!
What was the vertical distance traveled by the hammer?
Explore
Comparing Ratios of Sides in Similar Right Triangles
Similar triangles have congruent angles and proportional sides. In the following applet, some of the ratios of the side lengths of two similar right triangles are compared.
What conclusion can be made about the ratios of the side lengths of two similar triangles?
Discussion
Analyzing Ratios of Sides in Similar Right Triangles
Because all right angles are congruent, all right triangles have one pair of congruent angles. If they also have one pair of congruent acute angles, then the triangles have two pairs of congruent angles. Therefore, by the Angle-Angle Similarity Theorem, two triangles with one pair of congruent acute angles are similar.
Since corresponding sides of similar polygons are proportional, the ratios between corresponding sides of similar right triangles are the same.
B C A BB C A CA B A C=Q R P Q=Q R P R=P Q P R
Discussion
Trigonometric Ratios
The ratios between side lengths of right triangles depend on the acute angles of the triangle. Some of these ratios receive a special name.
A trigonometric ratio relates two side lengths of a right triangle. Consider the right triangle △A B C. One of its acute angles has been named θ.
Since it is opposite to the right angle, B C is the hypotenuse of the right triangle. The remaining sides — the legs — can be named relative to the marked angleθ. Because A B is next to ∠θ, it is called the adjacent side. Conversely, because A C lies across from ∠θ, it is called the opposite side.
The names of the three main ratios between side lengths are stated in the following table.
| Name | Definition | Notation |
---
| Sine of ∠θ | Hypotenuse Length of opposite side to∠θ | sin θ=hyp opp |
| Cosine of ∠θ | Hypotenuse Length of adjacent side to∠θ | cos θ=hyp adj |
| Tangent of ∠θ | Length of adjacent side to∠θ Length of opposite side to∠θ | tan θ=adj opp |
Example
Understanding Trigonometric Ratios in Right Triangles
Dominika is helping Tadeo understand trigonometric ratios. She drew three right triangles for him to write trigonometric ratios with respect to the acute angleθ. Help Tadeo grasp this topic by selecting the correct answers!
a
{"type":"choice","form":{"alts":["<\/span>sin<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","<\/span>sin<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","<\/span>sin<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
b
{"type":"choice","form":{"alts":["<\/span>cos<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>1<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","<\/span>cos<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>1<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>8<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","<\/span>cos<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span>8<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
c
{"type":"choice","form":{"alts":["<\/span>tan<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","<\/span>tan<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>1<\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","<\/span>tan<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>1<\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
Hint
a Identify the hypotenuse of the right triangle and the opposite side to ∠θ.
b Identify the hypotenuse of the right triangle and the adjacent side to ∠θ.
c Identify the opposite and adjacent sides to ∠θ.
Solution
a The sine of ∠θ is defined as the ratio of the length of the side opposite∠θ to the hypotenuse of the right triangle. Therefore, Tadeo needs to identify these two sides.
It can be seen above that the hypotenuse of the right triangle is 5 and that the length of the opposite side to ∠θ is 3. With this information, the sine of ∠θ can be written.
sin θ=5 3
b The cosine of ∠θ is the ratio of the length of the adjacent side to ∠θ to the hypotenuse of the right triangle. Therefore, Tadeo needs to identify these two sides.
The hypotenuse of the right triangle is 1 0 and the length of the adjacent side to ∠θ is 6. With this information, the cosine of ∠θ can be written.
cos θ=1 0 6
c The tangent of ∠θ is the ratio of the length of the side opposite∠θ to the length of the side adjacent∠θ. Therefore, Tadeo needs to identify these two sides.
It can be seen above that the lengths of the opposite and adjacent sides to ∠θ are 1 2 and 5, respectively. With this information, the tangent of ∠θ can be written.
tan θ=5 1 2
Example
Explaining Trigonometric Ratios in Right Triangles
Despite the awesome explanations Dominika provided, Tadeo still does not get how to find trigonometric ratios. To help his friend, Dominika thought of one more exercise.
This time, Dominika drew one right triangle and stated its three side lengths. She also labeled one of the triangle's acute angles.
Tadeo has now been asked to find the sine, cosine, and tangent of ∠θ. Help him find the answers!
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<\/span><\/span>"},"formTextBefore":"<\/span>sin<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["\dfrac{8}{17}","\frac{8}{17}","8\/17"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<\/span><\/span>"},"formTextBefore":"<\/span>cos<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["\dfrac{15}{17}","\frac{15}{17}","15\/17"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<\/span><\/span>"},"formTextBefore":"<\/span>tan<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["\dfrac{8}{15}","\frac{8}{15}","8\/15"]}}
Hint
Start by identifying the hypotenuse of the right triangle. Then identify the opposite and adjacent sides to ∠θ. Finally, recall the definitions of sine, cosine, and tangent of an acute angle of a right triangle.
Solution
To find the sine, cosine, and tangent of ∠θ, the hypotenuse of the right triangle needs to be identified. The opposite and adjacent sides to the angle also need to be identified.
It can be seen above that the hypotenuse is 1 7, and that the lengths of the opposite and adjacent sides to ∠θ are 8 and 1 5, respectively. This information can be substituted into the definitions for sine, cosine, and tangent.
| Definition | Substitute |
--- |
| sin θ=Hypotenuse Length of opposite side to∠θ | sin θ=1 7 8 |
| cos θ=Hypotenuse Length of adjacent side to∠θ | cos θ=1 7 1 5 |
| tan θ=Length of adjacent side to∠θ Length of opposite side to∠θ | tan θ=1 5 8 |
Example
Using Trigonometric Ratios to Find Side Lengths
Tadeo finally understands the topic! But wait, Dominika wants to level up and has let him know that trigonometric ratios can also be used to find missing side lengths of a right triangle. "Tell me more," Tadeo responds. An acute angle and the hypotenuse of a right triangle are given. To see whether Tadeo masters this topic, Dominika asked him to find the value of x, which is the length of the opposite side to the given angle.
Help Tadeo find the value of x. If necessary, round the answer to three significant figures.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<\/span><\/span>"},"formTextBefore":"<\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["8.66"]}}
Hint
Identify the trigonometric ratio that should be used according to the given and desired lengths. Then, with the help of a calculator, set and solve an equation.
Solution
The hypotenuse of the right triangle is given, and the length of the opposite side to the given angle is to be found.
The trigonometric ratio that relates these two sides and the acute angle is the sine ratio.
sin θ=Hypotenuse Length of opposite side to∠θ⇓sin 6 0∘=1 0 x
Finally, the equation can be solved for x.
sin 6 0∘=1 0 x
▼
Solve for x
MultEqn
LHS⋅1 0=RHS⋅1 0
sin 6 0∘⋅1 0=1 0 x⋅1 0
FracMultDenomToNumber
1 0 a⋅1 0=a
sin 6 0∘⋅1 0=x
Multiply
Multiply
1 0 sin 6 0∘=x
RearrangeEqn
Rearrange equation
x=1 0 sin 6 0∘
To find the value of sin 6 0∘, a calculator can be used. First, it must be set to degree mode. This is done by pushing M O D E and selecting Degree in the third row.
Next the value of sin 6 0∘ can be calculated by pushing S I N followed by the angle measure.
Now the value of x can be calculated.
x=1 0 sin 6 0∘
▼
UseCalc
Use a calculator
x=1 0(0.8 6 6 0 2 5…)
Multiply
Multiply
x=8.6 6 0 2 5 4…
RoundSigDig
Round to 3 significant digit(s)
x≈8.6 6
Pop Quiz
Practice Finding Side Lengths Using Trigonometric Ratios
In the right triangles below, one acute angle and one side length are given. By using the corresponding trigonometric ratio, find the length of the side labeled x. Round the answer to one decimal place.
Discussion
Pythagorean Identities
By using trigonometric ratios, an important property of angles can be derived.
For any angleθ, the following trigonometric identities hold true.
sin 2 θ+cos 2 θ=1
Proof
For Acute Angles
Consider a right triangle with a hypotenuse of 1.
By recalling the sine and cosine ratios, the lengths of the opposite and adjacent sides to ∠θ can be expressed in terms of the angle.
| | Definition | Substitute | Simplify |
--- --- |
| sin θ | Hypotenuse Length of opposite side to∠θ | 1 opp | opp |
| cos θ | Hypotenuse Length of adjacent side to∠θ | 1 adj | adj |
It can be seen that if the hypotenuse of a right triangle is 1, the sine of an acute angle is equal to the length of its opposite side. Similarly, the cosine of the angle is equal to the length of its adjacent side.
By the Pythagorean Theorem, the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. Therefore, for the above triangle, the sum of the squares of s i n θ and c o s θ is equal to the square of 1.
s i n 2 θ+c o s 2 θ⇓sin 2 θ+cos 2 θ=1 2=1
Therefore, for any acute angleθ, the sum of the squares of its sine and cosine equals 1. This property is also valid for any angle. The proof for angles whose measure is greater than or equal to 9 0∘ will be seen later in this course.
Example
Using Trigonometry to Determine the Cosine of an Angle
The property seen before can be used, among other things, to find the sine or cosine ratio of an acute angle in a right triangle.
Kriz and his friends plan to spend Saturday afternoon playing video games. To optimize the space, they decide to tidy up the basement to ensure the console, snacks, and beverages are placed in the form of a right triangle. Kriz decides to set the snacks and the beverages 3 and 5 meters away from the console, respectively.
The adjacent side to ∠θ connects the snacks and the beverages. To guarantee a good flow between the snacks and the beverages, Kriz wants to find the cosine of ∠θ. Help the gang have a good time by finding cos∠θ for them!
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<\/span><\/span>"},"formTextBefore":"<\/span>cos<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["\dfrac{4}{5}","\frac{4}{5}","4\/5"]}}
Hint
The hypotenuse of the right triangle is 5 and the measure of the opposite side to ∠θ is 3. With this information, the sine ratio can be found.
Solution
The hypotenuse of the right triangle is 5 and the measure of the opposite side to ∠θ is 3. With this information, the sine ratio can be found.
sin θ=Hypotenuse Length of opposite side to θ⇓sin θ=5 3
This value can be substituted in the equation sin 2 θ+cos 2 θ=1.
sin 2 θ+cos 2 θ=1
Substitute
sin θ=5 3
(5 3)2+cos 2 θ=1
▼
Solve for cos θ
PowQuot
(b a)m=b m a m
2 5 9+cos 2 θ=1
SubEqn
LHS−2 5 9=RHS−2 5 9
cos 2 θ=1−2 5 9
OneToFrac
Rewrite 1 as 2 5 2 5
cos 2 θ=2 5 2 5−2 5 9
SubFrac
Subtract fractions
cos 2 θ=2 5 1 6
SqrtEqn
LHS=RHS
cos θ=2 5 1 6
SqrtQuot
b a=ba
cos θ=5 4
Note that, when solving the equation for cos θ, only the principal root was considered. The reason is that the cosine of ∠θ is the ratio between two side lengths, and side lengths are always positive. Therefore, the quotient is also positive.
Explore
Using Inverse Trigonometric Ratios
Trigonometric ratios can also be used to find missing angles. Consider a right triangle△A B C where the hypotenuse and a leg are given.
Suppose now that the measure of ∠C is desired. Note that, apart from the hypotenuse, the side whose length is known is opposite to ∠C. The trigonometric ratio that relates the hypotenuse and the opposite side to an acute angle in a right triangle is the sine ratio.
sin∠C=h y p o t e n u s e length of opposite side to∠C⇓sin∠C=1 2 5
To find the measure of ∠C, the inverse of the sine ratio could be used.
sin∠C=1 2 5⇕m∠C=sin-1 1 2 5
Finally, to find the value of sin-1 1 2 5 and therefore the measure of ∠C, a calculator will help. In the following example, it will be shown how to use a calculator to find the value of an inverse trigonometric ratio.
Example
Calculating Angles of Right Triangles
Previously, it was said that apart from being useful to find side lengths of a right triangle, trigonometric ratios can also be used to find missing angle measures.
Before playing video games with his friends, Kriz wants to finish his math homework to have a care-free weekend. He wants to find the measure of an acute angle in three different right triangles. By using the corresponding trigonometric ratios, help Kriz find m∠θ in each triangle. Round the answer to the nearest degree.
a
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<\/span><\/span>"},"formTextBefore":"<\/span>m<\/span>\u2220<\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>","formTextAfter":"<\/span><\/span><\/span>\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","answer":{"text":["71"]}}
b
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<\/span><\/span>"},"formTextBefore":"<\/span>m<\/span>\u2220<\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>","formTextAfter":"<\/span><\/span><\/span>\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","answer":{"text":["74"]}}
c
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<\/span><\/span>"},"formTextBefore":"<\/span>m<\/span>\u2220<\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>","formTextAfter":"<\/span><\/span><\/span>\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","answer":{"text":["44"]}}
Hint
a The lengths of the adjacent and opposite sides to ∠θ are 1 2 and 3 5, respectively.
b The hypotenuse of the right triangle is 2 5 and the length of the adjacent side to ∠θ is 7.
c The hypotenuse of the right triangle is 2 9 and the length of the opposite side to ∠θ is 2 0.
Solution
a In the given diagram, it can be seen that the lengths of the opposite and adjacent sides to ∠θ are 3 5 and 1 2, respectively. The trigonometric ratio that relates these two sides is the tangent ratio.
tan θ=length of adjacent side to∠θ length of opposite side to∠θ⇓tan θ=1 2 3 5
To solve this equation, the inverse of the tangent function could be used.
tan θ=1 2 3 5⇕m∠θ=tan-1 1 2 3 5
To find the value of tan-1 1 2 3 5, a calculator should to be used. First, the calculator must be set in degree mode. This is done by pushing M O D E and selecting Degree in the third row.
Next the value of tan-1 1 2 3 5, can be calculated by pushing 2 N D, followed by T A N, and 3 5/1 2.
Thereofre, m∠θ≈7 1∘.
b In the given diagram, it is shown that the length of the adjacent side to ∠θ is 7 and that the hypotenuse of the right triangle is 2 5. The trigonometric ratio that relates these two sides is the cosine ratio.
cos θ=hypotenuse length of adjacent side to∠θ⇓cos θ=2 5 7
To solve this equation, the inverse of the cosine function is needed.
cos θ=2 5 7⇕m∠θ=cos-1 2 5 7
To find the value of cos-1 2 5 7, a calculator should be used. Just like before, the calculator must be set in degree mode. This is done by pushing M O D E and selecting Degree in the third row.
Next the value of cos-1 2 5 7, is calculated by pushing 2 N D, followed by C O S, and 7/2 5.
It was found that m∠θ≈7 4∘.
c In the diagram, it can be seen that the hypotenuse of the right triangle is 2 9 and that the length of the opposite side to ∠θ is 2 0. The trigonometric ratio that relates these two sides is the sine ratio.
sin θ=hypotenuse length of opposite side to∠θ⇓sin θ=2 9 2 0
To solve this equation, the inverse of the sine function can be used.
sin θ=2 9 2 0⇕m∠θ=sin-1 2 9 2 0
To find the value of sin-1 2 9 2 0, a calculator should be used. Just like in Parts A and B, the calculator must be set in degree mode by pushing M O D E and selecting Degree in the third row.
Next the value of sin-1 2 9 2 0, can be calculated by pushing 2 N D, followed by S I N, and 2 0/2 9.
Thereofre, m∠θ≈4 4∘.
Pop Quiz
Practice Finding Angles Using Trigonometric Ratios
In the following right triangles, two side lengths are given. By using the corresponding trigonometric ratio, find m∠θ. Round the answer to nearest degree.
Discussion
Reciprocal Trigonometric Ratios
Apart from the sine, cosine, and tangent ratios, there are three other trigonometric ratios that are worth mentioning.
Consider the right triangle △A B C.
The so called reciprocal ratios are written in the next table.
| Name | Definition | Notation |
---
| Cosecant of ∠θ | Length of opposite side to∠θ Hypotenuse | csc θ=opp hyp |
| Secant of ∠θ | Length of adjacent side to∠θ Hypotenuse | sec θ=adj hyp |
| Cotangent of ∠θ | Length of opposite side to∠θ Length of adjacent side to∠θ | cot θ=opp adj |
These ratios can be defined in terms of sine, cosine, and tangent.
Rule
Reciprocal Identities
The trigonometric ratios cosecant, secant, and cotangent are reciprocals of sine, cosine, and tangent, respectively.
csc θ=sin θ 1
sec θ=cos θ 1
cot θ=tan θ 1
Proof
Consider a right triangle with the three sides labeled with respect to an acute angleθ.
Next, the sine, cosine, tangent, cosecant, secant, and cotangent ratios are written.
sin θ=hyp oppcos θ=hyp adjtan θ=adj oppcsc θ=opp hypsec θ=adj hypcot θ=opp adj
The reciprocal of the sine ratio will now be calculated.
sin θ=hyp opp
▼
Solve for sin θ 1
MultEqn
LHS⋅sin θ 1=RHS⋅sin θ 1
1=hyp opp(sin θ 1)
MultEqn
LHS⋅opp hyp=RHS⋅opp hyp
opp hyp=sin θ 1
RearrangeEqn
Rearrange equation
sin θ 1=opp hyp
It has been found that s i n θ 1, which is the reciprocal of sin θ, is equal to opp hyp. By the definition, the cosecant of θ is also the ratio of the lengths of the hypotenuse and the opposite side to ∠θ. Therefore, by the Transitive Property of Equality, s i n θ 1 is equal to csc θ.
⎩⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎧sin θ 1=opp hyp.csc θ=opp hyp⇓csc θ=sin θ 1
By following a similar procedure, the other two identities for secant and cotangent can be proven.
Explore
Finding Reciprocal Identities
If the sine, cosine, and tangent ratios are known, then their reciprocals cosecant, secant, and cotangent can be calculated without too much effort.
LaShay is really good at her favorite subject, Geometry. She has been appointed by Jefferson High's principal to do some tutoring for some of her classmates after school. To do so, she drew a right triangle. She then asked her peers to find all six trigonometric ratios with respect to the marked angleθ.
Help LaShay's classmates find the trigonometric ratios!
{"type":"text","form":{"type":"math","options":{"comparison":"0","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<\/span><\/span>"},"formTextBefore":"<\/span>sin<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["\dfrac{20}{101}","\frac{20}{101}","20\/101"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"0","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<\/span><\/span>"},"formTextBefore":"<\/span>csc<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["\dfrac{101}{20}","\frac{101}{20}","101\/20"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"0","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<\/span><\/span>"},"formTextBefore":"<\/span>cos<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["\dfrac{99}{101}","\frac{99}{101}","99\/101"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<\/span><\/span>"},"formTextBefore":"<\/span>sec<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["101\/99","\dfrac{101}{99}","\frac{101}{99}"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"0","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<\/span><\/span>"},"formTextBefore":"<\/span>tan<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["\dfrac{20}{99}","\frac{20}{99}","20\/99"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"0","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<\/span><\/span>"},"formTextBefore":"<\/span>cot<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["\dfrac{99}{20}","\frac{99}{20}","99\/20"]}}
Hint
Identify the hypotenuse of the right triangle and the opposite and adjacent sides to ∠θ.
Solution
The hypotenuse of the right triangle and the opposite and adjacent sides to ∠θ will be identified.
It can be seen that the hypotenuse is 1 0 1 and the lengths of the opposite and adjacent sides to ∠θ are 2 0 and 9 9, respectively. With this information, the sine, cosine, and tangent ratios can be found.
sin θ cos θ tan θ=1 0 1 2 0=1 0 1 9 9=9 9 2 0
The reciprocals of the above ratios are the cosecant, secant, and cotangent of ∠θ.
sin θ=1 0 1 2 0cos θ=1 0 1 9 9tan θ=9 9 2 0⇒csc θ=2 0 1 0 1⇒sec θ=9 9 1 0 1⇒cot θ=2 0 9 9
Closure
Calculating a Side of a Right Triangle
With the topics learned in this lesson, the challenge presented at the onset can now be solved. Previously, it was learned that the Leaning Tower of Pisa has a tilt of 4 degrees. The hammer dropped by the worker landed 3 meters away from the base of the tower.
With the given information, the vertical distance traveled by the hammer can be calculated. Write the answer to the nearest tenth.
{"type":"text","form":{"type":"math","options":{"comparison":"0","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<\/span><\/span>"},"formTextBefore":"Vertical distance <\/span>=<\/span><\/span><\/span><\/span>","formTextAfter":"meters","answer":{"text":["42.9","42.9 m","42.9 meters","42.9m","42.9meters"]}}
Hint
Draw a right triangle and identify the given information.
Solution
If the tower has a tilt of 4∘, then the acute angle formed by the ground and the tower itself is the difference between 9 0∘ and 4∘.
9 0∘−4∘=8 6∘
A right triangle can be drawn with an acute angle whose measure is 8 6∘. Furthermore, the length of the a d j a c e n t side to this angle is 3 meters, and the length of its opposite side is unknown.
The trigonometric ratio that relates an angle of a right triangle with its opposite and a d j a c e n t sides is the tangent ratio.
tan θ=length of adjacent side to∠θ length of opposite side to∠θ⇓tan 8 6∘=3 x
This equation can be solved for x, which is the vertical distance traveled by the hammer.
tan 8 6∘=3 x
▼
Solve for x
MultEqn
LHS⋅3=RHS⋅3
tan 8 6∘⋅3=x
RearrangeEqn
Rearrange equation
x=tan 8 6∘⋅3
UseCalc
Use a calculator
x=4 2.9 0 1 9 9 8…
RoundDec
Round to 1 decimal place(s)
x≈4 2.9
The distance traveled by the hammer is about 4 2.9 meters.
{{ grade.displayTitle }}
Join the newsletter
News about pedagogy and AI
Continue
By joining the newsletter you agree to our terms and privacy policy
Mathleaks
It's time to change math education at its core. With innovative technology, we're making math easy and meaningful – for everyone. Our goal? To empower new generations to believe in themselves and become the problem-solvers of tomorrow. Welcome to the Math Revolution—are you in?
Follow us for more
We accept
Courses
Courses
Algebra 1
Algebra 2
Financial Literacy
Geometry
Pre-Algebra
Features
MathVizy AI
Textbook Solutions
Classroom Analytics
eCourse Tests+ Create Test
eCourse Worksheets+ Create Worksheet
eCourse Lessons+ Create Lesson
Cookie Preferences
Company
About Us
Help Center
FAQ
Privacy Policy
Terms of Use
News
Pressrum
Get Started
Premium Plans
Download the App
Courses
Algebra 1
Algebra 2
Financial Literacy
Geometry
Pre-Algebra
Textbook Solutions
eCourse Lessons
Cookie Preferences
About Us
Privacy Policy
Terms of Use
FAQ
Premium Plans
Download the App
Join the newsletter
Loading content |
13024 | https://math.stackexchange.com/questions/4218662/expressing-fx-fracx-1x1-as-a-sum-of-an-even-and-odd-function | solution verification - Expressing $f(x) = \frac{x-1}{x+1}$ as a sum of an even and odd function - Mathematics Stack Exchange
Join Mathematics
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Mathematics helpchat
Mathematics Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Expressing f(x)=x−1 x+1 f(x)=x−1 x+1 as a sum of an even and odd function
Ask Question
Asked 4 years, 1 month ago
Modified4 years, 1 month ago
Viewed 263 times
This question shows research effort; it is useful and clear
2
Save this question.
Show activity on this post.
I am trying to write
f(x)=1−x x+1 f(x)=1−x x+1
as a sum of an even and odd function. One solution, though messy, is to use the following derivation:
We work backwards. Let f=f e+f 0 f=f e+f 0, where f e f e is even and f 0 f 0 odd. Then f(x)=f e(x)+f 0(x)f(x)=f e(x)+f 0(x) for all x x in the domain of f f. But then
f(−x)=f e(−x)+f 0(−x)=f e(x)−f 0(x).f(−x)=f e(−x)+f 0(−x)=f e(x)−f 0(x).
Adding, we eliminate f 0(x)f 0(x):
f(x)+f(−x)=2 f e(x),f(x)+f(−x)=2 f e(x),
so
f e(x)=f(x)+f(−x)2.f e(x)=f(x)+f(−x)2.
Now, f 0=f(x)−f e f 0=f(x)−f e. Then:
f 0(x)=f(x)−f(x)+f(−x)2=2 f(x)−f(x)−f(−x)2=f(x)−f(−x)2.f 0(x)=f(x)−f(x)+f(−x)2=2 f(x)−f(x)−f(−x)2=f(x)−f(−x)2.
One easily checks that f e f e is even and f 0 f 0 odd.
Using this derivation, I get:
1−x 1+x=1 2(1−x 1+x+1+x 1−x)+1 2(1−x 1+x−1+x 1−x).1−x 1+x=1 2(1−x 1+x+1+x 1−x)+1 2(1−x 1+x−1+x 1−x).
Assuming I haven't made an algebra mistake, this seems to work, but it's not elegant. I did some research into this and found another proposed solution, but I believe it to be faulty. First, I'll just present it.
1−x 1+x=1−x 1+x⋅1−x 1−x=1−2 x+x 2 1−x 2=x 2+1 1−x 2+2 x 1−x 2.1−x 1+x=1−x 1+x⋅1−x 1−x=1−2 x+x 2 1−x 2=x 2+1 1−x 2+2 x 1−x 2.
One then checks that the first function is even and the second is odd.
My problem with this solution is that it is not valid for every x x in the domain of f f. By definition, the domain of f f is R∖{−1}R∖{−1}. But this solution, in multiplying by 1−x 1−x 1−x 1−x, presupposes that x≠1 x≠1, which does not seem to me to be allowed.
I'm back at square one, then, because the solution I found does not seem in any way elegant or natural, and I'm assuming there is some kind of a trick that I am missing. I tried polynomial long division, partial fractions, and so forth, and nothing, other than the above derivation, brought me any progress.
functions
solution-verification
even-and-odd-functions
Share
Share a link to this question
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this question to receive notifications
edited Aug 7, 2021 at 2:27
José Carlos Santos
442k 346 346 gold badges 299 299 silver badges 500 500 bronze badges
asked Aug 7, 2021 at 2:14
Brad G.Brad G.
2,268 8 8 silver badges 18 18 bronze badges
4
If you simplified your solution, it is exactly the same as the other one you found acat3 –acat3 2021-08-07 02:25:41 +00:00 Commented Aug 7, 2021 at 2:25
1 Your title and body ask slightly different questions.Thomas Andrews –Thomas Andrews 2021-08-07 02:46:08 +00:00 Commented Aug 7, 2021 at 2:46
3 Cosmetic question: should f 0 f 0 really be f o f o?2'5 9'2 –2'5 9'2 2021-08-07 04:34:35 +00:00 Commented Aug 7, 2021 at 4:34
@AmanKushwaha: I believe alex.jordan is suggesting that the numerical subscript f 0 f 0, which appears in the question, should be a letter subscript f o f o instead.robjohn –robjohn♦ 2021-08-07 07:19:53 +00:00 Commented Aug 7, 2021 at 7:19
Add a comment|
3 Answers 3
Sorted by: Reset to default
This answer is useful
6
Save this answer.
Show activity on this post.
In both cases, your solutions are undefined when x=±1 x=±1. And that's natural. After all, f(x)f(x) is undefined when x=−1 x=−1 and, if you express f(x)f(x) as f o(x)+f e(x)f o(x)+f e(x), with f o f o odd and f e f e even, then it is natural that at least one of the functions f o f o and f e f e is undefined when x=−1 x=−1. But then, since f o(−x)=−f o(x)f o(−x)=−f o(x) and f e(−x)=f e(x)f e(−x)=f e(x), if f o(x)f o(x) is undefined when x=−1 x=−1, then it is also undefined when x=1 x=1, and the same thing applies to f e(x)f e(x).
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this answer to receive notifications
edited Aug 7, 2021 at 2:37
answered Aug 7, 2021 at 2:24
José Carlos SantosJosé Carlos Santos
442k 346 346 gold badges 299 299 silver badges 500 500 bronze badges
7
There are just two things I don't understand. First, is my solution incorrect for this reason? If I'm trying to express f f as a sum of two functions, it should be true that f(x)=f e(x)+f 0(x)f(x)=f e(x)+f 0(x) for all x x, or I haven't actually done that, right? Second, why is it natural that both f 0 f 0 and f e f e are undefined when x=−1 x=−1 because f f is not? I can understand why at least one of them is not defined (because, if both were defined, then f f would be defined).Brad G. –Brad G. 2021-08-07 02:31:51 +00:00 Commented Aug 7, 2021 at 2:31
Your solution is correct. There is no solution such that both f e f e and f o f o are defined at −1−1. And, yes, it might well happen that only one of the functions f e f e and f o f o is undefined when x=−1 x=−1. If, say, f f was odd, you could take f e(x)=0 f e(x)=0 (and f o(x)=f(x)f o(x)=f(x), which is defined everywhere.José Carlos Santos –José Carlos Santos 2021-08-07 02:35:09 +00:00 Commented Aug 7, 2021 at 2:35
How is this a proper sum, though? If I want to write f(x)=(g+h)(x)f(x)=(g+h)(x), don't I need to do it for all x x in the domain of f f?Brad G. –Brad G. 2021-08-07 02:36:42 +00:00 Commented Aug 7, 2021 at 2:36
As I explained, there is no solution which is valid for every x x in the domain of f f.José Carlos Santos –José Carlos Santos 2021-08-07 02:38:06 +00:00 Commented Aug 7, 2021 at 2:38
@JoséCarlosSantos why at least one of the functions f o f o and f e f e need to be undefined when x=−1 x=−1? When you expressed f(x)f(x) as f o(x)+f e(x)f o(x)+f e(x) you did that for all x x belonging to the domain of f f only. As such, f(−1)≠f o(−1)+f e(−1)f(−1)≠f o(−1)+f e(−1) so what f o(−1)f o(−1) and f e(−1)f e(−1) are defined.Aman Kushwaha –Aman Kushwaha 2021-08-07 06:26:33 +00:00 Commented Aug 7, 2021 at 6:26
|Show 2 more comments
This answer is useful
1
Save this answer.
Show activity on this post.
You do have a sign error in your derivation. Your calculation
f(x)=1−x x+1⇒f(−x)=1−(−x)(−x)+1=1+x 1−x f(x)=1−x x+1⇒f(−x)=1−(−x)(−x)+1=1+x 1−x
is correct, but one of your symmetric functions is affected: f o d d f o d d should be 2 x x 2−1=−2 x 1−x 2.2 x x 2−1=−2 x 1−x 2. (Interestingly, f(−x)=1 f(x).f(−x)=1 f(x).)
[Also, the function expression in your title does not match the one in your discussion; the even and odd functions for that would be f e v e n=x 2+1 x 2−1 f e v e n=x 2+1 x 2−1 and f o d d=2 x 1−x 2 f o d d=2 x 1−x 2 . I am assuming you want the one under discussion here.]
We can check this by
1+x 2 1−x 2+−2 x 1−x 2=(1−x)2 1−x 2=1−x 1+x.1+x 2 1−x 2+−2 x 1−x 2=(1−x)2 1−x 2=1−x 1+x.
I noticed the error when I initially made a plot of your functions; the graph here shows 1−x 1+x 1−x 1+x in blue, and the even and (corrected) odd functions in green and red, respectively. As José Carlos Santos remarks, we cannot expect an asymmetric function with a vertical asymptote to be "reconstructed" perfectly by symmetric functions, since vertical asymptotes are arranged symmetrically about the y−y−axis in both even and odd functions. So there is actually a "hole" in the domain of f e+f o f e+f o at x=1 x=1 that is not present in the original function. The best that can be managed in such a sum then is to have the limit of the sum be equal to f(1).f(1). Indeed, while the two symmetric functions are not defined at x=1 x=1 , and moreover, the limits of the individual functions do not exist as x x approaches 1,1, we do have
lim x→1[1+x 2 1−x 2+−2 x 1−x 2]=lim x→1 1−x 1+x=0.lim x→1[1+x 2 1−x 2+−2 x 1−x 2]=lim x→1 1−x 1+x=0.
This sort of situation occurs in other kinds of summing of functions where there is a discontinuity in either the "component" functions or in the function to be constructed. An example that comes to mind is when a Fourier series (an infinite sum of continuous trigonometric functions) must "mimic" a jump discontinuity.
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this answer to receive notifications
edited Aug 7, 2021 at 7:43
answered Aug 7, 2021 at 7:06
user882145 user882145
Add a comment|
This answer is useful
1
Save this answer.
Show activity on this post.
Just to add a little more perspective. Others have explained why you should not be surprised that one or both of f o f o and f e f e are undefined at +1+1 as well as −1−1. However, this is not always the case.
Consider the function f(x)=x 2−1 x+1 f(x)=x 2−1 x+1. Like your function, this is defined on R∖{−1}R∖{−1}. However, it can be expressed as the sun of an odd function and an even function which are defined at {+1}{+1} and {−1}{−1}. f o(x)=x f o(x)=x and f e(x)=−1 f e(x)=−1.
In a sense, I have cheated since my function is f(x)=x−1 f(x)=x−1 in disguise except that it is not defined at −1−1. We could extend and clean up my function by saying that it is defined at −1−1 and that the value is −2−2. Borrowing a term that is more commonly used in complex analysis, this is a "removable singularity". We can get a nice function by giving it at a value at the missing point. "Nice" in this context means "continuous".
Your function is different in that the singularity is "essential". We can extend your function by defining a value at −1−1 but we cannot pick a value that will make it nice (continuous). In this case, one or both of f o f o and f e f e will be undefined at −1−1 and also +1+1.
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this answer to receive notifications
edited Aug 7, 2021 at 18:49
answered Aug 7, 2021 at 9:32
badjohnbadjohn
8,864 2 2 gold badges 19 19 silver badges 35 35 bronze badges
Add a comment|
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
functions
solution-verification
even-and-odd-functions
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Report this ad
Related
45How do I divide a function into even and odd sections?
2Is it possible for the sum of even and neither odd nor even function to be even or odd?
20Generalization of even / odd functions
0Prove the value of a function with odd and even function properties
0Is an odd/even function multiplied by i i still an odd/ even function?
4Proof organization: proving the set of functions f:R→R f:R→R is a direct sum of even and odd functions
0Determining if f(x)=x x+1 f(x)=x x+1 is Even or Odd
Hot Network Questions
Gluteus medius inactivity while riding
Clinical-tone story about Earth making people violent
Is it possible that heinous sins result in a hellish life as a person, NOT always animal birth?
What NBA rule caused officials to reset the game clock to 0.3 seconds when a spectator caught the ball with 0.1 seconds left?
RTC battery and VCC switching circuit
Copy command with cs names
An odd question
Childhood book with a girl obsessessed with homonyms who adopts a stray dog but gives it back to its owners
I have a lot of PTO to take, which will make the deadline impossible
With with auto-generated local variables
What is the feature between the Attendant Call and Ground Call push buttons on a B737 overhead panel?
Is encrypting the login keyring necessary if you have full disk encryption?
How long would it take for me to get all the items in Bongo Cat?
Determine which are P-cores/E-cores (Intel CPU)
How to home-make rubber feet stoppers for table legs?
How to locate a leak in an irrigation system?
Does the mind blank spell prevent someone from creating a simulacrum of a creature using wish?
Program that allocates time to tasks based on priority
What meal can come next?
The geologic realities of a massive well out at Sea
Can peaty/boggy/wet/soggy/marshy ground be solid enough to support several tonnes of foot traffic per minute but NOT support a road?
Discussing strategy reduces winning chances of everyone!
Numbers Interpreted in Smallest Valid Base
Non-degeneracy of wedge product in cohomology
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Mathematics
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
13025 | https://www.scribd.com/document/480441817/CE-8-Stability-of-Gravity-Dams | CE 8 Fluid Mechanics Stability of Gravity Dams: Lecturer S. Valle | PDF | Pressure | Dam
Opens in a new window Opens an external website Opens an external website in a new window
This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy
Open navigation menu
Close suggestions Search Search
en Change Language
Upload
Sign in
Sign in
Download free for 30 days
100%(2)100% found this document useful (2 votes)
5K views 16 pages
CE 8 Fluid Mechanics Stability of Gravity Dams: Lecturer S. Valle
The document describes the stability of gravity dams and the different failure modes of sliding and overturning. It provides the forces acting on dams, including the dam's weight, hydrostati…
Full description
Uploaded by
sairin park
AI-enhanced title and description
Go to previous items Go to next items
Download
Save Save CE_8_Stability_of_Gravity_Dams For Later
Share
100%100% found this document useful, undefined
0%, undefined
Print
Embed
Ask AI
Report
Download
Save CE_8_Stability_of_Gravity_Dams For Later
You are on page 1/ 16
Search
Fullscreen
N G A i u o j K g n d b m o n s^t b`o i o t y e a C r b v o t y J b k s
I g n t u r g r^.Q b i i g
adDownload to read ad-free
Crbvoty Jbk
tdg stb`oioty ea tdg jbk os jgpgmjgmt em tdg kbss ea tdg jbk
Aerngs bntomc em tdg Jbk<
d
A
D2
A
D0
y
np
C
y
np
A
Q
W Sgocdt ea tdg Jbk (C) Dyjrestbton Aerngs (AD & AQ) Dyjrestbton Wpioat (W)
adDownload to read ad-free
Kejgs ea Aboiurg ^iojomc
A
D2
A
D0
C A
Q
W a;
ᘄ
M
ennurs wdgm tdgrg os bm um`bibmngj derozemtbi aerng pusdomc tdg jbk aerwbrjs
ή
A
T
; A
D2
A
D0
a ; 8 A
D2
A
D0
; a A
D2
A
D0
;
M A
D2
A
D0
; P
D
ή
A
_
; M + W - C - A
Q
; 8 M ; C + A
Q
W ; P
Q
P
D
;
P
Q
A^siojomc ;
P
Q
= 2.8
P
D
M ketoem
adDownload to read ad-free
Kejgs ea Aboiurg Evgrturmomc
A
D2
A
D0
C A
Q
W
ennurs wdgm tdgrg os b rgsojubi neupig teppiomc tdg jbk aerwbrjs (povet bt tdg teg)
ή
K
teg
; P.K - E.K Evgrturmomc Kekgmt (E.K)<
A
D2
W Pocdtomc Kekgmt (P.K)<
C
A
Q
A
D0
A^siojomc ;
PK
= 2.8
EK
ketoem
adDownload to read ad-free
adDownload to read ad-free
adDownload to read ad-free
adDownload to read ad-free
adDownload to read ad-free
adDownload to read ad-free
adDownload to read ad-free
adDownload to read ad-free
adDownload to read ad-free
adDownload to read ad-free
adDownload to read ad-free
adDownload to read ad-free
Share this document
Share on Facebook, opens a new window
Share on LinkedIn, opens a new window
Share with Email, opens mail client
Copy link
Millions of documents at your fingertips, ad-free Subscribe with a free trial
You might also like
Design of Combined Footing 89% (9) Design of Combined Footing 29 pages
Chapter 10 Relative Equilibrium On Fluids No ratings yet Chapter 10 Relative Equilibrium On Fluids 42 pages
Design of Square Footing 50% (2) Design of Square Footing 13 pages
Bearing Capacity 100% (2) Bearing Capacity 13 pages
11 - Open Rotating Vessels 75% (4) 11 - Open Rotating Vessels 25 pages
HYDRAULICS Day 1 No ratings yet HYDRAULICS Day 1 13 pages
Dams 0% (2) Dams 3 pages
CE 8 ProbSet 1 PDF 0% (1) CE 8 ProbSet 1 PDF 3 pages
Body of The Book - Final No ratings yet Body of The Book - Final 95 pages
Fluid Mechanics & Hydraulics Civil Engineering - PRACTICE PROBLEMS - 2017 INSTRUCTION: Select The Correct Answer For Each of The Following No ratings yet Fluid Mechanics & Hydraulics Civil Engineering - PRACTICE PROBLEMS - 2017 INSTRUCTION: Select The Correct Answer For Each of The Following 36 pages
Stability of Floating Bodies 0% (3) Stability of Floating Bodies 12 pages
Fluid Mechanics Lesson 6 0% (3) Fluid Mechanics Lesson 6 1 page
Engineering Methodology 100% (1) Engineering Methodology 41 pages
Data Sheet - Pex 240DW No ratings yet Data Sheet - Pex 240DW 8 pages
ASCO 8316 Three-Way Solenoid Valves No ratings yet ASCO 8316 Three-Way Solenoid Valves 2 pages
Gravity Dam Stability Guide No ratings yet Gravity Dam Stability Guide 5 pages
Chapter - 3 Composite Manufacturing: Manufacturing Is A Very Broad Discipline and Encompasses Several Processes Such As No ratings yet Chapter - 3 Composite Manufacturing: Manufacturing Is A Very Broad Discipline and Encompasses Several Processes Such As 32 pages
Matrix-Vector Multiplication Using Falk Scheme 100% (1) Matrix-Vector Multiplication Using Falk Scheme 2 pages
Zelio Analog Voltage/Current Converter No ratings yet Zelio Analog Voltage/Current Converter 3 pages
Physical Science Paper 2 November 2009 No ratings yet Physical Science Paper 2 November 2009 10 pages
P6 3B Bernardo Kathryne 0% (1) P6 3B Bernardo Kathryne 4 pages
Mwe Unit Wise Preparation Plan: UNIT-1 MICROWAVE TUBES ..5 Topics No ratings yet Mwe Unit Wise Preparation Plan: UNIT-1 MICROWAVE TUBES ..5 Topics 3 pages
Design of Rectangular Footing 100% (2) Design of Rectangular Footing 9 pages
Faye Design No ratings yet Faye Design 13 pages
Advanced Separation Processes Notes No ratings yet Advanced Separation Processes Notes 8 pages
Minors Cofactors and Adjoint of A Matrix No ratings yet Minors Cofactors and Adjoint of A Matrix 6 pages
Matrix-Matrix Multiplication Using Falk Scheme No ratings yet Matrix-Matrix Multiplication Using Falk Scheme 2 pages
Kafuter K 704 100% (1) Kafuter K 704 2 pages
Matrix Basics and Definitions No ratings yet Matrix Basics and Definitions 8 pages
Root Finding Methods: Bisection: Solutions of Nonlinear Equations No ratings yet Root Finding Methods: Bisection: Solutions of Nonlinear Equations 8 pages
Deutz Diesel Generator SD46.5 Specs No ratings yet Deutz Diesel Generator SD46.5 Specs 4 pages
Piezoelectric Harvesting Energy No ratings yet Piezoelectric Harvesting Energy 23 pages
Equations of Motion in Terms of Radial and Transverse Components No ratings yet Equations of Motion in Terms of Radial and Transverse Components 5 pages
Abdoul Salam Issiaka Ibrahim No ratings yet Abdoul Salam Issiaka Ibrahim 70 pages
Determinant of A Matrix PDF No ratings yet Determinant of A Matrix PDF 6 pages
AvatarON E8331M DG G3 2 No ratings yet AvatarON E8331M DG G3 2 2 pages
Hoop Tension and Dam Safety Analysis 0% (1) Hoop Tension and Dam Safety Analysis 17 pages
Cramer S Rule - Linear System Two Unknowns No ratings yet Cramer S Rule - Linear System Two Unknowns 2 pages
A Adjoint If: A Where AA I Identity of A No ratings yet A Adjoint If: A Where AA I Identity of A 5 pages
02 Three Phase System No ratings yet 02 Three Phase System 8 pages
2 2-Analysisofgravitydam No ratings yet 2 2-Analysisofgravitydam 12 pages
Gaussian Elimination No ratings yet Gaussian Elimination 6 pages
Matrix Scalar Multiplication Guide No ratings yet Matrix Scalar Multiplication Guide 2 pages
Gauss Jordan Elimination Method No ratings yet Gauss Jordan Elimination Method 4 pages
Matrix Multiplication Explained No ratings yet Matrix Multiplication Explained 3 pages
Module 2 Sptopics PDF 0% (1) Module 2 Sptopics PDF 60 pages
Matrices and Its Definition No ratings yet Matrices and Its Definition 2 pages
Matrix Types Explained No ratings yet Matrix Types Explained 3 pages
Algebra Problem Solving Guide No ratings yet Algebra Problem Solving Guide 2 pages
Untitled No ratings yet Untitled 16 pages
Volume No ratings yet Volume 36 pages
P5 3B Gutierrez JohnPaul No ratings yet P5 3B Gutierrez JohnPaul 5 pages
Improving Wear Resistance Via Hardfacing of Cultivator Shovel No ratings yet Improving Wear Resistance Via Hardfacing of Cultivator Shovel 9 pages
Experiment 4 No ratings yet Experiment 4 5 pages
Laser Triggering of High Voltages No ratings yet Laser Triggering of High Voltages 119 pages
Lesson 4 No ratings yet Lesson 4 3 pages
Transpose of A Matrix PDF No ratings yet Transpose of A Matrix PDF 1 page
NEWKer 302 Manual V3.2 No ratings yet NEWKer 302 Manual V3.2 49 pages
Rethinking The Pivot Point (In English) No ratings yet Rethinking The Pivot Point (In English) 5 pages
Problem Set 10 Partition of Land1 PDF No ratings yet Problem Set 10 Partition of Land1 PDF 14 pages
Problem Set 10 Partition of Land1 PDF No ratings yet Problem Set 10 Partition of Land1 PDF 14 pages
4 Module Hydraulics Gravity Dam1 100% (1) 4 Module Hydraulics Gravity Dam1 27 pages
Stability of Floating Bodies No ratings yet Stability of Floating Bodies 15 pages
CE324 Activity1 CASUGAMED 100% (1) CE324 Activity1 CASUGAMED 6 pages
Problem Set No 5 PDF No ratings yet Problem Set No 5 PDF 14 pages
9.1 Relative Equilibrium (Rectilinear Motion) 100% (2) 9.1 Relative Equilibrium (Rectilinear Motion) 16 pages
P7 3B Bernardo Kathryne No ratings yet P7 3B Bernardo Kathryne 4 pages
Matrix Types Explained No ratings yet Matrix Types Explained 22 pages
MODULE 5.6 Orifice 100% (2) MODULE 5.6 Orifice 19 pages
ANS: (2.59807m/s2 Horizontal) (1.5m/s2 Vertical) (12.93725 Degree Angle That The Water Surface Makes With The Horizontal) No ratings yet ANS: (2.59807m/s2 Horizontal) (1.5m/s2 Vertical) (12.93725 Degree Angle That The Water Surface Makes With The Horizontal) 5 pages
Damsdocx PDF Free No ratings yet Damsdocx PDF Free 9 pages
Lecture 3 Buoyancy and Stability 100% (2) Lecture 3 Buoyancy and Stability 21 pages
Hydraulics Final No ratings yet Hydraulics Final 9 pages
Gravity Dam Stability Analysis 100% (1) Gravity Dam Stability Analysis 9 pages
Atmospheric Pressure Calculations 0% (1) Atmospheric Pressure Calculations 9 pages
Rotating Vessels Edited 100% (2) Rotating Vessels Edited 30 pages
Dams No ratings yet Dams 5 pages
Quickie 0% (1) Quickie 34 pages
EVALUATION 2 Solutions 100% (1) EVALUATION 2 Solutions 17 pages
Chapter 6 No ratings yet Chapter 6 32 pages
Hydrau Geotech Focusproblems3 2019 100% (1) Hydrau Geotech Focusproblems3 2019 7 pages
Fluid Mechanics Basics No ratings yet Fluid Mechanics Basics 52 pages
Fluid Statics 3 No ratings yet Fluid Statics 3 103 pages
Hydaulics Engineering Sample Problem No ratings yet Hydaulics Engineering Sample Problem 5 pages
Andre Migaea Bsce 3a Practice Problem No.1 100% (1) Andre Migaea Bsce 3a Practice Problem No.1 5 pages
Orifice Flow Coefficients Guide No ratings yet Orifice Flow Coefficients Guide 36 pages
Toaz - Info Dams PR No ratings yet Toaz - Info Dams PR 3 pages
Kolli-Liste Englisch No ratings yet Kolli-Liste Englisch 18 pages
Agacita John Paul No ratings yet Agacita John Paul 10 pages
Sie Sa Mccb-3va Conversion No ratings yet Sie Sa Mccb-3va Conversion 104 pages
Enhancement Review Fluid Mech. Set C No ratings yet Enhancement Review Fluid Mech. Set C 10 pages
Horizontal Motion : Rectilinear Translation (Moving Vessel) No ratings yet Horizontal Motion : Rectilinear Translation (Moving Vessel) 4 pages
Fluid Equilibrium for Engineering Students No ratings yet Fluid Equilibrium for Engineering Students 14 pages
LL Determination From Casagrande and Fall Cone Results Example No ratings yet LL Determination From Casagrande and Fall Cone Results Example 3 pages
Fluid Flow 5f155f8d95906 No ratings yet Fluid Flow 5f155f8d95906 13 pages
Hydraulics Chapter 5 Principles of Hydrodynamics Exercise Problems No ratings yet Hydraulics Chapter 5 Principles of Hydrodynamics Exercise Problems 12 pages
Chapter Six Orifices Weirs and Tubes Practice Problems No ratings yet Chapter Six Orifices Weirs and Tubes Practice Problems 14 pages
ACFrOgAwAlHRbAQ1kspyRR0wCaTP-ZOZDqFTOOHLLPDIYhp6sybMn8OW48XXCf63pqnjR1CliVQrz8n3e5w81 YbJMjPDM1xpBnLrsnX61i68Y6K5zQOLSnjJQBCW2xIbuWD9WrW ZhG6oosgWyg No ratings yet ACFrOgAwAlHRbAQ1kspyRR0wCaTP-ZOZDqFTOOHLLPDIYhp6sybMn8OW48XXCf63pqnjR1CliVQrz8n3e5w81 YbJMjPDM1xpBnLrsnX61i68Y6K5zQOLSnjJQBCW2xIbuWD9WrW ZhG6oosgWyg 6 pages
Civil Engineering Exam Review: Hydraulics & Geotechnics No ratings yet Civil Engineering Exam Review: Hydraulics & Geotechnics 5 pages
Hydraulics Analysis of Dam Assignment Solution No ratings yet Hydraulics Analysis of Dam Assignment Solution 3 pages
MODULE 5.1-5.5 Coefficients To Tubes No ratings yet MODULE 5.1-5.5 Coefficients To Tubes 18 pages
MODULE 4 Hydraulics Revised No ratings yet MODULE 4 Hydraulics Revised 30 pages
External Combustion No ratings yet External Combustion 9 pages
Hydrostatic Pressure On Surfaces: Engr. Angelica A. Losares No ratings yet Hydrostatic Pressure On Surfaces: Engr. Angelica A. Losares 26 pages
Method Statement For ICCP System Installation Presentation-04 100% (1) Method Statement For ICCP System Installation Presentation-04 53 pages
Gis Located Above B) 100% (1) Gis Located Above B) 10 pages
Chapter 4 To 6 No ratings yet Chapter 4 To 6 101 pages
Hydrostatic Force on Surfaces No ratings yet Hydrostatic Force on Surfaces 18 pages
Open Channel No ratings yet Open Channel 53 pages
Reactivity Practice Test No ratings yet Reactivity Practice Test 3 pages
Set 7 Fluid Properties and Hydrostatic Force No ratings yet Set 7 Fluid Properties and Hydrostatic Force 12 pages
Answered A Vertical Retaining Wall 3m Supports A... Bartleby... No ratings yet Answered A Vertical Retaining Wall 3m Supports A... Bartleby... 3 pages
MECH 105 Chapter 4 No ratings yet MECH 105 Chapter 4 91 pages
City Test-01: Arjuna JEE Phase-2 No ratings yet City Test-01: Arjuna JEE Phase-2 9 pages
Ri16rhb (O&m Manual) No ratings yet Ri16rhb (O&m Manual) 6 pages
ad
Footer menu
Back to top
About
About Scribd, Inc.
Everand: Ebooks & Audiobooks
Slideshare
Join our team!
Contact us
Support
Help / FAQ
Accessibility
Purchase help
AdChoices
Legal
Terms
Privacy
Copyright
Cookie Preferences
Do not sell or share my personal information
Social
Instagram Instagram
Facebook Facebook
Pinterest Pinterest
Get our free apps
About
About Scribd, Inc.
Everand: Ebooks & Audiobooks
Slideshare
Join our team!
Contact us
Legal
Terms
Privacy
Copyright
Cookie Preferences
Do not sell or share my personal information
Support
Help / FAQ
Accessibility
Purchase help
AdChoices
Social
Instagram Instagram
Facebook Facebook
Pinterest Pinterest
Get our free apps
Documents
Language:
English
Copyright © 2025 Scribd Inc.
We take content rights seriously. Learn more in our FAQs or report infringement here.
We take content rights seriously. Learn more in our FAQs or report infringement here.
Language:
English
Copyright © 2025 Scribd Inc.
576648e32a3d8b82ca71961b7a986505
scribd.scribd.scribd.scribd.scribd.scribd.scribd.scribd.scribd.scribd.scribd.scribd.scribd.scribd.scribd.scribd. |
13026 | https://x-engineer.org/calculate-tangent-function/ | How to calculate the tangent of a function – x-engineer.org
Typesetting math: 100%
Discover more
Math
Science
Mathematics
solid mechanics modeling software
Hybrid cars
OBD scanner tool buy
Engineering simulation software
Calculus textbook
Scilab programming services
Automotive parts
Menu
Mathematics and Science
Mathematics
Arithmetics
Algebra
Trigonometry
Advanced Mathematics
Statistics
Calculus I
Calculus II
Differential Equations
Numerical Methods
Chemistry
General Chemistry
Physics
Solid Mechanics
Thermodynamics
Electricity & Magnetism
Fluid Mechanics
General Engineering
Programming Languages
C
Scilab
Electronics
Circuits
Embedded Systems
Bus Communication
Microcontrollers
Signals & Systems
Control Systems
Signal Processing
System Identification
Modeling & Simulation
Systems Modeling
Model Based Design
Mechatronics
Machine Dynamics
Mechanisms
Sensors
CAD & CAE
Xcos
Automotive Engineering
Internal Combustion Engines
ICE Components & Systems
Diagnostics
Performance
Drivetrain
Coupling Devices
Transmissions
AWD Technologies
Chassis
Vehicle Dynamics
Vehicle
Hybrid
Electric Vehicles
Quiz !
News
Projects
Home
Mathematics and Science
Advanced Mathematics
Calculus I
How to calculate the tangent of a function
How to calculate the tangent of a function
A tangent line to a function at a point is a line that is in contact with the graphical representation of the function only in that particular point.
The tangent line is only touching the function (graph) in the specified point. In other words the tangent line is barely in contact with function (graph).
The line which is crossing (intersecting) a function graph it’s called a secant line.
Image: Tangent and secant to a function graph
In the example above the red line is the tangent. It’s tangent to the f(x) function in the point P(x 1, y 1). The blue line is the secant and as you can see it’s crossing the function f(x) in two points.
Given a function f(x) and a point P 1(x 1, y 1), how do we calculate the tangent? Finding the tangent means finding the equation of the line which is tangent to the function f(x) in the point P 1(x 1, y 1).
Example: Find the tangent of the function f(x) defined below, in the point x 1.
f(x)=2−x 3,x 1=1
Step 1: Calculate the (x, y) coordinates of the tangent point
The first thing to do is to evaluate the function in the tangent point x 1.
f(1)=2−1 3=1
We know that:
y 1=f(x 1)
This means that the result of the function evaluation in the point x 1 is the y 1 coordinate. We now have defined the point P 1(1, 1). Let’s plot the function f(x) for x = 0 … 2 and the point P 1 with x 1 and y 1 coordinates.
Image: Cubic function graph with tangent point
Step 2: Define two points P 0 and P 2, on the left and right of P 1.
We’ll chose a point P 0 at x 0 = 0.6 and P 2 at x 2 = 1.4. These point must be equally distributed on the left and right of the P 1 tangent point.
Step 3. Calculate the y coordinates of the P 0 and P 2 points
First let’s calculate the y coordinate of the P 0 point.
y 0 f(0.6)y 0=f(x 0)=2−0.6 3=1.784=1.784
We now have defined the point P 0 (0.6, 1.784).
Now we calculate the y coordinate of the P 2 point:
y 2 f(1.4)y 2=f(x 2)=2−1.4 3=−0.744=−0.744
We now have defined the point P 2 (1.4,-0.744).
Let’s now update our function graph with both P 0 and P 2 points and their coordinates.
Image: Cubic function graph with tangent point and two offset points
Step 4. Calculate the slopes of the lines defined between the points P 0-P 1 and P 1-P 2
Having two points defined we can easily draw a straight line between them. Also with the coordinates of the two points we can calculate the slope of the line.
In our case we’ll have two lines, one between point P 0 and P 1 and the second between P 1 and P 2. For both lines we can calculate their slope.
The usual notation of the slope of a line is the letter m.
For the first line, the slope m 0 is calculated as:
m 0 m 0=y 1−y 0 x 1−x 0=1−1.784 1−0.6=−1.96=−1.96
For the second line, the slope m 2 is calculated as:
m 2 m 2=y 2−y 1 x 2−x 1=−0.744−1 1.4−1=−4.36=−4.36
Below is the graphical representation of the two lines and the f(x) function. Both lines are secant to our f(x) function.
Image: Cubic function graph with tangent point and two secant lines
Step 5. Calculate the slope of the line tangent in the point P 1 (1, 1)
The slope of the line tangent in the point P 1 will be the arithmetic mean of the slopes of the two secant lines. This method of calculation is possible because we have chosen the x 0 and x 2 points at equal distance from x 1.
m 1 m 1=m 0+m 2 2=−1.96−4.36 2=−3.16=−3.16
Step 6 (final step). Calculate the equation of the tangent line in the point P 1.
Now that we know the coordinates of the tangent point P1 and the value of the slope, we can easily calculate the equation of the tangent line:
y(x)y(x)=y 1+m 1⋅(x–x 1)=1–3.16⋅(x–1)
Having the tangent line equation we can plot on the same graph the function f(x) and the tangent line y(x).
Image: Cubic function graph with tangent point and tangent line
As you can see the tangent line is “touching” the function f(x) in the point P 1 (1, 1).
Using the same method we can calculate the tangent line in the point (1, 1) for the following functions:
f(x)=x 2+2⋅x+1
Image: Quadratic function graph with tangent point and tangent line
f(x)=10⋅2⋅x−−−−√+3⋅x+2
Image: Square root function graph with tangent point and tangent line
Apply the same method to your function and let us now the result!
For any questions or observations regarding this tutorial please use the comment form below.
Don’t forget to Like, Share and Subscribe!
Next Article
One Response
Kerri
Thank you so much! Guys, if you are reading this without using the article, don’t X out of this. This is the way I wish my teacher taught!
Leave a Reply Cancel Reply
Discover more
Science
Mathematics
Math
statistics software purchase options
solid mechanics modeling software
Sensor modules
electrical circuits kit buy
Engineering courses online
×
Recent Comments
Istehque Khan on Automated Manual Transmission (AMT)
dam on Bisection method for root finding
naman on Air fuel ratio
Ali HAMMOUCHE on Modeling and simulation of nonlinear magnetic hysteresis
Marc Cunningham on Discretizing a transfer function from s-domain (continuous) to z-domain (discrete)
Patreon
Recent Posts
Specific Weight of Water
OBD Diagnostic Service (Mode) $04 – Clear/reset emission-related diagnostic information
OBD Diagnostic Service (Mode) $03 – Request emission-related diagnostic trouble codes (DTCs)
OBD Diagnostic Service (Mode) $02 – Request Powertrain Freeze Frame Data
Automotive Diagnostic Standards
Categories
Categories
Pages
About
Contact
Cookie Policy
Privacy Policy
Terms and Conditions
x-engineer.org
Engineering Tutorials
Copyright © 2025 x-engineer.org
Powered by Ubuntu and Scilab
Ad Blocker Detected
Dear user, Our website provides free and high quality content by displaying ads to our visitors. Please support us by disabling your Ad blocker for our site. Thank you!
Refresh |
13027 | https://pl.khanacademy.org/science/ap-physics-1/ap-mechanical-waves-and-sound/introduction-to-sound-waves-ap/a/intro-to-sound-ap1 | Use of cookies
Cookies are small files placed on your device that collect information when you use Khan Academy. Strictly necessary cookies are used to make our site work and are required. Other types of cookies are used to improve your experience, to analyze how Khan Academy is used, and to market our service. You can allow or disallow these other cookies by checking or unchecking the boxes below. You can learn more in our cookie policy
Privacy Preference Center
When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
More information
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
Certain cookies and other technologies are essential in order to enable our Service to provide the features you have requested, such as making it possible for you to access our product and information related to your account.
For example, each time you log into our Service, a Strictly Necessary Cookie authenticates that it is you logging in and allows you to use the Service without having to re-enter your password when you visit a new page or new unit during your browsing session.
Functional Cookies
These cookies provide you with a more tailored experience and allow you to make certain selections on our Service. For example, these cookies store information such as your preferred language and website preferences.
Targeting Cookies
These cookies are used on a limited basis, only on pages directed to adults (teachers, donors, or parents). We use these cookies to inform our own digital marketing and help us connect with people who are interested in our Service and our mission.
We do not use cookies to serve third party ads on our Service.
Performance Cookies
These cookies and other technologies allow us to understand how you interact with our Service (e.g., how often you use our Service, where you are accessing the Service from and the content that you’re interacting with). Analytic cookies enable us to support and improve how our Service operates.
For example, we use Google Analytics cookies to help us measure traffic and usage trends for the Service, and to understand more about the demographics of our users.
We also may use web beacons to gauge the effectiveness of certain communications and the effectiveness of our marketing campaigns via HTML emails.
Cookie List
Consent Leg.Interest
label
label
label |
13028 | https://jamanetwork.com/journals/jamaophthalmology/fullarticle/262606 | Parafoveal Telangiectasia in Association With CREST Syndrome | JAMA Ophthalmology | JAMA Network
[Skip to Navigation]
Our website uses cookies to enhance your experience. By continuing to use our site, or clicking "Continue," you are agreeing to our Cookie Policy|Continue
Navigation
HomeIssuesMultimediaFor Authors
JAMA+
AIWomen's HealthLearn More
JOURNALS
JAMAJAMA Network OpenJAMA CardiologyJAMA DermatologyJAMA Health ForumJAMA Internal MedicineJAMA NeurologyJAMA OncologyJAMA OphthalmologyJAMA Otolaryngology–Head & Neck SurgeryJAMA PediatricsJAMA PsychiatryJAMA SurgeryArchives of Neurology & Psychiatry (1919-1959)
JAMA NETWORK CMESUBSCRIBEJOBSINSTITUTIONS / LIBRARIESREPRINTS & PERMISSIONS
Search
Individual Sign In
Sign inCreate an Account
Access through your institution
Sign In
Search All
Search All
JAMA
JAMA Network Open
JAMA Cardiology
JAMA Dermatology
JAMA Forum Archive
JAMA Health Forum
JAMA Internal Medicine
JAMA Neurology
JAMA Oncology
JAMA Ophthalmology
JAMA Otolaryngology–Head & Neck Surgery
JAMA Pediatrics
JAMA Psychiatry
JAMA Surgery
Archives of Neurology & Psychiatry
Type text and use arrow keys to navigate suggestions
HomeIssuesMultimediaFor Authors
Home
JAMA Ophthalmology
Vol. 116, No. 6
Sections
Close
Top of Article
Report of a CaseCommentArticle InformationReferences
PDF
Share
Close
LinkedInXWhatsAppFacebookThreadsBlueskyWeChatCopy URLEmail
Case Reports and Small Case Series
Parafoveal Telangiectasia in Association With CREST Syndrome
Brad Proctor, BSc; Tom Chang, MD, FRCSC; Dawn Hay, RN
Article Information
Cite This### Citation
Proctor B, Chang T, Hay D. Parafoveal Telangiectasia in Association With CREST Syndrome. Arch Ophthalmol. 1998;116(6):814–815. doi:10.1001/archopht.116.6.814
Manage citations:
Select Format Download citation Copy citation Close
Permissions
View Metrics
Arch Ophthalmol
Published Online: June 1998
1998;116;(6):814-815. doi:10.1001/archopht.116.6.814
related icon Related Articlesmultimedia icon Mediafigure icon Figures
Close
More for You
Bilateral Infusion Pump Implants as Therapy for Refractory Corneal Ulcers in a Patient With CREST SyndromeJAMA Ophthalmology Review July 14, 2008
Surgical Debridement of Painful Fingertip Calcinosis Cutis in CREST SyndromeJAMA Dermatology Research February 1, 2009
Asymptomatic Symmetrical Telangiectasia on Lower ExtremitiesJAMA Dermatology Case Report June 18, 2025
Close
Close
Figure 1.
View LargeDownload
(opens in new tab)
Go to Figure in Article
Demonstration of parafoveal telangiectatic changes in the right (left) and left (right) eye.
Figure 2.
View LargeDownload
(opens in new tab)
Go to Figure in Article
Left, Midframe fluorescein angiography of right macula. Note evidence of leakage from telangiectatic vessels in parafoveal location in addition to enlargement of foveal avascular zone. Right, Midframe fluorescein angiography of left macula demonstrating staining in the temporal hemifovea.
Parafoveal telangiectasia has been described as a developmental or acquired vascular anomaly of the foveal avascular zone characterized by irregular capillary dilatation, serous exudation, and macular edema.1 The acquired form of this anomaly has been termed idiopathic juxtafoveal retinal telangiectasia and its etiology has not been established.2 Gass has divided this disorder into 3 groups based on ease of visibility of the telangiectatic vessels, presence of retinal exudation, and degree of capillary occlusion.1 We present a case that, to the best of our knowledge, is the first description of parafoveal telangiectasia seen in association with the syndrome of c alcinosis cutis, R aynaud phenomenon, e sophageal dysmotility, s clerodactyly, and t elangiectasia (CREST syndrome).
Report of a Case
In 1986, a 61-year-old white woman was seen by her general practitioner with symptoms indicative of the Raynaud phenomenon affecting both hands. Cutaneous erythema and telangiectatic vessels were observed and the facial skin had a foreshortened appearance with resulting tightness. From these features a diagnosis of scleroderma was established. Approximately 4 years after her initial diagnosis, she developed gastric motility symptoms consisting of dyspepsia and dysphagia. In January 1997, she complained of decreasing vision in her right eye. Examination by her ophthalmologist revealed macular edema and a retinal consultation was obtained. Examination at that time disclosed a visual acuity of 20/200 OD and 20/20 OS. Slitlamp examination results and intraocular pressure were unremarkable. Dilated fundus examination revealed telangiectatic vessels in both maculae with mild retinal thickening seen in the temporal parafoveal region (Figure 1). No evidence of diabetic retinopathy was seen. Fluorescein angiography disclosed a characteristic appearance of late staining involving the temporal hemifovea of both eyes, while the right macula displayed evidence of marked enlargement of the foveal avascular zone with an irregular "moth-eaten" appearance (Figure 2). No intraretinal crystals were noted. On general examination, dermal characteristics of scleroderma were noted on her face, extremities, and chest; sausage-like thickening of her fingers was also observed.
Figure 1.
View LargeDownload
(opens in new tab)
Go to Figure in Article
Demonstration of parafoveal telangiectatic changes in the right (left) and left (right) eye.
Figure 2.
View LargeDownload
(opens in new tab)
Go to Figure in Article
Left, Midframe fluorescein angiography of right macula. Note evidence of leakage from telangiectatic vessels in parafoveal location in addition to enlargement of foveal avascular zone. Right, Midframe fluorescein angiography of left macula demonstrating staining in the temporal hemifovea.
Comment
Although relatively uncommon, CREST syndrome consists of easily identifiable symptoms. Systemic vascular changes are responsible for the telangiectatic vessels, which seem to affect several systems within the body. These vascular abnormalities have been noted in the skin and gastric mucosa,3 but to date have not been identified with retinal findings. However, it is reasonable to suggest that such widespread vascular change could present itself in the eye. Our examination of the patient's retina and the discovery of parafoveal telangiectasia, in the absence of diabetic retinopathy or other causative factors, supports this argument and is important in identifying a potential systemic association. This finding is an important consideration for patients with scleroderma, who may also be seen with unexplained loss of vision.
According to the Gass classification,1 our patient would be categorized as group 3A. Similar cases have been described in the literature with systemic associations different from ours. Grand et al2 have identified a hereditary syndrome in which retinal telangiectasia with capillary nonperfusion is associated with frontoparietal lobe pseudotumor. The histopathologic features of the central nervous system lesions included unusual vasculopathy and necrosis of white matter.2 Lim and Bressler4 have reported a case of group 3A parafoveal telangiectasia associated with marked peripheral ischemia. Our patient did not display central nervous system dysfunction or peripheral retinal ischemia.
The prescence of both juxtafoveal telangiectasia and CREST syndrome in our patient suggests a possible common pathophysiologic basis for the 2 diseases. Investigation of more patients will assist in establishing this association.
Corresponding author: Tom Chang, MD, FRCSC, Section D, 2550 Willow St, Vancouver, British Columbia, Canada V5Z 3N9.
References
1.
Gass JDOyakawa RT Idiopathic juxtafoveal retinal telangiectasis.Arch Ophthalmol. 1982;100769-780
ArticleGoogle ScholarCrossref
2.
Grand MGKaine JFulling K et al.Cerebroretinal vasculopathy; a new hereditary syndrome.Ophthalmology. 1988;95649-659Google ScholarCrossref
3.
Binford R CRST syndrome with gastrointestinal bleeding.Arch Dermatol. 1968;97603-604Google ScholarCrossref
4.
Lim JIBressler NM Atypical parafoveal telangiectasis with subsequent anterior and posterior segment neovascularization.Retina. 1992;12351-354Google ScholarCrossref
More for You
Research Self-Supervised Learning for Improved OCT Detection of Macular Telangiectasia Type 2February 8, 2024
Case Report is learning Unilateral Exudative Retinal Lesion in a Previously Healthy Female TeenagerJuly 27, 2023
Case Report is learning Retinal Aneurysms in Myopic Traction MaculopathyApril 20, 2023
View Full TextDownload PDF
Button To Top
X
JAMA Ophthalmology
Content
HomeNew OnlineCurrent Issue
Podcast
JAMA Ophthalmology Author Interviews
Journal Information
For AuthorsEditors & PublishersRSSContact Us
JN Learning / CMEStoreAppsJobsInstitutionsReprints & Permissions
Subscribe
Go
JAMA Network
Publications
JAMA Network HomeOur Publications
Sites
Art and Images in PsychiatryEvidence-Based Medicine: An Oral HistoryFishbein FellowshipGenomics and Precision HealthJAMA Forum ArchiveTopics and CollectionsWar and Health
USPSTF Recommendation Statements
Primary Care Behavioral Counseling Interventions to Support BreastfeedingScreening for Food InsecurityScreening for Osteoporosis to Prevent FracturesScreening and Supplementation for Iron Deficiency and Iron Deficiency Anemia During PregnancyInterventions for High Body Mass Index in Children and AdolescentsInterventions to Prevent Falls in Community-Dwelling Older AdultsScreening for Breast CancerPrimary Care Interventions to Prevent Child MaltreatmentScreening for Speech and Language Delay and Disorders in Children
Information
For AuthorsFor Institutions & LibrariansFor AdvertisersFor Subscription AgentsFor Employers & Job SeekersFor the MediaEquity, Diversity, InclusionOnline Commenting PolicyPublic Access and Open Access PolicyStatement on Potentially Offensive Content
JAMA Network Products
AMA Manual of StyleAMA Style InsiderJAMA Network CMEJAMAevidencePeer Review Congress
Help
Subscriptions & RenewalsManage EmailsUpdate My AddressSupport CenterMy Account
JAMA Career Center
Physician Job Listings
© 2025 American Medical Association. All rights reserved, including those for text and data mining, AI training, and similar technologies.Terms of Use|Privacy Policy|Accessibility Statement|Cookie Settings
.
×
Access your subscriptions
Sign in | personal account
Access through your institution
Add or change institution
Free access to newly published articles
Create a free personal account
To register for email alerts, access free PDF, and more
Purchase access
Subscribe to journal
Get full journal access for 1 year
Buy article
Get unlimited access and a printable PDF ($45.00)—
Sign in or create a free account
Rent article
Rent this article from DeepDyve
Close
Access your subscriptions
Sign in | personal account
Access through your institution
Add or change institution
Free access to newly published articles
Create a free personal account
To register for email alerts, access free PDF, and more
Purchase access
Subscribe to journal
Get full journal access for 1 year
Buy article
Get unlimited access and a printable PDF ($45.00)—
Sign in or create a free account
Rent article
Rent this article from DeepDyve
Close
Sign in to access free PDF
Sign in | personal account
Access through your institution
Add or change institution
Free access to newly published articles
Create a free personal account
To register for email alerts, access free PDF, and more
Close
Save your search
Sign in | personal account
Free access to newly published articles
Create a free personal account
To register for email alerts, access free PDF, and more
Purchase access
Subscribe now
Close
Customize your interests
Sign in | personal account
Free access to newly published articles
Create a free personal account
To register for email alerts, access free PDF, and more
Create a personal account or sign in to:
Register for email alerts with links to free full-text articles
Access PDFs of free articles
Manage your interests
Save searches and receive search alerts
Privacy Policy
Close
Make a comment
Sign in | personal account
Free access to newly published articles
Create a free personal account
To register for email alerts, access free PDF, and more
Create a personal account or sign in to:
Register for email alerts with links to free full-text articles
Access PDFs of free articles
Manage your interests
Save searches and receive search alerts
Privacy Policy
Close
We use cookies and other technologies to collect information about your use of our websites and online apps. Some of these cannot be disabled. Unless you reject non-necessary cookies, we may also share your information with third-party advertising and analytics partners who may serve you with targeted ads. . To learn more about our practices, visit our Privacy Policy.
Cookie Settings
We have recognized your opt-out signal. To learn more about changing your opt-in signal preferences, please visit globalprivacycontrol.org
Cookie Settings
We use cookies and other technologies to collect information about your use of our websites and online apps. Some of these cannot be disabled. Please review and "Confirm Your Choices". Depending on your choices, we may also share your information with third-party advertising and analytics partners who may serve you with targeted ads. To learn more about our practices, visit our
Privacy Policy.
Allow All
Manage Consent Preferences
Targeted Advertising
[x] Targeted Advertising
We use third party technologies for analytics, personalization and marketing purposes. These technologies may be used by us or our partners to personalize the site or deliver relevant marketing to you on third party sites.
Strictly Necessary Cookies
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work.
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Confirm My Choices |
13029 | https://www.organic-chemistry.org/namedreactions/jones-oxidation.shtm | Jones Oxidation
Organic Chemistry Portal
Reactions>>Name Reactions
Further Information
Literature
Related Reactions
Corey-Kim Oxidation
Dess-Martin Oxidation
Swern Oxidation
Synthesis of aldehydes
Synthesis of carboxylic acids
Synthesis of ketones
Jones Oxidation
The Jones Oxidation allows a relatively inexpensive conversion of secondary alcohols to ketones and of most primary alcohols to carboxylic acids. The oxidation of primary allylic and benzylic alcohols gives aldehydes. Jones described for the first time a conveniently and safe procedure for a chromium (VI)-based oxidation, that paved the way for some further developments such as Collins Reaction and pyridinium dichromate, which also enabled the oxidation of primary alcohols to aldehydes.
Mechanism of the Jones Oxidation
The Jones Reagent is a mixture of chromic trioxide or sodium dichromate in diluted sulfuric acid, which forms chromic acid in situ.
The alcohol and chromic acid form a chromate ester that either reacts intramolecularly or intermolecularly in the presence of a base (water) to yield the corresponding carbonyl compound:
Aldehydes that can form hydrates in the presence of water are further oxidized to carboxylic acids:
Some alcohols such as benzylic and allylic alcohols give aldehydes that do not form hydrates in significant amounts; these can therefore be selectively oxidized with unmodified Jones Reagent to yield aldehydes. Although the reagent is very acidic, the substrate in acetone is essentially titrated with the oxidant solution and only very acid-sensitive groups are incompatible. For example esters, even tert-butyl esters, remain unchanged. The concentration of sulfuric acid can be decreased to minimize side reactions, although the oxidation power increases too.
Disproportionations and single electron transfers lead to chromium (V) acid and stable Cr(III) hydroxide. The chromium (V) acid promotes a two-electron oxidation of an alcohol and becomes Cr(III). Any residues of toxic Cr(V) and Cr(VI) compounds can be destroyed by the addition of an excess of 2-propanol once the intended reaction is complete. The formation of Cr(III) is indicated by a color change to green.
For the synthesis of aldehydes, the Collins Reaction or use of more modern - although more expensive - chromium (VI) reagents such as PCC and PDC can be an appropriate choice. If toxic chromium compounds must be avoided, there are some greener alternative procedures available. For example, molecular oxygen can be used in the presence of palladium as catalyst: synthesis of aldehydes, synthesis of ketones, synthesis of carboxylic acids.
Some newer protocols are available in which a catalytic amount of CrO 3 in aqueous solution is used in tandem with a strong stoichiometric oxidant, which is able to reoxidize Cr(IV) but does not interfere with the organic compounds. These are also interesting for the development of greener methods, and examples can be found in the recent literature section.
A full review of chromium-based reagents can be found in the book written by Tojo and Fernández (Oxidation of Alcohols to Aldehydes and Ketones, Springer Berlin, 2006, 1-97.).
Recent Literature
Selective oxidation of allylic alcohols with chromic acid
K. E. Harding, L. M. May, K. F. Dick, J. Org. Chem., 1975, 40, 1664-1665.
A novel chromium trioxide catalyzed oxidation of primary alcohols to the carboxylic acids
M. Zhao, J. Li, Z. Song, R. Desmond, D. M. Tschaen, E. J. J. Grabowski, P. J. Reider, Tetrahedron Lett., 1998, 39, 5323-5326.
Selective Oxidation of Benzylic Alcohols and TBDMS Ethers to Carbonyl Compounds with CrO 3-H 5 IO 6
S. Zhang, L. Xu, M. L. Trudell, Synthesis, 2005, 1757-1760.
Selective Oxidation of Benzylic Alcohols and TBDMS Ethers to Carbonyl Compounds with CrO 3-H 5 IO 6
S. Zhang, L. Xu, M. L. Trudell, Synthesis, 2005, 1757-1760.
Share:Follow:Support:💳💡🔗❤
Advertise here
Site Search
any all words
Main Categories
Organic Reactions
Org. Chem. Highlights
Abstracts
Chemicals
Job Market
Presentations
Chemistry Tools
Ads&Imprint
Popular Subcategories
Name Reactions
Protecting Groups
Organic Synthesis
Menu Open ▼Menu Close ▲ |
13030 | https://ocw.mit.edu/courses/5-12-organic-chemistry-i-spring-2005/e9f507ec7540428450eca35352bc3ed6_carbonyl1.pdf | TYPES OF CARBONYL COMPOUNDS General Name Name formula suffix O -al Aldehyde C H R O Ketone C -one R' R O -oic C acid Carboxylic H acid R O O Acid X=halogen -oyl C halide halide X R Acid O O -oic anhydride C O C anhydride R R Name General Name formula suffix O -oate Ester C O R' R O C Lactone (cyclic ester) O O R' -amide C Amide N R R" Lactam O C R' (cyclic amide) N hydrophobic hydrophilic grease Phospholipids - Fatty acid esters of glycerol glycerol a triol Phosphatidylcholine LIPID BILAYER MICELLE Detergents and "fatty acids" - amphiphilic molecules CO2 O O O O O P O O O NH3 OH OH OH H2O H2O H2O H2O H2O H2O CO2H omega-3-fatty acid Amides in Peptides and Proteins H2N O OH H2N O OH H2N O OH H2N O OH H2N O OH O NH2 O OH glycine alanine phenylalanine asparagine glutamic acid Five of the 20 amino acid building blocks that go into all proteins All -amino acids N H H N N H H N O O O O CO2H Proteins are polyamides Each protein is defined by a specific primary sequence of amino acids a a N S H 2C O H2N H H N S H 2C O H N H H O HO 6-Aminopenicillamic acid Penicillin G Other Penicillins varied at this position -Lactam Antibiotics HO b-lactam g-lactam d-lactam O O O NH NH NH g b O a a b d OH H2N OH H2N H2N OH b O g a O b-amino acid g-amino acid d-amino acid b SUMMARY - SYNTHESIS OF ALDEHYDES AND KETONES 1. Oxidation of alcohols N H 3 R1 O OH CrO Cl-Na2Cr2O7 will over-oxidize to carboxylic acid R1 H R1 R2 OH O O 1. R2MgBr R2 Na2Cr2O7 H2SO4 H 2. pH7 R1 OH R1 R1 R2 2. From alkynes O O Hg2 +, H2O R1 H 1. RBH2 R1 H H2SO4 R1 Me 2. H2O2, NaOH R1 H SUMMARY - SYNTHESIS OF ALDEHYDES AND KETONES 3. Friedel-Crafts Acylation O R 3 O R Cl AlCl 4. Ozonolysis 1. O3 2. Zn/H3O+ O O H Overview of Reactions 4/27-5/9 - Reactions with carbonyls 1. Nucleophilic Addition: Reactions with Aldehydes and Ketones (Chapter 19) A. O -O Nu H3O+ HO Nu R R' Nu Example: Grignard Reaction Section 19.8 3 O CH MgBr + HO Example: Cyanohydrin Formation Section 19.7 O HO CN H H HCN Overview of Reactions 4/27-5/9 - Reactions with carbonyls 1. Nucleophilic Addition A. Hydration and Acetal Formation OH O H2O HO OH O HO H3O+ Section 19.6 Section 19.11 O O O R H H+ H Electrophilic at C OH OH R No longer electrophilic at C R H R H R H O O O R H O R H H2O ( ) OH OH OH OH OH HO HO HO Protected masked aldehyde O O Acetals as Protecting Groups for Aldehydes H X o BrMg M g H Br O O H O O H+ H O H O O H O O O H o H2O/H+ OH OH Br OH OH BrMg Mg electrophile 1. 2. pH7 1. Nucleophilic Addition A.
Overview of Reactions 4/27-5/9 - Reactions with carbonyls Wittig Reaction triphenyl phosphine P CH3-Br P CH3 BuLi P CH2 O Ph3P CH2 O Ph = Phenyl Ph3PO Section 19.12 Overview of Reactions 4/27-5/9 - Reactions with carbonyls 1. Nucleophilic Addition H2O Nu Nu B.
O -O Nu H3O+ R R' R R' Nu Overview of Reactions 4/27-5/9 - Reactions with carbonyls 1.
Nucleophilic Addition B. continued Example: Imine Formation H O H3O+ H2N NH2 O H NH H2O H NH H2O H NH H N H Sometimes called a Schiff base in biochemistry Section 19.9 SUMMARY - Synthesis of Carboxylic Acids 1. Oxidative cleavage of alkene R' KMnO4 HO HO O O R R R' 2. Oxidation of 1 alcohols and aldehydes O O H Jones Oxidation OH R OH R H R H 3. Carboxylation of Grignard Reagents O CO2 RMgBr R OH Overview of Reactions 4/27-5/9 - Reactions with carbonyls 2. Nucleophilic Acyl Substitution: Reactions with Carboxylic Acid derivatives (Chapter 21) R Y O Y R O Y-Nu -O Nu Nu Y = Leaving Group O O LiAlH4 OH LiAlH4 Ester hydride reduction OR H H Chapter 17.5 RO-Li+ nucleophilic substitution nucleophilic addition MeMgBr O MeMgBr OH Ester grignard reaction O Me Section 21.6 Me Me OR RO-Li+ nucleophilic substitution nucleophilic addition Overview of Reactions 4/27-5/9 - Reactions with carbonyls 2. Nucleophilic Acyl Substitution Carboxylic Acid Derivatives Example:Acyl Halide Section 21.4 Example:Ester Section 21.6 O 2 O O 3 2 O OH SOCl Cl OH CH CH OH OEt HCl O NR2 O 2 Cl NHR Example:Amide Section 21.7 Overview of Reactions 4/27-5/9 - Reactions with carbonyls 3. Alpha Substitution: Enolate/Enol Chemistry H O ( ) O O O E E+ = must have " " H pKa 10-20 electrophile enolate OH OH enol 2 Br OH O Br Alpha Halogenation Section 22.3 Alpha Alkylation O OLi O Section 22.8 LDA (base) CH3I Li N LDA = H a Overview of Reactions 4/27-5/9 - Reactions with carbonyls 4. Condensation Reactions: Enolate/Enol Chemistry O O O OH O R' base R R' R'' R'' R R H Aldol Reaction Section 23.2-23.7 O O OH O 3 2 H C H2 H3C O CH base CH H H H3C H Claisen Condensation Section 23.8-23.9 O O 3 O 2 O C H2 H3C H3C O + -Key in fatty acid biosynthesis CH EtO base CH EtO OEt OEt OEt Overview of Reactions 4/27-5/9 - Reactions with carbonyls 4. Condensation Reactions: Enolate/Enol Chemistry continued Michael Reaction Section 23.11 O O O O H H H O O O H O O O H O H H H EtO NaOEt Me EtO Me Me EtO Me Me EtO Me HOEt Me |
13031 | https://www.quora.com/B-h-3-12-is-the-centroidal-moment-of-inertia-of-what-beam-cross-section | Something went wrong. Wait a moment and try again.
Area Moment of Inertia
Beam Section
Mechanics of Solids
Cross Section (physics)
Mechanical Engineering
Engineering and Technolog...
Civil Engineering
Structural Engineering
5
B(h^3) /12 is the centroidal moment of inertia of what beam cross section?
Ad by Coverage.com
Does switching car insurance really save you money, or is that just marketing hype?
This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up sav
(Continue reading)
Robert Toop
Author has 4.6K answers and 2.9M answer views
·
Feb 18
Given a rectangular plate with width B & height h, forming cross-section of a beam.
Analysis 1: Its area moment of inertia about its centroid (axis perpendicular to center of plate and parallel to beam length)
I = 1/12(B^2 + h^2) dimensions L^2/M.
Conclusion 1: Thanks to Wikipedia for the formula, but the result means the question’s use of the word “centroid” is wrong AND its lack of dimensions or units of measure is a mistake likely to lead to errors.
Analysis 2: Area moment about either edge of length B is
I = B (integrate y^2•dy for y = 0 to h)
I_indefinite = B•1/3y^3
I_definite = 1/3B(h^3
Given a rectangular plate with width B & height h, forming cross-section of a beam.
Analysis 1: Its area moment of inertia about its centroid (axis perpendicular to center of plate and parallel to beam length)
I = 1/12(B^2 + h^2) dimensions L^2/M.
Conclusion 1: Thanks to Wikipedia for the formula, but the result means the question’s use of the word “centroid” is wrong AND its lack of dimensions or units of measure is a mistake likely to lead to errors.
————————
Analysis 2: Area moment about either edge of length B is
I = B (integrate y^2•dy for y = 0 to h)
I_indefinite = B•1/3y^3
I_definite = 1/3B(h^3 - 0^3)
I = 1/3Bh^3 dimensions L^2/M
Conclusion 2: That’s not the correct axis.
————————
Analysis 3: Area moment about the line in the plate midway between edges of length B and parallel to them is
I = 2B(integrate y^2•dy for y = 0 to h/2)
I_indefinite B•2/3y^3
I_definite = 2/3B(h^3/8 - 0/8)
I = 1/12Bh^3 dimensions L^2/M.
Conclusion 2: This matches the question’s value and adds the correct units. However, I wonder what use this is for an actual beam, because each of beams thin cross-sections has a different, position-dependent moment of inertia relative to the one cross-section contain the axis.
Related questions
How do I find the area and moment of inertia (Ix & Iy) of an I-section beam?
Why must the moment of the area of the entire cross section of a beam with respect to its centroidal axis equal zero?
What is the moment of inertia of a beam section 200mm100mm about its top face?
What is the difference between section modulus & area moment of inertia & 2nd moment of inertia?
How is the cross sectional area of a beam calculated?
Rafiq Mian
Consulting Structural Engineer, Struct'rl Forensics Engineer
·
Author has 744 answers and 1.3M answer views
·
Feb 18
A homogeneous solid rectangular cross section of base width “B” and depth “h”.
Proviso - Centroidal: Baseline or reference line whereabouts the moment of inertia is formulated is the horizontal line through the centroid of the section parallel to base “B” of the section.
Bruce Alexander
Former Structural Engineer (Retired)
·
Author has 2K answers and 661.8K answer views
·
Feb 18
Ix = bh^3/12 is the moment of inertia about the ‘X’ axis of a rectangle having a base of ‘b’ and height of ‘h’.
Iy = hb^3/12 is the moment of inertia about the ‘Y’ axis of the same rectangle.
Sponsored by Stake
Stake: Online Casino games - Play & Win Online.
Play the best online casino games, slots & live casino games! Unlock VIP bonuses, bet with crypto & win.
Yerram Varun
Math and Phy Enthusiast
·
6y
Related
What is the moment of inertia of a rectangle with base 'b' and height 'h' about its base?
This is going to be fun. Let’s start
Consider the figure shown -
Here you can see the rectangular strip of base b and height h
Let the moment of Inertia about the specified axis be I and the mass of the strip be M.
Actually the OP doesn’t make it clear whether it’s a strip or a wound wire hence for a constructive discussion and better conceptual clarity let’s discuss the strip case.
Don’t worry I will give you the hint for the other case too.
We know the moment of inertia for a point and a rod, but this is a strip, so how do we go about it, then?
Yes, we will divide the strip in to rods, here again t
This is going to be fun. Let’s start
Consider the figure shown -
Here you can see the rectangular strip of base b and height h
Let the moment of Inertia about the specified axis be I and the mass of the strip be M.
Actually the OP doesn’t make it clear whether it’s a strip or a wound wire hence for a constructive discussion and better conceptual clarity let’s discuss the strip case.
Don’t worry I will give you the hint for the other case too.
We know the moment of inertia for a point and a rod, but this is a strip, so how do we go about it, then?
Yes, we will divide the strip in to rods, here again two cases arise, we can divide the strip into rods horizontally or vertically, how do we decide that?
Let 1 = horizontal and 2 = vertical, I am going to write those two numbers on two pieces of paper and pick one, give me a second….
Allright, I got 2 in the paper, so let’s divide the strip into vertical rods,
Let us consider a strip at a distance x from the right side,
this strip is of width dx and length h
Now, As you can see This is like piling up a number of rods around the axis which means that you have to add the moment of inertia contributed by each rod. So what is the moment of inertia contributed by each rod?
This one you can easily calculate using Parallel axis theorem as 13ML2 .
Remember that the M in this expression is the mass of the rod and not the mass of the strip, so we have to replace this M accordingly
The small moment of inertia contributed by the marked rod =dI
Now, dI=13(Mb⋅h⋅h⋅dx)h2
Understood? To find the mass of the rod, we take the complete mass, divide by the total area to find the unit mass, then multiply by the area of the rod (which is infact negligible) to find the mass of the rod, finally we multiply “L squared”
Now we integrate,
And ya, don’t forget the limits, here x varies from 0 to b ,
∫I0dI=13Mh2b∫b0dx
⟹I=13Mh2
So, here we have our answer, you get the same thing when you divide it horizontally but in that case the moment of inertial of the each subdivided rod will be different hence the solution would become slightly lengthy but maybe chance didn’t want me to take the lengthy way ;)
Oh yeah, what if the OP was infact asking about a rectangular wire?
Then the answer is too easy, consider this,
A,B,C and D mark the sections of the wire and the line represents the axis,
Now, we have the moment of inertial of D as zero as it is on the axis
The moment of inertial of B and C are just like that of stripy rods we have just calculated.
And the moment of inertia of A is.. Well…. exercise left to the reader.
Thank you
Sampath N
Former Asst. Professor at Velammal Engineering College '10 (2010–2015)
·
Author has 3.3K answers and 1.8M answer views
·
3y
Related
How do I find the area and moment of inertia (Ix & Iy) of an I-section beam?
Simplest method is as described below which eliminates the axis shifting
Formula for Ix of a rectangle, when axis is at base of rectangle is
Ix = b d^ / 3
This formula can be used to calculate for I section easily
The XX is shown in dotted line
Now moment of inertia for top region is
I xx top = (b d ^3 / 3) - 2(b1 d1^3/3)
The two shaded area is to be deducted, above equation is self explanatory
So Ixx = 2 Ixx top
If it is not symmetrical, you can calculate for bottom in same method and add bottom and top
Iyy can be calculated in similar method
Simplest method is as described below which eliminates the axis shifting
Formula for Ix of a rectangle, when axis is at base of rectangle is
Ix = b d^ / 3
This formula can be used to calculate for I section easily
The XX is shown in dotted line
Now moment of inertia for top region is
I xx top = (b d ^3 / 3) - 2(b1 d1^3/3)
The two shaded area is to be deducted, above equation is self explanatory
So Ixx = 2 Ixx top
If it is not symmetrical, you can calculate for bottom in same method and add bottom and top
Iyy can be calculated in similar method
Sponsored by CDW Corporation
How can AI help your teams make faster decisions?
CDW’s AI solutions offer retrieval-augmented generation (RAG) to expedite info with stronger insights.
Related questions
How do we estimate the cross section of a beam?
What is the difference between area moment of inertia and section modulus in bending moment equation?
Why we have to find moment of inertia in beams and columns?
What is the moment of inertia of a tapered cantilever of a rectangular cross section?
Which is the most efficient cross-section resisting bending moment?
Jordan Kull
Structural Engineer, designing Commercial buildings on the daily.
·
Author has 291 answers and 2.6M answer views
·
9y
Related
What is the moment of inertia of a tapered cantilever of a rectangular cross section?
As you state in the question the member tapers (changes in cross section), thus the moment of inertial also changes through the length of the member.
The moment of inertia will be I = (b x h^3)/12
where "b" is the width of the member
and "h" is the height of the member
It is unclear from the question title whether "b" varies along on the member or if "h" varies, or if both vary.
Above is an example where "h" varies while "b" remains constant.
To write I as a function of "x": "b" is a constant 300. "h" varies 200 (from 600 to 400) over the length of 3,000. thus h = 600 - 200x/3000.
I(x) = (b h^
As you state in the question the member tapers (changes in cross section), thus the moment of inertial also changes through the length of the member.
The moment of inertia will be I = (b x h^3)/12
where "b" is the width of the member
and "h" is the height of the member
It is unclear from the question title whether "b" varies along on the member or if "h" varies, or if both vary.
Above is an example where "h" varies while "b" remains constant.
To write I as a function of "x": "b" is a constant 300. "h" varies 200 (from 600 to 400) over the length of 3,000. thus h = 600 - 200x/3000.
I(x) = (b h^3)/12 = [300 (600 - 200x/3000 )^3]/12
If the "h" and "b" both vary see image below:
I(x) can still be written. Lets give "h" the same taper as above: h = 600 - 200x/3000. And we'll give "b" a similar taper: b = 500 - 100x/3000.
I(x) = (b h^3)/12 = { [500 - 100x/3000] [600 - 200x/3000]^3} / 12
Rambabu Dontu
PhD in Physics, Tata Institute of Fundamental Research (Graduated 1986)
·
Author has 3.6K answers and 1.3M answer views
·
Updated 1y
Related
What is the moment of inertia of a scalene triangle of mass M and side lengths a, b, c (vertices A, B, C) about an axis passing through vertex B?
Let the triangle ABC lie in the xy plane, with B as origin and the median BD as the positive x axis. Let BD = d.
Let m be the mass of the triangle and I = mR² be its moment of inertia about the z axis.
Take a point P on BD at a distance x from B. Consider a strip of slant width dx, parllel to AC at P whose length would be (x/d)b.
The mass dm of the strip is proportional to the area dS of the strip which, in turn, is proportional to (x/d)bdx
dm = k(x/d)bdx
where k is a proportionality constant.
Moment of inertia of the strip (considered as a thin rod) about an axis passing through P, and perpendicula
Let the triangle ABC lie in the xy plane, with B as origin and the median BD as the positive x axis. Let BD = d.
Let m be the mass of the triangle and I = mR² be its moment of inertia about the z axis.
Take a point P on BD at a distance x from B. Consider a strip of slant width dx, parllel to AC at P whose length would be (x/d)b.
The mass dm of the strip is proportional to the area dS of the strip which, in turn, is proportional to (x/d)bdx
dm = k(x/d)bdx
where k is a proportionality constant.
Moment of inertia of the strip (considered as a thin rod) about an axis passing through P, and perpendicular to the plane of the triangle is (dm)(xb/d)²/12. By parllel axis theorem, the moment of inertia of the strip about a parllel axis passing through B is
dI = (dm)(xb/d)²/12 + (dm)x² = (1+b²/12d²) x²dm
I = S (1+b²/12d²) x²dm
R² = I/m = S (1+b²/12d²) x²dm / S dm =
S (1+b²/12d²) x²(xdx) / S xdx
Note that the proportionality constant (kb/d) occuring in dm cancells out.
R² = (1+b²/12d²) S x³dx / S xdx
The indefinite integrals are S x³dx = x⁴/4 and S xdx = x²/2
Taking the limits from x=0 to x=d,
R² =(1/2)(1+b²/12d²)d² = (1/2)(d²+b²/12)
where d² = (2c²+2a²-b²)/4
R² = (1/8)(2c²+2a²-2b²/3) =(1/4)(c²+a²-b²/3)
Proof of value of d²:
vector BC = a; vector BA= c; vector CA = b; vector DA = b/2; vector DC = -b/2
Vector a = d+b/2 and vector c = d-b/2
a² = d² +b²/4 +d.b
c² = d² +b²/4 -d.b
a²+c² = 2d² +b²/2
d² = (1/4)(2c²+2a²-b²)
.
Sponsored by CDW Corporation
Ready for the latest in AI-powered capability?
AMD Ryzen™ and Windows 11-powered x86 PCs from CDW drive business productivity with AI intuition.
Puneet Kushwaha
Studied Mechanical Engineering at Rawal Institutions (Graduated 2019)
·
7y
Related
What is the moment of inertia of a Rod over an axis X'X passing through its center of mass at an angle 'Alpha'?
Fred Scuttle
BS in Physics & Mathematics, University of California, Davis
·
Author has 1.2K answers and 550.3K answer views
·
3y
Related
What is the moment of inertia of any rectangle with respect to the centroidal axis parallel to the base?
The moment of inertia is defined as: :
I=∫r2dm
We want to find the moment of inertia through the center of the rectangle parallel to the base.
Assuming this rectangle is of mass M, has a base of length b, and height h, and has uniform density σ we can find the mass element:
σ=massarea=Mbh
dm=σdxdy
Now we can work out the integral. If we set the centroidal axis parallel to the base as the x-axis, the edge of which is at zero so that x is going from 0 to b, and y is going from −h2 to +h2 the r2
The moment of inertia is defined as: :
I=∫r2dm
We want to find the moment of inertia through the center of the rectangle parallel to the base.
Assuming this rectangle is of mass M, has a base of length b, and height h, and has uniform density σ we can find the mass element:
σ=massarea=Mbh
dm=σdxdy
Now we can work out the integral. If we set the centroidal axis parallel to the base as the x-axis, the edge of which is at zero so that x is going from 0 to b, and y is going from −h2 to +h2 the r2 distance is going to be just the value of y.
The integral becomes:
I=∫r2dm=∫b0∫h2−h2 y2σdxdy
σ is a constant. We can pull that out of the expression. There is no dependency on x so we can just evaluate the integral and pull out the result which is just b. That leaves:
I=bσ∫h2−h2 y2dy
I=bσy33∣∣∣h2−h2=bσ((h2)33−(−h2)33)=bσ⎛⎝h383−−h383⎞⎠
I=bσ(h324+h324)=bσh312=σbh312
but we know:
σ=Mbh
I=(Mbh)bh312
I=Mh212
Subham Pal
Lecturer at Govt Polytechnic College (2018–present)
·
Author has 58 answers and 274.2K answer views
·
7y
Related
How is the moment of inertia defined for the transverse vibration of a beam?
I am so sorry for not noticing your request for my answer. It's been quite a hectic week for me packed with exam preparations and all.
Coming to your question , I guess , you are referring to ‘theory of machines’ problems.
Well , for transverse vibrations of beams ( whose mass is often neglected , it's dimensions , specially it's length and cross section being more significant ) , we define area moment of inertia for the beams unlike mass moment of inertia for other rotational dynamics.
It's the same area moment of inertia that we use for cacluating the transeverse deflectinon of beam dyring bend
I am so sorry for not noticing your request for my answer. It's been quite a hectic week for me packed with exam preparations and all.
Coming to your question , I guess , you are referring to ‘theory of machines’ problems.
Well , for transverse vibrations of beams ( whose mass is often neglected , it's dimensions , specially it's length and cross section being more significant ) , we define area moment of inertia for the beams unlike mass moment of inertia for other rotational dynamics.
It's the same area moment of inertia that we use for cacluating the transeverse deflectinon of beam dyring bending ( sagging or hogging )
Hope this note satisfies you !!
Happy pondering thoughts !!
Jason Hsia
M.S. in Civil Engineering & Structural Engineering, California Polytechnic State University, San Luis Obispo
·
Upvoted by , MS Structural Engineering, Northwestern University (1969)
·
6y
Related
How do I increase the moment of inertia of a beam cross-sectional area?
I see plenty of analytical answers below, all of them are great. Definitely see them for all the physics behind it.
Practically if we had an I beam (W- section), we can increase the moment of inertia and its carrying capacity by using 1 of 3 methods. There are others, but from experience these are the most commonly done.
Stitch weld channels to the web. This is not the most efficient but it takes
I see plenty of analytical answers below, all of them are great. Definitely see them for all the physics behind it.
Practically if we had an I beam (W- section), we can increase the moment of inertia and its carrying capacity by using 1 of 3 methods. There are others, but from experience these are the most commonly done.
Stitch weld channels to the web. This is not the most efficient but it takes advantage of space that is usually not used by MEP.
Weld a plate to the bottom flange. This is the easiest and most efficient. It may affect MEP and the architectural finish.
PJP a WT section to the bottom. This is very expensive, but greatly increases the carrying capacity. Thi...
John Barron
Mechanical and Aerospace Engineer
·
Author has 239 answers and 458K answer views
·
5y
Related
How do I calculate the second moment of inertia around the y-axis (Iy)? The gap between the 2 circular arcs and the thickness of the beam is negligible.
Second area moment of a shape about an axis y, or its moment of inertia about said axis, Iy, is defined as the integral of the square of the distance from said axis, with respect to area, over the entire shape. Since the shape is symmetric about the y axis, we only need to calculate the second area moment of one half and double it to get the second area moment of the full shape. The moment of inertia of the whole shape is the sum of the moments of inertia of distinct geometric pieces.
For the 45 degree straight piece:
dA=bdL, where dA is a small area element, and dL is a short length segment me
Second area moment of a shape about an axis y, or its moment of inertia about said axis, Iy, is defined as the integral of the square of the distance from said axis, with respect to area, over the entire shape. Since the shape is symmetric about the y axis, we only need to calculate the second area moment of one half and double it to get the second area moment of the full shape. The moment of inertia of the whole shape is the sum of the moments of inertia of distinct geometric pieces.
For the 45 degree straight piece:
dA=bdL, where dA is a small area element, and dL is a short length segment measured along the straight piece. Let L represent distance from the connection at the line of symmetry, measured along the straight piece. z is then distance from y-axis, and z=Lsin(45)=L/sqrt(2). I of the straight piece is given by the integral of z^2 with respect to area A from L=0 to L=R/sin(45)=Rsqrt(2). I1=integral of (L/sqrt(2))^2 b with respect to L from 0 to Rsqrt(2). The integrand integrates to b(L^3)/6. Evaluate at endpoints and difference: I=sqrt(2)/3bR^3. This is the second area moment of one straight piece.
For the circular arc quadrant piece:
Polar coordinates lend themselves well to evaluating this piece of the puzzle. Let theta represent angle measure from +y toward +z. dL=Rdtheta. dA=bdL=bRdtheta. z=Rsin(theta). z^2=R^2(sin(theta)^2). I2=integral of R^2(sin(theta)^2)bR with respect to theta from 0 to pi/2. Rewrite integrand as bR^3sin(theta)^2. Focus on sin(theta)^2. sin(theta)^2=1-cos(theta)^2=1-(1+cos(2theta))/2=1/2-(1/2)cos(2theta). This is now a straightforward integrand. It integrates to (1/2)theta-(1/4)sin(2theta). Evaluate at endpoints and difference: you get pi/4. Now I2=(pi/4)bR^3.
Finally, Itotal=2I1+2I2=(2sqrt(2)/3+2pi/4)bR^3=(4sqrt(2)+3pi)/6bR^3, or approximately 2.514bR^3
Steve Pomroy
PhD student in Aerospace Engineering
·
5y
Related
Why must the moment of the area of the entire cross section of a beam with respect to its centroidal axis equal zero?
Because that’s the definition of the centroid. The moment of area about any point is the integral of the area multiplied by the distance from a line through the point. For example, the first moment of area around the x-axis is:
Sx=∫AydA
By definition, the centroid is located on a line that makes this integral zero. Once you find two intersecting lines with zero first moment of area, the intersection point is the centroid.
Related questions
How do I find the area and moment of inertia (Ix & Iy) of an I-section beam?
Why must the moment of the area of the entire cross section of a beam with respect to its centroidal axis equal zero?
What is the moment of inertia of a beam section 200mm100mm about its top face?
What is the difference between section modulus & area moment of inertia & 2nd moment of inertia?
How is the cross sectional area of a beam calculated?
How do we estimate the cross section of a beam?
What is the difference between area moment of inertia and section modulus in bending moment equation?
Why we have to find moment of inertia in beams and columns?
What is the moment of inertia of a tapered cantilever of a rectangular cross section?
Which is the most efficient cross-section resisting bending moment?
What is the difference between the moment of resistance and moment of inertia in a beam?
Flexural rigidity, section modulus (Z), and deflection (∆) are all influenced by moment of inertia of the beam's cross section. Despite the solid rectangular beam having a higher moment of inertia, why I or T beams are preferred in construction?
What is the second moment of inertia for a rectangular beam?
What will be the bending moment at a beam cross section?
What is the difference between a section resisting moment and its moment of inertia?
About
·
Careers
·
Privacy
·
Terms
·
Contact
·
Languages
·
Your Ad Choices
·
Press
·
© Quora, Inc. 2025 |
13032 | https://support.microsoft.com/en-us/office/floor-function-14bb497c-24f2-4e04-b327-b0b4de5a8886 | Skip to main content
Microsoft
Support
Sign in to your account
Sign in
Sign in with Microsoft
Sign in or create an account.
Select a different account.
You have multiple accounts
Choose the account you want to sign in with.
FLOOR function
Applies To
Excel for Microsoft 365 Excel for Microsoft 365 for Mac Excel for the web Excel 2024 Excel 2024 for Mac Excel 2021 Excel 2021 for Mac Excel 2019 Excel 2016
This article describes the formula syntax and use of the FLOOR function in Microsoft Excel.
Description
The FLOOR function in Excel rounds a specified number down to the nearest specified multiple of significance. Negative numbers are rounded down (further negative) to nearest whole multiple below zero.
Syntax
FLOOR(number, significance)
The FLOOR function syntax has the following arguments:
Number Required. The numeric value you want to round.
Significance Required. The multiple to which you want to round.
Remarks
If either argument is nonnumeric, FLOOR returns the #VALUE! error value.
If Number is positive and Significance is negative, FLOOR returns the #NUM! error value.
If the sign of Number is positive, a value is rounded down toward zero. If the sign of Number is negative, a value is rounded down, adjusted away from zero (further negative).
Example
Copy all the fields in the following table. In a new Excel worksheet, paste all into cell A1 and adjacent cells. If you do not automatically see results, select and press F2 on the formulas. Adjust column widths if needed to see all.
| Formula | Description | Result |
---
| =FLOOR(3.7,2) | Rounds 3.7 down to nearest multiple of 2. | 2 |
| =FLOOR(-2.5,-2) | Rounds -2.5 down to nearest multiple of -2. | -2 |
| =FLOOR(2.5,-2) | Returns an error value, because 2.5 and -2 have different signs. | #NUM! |
| =FLOOR(1.58,0.1) | Rounds 1.58 down to the nearest multiple of 0.1. | 1.5 |
| =FLOOR(0.234,0.01) | Rounds 0.234 down to the nearest multiple of 0.01. | 0.23 |
Need more help?
Want more options?
Explore subscription benefits, browse training courses, learn how to secure your device, and more.
Microsoft 365 subscription benefits
Microsoft 365 training
Microsoft security
Accessibility center
Communities help you ask and answer questions, give feedback, and hear from experts with rich knowledge.
Ask the Microsoft Community
Microsoft Tech Community
Windows Insiders
Microsoft 365 Insiders
Thank you for your feedback!
× |
13033 | https://www.musicapalazzo.com/en/history-of-opera-in-venice/ | History of Opera in Venice - Musica Palazzo
Gestisci Consenso Cookie
Per fornire le migliori esperienze, utilizziamo tecnologie come i cookie per memorizzare e/o accedere alle informazioni del dispositivo. Il consenso a queste tecnologie ci permetterà di elaborare dati come il comportamento di navigazione o ID unici su questo sito. Non acconsentire o ritirare il consenso può influire negativamente su alcune caratteristiche e funzioni.
Funzionale- [x] Funzionale Always active
L'archiviazione tecnica o l'accesso sono strettamente necessari al fine legittimo di consentire l'uso di un servizio specifico esplicitamente richiesto dall'abbonato o dall'utente, o al solo scopo di effettuare la trasmissione di una comunicazione su una rete di comunicazione elettronica.
Preferences- [x] Preferences
L'archiviazione tecnica o l'accesso sono necessari per lo scopo legittimo di memorizzare le preferenze che non sono richieste dall'abbonato o dall'utente.
Statistiche- [x] Statistiche
L'archiviazione tecnica o l'accesso che viene utilizzato esclusivamente per scopi statistici.L'archiviazione tecnica o l'accesso che viene utilizzato esclusivamente per scopi statistici anonimi. Senza un mandato di comparizione, una conformità volontaria da parte del vostro Fornitore di Servizi Internet, o ulteriori registrazioni da parte di terzi, le informazioni memorizzate o recuperate per questo scopo da sole non possono di solito essere utilizzate per l'identificazione.
Marketing- [x] Marketing
L'archiviazione tecnica o l'accesso sono necessari per creare profili di utenti per inviare pubblicità, o per tracciare l'utente su un sito web o su diversi siti web per scopi di marketing simili.
Manage optionsManage servicesManage {vendor_count} vendorsRead more about these purposes
Accetta tutti Rifiuta tutti Visualizza preferenze Salva preferenzeVisualizza preferenze
Cookie PolicyDichiarazione sulla Privacy{title}
Intro
Shows
La Traviata
The Barber of Seville
Rigoletto
About
Association
History of Opera in Venice
Palazzo Barbarigo Minotto
Contact
FAQ
News
Book Now
English
Italiano
Français
Español
Deutsch
Intro
Shows
La Traviata
The Barber of Seville
Rigoletto
About
Association
History of Opera in Venice
Palazzo Barbarigo Minotto
Contact
FAQ
News
Book Now
English
Italiano
Français
Español
Deutsch
HISTORY OF OPERA IN VENICE
Opera in the palaces
In the Baroque era, Venetian Palaces were places where music was an integral part of parties and receptions. The Cantate or allegorical serenades were performed together with Opera although with a commemorative purpose: Monteverdihimself wrote the famous Combattimento di Tancredi e Clorindaperformed in 1624 at Palazzo Mocenigo. Six years later, in the same palazzo, Proserpina Rapitaby Monteverdi, with the text by Giulio Strozzi, was performed. Opera performances in the palaces were only for a selected audience and limited to the friends of the noble family who arranged the event.
The first public mucical theatre was created
In 1636 an extraordinary fact bound to change the history of opera occurred in Venice: the Tron family obtained the permission from the Council of Ten to rebuild the San Cassian Theatre (burnt in a fire). It was no longer a theatre for comedies and the family was allowed to turn it into an opera house.
The theatre was rebuilt in less than a year and inaugurated in 1637: for the first time in the history of the music drama a public opera house with fee was created.
It was so successful that soon other 16 Theatres were built and during the following century Venice boasted its enviable primacy for being the City of Opera par excellence in all Europe.
Between the opening of San Cassiano Theatre and the opening of the last theatre of the century, the San Giovanni Grisostomo in 1678, more than 150 operas were performed.
Francesco Cavalliand Crespi, as well as Monteverdi, are some of the most important composers in Venice. They had the privilege to launch a new formula of opera, then improved in the future by other composers, whose structural units harmonized the librettist’s and the composer’s contribution. Moreover, in the Seventeenth century in Venice, the da capo aria consolidated the dynamic relationship between the audience and the singers and the figure of the leading lady emerged.
The young Rossini at Teatro S. Moisè
The Teatro S. Moisé was one of the theatres where operas were performed for the first time, a small but very elegant theatre offering high quality music seasons. Here operas by Cavalli, Vivaldi, Albinoni, Galuppiand Rossini were performed.The young Gioacchino Rossini had his debut when he was eighteen years old at the Teatro S. Moisé with La cambiale di matrimonio [The Bill of Marriage]and, after the great success of this opera, he was applauded for L’Inganno felice; La Scala di seta; L’Occasione fa il ladro and Il Signor Bruschino.
In 1818 the Teatro S. Moisé was closed and sold.
“Had the King-Emperor Napoleon himself thought to honour Venice with his presence, the excitement of his arrival could have done nothing to tear the people away from Rossini” (Stendhal, Life of Rossini).
The teatro La Fenice
Foto: Pietro Tessarin
There were seven opera houses in Venice in the seventeenth century and in 1773 the greatest of these was S. Benedetto then destroyed in a fire. The judicial dispute between the Venier family (owner of the land on which it was built) and the association owner of the theatre followed the rebuilding and led the latter, after they failed the lawsuit, to rebuild a larger and more sumptuous theatre: the Teatro La Fenice.
The theatre was inaugurated in 1790 and, until 1836, the premieres of a lot of masterpieces like Tancrediand Semiramideby Rossini, I Capuleti e i Montecchi and Beatrice di Tenda by Bellini, Belisario, Pia de’ TolomeieMaria di Rudenz by Donizettiwere first performed.
In 1836 the best Veneto theatre was devastated in a fire.
The structure of the Teatro La Fenice changed during the reconstruction and became even more elegant and beautiful. The theatre will always be considered one of the best Italian and European theatres.
The Teatro La Fenice contributed to write the history of melodrama by hosting the highest number of premieres by Verdi, after the Teatro La Scala.
In the twentieth century the Teatro underwent a lot of restoration projects and in 1987 was rebuilt after a devastating fire.
During the twentieth century a lot of very important premieres like La carriera di un libertino by Igor Stravinskij, L’Angelo di fuoco by Serghei Prokofiev, Il Giro di vite by Benjamin Britten, Intolleranza 1960 by Luigi Nonowere here performed
Giuseppe Verdi opera’s in Venice
Giuseppe Verdi wrote five operas commissioned by the Teatro La Fenice: Ernani, Attila, Rigoletto, La TraviataandSimon Boccanegra.
Ernani, whose libretto was written by the Venetian Francesco Maria, was performed on the 10 th March 1844 and had a good success in spite of severe Austrian police censorship.
Attila,performed on the 17 th March 1846, thrilled the Venetian audience because it told the story of the foundation of the city itself: “It had such a delightful success during the first evening, and became fanaticism during the second”(G. Verdi).
Rigoletto,performed on the 11 th March 1851, was Giuseppe Verdi’s great success. For this reason he was called “a genius” by the press.
The premiere of La Traviatawas a complete failure though, probably because the performers were not up to the expectations and the libretto was pretty indecent. The opera, inspired by The Lady with the Camellias by Dumas fils, was performed on the 6 th March 1853: “La Traviata last night, fiasco. Is it my fault or the fault of the singers? Time will tell!”(G. Verdi)
When the opera was performed once again a year later, it was a great success. The last Verdi’s Venetian premiere was Simon Boccanegra, performed at La Fenice in March 1857. Like the Traviata, the premiere was not really successful. “It was a fiasco just like the Traviata”(G. Verdi) but the applauses were three times as much during the second performance and made it a great success.
Patrizia di Paolo
Dimensione Lirica - © 2020
Castello 5601, Venezia
c.f. e p.i. 03912670274
Designed by: Tomomot& Developed by ELAN42.com
Cookie Policy
Google Rating
4.7
Palace Music
4.7
×
Aidan Leach
7 years ago
Simply amazing. We saw the Barber of Seville, and you become part of the opera in its intimate setting. The singing and music was excellent, the old palace rooms that each scene were set in were lovely. Be warned if you sit in the front row, you actually will be involved. Figaro combed my hair, the Count left me with his wine bottle and used me to fend off the attack from the Doctor, and then made stink eyes at me in an overstated comical way for the rest of the night. I have never laughed so hard at the opera!
Ed Zito
7 years ago
Outstanding performance and experience!Usually not a fan of opera but this is different. Up close and personal with great view and seating. You feel like part of the show.A must do!
Kasey Kilgore
7 years ago
An absolutely fantastic experience! An intimate Opera as might have been seen/heard a century ago in the private salon of a Venetian noble.
Frank Levy
7 years ago
Up to the first intermission was good. Then after the second intermission the chairs became way too uncomfortable and it was way too hot. Had to leave as we were very, very uncomfortable. The first section is also the most familiar to non-expert opera attendees. If it was 1/3 the length it would have been perfect. Or better chairs or air conditioning. It was nice and cool at the start but quickly warms up.
Darrell Thomas
7 years ago
This was the highlight of our trip to Venice. Such a great experience. The performers were very talented, and simply approaching the entrance to the building had a feeling of exclusivity that you may not get at other shows. It's a bit pricey if you don't think you'll be returning to Venice to use your membership for future shows, but it's still worth every penny.
See All Reviews |
13034 | https://artofproblemsolving.com/wiki/index.php/2022_AIME_I_Problems/Problem_6?srsltid=AfmBOooOKyLvvWCNbamS_W-ikBioVA6wvFBDfLBKTBkSjQdlj70Nl2lI | Art of Problem Solving
2022 AIME I Problems/Problem 6 - AoPS Wiki
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical CampusVisit the Virtual Campus
Sign In
Register
online school
Class ScheduleRecommendationsOlympiad CoursesFree Sessions
books tore
AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates
community
ForumsContestsSearchHelp
resources
math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten
contests on aopsPractice Math ContestsUSABO
newsAoPS BlogWebinars
view all 0
Sign In
Register
AoPS Wiki
ResourcesAops Wiki 2022 AIME I Problems/Problem 6
Page
ArticleDiscussionView sourceHistory
Toolbox
Recent changesRandom pageHelpWhat links hereSpecial pages
Search
2022 AIME I Problems/Problem 6
Contents
[hide]
1 Problem
2 Solution 1
3 Solution 2 (Rigorous)
4 Solution 3
5 Solution 4
6 See Also
Problem
Find the number of ordered pairs of integers such that the sequenceis strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression.
Solution 1
Since and cannot be an arithmetic progression, or can never be . Since and cannot be an arithmetic progression, and can never be . Since , there are ways to choose and with these two restrictions in mind.
However, there are still specific invalid cases counted in these pairs . Since cannot form an arithmetic progression, . cannot be an arithmetic progression, so ; however, since this pair was not counted in our , we do not need to subtract it off. cannot form an arithmetic progression, so . cannot form an arithmetic progression, so . cannot form an arithmetic progression, ; however, since this pair was not counted in our (since we disallowed or to be ), we do not to subtract it off.
Also, the sequences , , , , and will never be arithmetic, since that would require and to be non-integers.
So, we need to subtract off progressions from the we counted, to get our final answer of .
~ ihatemath123
Solution 2 (Rigorous)
We will follow the solution from earlier in a rigorous manner to show that there are no other cases missing.
We recognize that an illegal sequence (defined as one that we subtract from our 231) can never have the numbers {3, 4} and {4,5} because we have not included a 6 in our count. Similarly, sequences with {30,40} and {40,50} will not give us any subtractions because those sequences must all include a 20. Let's stick with the lower ones for a minute: if we take them two at a time, then {3,5} will give us the subtraction of 1 sequence {3,5,7,9}. We have exhausted all pairs of numbers we can take, and if we take the triplet of single digit numbers, the only possible sequence must have a 6, which we already don't count. Therefore, we subtract from the count of illegal sequences with any of the single-digit numbers and none of the numbers 30,40,50. (Note if we take only 1 at a time, there will have to be 3 of , which is impossible.)
If we have the sequence including {30,50}, we end up having negative values, so these do not give us any subtractions, and the triplet {30,40,50} gives us a 20. Hence by the same reasoning as earlier, we have 0 subtractions from the sequences with these numbers and none of the single digit numbers {3,4,5}.
Finally, we count the sequences that are something like (one of 3,4,5,), , (one of 30, 40, 50). If this is to be the case, then let be the starting value in the sequence. The sequence will be ; We see that if we subtract the largest term by the smallest term we have , so the subtraction of one of (30,40,50) and one of (3,4,5) must be divisible by 3. Therefore the only sequences possible are . Of these, only the last is invalid because it gives , larger than our bounds . Therefore, we subtract from this case.
Our final answer is
~KingRavi
Solution 3
Denote .
Denote by a subset of , such that there exists an arithmetic sequence that has 4 terms and includes but not .
Denote by a subset of , such that there exists an arithmetic sequence that has 4 terms and includes but not .
Hence, is a subset of , such that there exists an arithmetic sequence that has 4 terms and includes both and .
Hence, this problem asks us to compute
First, we compute .
We have .
Second, we compute .
: .
We have . Thus, the number of solutions is 22.
: .
We have . Thus, the number of solutions is 9.
Thus, .
Third, we compute .
In , we have . However, because , we have . Thus, .
This implies . Note that belongs in .
Thus, .
Fourth, we compute .
: In the arithmetic sequence, the two numbers beyond and are on the same side of and .
Hence, . Therefore, the number solutions in this case is 3.
: In the arithmetic sequence, the two numbers beyond and are on the opposite sides of and .
: The arithmetic sequence is .
Hence, .
: The arithmetic sequence is .
Hence, .
: The arithmetic sequence is .
Hence, . However, the sequence is not strictly increasing.
Putting two cases together,
Therefore,
~Steven Chen (www.professorchenedu.com)
Solution 4
divide cases into .(Notice that can't be equal to , that's why I divide them into two parts. There are three cases that arithmetic sequence forms: .(NOTICE that IS NOT A VALID SEQUENCE!) So when , there are possible ways( 3 means the arithmetic sequence and 13 means there are 13 "a" s and b cannot be 20)
When , there are ways.
In all, there are possible sequences.
~bluesoul
See Also
2022 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 5Followed by
Problem 7
1•2•3•4•5•6•7•8•9•10•11•12•13•14•15
All AIME Problems and Solutions
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.
Retrieved from "
Category:
Intermediate Combinatorics Problems
Art of Problem Solving is an
ACS WASC Accredited School
aops programs
AoPS Online
Beast Academy
AoPS Academy
About
About AoPS
Our Team
Our History
Jobs
AoPS Blog
Site Info
Terms
Privacy
Contact Us
follow us
Subscribe for news and updates
© 2025 AoPS Incorporated
© 2025 Art of Problem Solving
About Us•Contact Us•Terms•Privacy
Copyright © 2025 Art of Problem Solving
Something appears to not have loaded correctly.
Click to refresh. |
13035 | https://www.quora.com/How-do-I-find-integer-solutions-to-3a-2-b-2+2 | Something went wrong. Wait a moment and try again.
Integral Solutions
Problem Solving
Integer Systems
Algebraic Equations
Mathematical Problems
Gaussian Integer Solution...
Solving Equations
5
How do I find integer solutions to
3
a
2
=
b
2
+
2
?
Marco Vona
Trinity Wrangler (Cambridge Maths Tripos 2021)
·
Author has 99 answers and 423.1K answer views
·
Updated 8y
Your equation has infinitely many integer solutions. Let us prove it by induction:
Base case
Note that
{a=1b=1
so the equation has at least one solution,
Inductive step
We want to prove that if (a,b) is a solution, then (2a+b,2b+3a) is also a solution.
We can just substitute
3(2a+b)2=?(2b+3a)2+2
12a2+12ab+3b2=?4b2+12ab+9a2+2
3a2=?b2+2
which is true because we are assuming that (a,b) is a
Your equation has infinitely many integer solutions. Let us prove it by induction:
Base case
Note that
{a=1b=1
so the equation has at least one solution,
Inductive step
We want to prove that if (a,b) is a solution, then (2a+b,2b+3a) is also a solution.
We can just substitute
3(2a+b)2=?(2b+3a)2+2
12a2+12ab+3b2=?4b2+12ab+9a2+2
3a2=?b2+2
which is true because we are assuming that (a,b) is a solution.
Iterating this procedure yields infinitely many solutions. To be a bit more formal since
{b+2a>a2b+3a>b
the recursion will never enter a loop, because each new solution is strictly larger than the previous one.
Now we would like to find a closed expression for an and bn (the n-th a and b solutions). Let’s prove by induction that
{an+1=4an−an−1bn+1=4bn−bn−1(1)
Base step
The first three solutions are (1,1),(3,5) and (11,19), and we have that 11=4⋅3−1 and 19=4⋅5−1 ■
Inductive step
We want to show that if (1) holds for (an,bn) it also holds for (an+1,bn+1).
We know that
{an=4an−1−an−2bn=4bn−1−bn−2(2)
by hypotesis. Keeping in mind that an+1=2an+bn we have by (2)
an+1=2an+bn=4(2an−1+bn−1)−(2an−2+bn−2)=4an−an−1■
Similarly, since bn+1=3an+2bn we have by (2)
bn+1=3an+2bn=4(3an−1+2bn−1)−(3an−2+2bn−2)=4bn−bn−1■
which completes the proof.
Knowing that
{an+1=4an−an−1bn+1=4bn−bn−1(3)
we can now find an expression for an and bn.
We are going to use the following theorem
Theorem Let x0,x1,x2... a series defined recursively by
⎧⎨⎩x0=px1=qxn+1=axn+bxn−1
Then the general term xn is given by
xn=kλn1+hλn2(4)
where λ1 and λ2 are the two distinct roots of the equation
x2−ax−b=0
and h, k are coefficients that should be determined according to the values of x0 and x1.
In our case, the characteristic polynomial for an and bn is
x2−4x+1
which has 2±√3 for roots. Also, we have a0=1, a1=3 and b0=1, b1=5, so the coefficients h and k can be computed.
This gives as solutions
an=±16((√3−3)(2+√3)n−(3+√3)(2−√3)n)
bn=±12((1+√3)(2−√3)n−(√3−1)(2+√3)n)
Related questions
What are the integer solutions of
a
2
+
7
=
2
b
2
?
How do I find all the integer solutions of
y
2
=
x
(
x
+
6
)
(
x
+
12
)
?
Why does
a
2
+
2
b
2
=
c
2
have no integer solutions?
What are the integer solutions to
a
+
b
=
a
b
?
How do you find the integer solutions for
12
a
b
2
+
21
a
b
−
4
b
2
−
6
a
−
7
b
=
178
?
Griffin Macris
number theory enthusiast.
·
8y
Originally Answered: How do I find integer solutions to 3a^2=b^2+2?
·
The solutions seem to be contained within the convergents to the continued fraction for √3.
Here are the convergents: {1,2,53,74,1911,…}
Now notice that the numerator is b, and the denominator is a. Not all of them work, but all of the solutions are in there.
One known theorem about rationals p/q with relation to an irrational number α is that p/q is a convergent to α if and only if |α−pq|<1q2.
So now let’s use our equation:
3=b2/a2+2/a2
3−b2/a2=2/a2
|(√3−b/a)(√3+b/a)|=2/a2
At this point
The solutions seem to be contained within the convergents to the continued fraction for √3.
Here are the convergents: {1,2,53,74,1911,…}
Now notice that the numerator is b, and the denominator is a. Not all of them work, but all of the solutions are in there.
One known theorem about rationals p/q with relation to an irrational number α is that p/q is a convergent to α if and only if |α−pq|<1q2.
So now let’s use our equation:
3=b2/a2+2/a2
3−b2/a2=2/a2
|(√3−b/a)(√3+b/a)|=2/a2
At this point we assume b/a is approximately √3, so we can divide the whole thing by 3.4641…, approximately, and get:
|√3−b/a|≈11.73205a2
Clearly now we have that b/a is very close to √3, confirming your suspicion. I believe this inequality forces it to be a convergent, but I’m not able to find a source on that at the moment.
Amitabha Tripathi
five decades of high school Algebra
·
Author has 4.7K answers and 13.9M answer views
·
8y
Read my answer to How do I solve x^2-8y^2=1 where, x,y\in\mathbb {N} using the continued fraction of \sqrt{8}?
The fundamental unit ϵ3 is obtained from the continued fraction of √3=[1,¯¯¯¯¯¯¯¯1,2]. That this is the continued fraction expansion for √3 is along the same lines as the one worked out for √8 in my above post. This shows that
ϵ3=2+√3.
To find all integer solutions to x2–3y2=1, simply find all solutions in positive integers (x,y), then get the four combinations (±x,±y). Therefore it is enough to focus on the positive soluti
Read my answer to How do I solve x^2-8y^2=1 where, x,y\in\mathbb {N} using the continued fraction of \sqrt{8}?
The fundamental unit ϵ3 is obtained from the continued fraction of √3=[1,¯¯¯¯¯¯¯¯1,2]. That this is the continued fraction expansion for √3 is along the same lines as the one worked out for √8 in my above post. This shows that
ϵ3=2+√3.
To find all integer solutions to x2–3y2=1, simply find all solutions in positive integers (x,y), then get the four combinations (±x,±y). Therefore it is enough to focus on the positive solutions. These are given by x=xn, y=yn, where
xn+√3yn=(2+√3)n,n∈Z≥0…(1)
Since (1,1) is a solution to x2–3y2=−2, a class of solutions in positive integers to x2–3y2=−2 is given by x=Xn, y=Yn, where
Xn+√3Yn=(1+√3)⋅(2+√3)n,n∈Z≥0…(2)
The easiest way to see this is by multiplying both sides of eqn. (1) or of eqn. (2) by their respective conjugates. I leave this as an exercise for the interested reader. ■
John Calligy
Author has 2.3K answers and 1.6M answer views
·
8y
Originally Answered: How do I find integer solutions to 3a^2=b^2+2?
·
You’ve done very well with your findings. Let me suggest that you try b=[(1+√3)∗(2+√3)n]/2.
Update: I see in one of your comments that you’re trying to avoid floor functions. That’s possible, and I’ll give you a clue as to how it’s done: (2+√3)n+(2−√3)n is always an integer, and (2−√3)n is always very small.
Try here for information on solving problems of this type.
Promoted by The Penny Hoarder
Lisa Dawson
Finance Writer at The Penny Hoarder
·
Updated Sep 16
What's some brutally honest advice that everyone should know?
Here’s the thing: I wish I had known these money secrets sooner. They’ve helped so many people save hundreds, secure their family’s future, and grow their bank accounts—myself included.
And honestly? Putting them to use was way easier than I expected. I bet you can knock out at least three or four of these right now—yes, even from your phone. Don’t wait like I did.
Cancel Your Car Insurance
You might not even realize it, but your car insurance company is probably overcharging you. In fact, they’re kind of counting on you not noticing. Luckily, this problem is easy to fix.
Don’t waste your time
Here’s the thing: I wish I had known these money secrets sooner. They’ve helped so many people save hundreds, secure their family’s future, and grow their bank accounts—myself included.
And honestly? Putting them to use was way easier than I expected. I bet you can knock out at least three or four of these right now—yes, even from your phone. Don’t wait like I did.
Cancel Your Car Insurance
You might not even realize it, but your car insurance company is probably overcharging you. In fact, they’re kind of counting on you not noticing. Luckily, this problem is easy to fix.
Don’t waste your time browsing insurance sites for a better deal. A company calledInsurify shows you all your options at once — people who do this save up to $996 per year.
If you tell them a bit about yourself and your vehicle, they’ll send you personalized quotes so you can compare them and find the best one for you.
Tired of overpaying for car insurance? It takes just five minutes to compare your options with Insurify andsee how much you could save on car insurance.
Ask This Company to Get a Big Chunk of Your Debt Forgiven
A company calledNational Debt Relief could convince your lenders to simply get rid of a big chunk of what you owe. No bankruptcy, no loans — you don’t even need to have good credit.
If you owe at least $10,000 in unsecured debt (credit card debt, personal loans, medical bills, etc.), National Debt Relief’s experts will build you a monthly payment plan. As your payments add up, they negotiate with your creditors to reduce the amount you owe. You then pay off the rest in a lump sum.
On average, you could become debt-free within 24 to 48 months. It takes less than a minute to sign up and see how much debt you could get rid of.
Set Up Direct Deposit — Pocket $300
When you set up direct deposit withSoFi Checking and Savings (Member FDIC), they’ll put up to $300 straight into your account. No… really. Just a nice little bonus for making a smart switch.
Why switch? With SoFi, you can earn up to 3.80% APY on savings and 0.50% on checking, plus a 0.20% APY boost for your first 6 months when you set up direct deposit or keep $5K in your account. That’s up to 4.00% APY total. Way better than letting your balance chill at 0.40% APY.
There’s no fees. No gotchas.Make the move to SoFi and get paid to upgrade your finances.
You Can Become a Real Estate Investor for as Little as $10
Take a look at some of the world’s wealthiest people. What do they have in common? Many invest in large private real estate deals. And here’s the thing: There’s no reason you can’t, too — for as little as $10.
An investment called the Fundrise Flagship Fund lets you get started in the world of real estate by giving you access to a low-cost, diversified portfolio of private real estate. The best part? You don’t have to be the landlord. The Flagship Fund does all the heavy lifting.
With an initial investment as low as $10, your money will be invested in the Fund, which already owns more than $1 billion worth of real estate around the country, from apartment complexes to the thriving housing rental market to larger last-mile e-commerce logistics centers.
Want to invest more? Many investors choose to invest $1,000 or more. This is a Fund that can fit any type of investor’s needs. Once invested, you can track your performance from your phone and watch as properties are acquired, improved, and operated. As properties generate cash flow, you could earn money through quarterly dividend payments. And over time, you could earn money off the potential appreciation of the properties.
So if you want to get started in the world of real-estate investing, it takes just a few minutes tosign up and create an account with the Fundrise Flagship Fund.
This is a paid advertisement. Carefully consider the investment objectives, risks, charges and expenses of the Fundrise Real Estate Fund before investing. This and other information can be found in the Fund’s prospectus. Read them carefully before investing.
Cut Your Phone Bill to $15/Month
Want a full year of doomscrolling, streaming, and “you still there?” texts, without the bloated price tag? Right now, Mint Mobile is offering unlimited talk, text, and data for just $15/month when you sign up for a 12-month plan.
Not ready for a whole year-long thing? Mint’s 3-month plans (including unlimited) are also just $15/month, so you can test the waters commitment-free.
It’s BYOE (bring your own everything), which means you keep your phone, your number, and your dignity. Plus, you’ll get perks like free mobile hotspot, scam call screening, and coverage on the nation’s largest 5G network.
Snag Mint Mobile’s $15 unlimited deal before it’s gone.
Get Up to $50,000 From This Company
Need a little extra cash to pay off credit card debt, remodel your house or to buy a big purchase?
We found a company willing to help.
Here’s how it works: If your credit score is at least 620, AmONE can help you borrow up to $50,000 (no collateral needed) with fixed rates starting at 6.40% and terms from 6 to 144 months.
AmONE won’t make you stand in line or call a bank. And if you’re worried you won’t qualify, it’s free tocheck online. It takes just two minutes, and it could save you thousands of dollars.
Totally worth it.
Get Paid $225/Month While Watching Movie Previews
If we told you that you could get paid while watching videos on your computer, you’d probably laugh.
It’s too good to be true, right?
But we’re serious. By signing up for a free account with InboxDollars, you could add up to $225 a month to your pocket. They’ll send you short surveys every day, which you can fill out while you watch someone bake brownies or catch up on the latest Kardashian drama.
No, InboxDollars won’t replace your full-time job, but it’s something easy you can do while you’re already on the couch tonight, wasting time on your phone.
Unlike other sites, InboxDollars pays you in cash — no points or gift cards. It’s already paid its users more than $56 million.
Signing up takes about one minute, and you’ll immediately receive a $5 bonus to get you started.
Earn $1000/Month by Reviewing Games and Products You Love
Okay, real talk—everything is crazy expensive right now, and let’s be honest, we could all use a little extra cash. But who has time for a second job?
Here’s the good news. You’re already playing games on your phone to kill time, relax, or just zone out. So why not make some extra cash while you’re at it?
WithKashKick, you can actually get paid to play. No weird surveys, no endless ads, just real money for playing games you’d probably be playing anyway. Some people are even making over $1,000 a month just doing this!
Oh, and here’s a little pro tip: If you wanna cash out even faster, spending $2 on an in-app purchase to skip levels can help you hit your first $50+ payout way quicker.
Once you’ve got $10, you can cash out instantly through PayPal—no waiting around, just straight-up money in your account.
Seriously, you’re already playing—might as well make some money while you’re at it.Sign up for KashKick and start earning now!
Related questions
What are the integer solutions to
(
a
2
−
1
)
(
b
2
−
1
)
=
c
2
?
What are the integer solutions to
a
3
+
b
3
=
m
!
+
4
?
What are the integer solutions to
x
2
+
x
y
+
2
y
2
=
29
?
How do you find integer solutions to
2
x
2
+
27
x
+
91
=
y
2
?
What are the integer solutions to
x
2
−
1
=
2
y
2
?
Bill Crean
Solved the Holey Cube problem without calculus.
·
Author has 5.1K answers and 6.4M answer views
·
Updated 8y
For questions such as these, what I have done in the past is to write a visual basic routine, something like the following coding:
for I = 1 to 1000
for j = 1 to 1000
if (3iI = jj +2) then
print, “a =”, I ; “b = “, j
next j
next i
If that does not catch the solutions try another range.
An obviou...
Brian Sittinger
PhD in Mathematics, University of California, Santa Barbara (Graduated 2006)
·
Upvoted by
Michael Jørgensen
, PhD in mathematics and
Michael Lamar
, PhD in Applied Mathematics · Author has 8.5K answers and 21.1M answer views
·
May 20
Related
How do I solve the following equation over integers :
x
3
=
y
2
+
2
?
We want to find the integer solutions to the Diophantine equation x3=y2+2. If x is even, then which has no solutions. Therefore [math]x[/math] must be odd, and consequently [math]y[/math] is odd as well.
Next, we exploit the arithmetic in the algebraic number ring [math]\mathbb{Z}[\sqrt{-2}][/math] (which is a PID with unit group [math]{\pm 1}[/math]) to factor the right side of the equation:
[math]x^3 = (y + \sqrt{-2})(y - \sqrt{-2}). \tag{}[/math]
We claim that the factors on the right side of the equation above are relatively prime in [math]\mathbb{Z}[\sqrt{-2}][/math]. To this end, we let [math]\delta = \gcd(y + \sqrt{-2}, y - \sqrt{-2})[/math]. Then,
We want to find the integer solutions to the Diophantine equation [math]x^3 = y^2 + 2[/math]. If [math]x[/math] is even, then [math]y^2 \equiv 2 \bmod 4[/math] which has no solutions. Therefore [math]x[/math] must be odd, and consequently [math]y[/math] is odd as well.
Next, we exploit the arithmetic in the algebraic number ring [math]\mathbb{Z}[\sqrt{-2}][/math] (which is a PID with unit group [math]{\pm 1}[/math]) to factor the right side of the equation:
[math]x^3 = (y + \sqrt{-2})(y - \sqrt{-2}). \tag{}[/math]
We claim that the factors on the right side of the equation above are relatively prime in [math]\mathbb{Z}[\sqrt{-2}][/math]. To this end, we let [math]\delta = \gcd(y + \sqrt{-2}, y - \sqrt{-2})[/math]. Then, we see that [math]\delta[/math] divides their difference [math]2\sqrt{-2}[/math], and taking norms yields [math]N(\delta) \mid N(2\sqrt{2}) = 8[/math]. Moreover, [math]N(\delta) \mid N(y \pm \sqrt{-2}) = y^2 + 2[/math] which we know is odd (since [math]x[/math] and [math]y[/math] are odd). Therefore, it follows that [math]N(\delta) = 1[/math] which implies that [math]\delta[/math] is a unit in [math]\mathbb{Z}[\sqrt{-2}][/math], and thus the claim follows.
Since we now know that [math]y \pm \sqrt{-2}[/math] are relatively prime in [math]\mathbb{Z}[\sqrt{-2}][/math] and relatively prime in [math]\mathbb{Z}[\sqrt{-2}][/math] is a unique factorization domain (due to its being a PID), [math]y \pm \sqrt{-2}[/math] are both cubes, up to unit multiples. However since the units in [math]\mathbb{Z}[\sqrt{-2}][/math] are [math]\pm 1[/math], both of which are cubes, it follows that [math]y \pm \sqrt{-2}[/math] are both cubes.
In particular, we can write for some [math]a, b \in \mathbb{Z}[/math]:
[math]y + \sqrt{-2} = (a + b\sqrt{-2})^3. \tag{}[/math]
Then expanding the right side and equating real and imaginary ([math]\sqrt{-2}[/math]) parts yields the system of equations (in the set of integers)
[math]y = a^3 - 6ab^2 = a(a^2 - 6b^2), \text{ and } 1 = 3a^2 b - 2b^3 = b(3a^2 - 2b^2). \tag{}[/math]
The second equation immediately implies that [math]b = \pm 1[/math]; we now have two cases to consider. If [math]b = 1[/math], then the second equation yields [math]3a^2 - 2 = 1[/math], and thus [math]a = \pm 1[/math]. Substituting this into the equation for [math]y[/math] then gives us [math]y = \pm 5[/math] and thus [math]x = 3[/math]. If [math]b = -1[/math], then the second equation yields [math]3a^2 - 2 = -1[/math] which has no integer solution.
Therefore, we conclude that the only integer solutions to the Diophantine equation [math]x^3 = y^2 + 2[/math] are
[math]\boxed{(x, y) = (3, \pm 5)}. \; \blacksquare \tag{}[/math]
Promoted by Almedia
Charlee Anthony
Go-to Resource for Realistic, Side Hustle Ideas
·
Sep 22
How can I make an extra $200 a week online?
This one good method to make extra income brought me over $3,000, and I still make $150–$200 every single week from Freecash. As a gig hunter, this app came natural to me.
I hacked this Freecash platform so you don't have to. These GPT platforms can be hell to learn, so learn my top methods, rent free.👇
How I consistently make $200 a week online
Here’s exactly how I made $200 a week online with zero upfront investment. And yeah, it may have involved 10 hours a week 😂
signed up for free - got a $5 welcome bonus instantly just right after I started my first offer.
played games and completed offers
This one good method to make extra income brought me over $3,000, and I still make $150–$200 every single week from Freecash. As a gig hunter, this app came natural to me.
I hacked this Freecash platform so you don't have to. These GPT platforms can be hell to learn, so learn my top methods, rent free.👇
How I consistently make $200 a week online
Here’s exactly how I made $200 a week online with zero upfront investment. And yeah, it may have involved 10 hours a week 😂
signed up for free - got a $5 welcome bonus instantly just right after I started my first offer.
played games and completed offers - chose the highest-paying tasks first like monopoly GO and raid shadow legends.
made an in-app purchase - spent $3 on a power-up in one mobile game, which unlocked a $25 reward.
cashed out fast - once I hit my goal, I withdrew directly to PayPal. The money was there in minutes.
repeated daily - by consistently putting in 1–2 hours a day, I reached $200 each week.
Use my referral code and test these hacks for yourself 👉 Freecash Referral Code
Not sure yet? Keep reading - and you will catch on to what I am really doing to earn this $200 a week.
My secrets to hit $200 a week
Not your average tips - so consider this exclusive. If you follow this well, your chances of hitting your first $100 in a week on Freecash will feel like a piece of cake.
sort tasks by highest payout; go for the ones that give the biggest returns first.
check the “hot offers” tab daily; these change fast, and they often have short deadlines.
don’t ignore games; some game offers can pay $20–$50 each, especially with in-app upgrades (if you are into that)
check for giveaways; Freecash has social giveaways regularly. A friend of mine won $100 in the last event!
use your referral code; it sounds dumb, but I’ve about $100 from inviting around 10 friends.
To be honest, earning on Freecash is so easy it makes debt complainers look lazy. If I can sit on the couch and earn $200 a week and pay off debt, so can you.
How to start right now
sign up and grab your $5 welcome bonus instantly
complete your first high-paying offer tonight and cash out within hours.
reinvest small amounts into in-app purchases to unlock bigger rewards.
withdraw your first payout to PayPal before the week’s over.
Freecash has become my go-to extra income stream. If you want to start seeing $200 weeks, then stop your doom scroll session and try Freecash now.
Alon Amit
PhD in Mathematics; Mathcircler.
·
Upvoted by
Paul King
, Data Scientist, Computational Neuroscientist and
Ermal Feleqi
, PhD Mathematics, University of Padua (2010) · Author has 8.8K answers and 173.8M answer views
·
7y
Related
What are the integer solutions to
[math]a^3+b^3=m!+4[/math]
?
(Your internal monologue, transcribed:)
“Modulo… modulo something. I remember reading something on Quora about how looking modulo something can help show that certain things are impossible. In integers. Right. But modulo what?
Modulo [math]2[/math]? Nah. Cubes can be either even or odd. I want something where [math]a^3[/math] only has very few options modulo [math]M[/math], and with luck even [math]a^3+b^3[/math] will have few options, and with even more luck those options won’t fit [math]m!+4[/math] so we can limit the number of possible solutions.
Modulo [math]4[/math]? [math]8[/math]? No, those are good for squares. We have cubes here. How about modulo [math]5[/math]? [math]0^3=0[/math], [math]1^3=1[/math], [math]2^3=3[/math], [math]3^3=2[/math],
(Your internal monologue, transcribed:)
“Modulo… modulo something. I remember reading something on Quora about how looking modulo something can help show that certain things are impossible. In integers. Right. But modulo what?
Modulo [math]2[/math]? Nah. Cubes can be either even or odd. I want something where [math]a^3[/math] only has very few options modulo [math]M[/math], and with luck even [math]a^3+b^3[/math] will have few options, and with even more luck those options won’t fit [math]m!+4[/math] so we can limit the number of possible solutions.
Modulo [math]4[/math]? [math]8[/math]? No, those are good for squares. We have cubes here. How about modulo [math]5[/math]? [math]0^3=0[/math], [math]1^3=1[/math], [math]2^3=3[/math], [math]3^3=2[/math], [math]4^3=4[/math]. Ewww. Everything is a cube. Bad.
Modulo [math]7[/math]? Let’s see: [math]0^3=0[/math], [math]1^3=1[/math], [math]2^3=1[/math], [math]3^3=6[/math], [math]4^3=-3^3=1[/math], [math]5^3=-2^3=6[/math], [math]6^3=-1^3=6[/math]. Hooray! The cubes modulo [math]7[/math] are very restricted. Any cube is [math]0[/math] or [math]\pm 1[/math] modulo [math]7[/math], so a sum of two cubes is [math]-2,-1,0,1,2[/math] modulo [math]7[/math], which means the residues [math]3[/math] and [math]4[/math] are ruled out.
Wait! [math]4[/math]…! We have [math]m!+4[/math], and [math]m![/math] is almost always divisible by [math]7[/math], so [math]m!+4[/math] must then leave a remainder of [math]4[/math] mod [math]7[/math]. Yes! We’ve just shown that as soon as [math]m \ge 7[/math], [math]m!+4[/math] cannot possibly be the sum of two cubes!”
(End of transcript)
This is the heart of the matter. Instead of or in addition to mod [math]7[/math], you should also try mod [math]9[/math], which has the small advantage that [math]m![/math] is divisible by [math]9[/math] already starting at [math]m=6[/math]. Also, cubes modulo [math]9[/math] are very restricted, also being [math]0,\pm 1[/math], so there are four residue classes mod [math]9[/math] which aren’t a sum of two cubes. The residues [math]3,4,5,6[/math] are all ruled out mod [math]9[/math].
(PS: The reason [math]7[/math] is helpful for cubes and [math]5[/math] is not is that [math]7 \equiv 1 \pmod{3}[/math]. If you know that the nonzero residues mod [math]p[/math] form a multiplicative group, you should be able to see why).
So:
[math]0!+4=1!+4=5[/math] cannot be the sum of two cubes, because it’s [math]5[/math] mod [math]9[/math].
[math]2!+4=6[/math] is also ruled out mod [math]9[/math].
[math]3!+4=10[/math] is [math]3[/math] mod [math]7[/math], so it’s ruled out by the [math]7[/math].
[math]4!+4=28[/math] is not ruled out. We’ll analyze it in a moment.
[math]5!+4=124[/math] is not ruled out. We’ll analyze it in a moment.
[math]m!+4[/math] for [math]m \ge 6[/math] don’t need to be checked. They are all [math]4[/math] modulo [math]9[/math].
All right. We have narrowed down the options for [math]m[/math] to just two alternatives: [math]m=4[/math] and [math]m=5[/math]. Let’s take a moment to recognize the magnitude of this feat: Without this analysis, there are infinitely many possibilities for [math]m[/math]. Now there are only two. That’s progress.
First, let’s solve [math]a^3+b^3=28[/math]. One important realization is that this isn't completely a trivial task: we are asked for solutions in integers, not positive integers, so [math]a[/math] or [math]b[/math] may well be negative, which means we can’t immediately rule out some two huge numbers. The search space doesn’t seem to be finite.
But you see, it is, because [math]a^3+b^3[/math] factors. I prefer to think of it this way: Either [math]a[/math] and [math]b[/math] are both positive, in which case they must both be very small (the sum of their cubes is [math]28[/math], how large can they be?!), or one of them is positive and the other negative. In the latter case we may as well call the positive one [math]a[/math] and the negative one [math]-b[/math], so we have [math]a^3-b^3[/math] for two positive numbers [math]a,b[/math].
But [math]a^3-b^3=(a-b)(a^2+ab+b^2)[/math]. The second factor is visibly positive, so the first one must be as well, and we have a factorization of [math]28[/math] into positive factors. There aren't that many options to check. We’re not done, but we know that the task ahead of us is finite.
So: if [math]a,b[/math] are positive we quickly find [math]a=3[/math], [math]b=1[/math] (or vice versa) and nothing else. In the other case, we check [math]a-b=1[/math] which fails for [math]a=1,2,3,4[/math] and there’s no need to check [math]5[/math] and up because [math]a^2+ab+b^2[/math] is going to be too big. Trying [math]a-b=4[/math] we fail even faster, and [math]a-b=7[/math] or more are ridiculously out of the question.
So, the only two solutions when [math]m=4[/math] are [math]3^3+1^3=4!+4[/math] and its sibling [math]1^3+3^3=4!+4[/math].
For [math]m=5[/math] we don’t find any positive solutions, which is a quick check. Looking at the factors, we try [math]a-b=1[/math] (fail, though [math]7^3-6^3[/math] is damn close), [math]a-b=2[/math] (fail) and then [math]a-b=4[/math] success! [math]5^3+(-1)^3=(5-1)(5^2+5+1)=5!+4[/math]. Continuing on with other factors doesn’t yield anything new.
So, we’re done.
The only solutions math[/math] to [math]a^3+b^3=m!+4[/math] in integers are:
math[/math], math[/math]
math[/math], math[/math]
and there are no others. [math]\blacksquare[/math]
Rik Bos
Ph.D. Mathematics from Utrecht University (Graduated 1979)
·
Upvoted by
Horst H. von Brand
, PhD Computer Science & Mathematics, Louisiana State University (1987) and
Bernard Montaron
, PhD Mathematics & Discrete Mathematics, Université Pierre Et Marie Curie Paris VI (1980) · Author has 1.4K answers and 1.3M answer views
·
1y
Related
Can you find all integer solutions of
[math]x^4+3y^4=z^2[/math]
?
Given any integral solution math[/math] of the equation
[math]x^4+3y^4=z^2\tag{1}[/math]
a prime divisor [math]p[/math] of both [math]x[/math] and [math]y[/math] will give a “reduced” solution math[/math], and conversely any solution math[/math] will give a new solution math[/math], where [math]k[/math] is any integer. So the real problem is to find primitive solutions, i.e. solutions where [math]\gcd(x,y)=1[/math]. Moreover, given a solution math[/math], we may change the sign of any of the coordinates and still keep a solution, so we may as well assume [math]x,y,z\ge 0[/math]. In fact, we can assume [math]x,y,z>0[/math], since the only primitive solutions with a zero coordinate are math[/math] and math[/math].
Given any integral solution math[/math] of the equation
[math]x^4+3y^4=z^2\tag{1}[/math]
a prime divisor [math]p[/math] of both [math]x[/math] and [math]y[/math] will give a “reduced” solution math[/math], and conversely any solution math[/math] will give a new solution math[/math], where [math]k[/math] is any integer. So the real problem is to find primitive solutions, i.e. solutions where [math]\gcd(x,y)=1[/math]. Moreover, given a solution math[/math], we may change the sign of any of the coordinates and still keep a solution, so we may as well assume [math]x,y,z\ge 0[/math]. In fact, we can assume [math]x,y,z>0[/math], since the only primitive solutions with a zero coordinate are math[/math] and math[/math].
Here are a few solutions. math=(1,1,2)[/math] and math[/math] are easy to find, but less easy are math=(11,3,122)[/math], math[/math] and math[/math].
To find all solutions (positive and primitive), we will make use of elliptic curves, though at the end we will mention a more elementary approach that is also more restricted.
The elliptic curve approach
We can convert our quartic equation to an elliptic curve as follows. Multiplying both sides of the equation by [math]x^2/y^6[/math] we obtain [math]x^6/y^6+3x^2/y^2=x^2z^2/y^6[/math]. Substituting [math]X=x^2/y^2[/math] and [math]Y=xz/y^3[/math] we get the elliptic curve [math]E[/math] defined by
[math]Y^2=X^3+3X\tag{2}[/math]
It turns out that the substitutions above set up a correspondence between rational points of [math]E[/math] whose first coordinate is a rational square and rational solutions of the equation [math]x^4+3y^4=z^2[/math] up to a certain equivalence (math[/math] is equivalent to math[/math] for all [math]t\in \Q^[/math]; in particular the equivalence class contains integral solutions and therefore also a primitive solution). For details, see [1,p. 392–393].
Examples of solutions to (1): we start with the generator [math]P=(1,2)[/math] for the elliptic curve [math]E[/math] . Note that [math]P[/math] corresponds to the equivalence class of math[/math].
One can compute multiples of [math]P[/math], for instance [math]2P=(1/4,-7/8)[/math] and by the correspondence mentioned above conclude that [math]x^2/y^2=1/4[/math], so that a primitive solution to (1) has coordinates [math]x=1,y=2[/math] and [math]z=7[/math] (remember we restrict ourselves to nonnegative solutions). Similarly, [math]3P=(121/9,-1342/27)[/math], so since [math]121/9=(11/3)^2[/math] we find the primitive solution math=(11,3,122)[/math] already mentioned above. We can go on and find [math]4P[/math] is [math]2209/784=(47/28)^2[/math], corresponding to the solution math[/math] above, while [math]5P[/math] gives math[/math] and [math]6P[/math] gives math[/math].
Explicit formulas
Instead of giving numerical values, we can try to give formulas for the multiples of [math]P[/math]. Using the correspondence above, it then should be possible to generate all primitive solutions of (1). The idea is to express the coordinates of mathP[/math] in terms of the coordinates of [math]nP[/math].
Before we go into details, here are the formulas to go from primitive solution math[/math] of (1) to a “larger” solution (we will explain below how to deal with the [math]\pm[/math] sign in front of [math]z[/math]):
[math]x'=|2xy-z|[/math], [math]y'=|x^2-y^2|[/math] and [math]z'=2|xy(x^2+3y^2)-(x^2+y^2)z|[/math]
It’s very well possible that [math]\gcd(x',y')>1[/math] and in this case we have to factor out the gcd to obtain a primitive solution.
Ok, let’s start, so let [math]Q=(u,v)\ne P[/math] be a multiple of [math]P[/math]. Then we can determine [math]P+Q=(r,s)[/math] in terms of [math]u[/math] and [math]v[/math]. Keep in mind that math[/math] (note the minus sign!) is the intersection of the elliptic curve [math]E[/math] with the line [math]L[/math] through [math]P[/math] and [math]Q[/math]. Since [math]L[/math] can be described by the equation [math]Y-2=\lambda(X-1)[/math] where [math]\lambda=\frac{v-2}{u-1}[/math] we find, after substitution this in [math]E[/math], that
[math]X^3-\lambda^2X^2+(3-\lambda(2-\lambda))X-(2-\lambda)^2=0\tag{3}[/math]
As [math]P,Q[/math] and [math]-(P+Q)[/math] are intersection points of [math]E[/math] and [math]L[/math] we know that [math]1,u[/math] and [math]r[/math] are solutions to (3), so the polynomial can be factored as math(X-u)(X-r)[/math] and therefore [math]ur=(2-\lambda)^2[/math]. From this it is immediate that if [math]u[/math] is a rational square, the same holds for [math]r[/math]. In fact, [math]2-\lambda=(2u-v)/(u-1)[/math] and if we use the correspondence [math]u=x^2/y^2,v=xz/y^3[/math], then [math]r[/math] turns out to be
[math]r=\frac{(2xy-z)^2}{(x^2-y^2)^2}\tag{4}[/math]
Note that [math]\sqrt r=\frac{|2xy-z|}{|x^2-y^2|}[/math], so to find a new solution to (1) we can use the correspondence above and put [math]x'=|2xy-z|[/math] and [math]y'=|x^2-y^2|[/math] though we should be careful concerning the sign of [math]z[/math]. For instance, based on the solution math=(1,2,7)[/math] we get [math]x'=3,y'=3[/math]. To get a primitive solution we should factor out the gcd, so that would give us the old solution math[/math] corresponding to the point [math]P[/math] on [math]E[/math]. But if instead we use math=(1,2,-7)[/math] we do get a new solution with [math]x'=11,y'=3[/math] (this change in sign from [math]7[/math] to [math]-7[/math] actually corresponds to replacing the point [math]-2P[/math] to [math]2P[/math]).
We still need to express [math]s[/math], the second coordinate of [math]P+Q[/math], in terms of [math]x,y,z[/math], because then we could deduce [math]z'[/math] from [math]s=x'z'/y'^3[/math]. Since math[/math] belongs to [math]L[/math] we have [math]-s=\lambda r+2-\lambda[/math] and this turns out to be
[math]-s=\frac{2(v-2u)(u^2-uv+3u-v)}{u(u-1)^3}[/math]
Again using the correspondence [math]u=x^2/y^2,v=xz/y^3[/math], we find
[math]s=\frac{2(2xy-z)(xy(x^2+3y^2)-(x^2+y^2)z}{(x^2-y^2)^3}\tag{5}[/math]
Then [math]z'=2|xy(x^2+3y^2)-(x^2+y^2)z|[/math]
COHEN, Henri; Number theory: Volume I: Tools and Diophantine Equations. Springer New York, 2007.
A more elementary (and more restricted) approach
Of course, this reduction to an elliptic curve is a beautiful result, but my first attempt was more elementary. I succeeded to derive explicit formulas in case [math]y[/math] is even. More precisely, if math[/math] is a primitive solution with [math]y[/math] even and positive, then [math]x,y,z[/math] can be expressed in terms of a smaller solution:
Suppose we have a primitive solution math[/math], i.e. [math]M^2=R^4+3S^4[/math] and [math]\gcd(R,S)=1[/math].
If [math]R[/math] and [math]S[/math] have different parity then the triple math[/math] with
[math]x=|R^4-3S^4|,y=2MRS[/math] and [math]z=M^4+12R^4S^4[/math] is also a primitive solution.
If [math]R[/math] and [math]S[/math] are both odd, then the triple math[/math] with
[math]x=|R^4-3S^4|/2,y=MRS[/math] and [math]z=M^4/4+3R^4S^4[/math] is a primitive solution.
Moreover, if math[/math] is a primitive solution with [math]y[/math] even and positive, then there is a primitive solution math[/math] such that by applying 1 or 2 above we obtain math[/math].
If we translate the above in terms of the elliptic curve, it turns out that going from math[/math] to math[/math] corresponds to doubling the point on the elliptic curve belonging to math[/math].
Unfortunately, if [math]y[/math] is odd, there doesn’t seem to be such a nice reduction to a smaller solution of equation (1).
Details of the proof are below the line.
We start with a primitive solution math[/math] and let [math]X=x^2,Y=y^2[/math]. Then
[math]X^2+3Y^2=z^2\tag{6}[/math]
Since [math]\gcd(X,Y)=1[/math] we have to find - as an intermediate step - all primitive nonnegative solutions to (6) and later require that [math]X,Y[/math] are perfect squares.
Fortunately, the primitive solutions to (6) can be completely described as two mutually exclusive parameterized formulas given below.
If [math]Y[/math] is even, then [math]X=|m^2-3n^2|,Y=2mn[/math] and [math]z=m^2+3n^2[/math] where [math]m,n[/math] are coprime nonnegative integers of different parity and [math]m[/math] not divisible by [math]3[/math].
If [math]Y[/math] is odd, then [math]X=|m^2-3n^2|/2,Y=mn[/math] and [math]z=(m^2+3n^2)/2[/math], where [math]m[/math] and [math]n[/math] are coprime odd nonnegative integers and [math]m[/math] not divisible by [math]3[/math].
The way to find these solutions is to solve the equation
[math]u^2+3v^2=1\tag{7}[/math]
over the rationals (note that if we divide both sides of (6) by [math]z^2[/math] we get equation (7) with [math]u=X/z,v=Y/z[/math]).
Elsewhere I have shown how to solve the equation [math]u^2+kv^2=1[/math] over the rationals (where [math]k[/math] is any positive integer), so here I will only sketch how to solve (7). The idea is to choose one particular rational solution, for instance math[/math] and draw lines through this point with a rational slope. Such a line will have one other intersection point with the geometric figure defined by (7) (which is an ellipse). If the slope of the line is [math]m/n[/math] where [math]m[/math] and [math]n[/math] are coprime integers, then one finds after some calculations that
[math]u=\frac{m^2-3n^2}{m^2+3n^2},v=\frac{2mn}{m^2+3n^2}[/math]
From this it is not hard to obtain the two parameterizations above.
Now let’s go back to a primitive solution math[/math] of (1).
So suppose [math]x^4+3y^4=z^2[/math] where [math]x[/math] and [math]y[/math] are coprime, [math]y[/math] is even and [math]y>0[/math]. Then by the first parameterization above we have [math]x^2=|m^2-3n^2|,y^2=2mn[/math] and [math]z=m^2+3n^2[/math] for suitable [math]m,n\ge 1[/math]. Since [math]2mn[/math] is a perfect square, we know that either [math]m[/math] or [math]n[/math] is even. Also note we cannot have [math]x^2=-m^2+3n^2[/math] by a mod [math]4[/math] argument. Therefore [math]x^2=m^2-3n^2[/math], so [math]x^2+3n^2=m^2[/math] is the same type of equation as (6). This turns out to be crucial.
Let’s now first assume that [math]n[/math] is even. From [math]y^2=2mn[/math], it follows that [math]m[/math] and [math]2n[/math] are perfect squares. Also [math]x^2=m^2-3n^2[/math] implies that [math]x^2+3n^2=m^2[/math] and we know this is a primitive solution to (6), so the first parameterization above applies: there are suitable [math]r,s[/math] with [math]x=|r^2-3s^2|,n=2rs,m=r^2+3s^2[/math]. But recall that [math]2n[/math] and [math]m[/math] are perfect squares from which we deduce that [math]r[/math] and [math]s[/math] are also perfect squares. Writing [math]m=M^2,r=R^2,s=S^2[/math] where [math]M,R,S\ge 0[/math], the equation [math]m=r^2+3s^2[/math] becomes [math]M^2=R^4+3S^4[/math] which is the same type of equation as [math]z^2=x^4+3y^4[/math]. Since [math]y^2=2mn[/math] and [math]n=2rs[/math], so [math]2n=4R^2S^2[/math], while [math]m=M^2[/math], we find [math]y=2MRS[/math]. Also [math]x=|r^2-3s^2|=|R^4-3S^4|[/math]. Finally, since [math]n=2R^2S^2[/math], we have [math]z=m^2+3n^2=M^4+12R^4S^4[/math]
As an example of a solution to [math]M^2=R^4+3S^4[/math] where [math]r=R^2[/math] and [math]s=S^2[/math] are coprime and of different parity we have math=(1,2,7)[/math] and therefore math=(47,28,2593)[/math]. One can check this is indeed a primitive solution.
Next suppose [math]n[/math] is odd (while we still assume [math]y>0[/math] and [math]y[/math] is even). Then from [math]y^2=2mn[/math] and [math]n[/math] is odd, it follows that [math]2m[/math] and [math]n[/math] are perfect squares. And from [math]x^2+3n^2=m^2[/math] and [math]n[/math] odd, it follows that the second parameterization above applies, so focusing on [math]m[/math] and [math]n[/math] we find there are odd coprime integers [math]r,s[/math] with [math]n=rs[/math] and [math]m=(r^2+3s^2)/2[/math]. Since both [math]2m=r^2+3s^2[/math] and [math]n=rs[/math] are perfect squares and since [math]r[/math] and [math]s[/math] are coprime, we can write [math]r=R^2,s=S^2[/math]. Also, [math]2m=M^2[/math], for some positive integer [math]M[/math] and so [math]R^4+3S^4=M^2[/math]. As above, we find [math]y^2=2mn=M^2R^2S^2[/math], so [math]y=MRS[/math]. Moreover, [math]x=|r^2-3s^2|/2=|R^4-3S^4|/2[/math] and [math]z=m^2+3n^2=M^4/4+3R^4S^4[/math].
As an example of a solution to [math]M^2=R^4+3S^4[/math] where [math]r=R^2[/math] and [math]s=S^2[/math] are coprime and odd, we have math=(1,1,2)[/math] and so math=(1,2,7)[/math]. Similarly, math[/math] gives math[/math].
So this solves the original equation in case [math]y[/math] is even.
[For completeness, here are a few equations that can be derived in case [math]y[/math] is odd. Using a similar approach as above, there doesn’t seem to be a way to construct a smaller solution from which math[/math] can be constructed. Of course, using the elliptic curve approach, we know there actually is such a smaller solution, but it doesn’t show up in the more elementary approach.]
Let’s now suppose [math]y[/math] is odd. Then by the second parameterization above we have [math]x^2=|m^2-3n^2|/2,y^2=mn[/math] and [math]z=(m^2+3n^2)/2[/math] for odd coprime [math]m,n\ge 1[/math]. Anyway, we see that [math]m[/math] and [math]n[/math] are perfect squares.
However, we now get a different type of equation, namely from [math]x^2=|m^2-3n^2|/2[/math] we obtain either [math]2x^2+3n^2=m^2[/math] or [math]3n^2=m^2+2x^2[/math]. But the first equation cannot be solved as [math]m,n[/math] are odd, so [math]m^2[/math] and [math]n^2[/math] are [math]1\pmod 8[/math] and then [math]2x^2\equiv -2\pmod 8[/math], so [math]x^2\equiv -1\pmod 4[/math], which is impossible. The second one can be solved: we have solutions to [math]3n^2=m^2+2x^2[/math] with
[math]m=|r^2+4rs-2s^2|,x=|-r^2+2rs+2s^2|[/math] and [math]n=r^2+2s^2[/math]
However, from here, there seems to be no way to retrieve a smaller solution to (1) that depends somehow on [math]r,s,m[/math].
Sponsored by CDW Corporation
Want document workflows to be more productive?
The new Acrobat Studio turns documents into dynamic workspaces. Adobe and CDW deliver AI for business.
Jos van Kan
Knows a thing or two about mathematics
·
Upvoted by
Shubhankar Datta
, Master of Science Mathematics, Jadavpur University (2022) · Author has 2.5K answers and 2.1M answer views
·
Updated 7y
Related
How do I find all the integer solutions of
[math]y^2=x(x+6)(x+12)[/math]
?
You mean every one of them? You need a big sheet of paper, because what we have here is an elliptic curve and if you find one non trivial integer point on that curve (the trivial ones are (0,0),(-6,0) and (-12,0)) you immediately get another one by drawing a straight line through two points with different [math]x-[/math]coordinates and find the intersection with the curve.. (Not quite, see EDIT)
Let’s try an intersection with a straight line through the origin:
[math]y=mx\m^2x^2=x^3+18x^2+72x\x^2+(18-m^2)x+72=0[/math]
and we need math^2-288[/math] to be a perfect square in order to have a possible integer solution.
[math]m^4-36m^[/math]
You mean every one of them? You need a big sheet of paper, because what we have here is an elliptic curve and if you find one non trivial integer point on that curve (the trivial ones are (0,0),(-6,0) and (-12,0)) you immediately get another one by drawing a straight line through two points with different [math]x-[/math]coordinates and find the intersection with the curve.. (Not quite, see EDIT)
Let’s try an intersection with a straight line through the origin:
[math]y=mx\m^2x^2=x^3+18x^2+72x\x^2+(18-m^2)x+72=0[/math]
and we need math^2-288[/math] to be a perfect square in order to have a possible integer solution.
[math]m^4-36m^2+36[/math] is perfect square readily yield 4 solutions: [math]m=\pm1[/math] and [math]m=\pm6.[/math] Yes, [math]m=0[/math] is a solution too, but that gives us back the trivial solutons.
The problem is symmetric with respect to that [math]x-[/math] axis, so whenever math[/math] is a solution, so is math.[/math]
[math]m=1[/math] yields the points math[/math] and[math] (-9,\pm9)[/math]
[math]m=6 [/math]yields the points math[/math] and math[/math]
Just to show that I am not bluffing let us draw a line through math [/math] and math [/math] and find the intersection:
math(-9+8)-(x+8)(-9-8)=0\y=17x+144[/math]
Substitute
math^2=x^3+18x^2+72x\x^3-271x^2-(288\cdot17-72)x+144^2=0[/math]
Fortunately we already know two solutions to that equation, so divide by math(x+9)=x^2+17x+72,[/math] the quotient is [math]x-288.[/math]
Modulo calculation errors. So we have a new point math.[/math]
If you continue the numbers are growing very fast and you better get a computer program. If I remember correctly Alon Amit knows someone who has that.
EDIT It is true that a line through two integer points generates a new rational point, but for a new integer point you must satisfy an additional condition. In this particular situation you must satisfy the condition that
[math]\left(\dfrac{x_0y_1-x_1y_0}{x_1-x_0}\right)^2[/math] is divisible by [math]x_0x_1.[/math]
In our example this condition is easily satisfied, because [math]x_1-x_0=1[/math] and [math]y_k=\pm m x_k,[/math] but for any other combination of the known integer points it fails. So I think we have got them all.
Amitabha Tripathi
have more than a working knowledge of Z
·
Upvoted by
Alon Amit
, Lover of math. Also, Ph.D. and
Parth Trivedi
, M.Sc. Mathematics, Bhavnagar University (2008) · Author has 4.7K answers and 13.9M answer views
·
7y
Related
What are the integer solutions to
[math]a^3+b^3=m!+4[/math]
?
Observe that [math]n^3[/math] is congruent to [math]0[/math] or [math]\pm 1[/math] modulo [math]9[/math]. math[/math] Thus, [math]a^3+b^3[/math] modulo [math]9[/math] can only be [math]0[/math], [math]\pm 1[/math], [math]\pm 2[/math]. However, if [math]m \ge 6[/math], then [math]3 \cdot 6 \mid m![/math], and so [math]m!+4 \equiv 4\pmod{9}[/math].
So [math]a^3+b^3=m!+4[/math] is only possible with [math]m \in {0,1,2,3,4,5}[/math]. We may verify that [math]0!+4=1!+4=5[/math], [math]2!+4=6[/math], and [math]3!+4=10[/math] cannot be written as a sum of two integral cubes, while [math]4!+4=28=3^3+1^3[/math] and [math]5!+4=124=5^3–1^3[/math].
We claim that the only solution to [math]a^3+b^3=28[/math] is [math]{a,b}={1,3}[/math] and that the only solution to [math]a^3+b^3=124[/math] is [math]{a,b}={-1,5}[/math].
To see th
Observe that [math]n^3[/math] is congruent to [math]0[/math] or [math]\pm 1[/math] modulo [math]9[/math]. math[/math] Thus, [math]a^3+b^3[/math] modulo [math]9[/math] can only be [math]0[/math], [math]\pm 1[/math], [math]\pm 2[/math]. However, if [math]m \ge 6[/math], then [math]3 \cdot 6 \mid m![/math], and so [math]m!+4 \equiv 4\pmod{9}[/math].
So [math]a^3+b^3=m!+4[/math] is only possible with [math]m \in {0,1,2,3,4,5}[/math]. We may verify that [math]0!+4=1!+4=5[/math], [math]2!+4=6[/math], and [math]3!+4=10[/math] cannot be written as a sum of two integral cubes, while [math]4!+4=28=3^3+1^3[/math] and [math]5!+4=124=5^3–1^3[/math].
We claim that the only solution to [math]a^3+b^3=28[/math] is [math]{a,b}={1,3}[/math] and that the only solution to [math]a^3+b^3=124[/math] is [math]{a,b}={-1,5}[/math].
To see this, note that [math]a^3+b^3=(a+b)(a^2-ab+b^2)[/math] and [math]4(a^2-ab+b^2)=(2a-b)^2+3b^2>0[/math] together imply math \mid (a^3+b^3)[/math] and [math]a^3+b^3>0 \Leftrightarrow a+b>0[/math]. So if [math]a^3+b^3>0[/math], we may assume [math]a>|b|[/math]. Now math-(a+b)=a(a-b-1)+b(b-1) \ge a(a-b-1)>0[/math]. Thus, if [math]a^3+b^3>0[/math], we have [math]a+b<a^2-ab+b^2=(a+b)^2–3ab[/math].
math[/math] If [math]a^3+b^3=28[/math], then math \in {(2,14),(4,7)}[/math]. In the first case, [math]ab=\frac{1}{3}\big((a+b)^2-(a^2-ab+b^2)\big) \notin \mathbb Z[/math]; in the second case, [math]ab=3[/math] and so [math]a=3[/math], [math]b=1[/math].
math[/math] If [math]a^3+b^3=124[/math], then math \in {(2,62),(4,31)}[/math]. In the first case, [math]ab=\frac{1}{3}\big((a+b)^2-(a^2-ab+b^2)\big) \notin \mathbb Z[/math]; in the second case, [math]ab=-5[/math] and so [math]a=5[/math], [math]b=-1[/math].
The only solutions to [math]a^3+b^3=m!+4[/math] are [math]3^3+1^3=4!+4[/math] and [math]5^3–1^3=5!+4[/math]. [math]\blacksquare[/math]
Alon Amit
PhD in Mathematics; Mathcircler.
·
Upvoted by
James McElhatton Ph.D. (Glasgow, 1976)
, B.Sc. Mathematics & Chemistry, University of Malta (1967) and
Yair Livne
, Master's Mathematics, Hebrew University of Jerusalem (2007) · Author has 8.8K answers and 173.8M answer views
·
6y
Related
What are the integer solutions to
[math]x^2+5=y^3[/math]
?
There aren’t any.
This equation is usually written [math]y^2=x^3-5[/math] (switching the roles of [math]x[/math] and [math]y[/math]). Viewed this way, it’s an elliptic curve in standard Weierstrass form, and in fact a Mordell equation. Mordell’s equation is simply any equation that looks like [math]y^2=x^3+k[/math], and despite their apparent simplicity, many of those equations are very difficult.
Luckily, this one isn’t. It can be shown to have no solution in integers using elementary considerations involving quadratic residues; if you’re not familiar with those, that’s a good reason to learn! Quadratic residues are fundamental for many problems
There aren’t any.
This equation is usually written [math]y^2=x^3-5[/math] (switching the roles of [math]x[/math] and [math]y[/math]). Viewed this way, it’s an elliptic curve in standard Weierstrass form, and in fact a Mordell equation. Mordell’s equation is simply any equation that looks like [math]y^2=x^3+k[/math], and despite their apparent simplicity, many of those equations are very difficult.
Luckily, this one isn’t. It can be shown to have no solution in integers using elementary considerations involving quadratic residues; if you’re not familiar with those, that’s a good reason to learn! Quadratic residues are fundamental for many problems in Number Theory.
One good thing to do with any Diophantine equation is factor it. [math]x^3-5[/math] cannot be factored over the integers, nor can [math]y^2+5[/math] if we tried moving the [math]5[/math] to the other side.
But we can do this, for example:
That’s the same equation, but now it can be factored:
The [math]+4[/math] on the left beckons us to consider what happens modulo [math]4[/math], which is always a good idea anyway. So let’s do that.
Modulo [math]4[/math], [math]y^2[/math] is [math]0[/math] or [math]1[/math], so [math]y^2+4[/math] is the same. On the other hand, [math]x^3[/math] is [math]0[/math], [math]1[/math] or [math]3[/math], so [math]x^3-1[/math] is [math]-1=3[/math], [math]0[/math] or [math]2[/math]. The only common value in those lists is [math]0[/math], so we conclude that [math]x[/math] is congruent to [math]1[/math] mod [math]4[/math].
Ok. We now know a little bit about our unknown variables [math]x,y[/math], but it’s still unclear what sort of contradiction arises from our equation.
The trick is to consider quadratic residues. The equation can be rearranged again to say
[math]-4 = y^2 - (x-1)(x^2+x+1)[/math]
Now if a prime [math]p[/math] divides [math]x^2+x+1[/math] then [math]-4[/math] is a quadratic residue modulo [math]p[/math]. Since [math]4[/math] is a square, this simply says that [math]-1[/math] is a quadratic residue modulo [math]p[/math], and this says that [math]p[/math] is either [math]2[/math] or itself congruent to [math]1[/math] mod [math]4[/math].
But lo, [math]x^2+x+1[/math] is congruent to [math]3[/math] mod [math]4[/math] (since [math]x[/math] is [math]1[/math] mod [math]4[/math]), and it’s also clearly positive, so its prime factors cannot all be [math]2[/math] and things congruent to [math]1[/math] mod [math]4[/math]. There must be at least one prime factor [math]p[/math] of [math]x^2+x+1[/math] which is congruent to [math]3[/math] mod [math]4[/math], which contradicts [math]-1[/math] being a quadratic residue mod [math]p[/math]. [math]\blacksquare[/math]
Lev Borisov
mathematics professor
·
Author has 1.9K answers and 1.7M answer views
·
1y
Related
How can you find all integer solutions of
[math]2^n+3^k=5^m[/math]
?
I can only get a partial result.
Clearly, we want the integers to be nonnegative, denominators can’t cancel here. If [math]k=0[/math] then we have an equation [math]5^m-2^n=1[/math]. For [math]0\leq n<3[/math] a simple search finds [math]5^1=2^2+3^0[/math] as the only option. Then if [math]n\geq 3[/math], working modulo [math]8[/math] we see that [math]m[/math] is even. This gives math(5^r+1) = 2^n[/math]. Both factors must be powers of [math]2[/math] but their gcd is [math]2[/math], which leads to [math]5^r-1=2[/math] which is impossible.
So we can now focus on the [math]k>0[/math] case. Looking at this modulo [math]3[/math], to see that [math]n[/math] and [math]m[/math] are either both even or are both odd.
If they are both even, then the situation is fairly straightforward,
I can only get a partial result.
Clearly, we want the integers to be nonnegative, denominators can’t cancel here. If [math]k=0[/math] then we have an equation [math]5^m-2^n=1[/math]. For [math]0\leq n<3[/math] a simple search finds [math]5^1=2^2+3^0[/math] as the only option. Then if [math]n\geq 3[/math], working modulo [math]8[/math] we see that [math]m[/math] is even. This gives math(5^r+1) = 2^n[/math]. Both factors must be powers of [math]2[/math] but their gcd is [math]2[/math], which leads to [math]5^r-1=2[/math] which is impossible.
So we can now focus on the [math]k>0[/math] case. Looking at this modulo [math]3[/math], to see that [math]n[/math] and [math]m[/math] are either both even or are both odd.
If they are both even, then the situation is fairly straightforward, because we can write [math]3^k = (5^a+2^b)(5^a-2^b)[/math] and each of these factors must be a power of [math]3[/math]. However, their sum is coprime to [math]3[/math], which means that [math]5^a-2^b = 1, 5^a + 2^b = 3^k[/math]. We already solved the first equation to get [math]a=1,b=2[/math] which happens to fit the second equation and lead to [math]2^4+3^2=5^2[/math].
So now we are in the situation where [math]n[/math] and [math]m[/math] are both odd. If [math]n[/math] is at least [math]3[/math] then working modulo [math]8[/math] we get a contradiction, since [math]3^k [/math] can only be [math]1[/math] or [math]3[/math] but [math]5^m[/math] gives [math]5[/math]. If [math]n=0,2[/math] we still get trouble modulo [math]8[/math] so it remains to look at [math]n=1[/math]. We get [math]2+3^k = 5^m[/math]. We again argue mod [math]8 [/math]that [math]k[/math] must be odd. If [math]k=1[/math] we get a solution [math]2^1 + 3^1 = 5^1[/math].
It gets hard now.
If [math]k>1[/math] my inclination it to write this as [math] 10 = X^2 - 15 Y^2[/math] where [math]X,Y[/math] are powers of [math]5[/math] and [math]3[/math] respectively and somehow work in [math]\mathbb Z[\sqrt{15}][/math]. The class group is unfortunately nontrivial, but still one should be able to write math[/math] as some constant times a power of the fundamental unit math[/math]. I am not really sure how to proceed beyond that, though.
Edit: Inspired by Bernard Montaron’s comment, I can look at [math]2+3^k=5^m[/math] modulo [math]5[/math] to see that [math]k=1 \mod 4[/math]. This gives [math]2+3 X^4 = 5 Y^2[/math] for [math]X[/math] and [math]Y[/math] powers of [math]3[/math] and [math]5[/math] respectively. Importantly, the equation [math]2+3 X^4 = 5 Y^2[/math] describes an elliptic curve, so if it happens to have rank zero (which is verifiable) then one can find all rational solutions and check.
This is not something I was willing to do by hand, but with the help of various software, I got the following. If we substitute [math]y=(1/5)(1 + t - 6 x - 2 t x + t x^2)[/math] then we get a quadratic equation on [math]x[/math] whose discriminant has to be a square of the rational number. This gives us that [math]10(t+3)(3+6t+t^2)[/math] is a square of a rational number. Alas, this curve has rank at least 2.
Upvoted by
Patrick T
, PhD Mathematics, University of California, Davis (2018) and
Utkarsh Priyam
, studied Mathematics at Harker School (2021)
·
7y
Related
What are the integer solutions to
[math] a + b = ab [/math]
?
For the equation a+b=ab, it is easy to transform to ab-a-b=0.
Plus 1 on
BOTH
sides→ ab-a-b+1=1
Do Factorization→ (a-1)(b-1)=1
a-1=1, b-1=1→a=2, b=2
a-1=-1,b-1=-1→a=0, b=0
The solution set, (2,2) (0,0)
Related questions
What are the integer solutions of
a
2
+
7
=
2
b
2
?
How do I find all the integer solutions of
y
2
=
x
(
x
+
6
)
(
x
+
12
)
?
Why does
a
2
+
2
b
2
=
c
2
have no integer solutions?
What are the integer solutions to
a
+
b
=
a
b
?
How do you find the integer solutions for
12
a
b
2
+
21
a
b
−
4
b
2
−
6
a
−
7
b
=
178
?
What are the integer solutions to
(
a
2
−
1
)
(
b
2
−
1
)
=
c
2
?
What are the integer solutions to
a
3
+
b
3
=
m
!
+
4
?
What are the integer solutions to
x
2
+
x
y
+
2
y
2
=
29
?
How do you find integer solutions to
2
x
2
+
27
x
+
91
=
y
2
?
What are the integer solutions to
x
2
−
1
=
2
y
2
?
What are the integer solutions to
12
≥
2
x
+
y
−
2
x
?
How do you find the integer solutions to
y
=
√
x
2
−
n
if
n
is known?
How do I find the integer solutions for
cos
(
(
π
x
2
+
n
π
)
x
)
−
1
if
n
is known?
How does one find all integer solutions to
x
3
−
27
y
3
=
2
?
If
a
2
+
b
2
=
c
2
and
b
2
+
c
2
=
d
2
, what is the integer solution for a, b, c, and d?
About
·
Careers
·
Privacy
·
Terms
·
Contact
·
Languages
·
Your Ad Choices
·
Press
·
© Quora, Inc. 2025 |
13036 | https://ehtrust.org/educate-yourself__trashed/sar-of-cell-phones-specific-absorption-rate/ | SAR of Cell Phones Specific Absorption Rate
The American Academy of Pediatrics has called for a re-evaluation of the SAR to better assess real world exposures because, “the current metric of RF exposure available to consumers, the Specific Absorption Rate, is not an accurate predictor of actual exposure.”
Why? Because the SAR is an outdated irrelevant test for cell phone radiation.
When did the SAR become part of US cell phone regulations?
The Specific Absorption Rate (SAR) for cellphones was set in 1993 when less than 10% of the population used cellphones. Phones then were used chiefly for business and military purposes and cost $900 at that time and were used for a few minutes a month.
Why do scientists state that SAR is an inadequate metric?
The SAR is a measure of the maximum amount of microwave radiation absorbed by a test dummy, not the amount of microwave radiation emitted by a wireless device.
The SAR was based on a model of the human head taken from a military recruit at the top 10% in height and weight, and thus falls in the top 3% of all persons using phones today, when millions of children and smaller adults use these devices. Today, most adults and growing numbers of young children use phones for thousands of minutes each month, according to PEW Foundation surveys and reports from the CTIA.
The SAR is a measure of maximum power emitted under one frequency. Most smartphones today use several different frequencies in a single call, or with multiple uses of the phone taking place simultaneously.
The sole criterion for the SAR rests on the level of induced heat to be avoided and does not reflect growing scientific reports that levels that do not induce measurable changes in heat can have impacts on biological systems. Thus, cell-phone like radiation is now used in FDA approved medical applications to treat liver and brain cancer ( hepatocellular carcinoma and glioblastoma multiforme). Other measured impacts of cellphone radiation – not included in the SAR- include increasing the production of biological markers in living systems that have been linked with increased risks of cancer, reproductive damage and neurodegenerative diseases, including enhanced production of damaging free radicals, reactive oxygen species, membrane weakening, and heat shock proteins involved in cellular repair and defense.
What are the SAR limits in the US?
Ears, hands, feet, wrists, and ankles can be exposed to 4W/kg averaged over 10g. (In 2013, ears were reclassified as appendages). Prior to that, exposure limit for ears was at 1.6W/kg, but when phones are held to the ears, the 1.6W/kg was exceeded. So the limit for ears was raised because the 1.6W/kg exposure limit could not be met for ears. All other parts of the body (including the head) can be exposed to a SAR of 1.6W/kg averaged over a 1 g sample of tissue for 30 minutes (established in 1996).
Can humans be exposed to higher SAR levels that the US government allows?
Yes. For example, the iphone 6 plus is rated at 1.59 W/kg when all antennas are on but this rating is based on a distance of 5mm from the body. Read fine print warnings at this link. So, if you are holding the iphone in your hand or pressing it to your head (this is all less than 5mm from the body), your hand or head are exposed to higher than 1.59W/kg
Oh Sam What a Man from Kate Corcoran on Vimeo.
What should be done to protect cell phone users?
The French Health Agency’s 2016 Report states that reliance on SAR to set human exposure limits should be re-evaluated and replaced through the development of an indicator to assess real exposures for mobile phone users that applies to various conditions: signal type, good or bad reception, mode of use (call, data loading, etc.), location device is used on the body.
These scientifically driven regulatory developments in France, Israel and countries worldwide should be considered in reviewing the scientific grounds for US standards for cellphones today.
The fact is that SAR does not ensure safety. Anatomical modeling of brains of various ages shows us how RF-EMF radiation penetrates tissue with deeper penetrations into children. Computer simulations that show peak SAR rates into the brain should be a part of the evaluation as hotspots can occur in tissue and this is not accounted for in the SAR metric.
We need to adopt biologically-based safety limits and an appropriate testing procedure. Cell phone/wireless device emission regulations and standards should be revised to reflect current science, incorporating information developed and reviewed by the International Agency for Research on Cancer in its monograph on cellphone and other wireless radiation, which determined in 2011 that cellphone and other wireless radiation is classified as a “possible human carcinogen,” a category shared with engine exhausts and some pesticides such as DDT and Kepone that have received concerted regulatory attention around the world.
Research results from the National Institute of Environmental Health Sciences National Toxicology Program, in addition to post 2011 human studies, now clearly link long term cell phone radiation exposure to cancer. The $25 million US government study showed a statistically significant increase in glioma and schwannomas in rats exposed to cell phone radiation at non-thermal radiation levels.
If the International Agency for Research on Cancer were to meet now, cellphone and wireless radiation would classified as a “probable human carcinogen,” and some scientists state the evidence nw meets criteria for Group 1: “Carcinogenic to humans- There is enough evidence to conclude that it can cause cancer in humans”.
The current FCC website does not reflect state of the art scientific information. The FCC inaccurately states that:
“ALL cell phones must meet the FCC’s RF exposure standard, which is set at a level well below that at which laboratory testing indicates, and medical and biological experts generally agree, adverse health effects could occur.”
The FCC falsely describes the opinion of scientists. Over 220 scientists agree that current standards do not protect health. The FCC does not include a single employee with expertise in health. There are no ongoing monitoring studies underway to evaluate potential health impacts of cellphones in the department of Health and Human Services or at the National Institute of Health.
Based on the body of research associating cell phones to cancer, Consumer Reports believes”swift and decisive action by the government and industry” is required.
Specifically:
The National Institutes of Health should commission another animal study using current cell phone technology to determine if it poses the same risks as found in this new study.
The Federal Communications Commission should update its requirements for testing the effect of cell phone radiation on human heads. The agency’s current test is based on the devices’ possible effect on large adults, though research suggests that children’s thinner skulls mean they may absorb more radiation. The FCC should develop new tests that take into account the potential increased vulnerability of children.
The Food and Drug Administration and the FCC should determine whether the maximum specific absorption rate of 1.6 W/kg over a gram of tissue is an adequate maximum limit of radiation from cell phones.
The Centers for Disease Control and Prevention should repost it’s advice on the potential hazard of cell phone radiation and cautionary advice that was taken down in August 2014.
Cell phone manufacturers should prominently display advice on steps that cell phone users can take to reduce exposure to cell phone radiation.
Consumer Reports recommends that you:
Try to keep the cell phone away from your head and body. Keeping it an arm’s distance away significantly reduces exposure…
Text or video call when possible…
When speaking, use the speakerphone or a hands-free headset.
Based on this study, an article from Consumer Reports was published a few months ago.
Don’t stow your phone in your pants or shirt pocket. Instead, carry it in a bag or use a belt clip.
For more tips read EHT’s 10 Things You Can Do To Reduce The Health Risk From Cell Phones.
Arabic Chinese (Simplified) Dutch English French German Italian Portuguese Russian Spanish |
13037 | https://allen.in/dn/qna/646658570 | A(2, 0) and B(4, 0) are two given points. A point P moves so that P A 2 + P B 2 = 10 . Find the locus of P.
Courses
NEET
Class 11th
Class 12th
Class 12th Plus
JEE
Class 11th
Class 12th
Class 12th Plus
Class 6-10
Class 6th
Class 7th
Class 8th
Class 9th
Class 10th
View All Options
Online Courses
Distance Learning
Hindi Medium Courses
International Olympiad
Test Series
NEET
Class 11th
Class 12th
Class 12th Plus
JEE (Main+Advanced)
Class 11th
Class 12th
Class 12th Plus
JEE Main
Class 11th
Class 12th
Class 12th Plus
Classroom
Results
NEET
2025
2024
2023
2022
JEE
2025
2024
2023
2022
Class 6-10
Scholarships NEW
TALLENTEX
AOSAT
ALLEN E-Store
More
ALLEN for Schools
About ALLEN
Blogs
News
Careers
Request a call back
Book home demo
Login
HomeClass 12MATHSA(2, 0) and B(4, 0) are two given points...
A(2, 0) and B(4, 0) are two given points. A point P moves so that P A 2 + P B 2 = 10 . Find the locus of P.
To view this video, please enable JavaScript and consider upgrading to a web browser thatsupports HTML5 video
Video Player is loading.
Play Video
Play Skip Backward Skip Forward
Mute
Current Time 0:00
/
Duration-:-
Loaded: 0%
Stream Type LIVE
Seek to live, currently behind live LIVE
Remaining Time-0:00
1x
Playback Rate
2x
1.5x
1x, selected
0.5x
0.25x
Chapters
Chapters
Descriptions
descriptions off, selected
Captions
captions settings, opens captions settings dialog
captions off, selected
Audio Track
Picture-in-Picture Fullscreen
This is a modal window.
Beginning of dialog window. Escape will cancel and close the window.
Text Color Opacity Text Background Color Opacity Caption Area Background Color Opacity
Font Size Text Edge Style Font Family
Reset Done
Close Modal Dialog
End of dialog window.
Text Solution
AI Generated Solution
|
Share
Topper's Solved these Questions
TEST PAPERS
ALLEN|Exercise CHEMISTRY|18 Videos View Playlist
TEST PAPERS
ALLEN|Exercise MATHEMATICS|65 Videos View Playlist
TEST PAPERS
ALLEN|Exercise part-2 Mathematic|18 Videos View Playlist
TEST PAPER
ALLEN|Exercise CHEMISTRY SECTION-II|3 Videos View Playlist
Similar Questions
Explore conceptually related problems
A(2, 0) and B(4, 0) are two given points. A point P moves so that PA^(2)+PB^(2)=10 . Find the locus of P.
Watch solution
A (a,0) , B (-a,0) are two points . If a point P moves such that anglePAB - anglePBA = pi//2 then find the locus of P.
Watch solution
A(a,0) , B(-a,0) are two point . If a point P moves such that anglePAB - anglePBA = 2alpha then find the locus of P .
Watch solution
A and B are two given points whose coordinates are (-5, 3) and (2, 4) respectively. A point P moves in such a manner that PA : PB = 3:2 . Find the equation to the locus traced out by P .
Watch solution
If A and B are two fixed points and P is a variable point such that PA + PB = 4 , the locus of P is
Watch solution
A (c,0) and B (-c,0) are two points . If P is a point such that PA + PB = 2a where 0 lt c lt a, then find the locus of P.
Watch solution
A(1,1) , B (-2,3) are two points . If a point P forms a triangle of are 2 square units with A, B then find the locus of P.
Watch solution
If P is the middle point of the straight line joining a given point A (1, 2) and Q , where Q is a variable point on the curve x^2 + y^2 + x + y=0 . Find the locus of P .
Watch solution
If A(1, 1) and B (-2, 3) are two fixed points, find the locus of a point P so that area of DeltaPAB is 9 units.
Watch solution
The coordinates of the point A and B are (a,0) and (-a ,0), respectively. If a point P moves so that P A^2-P B^2=2k^2, when k is constant, then find the equation to the locus of the point P .
Watch solution
ALLEN-TEST PAPERS-part-2 Mathematics
If x=2 is a real root of cubic equation f (x) = 3x^(3)+bx^(2)+bx+3=0. ...Text Solution | Play
The solution set of inequality (tan^(-1)x)(cot^(-1)x)-(tan^(-1)x)(1+...05:12 | Play
A(2, 0) and B(4, 0) are two given points. A point P moves so that P A...03:49 | Playing Now
Let A(3, 2) and B(6, 5) be two points and a point C (x,y) is chosen on...04:13 | Play
Let f and g be two real values functions defined by f ( x ) = x + 1 ...01:34 | Play
The number of injections possible from A= {1,3,5,6} to B= {2,8,11}...02:07 | Play
Let f:R→R:f(x)=x ^2+1. find f^(−1) {10}01:38 | Play
If f:C→C is defined by f(x)=x^4 then find f^(-1){1}.02:39 | Play
Prove that the function f defined by f(x)=x^2−x+1 is neither increasin...02:01 | Play
If sinΘ+cosec Θ=2 find the value of cotΘ+cosΘ 01:34 | Play
If 2sin^(-1)((9-4x^(2))/(9+4x^(2)))+3cos^(-1)((12x)/(9+4x^(2)))+tan^(-...Text Solution | Play
The value of cos1^o.cos2 ^o.cos3^o.....cos179^o is equal to 01:05 | Play
If A=15 deg, verify that 4sin2A.cos4A.sin6A=1 01:48 | Play
If cos(α+β)=0 , then sin(α−β), can be reduced to 01:39 | Play
A(x(1),y(1)) and B(x(2),y(2)) are any two distinct points on the parab...05:31 | Play
If f:R rarr R is continous function satisying f(0) =1 and f(2x) -f(x)...Text Solution | Play
Consider the parabola y^(2)=8x, if the normal at a point P on the par...07:08 | Play
If sinθ1+sinθ 2 +sinθ 3=3, then cosθ 1+cosθ 2+cosθ3 is equal to 01:48 | Play
The equations of the transverse and conjugate axes of a hyperbola are...Text Solution | Play
Find the locus of the midpoints of chords of hyperbola 3x^(2)-2y^(2)+4...04:59 | Play
HomeProfile |
13038 | https://www.accessdata.fda.gov/drugsatfda_docs/label/2021/021383Orig1s010lbl.pdf | Prilocaine HCI 4% with Epinephrine 1:200,000 Injection (prilocaine HCl and epinephrine injection, USP) For Local Anesthesia in Dentistry Rx Only DESCRIPTION Prilocaine HCI 4% with Epinephrine 1:200,000 Injection is a sterile, non-pyrogenic, isotonic solution that contains a local anesthetic agent with epinephrine (as bitartrate) and is administered parenterally by injection. See INDICATIONS AND USAGE for specific uses. The quantitative composition is shown in Table 1. Prilocaine HCl 4% with Epinephrine 1:200,000 contains prilocaine HCl, which is chemically designated as propanamide, N-(2-methyl-phenyl) -2- (propylamino)-, monohydrochloride and has the following structural formula: And epinephrine, which is (-) -3, 4-Dihydroxy- ∝-[(methylamino) methyl] benzyl alcohol and has the following structural formula: Parenteral drug products should be inspected visually for particulate matter and discoloration prior to administration. The specific quantitative composition is shown in Table 1. Table 1. COMPOSITION Product Identification Formula (mg/mL) Prilocaine HCl Epinephrine (as the bitartrate) Citric Acid Sodium Metabisulfite pH Prilocaine HCI 4% with Epinephrine Injection 40.0 0.005 0.2 0.5 3.3 to 5.5 Note: Sodium hydroxide and/or hydrochloric acid may be used to adjust the pH of Prilocaine HCl 4% with Epinephrine. CLINICAL PHARMACOLOGY Mechanism of Action: Prilocaine stabilizes the neuronal membrane by inhibiting the ionic fluxes required for the initiation and conduction of impulses, thereby effecting local anesthetic action. Onset and Duration of Action: When used for infiltration injection in dental patients, the time of onset of anesthesia averages less than 2 minutes with an average duration of soft tissue anesthesia of approximately 2¼ hours. When used for inferior alveolar nerve block, the time of onset averages less than three minutes with an average duration of soft tissue anesthesia of approximately 3 hours. Reference ID: 4883113 This label may not be the latest approved by FDA. For current labeling information, please visit Hemodynamics: Excessive blood levels may cause changes in cardiac output, total peripheral resistance, and mean arterial pressure. These changes may be attributable to a direct depressant effect of the local anesthetic agent on various components of the cardiovascular system and/or the beta-adrenergic receptor stimulating action of epinephrine. Pharmacokinetics and Metabolism: Information derived from diverse formulations, concentrations and usages reveals that prilocaine is completely absorbed following parenteral administration, its rate of absorption depending, for example, upon such factors as the site of administration and the presence or absence of a vasoconstrictor agent. Prilocaine is metabolized in both the liver and the kidney and excreted via the kidney. It is not metabolized by plasma esterases. Hydrolysis of prilocaine by amidases yields ortho-toluidine and n-propylalanine. Both of these compounds may undergo ring hydroxylation. O-toluidine has been found to produce methemoglobin, both in vitro and in vivo (see ADVERSE REACTIONS). Because prilocaine is metabolized in both the liver and kidneys, hepatic and renal dysfunction may alter prilocaine kinetics. As with other local anesthetic agents, the plasma binding of prilocaine may be dependent on drug concentration. At 0.5 to 1.0 mg/mL, it is 55% protein bound. Prilocaine crosses the blood-brain and placental barriers, presumably by passive diffusion. Factors such as acidosis and the use of CNS stimulants and depressants affect the CNS levels of prilocaine required to produce overt systemic effects. In the rhesus monkey, arterial blood levels of 20 mg/mL have been shown to be the threshold for convulsive activity. INDICATIONS AND USAGE Prilocaine HCl 4% with Epinephrine 1:200,000 is indicated for the production of local anesthesia in dentistry by nerve block or infiltration techniques. Only accepted procedures for these techniques as described in standard textbooks are recommended. CONTRAINDICATIONS Prilocaine is contraindicated in patients with a known history of hypersensitivity to local anesthetics of the amide type and in those rare patients with congenital or idiopathic methemoglobinemia. WARNINGS DENTAL PRACTITIONERS WHO EMPLOY LOCAL ANESTHETIC AGENTS SHOULD BE WELL VERSED IN DIAGNOSIS AND MANAGEMENT OF EMERGENCIES THAT MAY ARISE FROM THEIR USE. RESUSCITATIVE EQUIPMENT, OXYGEN AND OTHER RESUSCITATIVE DRUGS SHOULD BE AVAILABLE FOR IMMEDIATE USE. Methemoglobinemia Cases of methemoglobinemia have been reported in association with local anesthetic use. Although all patients are at risk for methemoglobinemia, patients with glucose-6-phosphate dehydrogenase deficiency, congenital or idiopathic methemoglobinemia, cardiac or pulmonary compromise, infants under 6 months of age, and concurrent exposure to oxidizing agents or their metabolites are more susceptible to developing clinical manifestations of the condition. If local anesthetics must be used in these patients, close monitoring for symptoms and signs of methemoglobinemia is recommended. Signs of methemoglobinemia may occur immediately or may be delayed some hours after exposure and are characterized by a cyanotic skin discoloration and/or abnormal coloration of the blood. Methemoglobin levels may continue to rise; therefore, immediate treatment is required to avert more serious central nervous system Reference ID: 4883113 This label may not be the latest approved by FDA. For current labeling information, please visit and cardiovascular adverse effects, including seizures, coma, arrhythmias, and death. Discontinue Prilocaine HCI 4% with Epinephrine 1:200,000 and any other oxidizing agents. Depending on the severity of the signs and symptoms, patients may respond to supportive care, i.e., oxygen therapy, hydration. A more severe clinical presentation may require treatment with methylene blue, exchange transfusion, or hyperbaric oxygen. To minimize the likelihood of intravascular injection, aspiration should be performed before the local anesthetic solution is injected. If blood is aspirated, the needle must be repositioned until no return of blood can be elicited by aspiration. Note, however, that the absence of blood in the syringe does not assure that intravascular injection will be avoided. Prilocaine HCl 4% with Epinephrine 1:200,000 contains sodium metabisulfite, a sulfite that may cause allergic-type reactions including anaphylactic symptoms and life-threatening or less severe asthmatic episodes in certain susceptible people. The overall prevalence of sulfite sensitivity in the general population is unknown and probably low. Sulfite sensitivity is seen more frequently in asthmatic than non-asthmatic people. PRECAUTIONS General: The safety and effectiveness of prilocaine depend on proper dosage, correct technique, adequate precautions, and readiness for emergencies. Standard textbooks should be consulted for specific techniques and precautions for various regional anesthetic procedures. Resuscitative equipment, oxygen, and other resuscitative drugs should be available for immediate use. (See WARNINGS and ADVERSE REACTIONS.) The lowest dosage that results in effective anesthesia should be used to avoid high plasma levels and serious adverse effects. Repeated doses of prilocaine may cause significant increases in blood levels with each repeated dose because of slow accumulation of the drug or its metabolites. Tolerance to elevated blood levels varies with the status of the patient. Debilitated, elderly patients, acutely ill patients, and children should be given reduced doses commensurate with their age and physical status. Prilocaine should also be used with caution in patients with severe shock or heart block. Local anesthetic injections containing a vasoconstrictor should be used cautiously in areas of the body supplied by end arteries or having otherwise compromised blood supply. Patients with peripheral vascular disease and those with hypertensive vascular disease may exhibit exaggerated vasoconstrictor response. Ischemic injury or necrosis may result. Preparations containing a vasoconstrictor should be used with caution in patients during or following the administration of potent general anesthetic agents, since cardiac arrhythmias may occur under such conditions. Cardiovascular and respiratory (adequacy of ventilation) vital signs and the patient’s state of consciousness should be monitored after each local anesthetic injection. Restlessness, anxiety, tinnitus, dizziness, blurred vision, tremors, depression, or drowsiness should alert the practitioner to the possibility of central nervous system toxicity. Signs and symptoms of depressed cardiovascular function may commonly result from a vasovagal reaction, particularly if the patient is in an upright position. (See ADVERSE REACTIONS, Cardiovascular System). Since amide-type local anesthetics are metabolized by the liver, prilocaine should be used with caution in patients with hepatic disease. Patients with severe hepatic disease, because of their inability to metabolize local anesthetics normally, are at greater risk of developing toxic plasma concentrations. Prilocaine should also be used with caution in patients with impaired cardiovascular function since they may be less able to compensate for functional changes associated with the prolongation of A-V conduction produced by these drugs. Many drugs used during the conduct of anesthesia are considered potential triggering agents for familial malignant hyperthermia. Since is not known whether amide-type local anesthetics may trigger this reaction and since the need for supplemental general anesthesia cannot be predicted in advance, it is suggested that a standard protocol for the management of malignant hyperthermia should be available. Early unexplained signs of tachycardia, tachypnea, labile blood pressure and metabolic acidosis may precede temperature elevation. Successful outcome is dependent on early diagnosis, prompt discontinuance of the suspect triggering agent(s) Reference ID: 4883113 This label may not be the latest approved by FDA. For current labeling information, please visit and institution of treatment, including oxygen therapy, indicated supportive measures and dantrolene (consult dantrolene sodium intravenous package insert before using). Prilocaine should be used with caution in persons with known drug sensitivities. Patients allergic to para-aminobenzoic acid derivatives (procaine, tetracaine, benzocaine, etc.) have not shown cross sensitivity to prilocaine. Use in the Head and Neck Area: Small doses of local anesthetics injected into the head and neck area, including retrobulbar, dental and stellate ganglion blocks, may produce adverse reactions similar to systemic toxicity seen with unintentional intravascular injections of larger doses. Confusion, convulsions, respiratory depression and/or respiratory arrest, and cardiovascular stimulation or depression have been reported. These reactions may be due to intra-arterial injection of the local anesthetic with retrograde flow to the cerebral circulation. Patients receiving these blocks should have their circulation and respiration monitored and be constantly observed. Resuscitative equipment and personnel for treating adverse reactions should be immediately available. Dosage recommendations should not be exceeded. (See DOSAGE AND ADMINISTRATION.) Information for Patients: Inform patients that use of local anesthetics may cause methemoglobinemia, a serious condition that must be treated promptly. Advise patients or caregivers to seek immediate medical attention if they or someone in their care experience the following signs or symptoms: pale, gray, or blue colored skin (cyanosis); headache; rapid heart rate; shortness of breath; lightheadedness; or fatigue. The patient should be informed of the possibility of temporary loss of sensation and muscle function following infiltration or nerve block injections. The patient should be advised to exert caution to avoid inadvertent trauma to the lips, tongue, cheek mucosae or soft palate when these structures are anesthetized. The ingestion of food should therefore be postponed until normal function returns. The patient should be advised to consult the dentist if anesthesia persists, or if a rash develops. Clinically Significant Drug Interactions: Patients who are administered local anesthetics are at increased risk of developing methemoglobinemia when concurrently exposed to the following drugs, which could include other local anesthetics: Examples of Drugs Associated with Methemoglobinemia: Class Examples Nitrates/Nitrites nitric oxide, nitroglycerin, nitroprusside, nitrous oxide Local anesthetics articaine ,benzocaine, bupivacaine, lidocaine, mepivacaine, prilocaine, procaine, ropivacaine, tetracaine Antineoplastic agents cyclophosphamide, flutamide, hydroxyurea, ifosfamide, rasburicase Antibiotics dapsone, nitrofurantoin, para-aminosalicylic acid, sulfonamides Antimalarials chloroquine, primaquine Anticonvulsants Phenobarbital, phenytoin, sodium valproate Other drugs acetaminophen, metoclopramide, quinine, sulfasalazine Reference ID: 4883113 This label may not be the latest approved by FDA. For current labeling information, please visit The administration of local anesthetic injections containing epinephrine or norepinephrine to patients receiving monoamine oxidase inhibitors, tricyclic antidepressants or phenothiazines may produce severe, prolonged hypotension or hypertension. Concurrent use of these agents should generally be avoided. In situations when concurrent therapy is necessary, careful patient monitoring is essential. Concurrent administration of vasopressor drugs and ergot-type oxytocic drugs may cause severe, persistent hypertension or cerebrovascular accidents. Drug/Laboratory Test Interactions: The intramuscular injection of prilocaine may result in an increase in creatine phosphokinase levels. Thus, the use of this enzyme determination, without isoenzyme separation, as a diagnostic test for the presence of acute myocardial infarction may be compromised by the intramuscular injection of prilocaine. Carcinogenesis, Mutagenesis, Impairment of Fertility: Studies of prilocaine in animals to evaluate the carcinogenic and mutagenic potential or the effect on fertility have not been conducted. Chronic oral toxicity studies of ortho-toluidine, a metabolite of prilocaine, in mice (150 to 4800 mg/kg) and rats (150 to 800 mg/kg) have shown that ortho-toluidine is a carcinogen in both species. The lowest dose corresponds to approximately 50 times the maximum amount of ortho-toluidine to which a 50 kg subject would be expected to be exposed following a single injection (8 mg/kg) of prilocaine. Ortho-toluidine (0.5 mg/mL) showed positive results in Escherichia coli DNA repair and phage-induction assays. Urine concentrates from rats treated with ortho-toluidine (300 mg/kg, orally) were mutagenic for Salmonella typhimurium with metabolic activation. Several other tests, including reverse mutations in five different Salmonella typhimurium strains with or without metabolic activation and single strand breaks in DNA of V79 Chinese hamster cells, were negative. Use in Pregnancy: . Reproduction studies have been performed in rats at doses up to 30 times the human dose and revealed no evidence of impaired fertility or harm to the fetus due to prilocaine. There are, however, no adequate and well-controlled studies in pregnant women. Animal reproduction studies are not always predictive of human response. General consideration should be given to this fact before administering prilocaine to women of childbearing potential, especially during early pregnancy when maximum organogenesis takes place. Nursing Mothers: It is not known whether this drug is excreted in human milk. Because many drugs are excreted in human milk, caution should be exercised when prilocaine is administered to a nursing woman. Pediatric Use: Dosages in children should be reduced, commensurate with age, body weight, and physical condition. (See DOSAGE AND ADMINISTRATION.) ADVERSE REACTIONS Swelling and persistent paresthesia of the lips and oral tissues may occur. Persistent paresthesia lasting weeks to months, and in rare instances paresthesia lasting greater than one year have been reported. Adverse experiences following the administration of prilocaine are similar in nature to those observed with other amide local anesthetic agents. These adverse experiences are, in general, dose-related and may result from high plasma levels caused by excessive dosage, rapid absorption or unintentional intravascular injection, or may result from a hypersensitivity, idiosyncrasy or diminished tolerance on the part of the patient. Serious adverse experiences are generally systemic in nature. The following types are those most commonly reported: Central Nervous System: CNS manifestations are excitatory and/or depressant and may be characterized by lightheadedness, nervousness, apprehension, euphoria, confusion, dizziness, drowsiness, tinnitus, blurred or double vision, vomiting, sensations of heat, cold or numbness, twitching, tremors, convulsions, unconsciousness, respiratory depression, and arrest. The excitatory manifestations may be very brief or may Reference ID: 4883113 This label may not be the latest approved by FDA. For current labeling information, please visit not occur at all, in which case the first manifestation of toxicity may be drowsiness merging into unconsciousness and respiratory arrest. Drowsiness following the administration of prilocaine is usually an early sign of a high blood level of the drug and may occur as a consequence of rapid absorption. Cardiovascular System: Cardiovascular manifestations are usually depressant and are characterized by bradycardia, hypotension, and cardiovascular collapse, which may lead to cardiac arrest. Signs and symptoms of depressed cardiovascular function may commonly result from a vasovagal reaction, particularly if the patient is in an upright position. Less commonly, they may result from a direct effect of the drug. Failure to recognize the premonitory signs such as sweating, a feeling of faintness, and changes in pulse or sensorium may result in progressive cerebral hypoxia and seizure or serious cardiovascular catastrophe. Management consists of placing the patient in the recumbent position and ventilation with oxygen. Supportive treatment of circulatory depression may require the administration of intravenous fluids, and, when appropriate, a vasopressor (e.g., ephedrine) as directed by the clinical situation. Allergic: Allergic reactions are characterized by cutaneous lesions, urticaria, edema or anaphylactoid reactions. Allergic reactions as a result of sensitivity to prilocaine are extremely rare and, if they occur, should be managed by conventional means. The detection of sensitivity by skin testing is of doubtful value. Neurologic: The incidences of adverse reactions (e.g., persistent neurologic deficit) associated with the use of local anesthetics may be related to the technique employed, the total dose of local anesthetic administered, the particular drug used, the route of administration, and the physical condition of the patient. OVERDOSAGE Acute emergencies from local anesthetics are generally related to high plasma levels encountered during therapeutic use of local anesthetics (see ADVERSE REACTIONS, WARNINGS, and PRECAUTIONS). Management of Local Anesthetic Emergencies: The first consideration is prevention, best accomplished by careful and constant monitoring of cardiovascular and respiratory vital signs and the patient’s state of consciousness after each local anesthetic injection. At the first sign of change, oxygen should be administered. The first step in the management of convulsions consists of immediate attention to the maintenance of a patent airway and assisted or controlled ventilation with oxygen and a delivery system capable of permitting immediate positive airway pressure by mask. Immediately after the institution of these ventilatory measures, the adequacy of the circulation should be evaluated, keeping in mind that drugs used to treat convulsions sometimes depress the circulation when administered intravenously. Should convulsions persist despite adequate respiratory support, and if the status of the circulation permits, small increments of an ultra-short acting barbiturate (such as thiopental or thiamylal) or a benzodiazepine (such as diazepam) may be administered intravenously. The clinician should be familiar, prior to use of local anesthetics, with these anticonvulsant drugs. Supportive treatment of circulatory depression may require administration of intravenous fluids and, when appropriate, a vasopressor ad directed by the clinical situation (e.g., ephedrine). If not treated immediately, both convulsions and cardiovascular depression can result in hypoxia, acidosis, bradycardia, arrhythmias, and cardiac arrest. If cardiac arrest should occur, standard cardiopulmonary resuscitative measures should be instituted. Endotracheal intubation, employing drugs and techniques familiar to the clinician, may be indicated, after initial administration of oxygen by mask, if difficulty is encountered in the maintenance of a patent airway or if prolonged ventilatory support (assisted or controlled) is indicated. Dialysis is of negligible value in the treatment of acute overdosage with prilocaine. The development of methemoglobinemia is generally dose related but may occur at any dose in susceptible individuals. While methemoglobin values of less than 20% do not generally produce any clinical symptoms, the Reference ID: 4883113 This label may not be the latest approved by FDA. For current labeling information, please visit appearance of cyanosis at 2 to 4 hours following administration should be evaluated in terms of the general health status of the patient. Methemoglobinemia can be reversed when indicated by intravenous administration of methylene blue at a dosage of 1 to 2 mg/kg given over a five-minute period. The subcutaneous LD50 of prilocaine HCl in female mice is 550 (359 to 905) mg/kg. DOSAGE AND ADMINISTRATION The dosage of Prilocaine HCl 4% with Epinephrine 1:200,000 varies and depends on the physical status of the patient, the area of the oral cavity to be anesthetized, the vascularity of the oral tissues, and the technique of anesthesia. The least volume of injection that results in effective local anesthesia should be administered. For specific techniques and procedures of local anesthesia in the oral cavity, refer to standard textbooks. Inferior Alveolar Block: There are no practical clinical differences between prilocaine with and without epinephrine when used for inferior alveolar blocks. Maxillary Infiltration: Prilocaine HCl 4% (without epinephrine) is recommended for use in maxillary infiltration anesthesia for procedures in which the painful aspects can be completed within 15 minutes after the injection. Prilocaine HCl 4% is therefore especially suited to short procedures in the maxillary anterior teeth. For long procedures, or those involving maxillary posterior teeth where soft tissue numbness is not troublesome to the patient, Prilocaine HCl 4% is recommended. For most routine procedures, initial dosages of 1 to 2 mL of Prilocaine HCl 4% with Epinephrine 1:200,000 will usually provide adequate infiltration or major nerve block anesthesia. The maximum recommended dose that should ever be administered within a two-hour period in normal healthy adults should be calculated based upon the patient’s weight as follows: Weight Maximum recommended dose <150 lbs. (<70 kg) 4 mg/lb. (8 mg/kg) ≥150 lbs. (≥70 kg) 600 mg (15 mL) or 8 glass single-dose cartridges In children under 10 years of age it is rarely necessary to administer more than one-half cartridge (40 mg) of Prilocaine HCl 4% with Epinephrine 1:200,000 per procedure to achieve local anesthesia for a procedure involving a single tooth. In maxillary infiltration, this amount will often suffice to the treatment of two or even three teeth. In the mandibular block, however, satisfactory anesthesia achieved with this amount of drug will allow treatment of the teeth in an entire quadrant. ASPIRATION PRIOR TO INJECTION IS RECOMMENDED, since it reduces the possibility of intravascular injection, thereby keeping the incidence of side effects and anesthetic failure to a minimum. NOTE: Parenteral drug products should be inspected visually for particulate matter and discoloration prior to administration whenever the solution and container permit. Solutions that are discolored and/or contain particulate matter should not be used. Any unused portion of a cartridge should be discarded. Maximum Recommended Dosages: In patients weighting <150 lbs. (70 kg), no more than 4 mg/lb. (8 mg/kg) should be administered. In patients weighing ≥150 lbs., no more than 600 mg (8 glass single-dose cartridges) of prilocaine HCl should be administered as a single injection. Children: It is difficult to recommend a maximum dose of any drug for children since this varies as a function of age and weight. For children of less than ten years who have a normal lean body mass and normal body Reference ID: 4883113 This label may not be the latest approved by FDA. For current labeling information, please visit development, the maximum dose may be determined by the application of one of the standard pediatric drug formulas (e.g., Clark’s rule). For example, in a child of five years weighing 50 lbs., the dose of prilocaine HCl should not exceed 150 to 200 mg (6.6 to 8.8 mg/kg or 3 to 4 mg/lb. of body weight) when calculated according to Clark’s rule. HOW SUPPLIED Prilocaine HCI 4% with Epinephrine 1:200,000 (NDC 66312-580-16) are dispensed in 1.8 mL single-dose glass cartridges, packed 50 or 100 per box. Not all pack sizes may be marketed. Sterilization, Storage and Technical Procedures: 1. Glass Cartridges should not be autoclaved, because solutions of epinephrine and the closures employed in glass cartridges cannot withstand autoclaving temperatures and pressures. 2. If chemical disinfection of anesthetic glass cartridges is desired, either 91% isopropyl alcohol or 70% ethyl alcohol is recommended. Many commercially available brands of rubbing alcohol, as well as solutions of ethyl alcohol not of USP grade, contain denaturants that are injurious to rubber and, therefore, are not to be used. It is recommended that chemical disinfection be accomplished by wiping the cartridge cap thoroughly with a pledget of cotton that has been moistened with the recommended alcohol just prior to use. IMMERSION IS NOT RECOMMENDED. 3. Certain metallic ions (mercury, zinc, copper, etc.) have been related to swelling and edema after local anesthesia in dentistry. Therefore, chemical disinfectants containing or releasing those ions are not recommended. Antirust tablets usually contain metal ions. Accordingly, aluminum sealed glass cartridges should not be kept in such solutions. 4. Quaternary ammonium salts, such as benzalkonium chloride, are electrolytically incompatible with aluminum. Cartridges are sealed with aluminum caps and therefore should not be immersed in any solution containing these salts. 5. To avoid leakage of solutions during injection, be sure to penetrate the center of the rubber diaphragm when loading the syringe. An off-center penetration produces an oval shaped puncture that allows leakage around the needle. Other causes of leakage and breakage included badly worn syringes, aspirating syringes with bent harpoons, the use of syringes not designed to take 1.8 mL single-dose glass cartridges, and inadvertent freezing. 6. Cracking of glass cartridges is most often the result of an attempt to use a cartridge with an extruded plunger. An extruded plunger loses its lubrication and can be forced back into the cartridge only with difficulty. Cartridges with extruded plungers should be discarded. 7. Store at 20°C to 25°C (68°F to 77°F); excursions permitted between 15°C to 30°C (59°F to 86°F) [see USP Controlled Room Temperature]. 8. Solutions containing epinephrine should be protected from light. Manufactured for Dentsply Pharmaceutical, York, PA 17404 by Novocol Pharmaceutical of Canada, Inc. Rev 11/2021(2659-5) Reference ID: 4883113 This label may not be the latest approved by FDA. For current labeling information, please visit Each mL contains: Prilocaine HCI.......................................40.0 mg Epinephrine (as bitartrate).......0.005 mg Sodium Metabisulfite..........................0.5 mg Citric Acid..................................................0.2 mg Sodium Hydroxide and/or Hydrochloric Acid to adjust pH Water for injection............................q.s. 1 mL Reorder #: 48816 NDC 66312-580-16 2551-1 Store at 20 to 25° C (68 to 77° F) (See USP Controlled Room T emperature) DO NOT PERMIT TO FREEZE Consult package insert for dosage and full prescribing information.
with Epinephrine 1:200,000 Prilocaine HCl 4% (prilocaine HCI and epinephrine Injection, USP) with Epinephrine 1:200,000 Prilocaine HCl 4% (prilocaine HCI and epinephrine Injection, USP) with Epinephrine 1:200,000 Prilocaine HCl 4% (prilocaine HCI and epinephrine Injection, USP) 50 Single Dose cartridges, 1.8mL each Rx only Manufactured for Dentsply Pharmaceutical, York, PA 17404 by Novocol Pharmaceutical of Canada, Inc.
www.dentsplysirona.com 1-800-989-8826 Made in Canada Reorder #: 48816 NDC 66312-580-16 DIRECTIONS FOR USE • To be sold only as unbroken package.
• DO NOT INJECT INTRAVENOUSLY.
• A local anesthetic for dental block and infiltration injections only.
• USUAL DOSAGE: See package insert. Read directions enclosed concerning possible side effects, precautions and contraindications.
• Store at 20 to 25° C (68 to 77° F) (See USP Controlled Room Temperature).
• Any unused portion of a cartridge should be discarded.
• DO NOT PERMIT TO FREEZE.
• PROTECT FROM LIGHT. KEEP IN CARTON UNTIL READY TO USE.
• This injection is not to be used if its color is pinkish or darker than slightly yellow or it contains a precipitate.
Rx Only Made in Canada Manufactured for Dentsply Pharmaceutical, York, PA 17404 by Novocol Pharmaceutical of Canada, Inc.
Cambridge, Ontario N1R 6X3 NDC Barcode Placement GTIN (01) LOT (10) EXP (17) SERIAL (21) This label may not be the latest approved by FDA. For current labeling information, please visit Manufactured for Dentsp y Pharmaceutical York PA 17404 by Novocol Pharmaceutical of Canada Inc 2 5 5 0 - 1 1.8 mL with Epinephrine 1:200,000 Prilocaine HCl 4% (prilocaine HCI and epinephrine Injection USP) Single dose LOT EXP.
This label may not be the latest approved by FDA. current labeling information, please visit --------------------------------------------------------------------------------------------This is a representation of an electronic record that was signed electronically. Following this are manifestations of any and all electronic signatures for this electronic record.
--------------------------------------------------------------------------------------------/s/ ------------------------------------------------------------RIGOBERTO A ROCA 11/03/2021 03:39:04 PM Signature Page 1 of 1 Reference ID: 4883113 ( This label may not be the latest approved by FDA. For current labeling information, please visit |
13039 | https://mathworld.wolfram.com/HexagonalGrid.html | Hexagonal Grid -- from Wolfram MathWorld
TOPICS
AlgebraApplied MathematicsCalculus and AnalysisDiscrete MathematicsFoundations of MathematicsGeometryHistory and TerminologyNumber TheoryProbability and StatisticsRecreational MathematicsTopologyAlphabetical IndexNew in MathWorld
Discrete Mathematics
Point Lattices
Hexagonal Grid
Download Wolfram Notebook
A hexagonal grid is a grid formed by a tessellation of regular hexagons. Boards made of hexagonal grids are often found in strategy and role-playing games because of the lack of single points of contact characteristic of isometric and orthogonal grids.
See also
Clean Tile Problem, Grid, Hexagon, Hexagonal Grid Graph, Hexagon Tiling, Square Grid, Tessellation, Triangular Grid
Explore with Wolfram|Alpha
More things to try:
hexagonal grid
1, 4, 9, 16, 25, ...
expand sin 4x
Cite this as:
Weisstein, Eric W. "Hexagonal Grid." From MathWorld--A Wolfram Resource.
Subject classifications
Discrete Mathematics
Point Lattices
About MathWorld
MathWorld Classroom
Contribute
MathWorld Book
wolfram.com
13,278 Entries
Last Updated: Sun Sep 28 2025
©1999–2025 Wolfram Research, Inc.
Terms of Use
wolfram.com
Wolfram for Education
Created, developed and nurtured by Eric Weisstein at Wolfram Research
Created, developed and nurtured by Eric Weisstein at Wolfram Research |
13040 | https://math.stackexchange.com/questions/843684/prove-an-inequality-using-taylor-expansion | Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Prove an inequality (Using Taylor expansion)
Ask Question
Asked
Modified 6 years, 3 months ago
Viewed 4k times
6
$\begingroup$
Prove: $\frac{x}{1+x} < \ln(1+x) < x$.
I thought a good practice would be to prove it using Taylor Expansion.
Here's my try:$$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3}...$$
The n=1 Taylor polynomial is: $$T_1(x) = x$$ and $$ ln(1+x) = T_1(x) + R_1(x)$$
Lets evaluate $R_1(x)$ by Cauchy's remainder formula:
$$R_1(x) = \frac{f^{(2)}(\xi)}{2!}\cdot x^2 = \frac{\frac{-1}{(\xi+1)^2}}{2!}\cdot x^2 = \frac{-x^2}{2(\xi+1)^2} < 0$$
Now, it does prove the right-hand side because $x + R_1(x) < x$ ($R_1(x)$ is negative). I'm not so sure what should I do for the left-hand side. I'd also like to get general critique for my current work.
Thanks!
calculus
real-analysis
polynomials
derivatives
taylor-expansion
Share
asked Jun 22, 2014 at 15:33
AnnieOKAnnieOK
2,99033 gold badges2222 silver badges3838 bronze badges
$\endgroup$
Add a comment |
3 Answers 3
Reset to default
5
$\begingroup$
We apply the mean value theorem on the function $t\mapsto \ln t$ on the interval $[1,1+x]$: there's $\zeta\in(1,1+x)$ such that $$\ln(1+x)=\frac x\zeta$$ and notice that $$\frac1{1+x}<\frac1\zeta<1$$
Share
answered Jun 22, 2014 at 15:41
user63181user63181
$\endgroup$
2
$\begingroup$ Very nice, thank you! $\endgroup$
AnnieOK
– AnnieOK
2014-06-22 16:00:58 +00:00
Commented Jun 22, 2014 at 16:00
$\begingroup$ You're welcome. $\endgroup$
user63181
– user63181
2014-06-22 16:02:38 +00:00
Commented Jun 22, 2014 at 16:02
Add a comment |
1
$\begingroup$
Using the elementary inequality $$ 1+x\le e^x $$ one directly obtains one side of the inequality chain. Replace $x$ by $-y$ and invert to obtain $$ \frac1{1-y}\ge e^y $$ and then set $1+x=\frac1{1-y}=1+\frac{y}{1-y}$ or $y=\frac{x}{1+x}$ to obtain $$ 1+x\ge e^{\frac{x}{1+x}} $$ for the other part of the inequality chain.
Share
answered Jul 9, 2014 at 16:42
Lutz LehmannLutz Lehmann
132k77 gold badges4343 silver badges133133 bronze badges
$\endgroup$
Add a comment |
1
$\begingroup$
Proof:
Assuming you argued and justified that $ f (x) = Ln (1 + x) $ is continuous and derivable $ \forall x> 0 $.
This is equivalent to writing the interval $ (0, x) $.
By the Mean Value theorem $\exists c \in (0,x)$ such that $f'(c)=\frac{f(b)-f(a)}{b-a}$, i.e. $\frac{1}{1+c}=\frac{Ln(1+x)-Ln(1)}{x-0}=\frac{Ln(1+x)}{x}$.
We can delimit $\frac{1}{1+c}=f'(c)$ (), then:
$\frac{1}{1+x}
But we know that $f'(c)=\frac{1}{1+c}=\frac{Ln(1+x)}{x}$ ()
Replacing in () we get:
$\frac{1}{1+x}<\frac{Ln(1+x)}{x}<1$
Multiplying the inequality by $x$, we obtain the desired result:
$\frac{x}{1+x}.
Share
answered Jun 12, 2019 at 14:13
Lorenzo CastagnoLorenzo Castagno
20133 silver badges99 bronze badges
$\endgroup$
Add a comment |
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
calculus
real-analysis
polynomials
derivatives
taylor-expansion
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Related
5 A Taylor series expansion of $e^{ix}$
Taylor series with initial value of infinity
Lagrange Error Value for $f(x)=\frac{1}{x}$
1 Finding Taylor polynomial of the right degree and checking the remainder
1 What is the vector form of Taylor series expansion?
6 Deducing an inequality from a Taylor series expansion
1 How to compute the second degree Taylor polynomial of $\frac{1}{x^2}$ using formula for Taylor polynomial of function composition?
Hot Network Questions
Spectral Leakage & Phase Discontinuites
manage route redirects received from the default gateway
Why is the definite article used in “Mi deporte favorito es el fútbol”?
Origin of Australian slang exclamation "struth" meaning greatly surprised
Identifying a thriller where a man is trapped in a telephone box by a sniper
A time-travel short fiction where a graphologist falls in love with a girl for having read letters she has not yet written… to another man
What is the meaning of 率 in this report?
Does the mind blank spell prevent someone from creating a simulacrum of a creature using wish?
Another way to draw RegionDifference of a cylinder and Cuboid
Determine which are P-cores/E-cores (Intel CPU)
Survival analysis - is a cure model a good fit for my problem?
Storing a session token in localstorage
How do trees drop their leaves?
Calculating the node voltage
Implications of using a stream cipher as KDF
Where is the first repetition in the cumulative hierarchy up to elementary equivalence?
How to home-make rubber feet stoppers for table legs?
How to design a circuit that outputs the binary position of the 3rd set bit from the right in an 8-bit input?
How to use \zcref to get black text Equation?
Quantizing EM field by imposing canonical commutation relations
What’s the usual way to apply for a Saudi business visa from the UAE?
How can blood fuel space travel?
Analog story - nuclear bombs used to neutralize global warming
Is existence always locational?
more hot questions
Question feed |
13041 | https://en.wikipedia.org/wiki/35_mm_equivalent_focal_length | Jump to content
Search
Contents
(Top)
1 Calculation
2 Depth of field equivalent
3 Conversions
4 References
5 External links
35 mm equivalent focal length
Français
日本語
Русский
Svenska
Українська
中文
Edit links
Article
Talk
Read
Edit
View history
Tools
Actions
Read
Edit
View history
General
What links here
Related changes
Upload file
Permanent link
Page information
Cite this page
Get shortened URL
Download QR code
Print/export
Download as PDF
Printable version
In other projects
Wikidata item
Appearance
From Wikipedia, the free encyclopedia
Camera setup
Not to be confused with effective focal length.
In photography, the 35 mm equivalent focal length is a measure of the angle of view for a particular combination of a camera lens and film or image sensor size. The term is popular because in the early years of digital photography, most photographers experienced with interchangeable lenses were most familiar with the 35 mm film format.
On any 35 mm film camera, a 28 mm lens is a wide-angle lens, and a 200 mm lens is a long-focus lens. Because digital cameras have mostly replaced film cameras and the image sensor size that also determines the angle of view is not standardized as the film size was, there is no uniform relation between the lens focal length and the angle of view due to possibilities of using various image sensor sizes at the same focal length (i.e., a different image sensor size resulting in a different angle of view at the same lens focal length). The 35 mm equivalent focal length of a particular lens–sensor combination is the focal length that one would need for a 35 mm film camera to obtain the same angle of view. Two lens-sensor combinations with the same 35 mm equivalent focal length are expected to have the same angle of view.
Most commonly, the 35 mm equivalent focal length is based on equal diagonal angle of view. This definition is also in the CIPA guideline DCG-001. Alternatively, it may sometimes be based on horizontal angle of view. Since 35 mm film is normally used for images with an aspect ratio (width-to-height ratio) of 3:2, while many digital cameras have a 4:3 aspect ratio, which have different diagonal-to-width ratios, these two definitions are often not equivalent, i.e., the equivalent focal length based on the diagonal angle of view differs from the equivalent focal length based on the horizontal angle of view.
Calculation
[edit]
35 mm equivalent focal lengths are calculated by multiplying the actual focal length of the lens by the crop factor of the sensor. Typical crop factors are 1.26× – 1.29× for Canon (1.35× for Sigma "H") APS-H format, 1.5× for Nikon APS-C ("DX") format (also used by Sony, Pentax, Fuji, Samsung and others), 1.6× for Canon APS-C format, 2× for Micro Four Thirds format, 2.7× for 1-inch sensors (used in Nikon 1 cameras and some Sony RX cameras), 5× to 6× for compact digital cameras, and even higher for built-in cameras of mobile devices like cell phones or tablets.
According to CIPA guidelines, 35 mm equivalent focal length is to be calculated like this: "Converted focal length into 35 mm camera" = (Diagonal distance of image area in the 35 mm camera (43.27 mm) / Diagonal distance of image area on the image sensor of the DSC) × focal length of the lens of the DSC.
Depth of field equivalent
[edit]
Quoted 35 mm equivalent focal lengths typically ignore depth of field (DOF), which depends on both focal length and aperture. The DOF of smaller sensors is deeper at a given f number, due to the smaller absolute aperture diameters corresponding to shorter focal length lenses.
Equivalent depth of field can be calculated the same way using the crop factor. For example, a 50mm f/2 lens on a 2× crop factor Micro Four Thirds camera would be equivalent to a 100 mm (= 2×50 mm) f/4 (= f/(2×2)) lens on a full-frame digital SLR in terms of field of view, depth of field, total light gathered, and diffraction effects. However for the purposes of exposure calculations the aperture does not change for different sensor sizes.
Conversions
[edit]
A standard 35 mm film image is 36 mm wide by 24 mm tall (35 mm refers to the height of the film including the perforations for film transport), and the diagonal is 43.3 mm. This leads to the following conversion formulas for a lens with a true focal length f:
| Image size | diagonal-based EFL | width-based EFL |
---
| 4:3 (sensor width w) | f35 = 34.6 f /w mm | f35 = 36.0 f /w mm |
| 4:3 (sensor diagonal d) | f35 = 43.3 f /d mm | f35 = 45.0 f /d mm |
| 3:2 (sensor width w) | f35 = 36.0 f /w mm | f35 = 36.0 f /w mm |
| 3:2 (sensor diagonal d) | f35 = 43.3 f /d mm | f35 = 43.3 f /d mm |
For historical reasons, sensor size specifications such as 1/2.5" do not match the actual sensor size, but are a bit larger (typically about a factor of 1.5) than the actual sensor diagonal. This is because these sensor size specifications refer to the size of a camera tube, while the usable sensor size is about 2/3 of the size of the tube. Tubes are not used on digital cameras, but the same specifications are used.
Apart from the width- and diagonal-based 35 mm equivalent focal length definitions, there is a third definition: EFL = 50 f /d mm. However, it is not clear to what extent this definition is used.[citation needed]
References
[edit]
^ a b What is "35 mm equivalent focal length?" The Panorama Factory, 2004.
^ a b "CIPA DCG-001-Translation-2005 Guideline for Noting Digital Camera Specifications in Catalogs" (PDF). Archived from the original (PDF) on 2017-02-02. Retrieved 2015-10-19.
^ Atkins, Bob. "Digital Depth of Field". Retrieved 23 May 2012.
^ Butler, Richard. "What is equivalence and why should I care?". DPReview. Retrieved 22 August 2019.
^ Vincent Bockaert, Sensor sizes. DPreview.com.
External links
[edit]
Focal Length Conversion for medium format and large format, at photo.net
Focal Length at dpreview
| v t e Photography |
| Equipment | Camera + light-field + digital + field + instant + phone + pinhole + press + rangefinder + SLR + still + TLR + toy + view Darkroom + enlarger + safelight Film + base + format + holder + stock + available films + discontinued films Filter Flash + beauty dish + cucoloris + gobo + hot shoe + lens hood + monolight + reflector + snoot + softbox Lens + long-focus + prime + zoom + wide-angle + fisheye + swivel + telephoto Manufacturers Monopod Movie projector Slide projector Tripod + head Zone plate |
| Terminology | 35 mm equivalent focal length Angle of view Aperture Backscatter Black-and-white Chromatic aberration Circle of confusion Clipping Color balance Color temperature Depth of field Depth of focus Exposure Exposure compensation Exposure value Zebra patterning F-number Film format + large + medium Film speed Focal length Guide number Hyperfocal distance Lens flare Metering mode Perspective distortion Photograph Photographic printing + Albumen Photographic processes Reciprocity Red-eye effect Science of photography Shutter speed Sync Zone System |
| Genres | Abstract Aerial Aircraft Architectural Astrophotography Banquet Candid Conceptual Conservation Cloudscape Documentary Eclipse Ethnographic Erotic Fashion Fine-art Fire Forensic Glamour High-speed Landscape Monochrome Nature Neues Sehen Nude Photojournalism Pictorialism Pornography Portrait Post-mortem Ruins Selfie + space selfie Social documentary Sports Still life Stock Straight photography Street Underwater Vernacular Wedding Wildlife |
| Techniques | Afocal Blur Bokeh Brenizer Burst mode Contre-jour Crittercam (Pigeon photography) ETTR Fill flash Fireworks Hand-colouring Harris shutter High-speed Holography Infrared Intentional camera movement Kirlian Kite aerial Lo-fi photography Lomography Long-exposure Luminogram Macro Mordançage Multiple exposure Multi-exposure HDR capture Night Panning Panoramic Photogram Print toning Redscale Rephotography Rollout Scanography Schlieren photography Sabattier effect Slow motion Stereoscopy Stopping down Strip + Slit-scan Sprocket hole Sun printing Tilt–shift + Miniature faking Time-lapse Ultraviolet Vignetting Xerography Zoom burst |
| Composition | Diagonal method Framing Headroom Lead room Rule of thirds Simplicity Golden triangle (composition) |
| History | Timeline of photography technology Ambrotype Film photography Autochrome Lumière Box camera Calotype Camera obscura Daguerreotype Dufaycolor Heliography Lippmann plate Painted photography backdrops Photography and the law Glass plate Tintype Visual arts |
| Regional | Albania Bangladesh Canada China Denmark Greece India Japan Korea Luxembourg Norway Philippines Serbia Slovenia Sudan Taiwan Turkey Ukraine United States Uzbekistan Vietnam |
| Digital photography | Digital camera + D-SLR + MILC + camera back Digiscoping Comparison of digital and film photography Film scanner Image sensor + CMOS APS + CCD + Three-CCD camera + Foveon X3 sensor Image sharing Pixel |
| Color photography | Print film + Chromogenic print Reversal film Color management + color space + primary color + CMYK color model + RGB color model |
| Photographicprocessing | Bleach bypass C-41 process Collodion process Cross processing Cyanotype Developer Digital image processing Dye coupler E-6 process Fixer Gelatin silver process Gum printing Instant film K-14 process Print permanence Push processing Stop bath |
| Lists | Largest photographs Most expensive photographs Museums devoted to one photographer Photographs considered the most important Photographers + Norwegian + Polish + street + women Photography periodicals |
| Related | Conservation and restoration of photographs + film + photographic plates Polaroid art Stereoscopy Say cheese |
Retrieved from "
Category:
Science of photography
Hidden categories:
Articles with short description
Short description matches Wikidata
All articles with unsourced statements
Articles with unsourced statements from February 2008
35 mm equivalent focal length
Add topic |
13042 | https://foodb.ca/compounds/FDB030735 | Showing Compound citrate (FDB030735) - FooDB
Toggle navigation
Browse
FoodsCompoundsNutrientsContentsPathways
Search
ChemQuery Structure Search
Advanced Search
LC-MS Search
LC-MS/MS Search
GC-MS Search
NMR Search
Downloads
Reports
Examples
About
About the FooDB
Other Databases
Database Schema
Foodb API
FAIR Compliance
Wishart Research Group
TMIC Wishart Node
Contact Us
Search
Quantitative metabolomics services for biomarker discovery and validation. Specializing in ready to use metabolomics kits. Your source for quantitative metabolomics technologies and bioinformatics.
×
User Survey Request
Can we take 30 seconds of your time?
We hope that this free tool has been helpful for you and your research program. It is part of TMIC’s mission to provide enabling technologies to the Canadian and international metabolomics communities, and we’d really appreciate it if you could fill out the survey link below - it should take less than a minute of your time, and will help us continue to provide this service to the community.
Take the SurveyContinue to Site
Back to Compounds
Showing Compound citrate (FDB030735)
Jump To Section:
Record informationChemical informationClassificationOntologyPhysico chemical propertiesSpectraExternal linksAssociated foodsBiological effects and interationsOrganoleptic propertiesReferences
| Record Information |
| Version | 1.0 |
| Creation date | 2015-05-07 19:57:49 UTC |
| Update date | 2019-11-26 03:21:52 UTC |
| Primary ID | FDB030735 |
| Secondary Accession Numbers | FDB012586 |
| Chemical Information |
| FooDB Name | citrate |
| Description | Citrate, also known as anhydrous citric acid or 2-hydroxy-1,2,3-propanetricarboxylic acid, belongs to tricarboxylic acids and derivatives class of compounds. Those are carboxylic acids containing exactly three carboxyl groups. Citrate is soluble (in water) and a weakly acidic compound (based on its pKa). Citrate can be found in a number of food items such as ucuhuba, loquat, bayberry, and longan, which makes citrate a potential biomarker for the consumption of these food products. Citrate can be found primarily in most biofluids, including saliva, sweat, feces, and blood, as well as throughout all human tissues. Citrate exists in all living species, ranging from bacteria to humans. In humans, citrate is involved in several metabolic pathways, some of which include the oncogenic action of succinate, the oncogenic action of fumarate, the oncogenic action of 2-hydroxyglutarate, and congenital lactic acidosis. Citrate is also involved in several metabolic disorders, some of which include 2-ketoglutarate dehydrogenase complex deficiency, pyruvate dehydrogenase deficiency (E2), fumarase deficiency, and glutaminolysis and cancer. Moreover, citrate is found to be associated with lung Cancer, tyrosinemia I, maple syrup urine disease, and propionic acidemia. A citrate is a derivative of citric acid; that is, the salts, esters, and the polyatomic anion found in solution. An example of the former, a salt is trisodium citrate; an ester is triethyl citrate. When part of a salt, the formula of the citrate ion is written as C6H5O73− or C3H5O(COO)33− . |
| CAS Number | 77-92-9 |
| Structure | MOLSDFPDBSMILESInChI × #### Structure for FDB030735 (citrate) Close |
| Synonyms | Show entries Search: | Synonym | Source | --- | | 2-Hydroxy-1,2,3-propanetricarboxylate | Generator | | 2-Hydroxy-1,2,3-propanetricarboxylic acid | ChEBI | | 2-Hydroxytricarballylate | Generator | | 2-Hydroxytricarballylic acid | ChEBI | | 3-Carboxy-3-hydroxypentane-1,5-dioate | Generator | | 3-Carboxy-3-hydroxypentane-1,5-dioic acid | ChEBI | | Acid monohydrate, citric | HMDB | | Aciletten | HMDB | | Anhydrous citrate | Generator | | Anhydrous citric acid | Kegg | Showing 1 to 10 of 31 entries Previous 1 2 3 4 Next |
| Predicted Properties | | Property | Value | Source | --- | Water Solubility | 106 g/L | ALOGPS | | logP | -1.3 | ALOGPS | | logP | -1.3 | ChemAxon | | logS | -0.26 | ALOGPS | | pKa (Strongest Acidic) | 3.05 | ChemAxon | | pKa (Strongest Basic) | -4.2 | ChemAxon | | Physiological Charge | -3 | ChemAxon | | Hydrogen Acceptor Count | 7 | ChemAxon | | Hydrogen Donor Count | 4 | ChemAxon | | Polar Surface Area | 132.13 Ų | ChemAxon | | Rotatable Bond Count | 5 | ChemAxon | | Refractivity | 35.62 m³·mol⁻¹ | ChemAxon | | Polarizability | 15.54 ų | ChemAxon | | Number of Rings | 0 | ChemAxon | | Bioavailability | Yes | ChemAxon | | Rule of Five | Yes | ChemAxon | | Ghose Filter | No | ChemAxon | | Veber's Rule | No | ChemAxon | | MDDR-like Rule | No | ChemAxon | |
| Chemical Formula | C6H8O7 |
| IUPAC name | 2-hydroxypropane-1,2,3-tricarboxylic acid |
| InChI Identifier | InChI=1S/C6H8O7/c7-3(8)1-6(13,5(11)12)2-4(9)10/h13H,1-2H2,(H,7,8)(H,9,10)(H,11,12) |
| InChI Key | KRKNYBCHXYNGOX-UHFFFAOYSA-N |
| Isomeric SMILES | OC(=O)CC(O)(CC(O)=O)C(O)=O |
| Average Molecular Weight | 192.1235 |
| Monoisotopic Molecular Weight | 192.02700261 |
| Classification |
| Description | Belongs to the class of organic compounds known as tricarboxylic acids and derivatives. These are carboxylic acids containing exactly three carboxyl groups. |
| Kingdom | Organic compounds |
| Super Class | Organic acids and derivatives |
| Class | Carboxylic acids and derivatives |
| Sub Class | Tricarboxylic acids and derivatives |
| Direct Parent | Tricarboxylic acids and derivatives |
| Alternative Parents | Alpha hydroxy acids and derivatives Tertiary alcohols Carboxylic acids Organic oxides Hydrocarbon derivatives Carbonyl compounds |
| Substituents | Tricarboxylic acid or derivatives Hydroxy acid Alpha-hydroxy acid Tertiary alcohol Carboxylic acid Organic oxygen compound Organic oxide Hydrocarbon derivative Organooxygen compound Carbonyl group Alcohol Aliphatic acyclic compound |
| Molecular Framework | Aliphatic acyclic compounds |
| External Descriptors | tricarboxylic acid (CHEBI:30769) |
| Ontology |
| |
| Physiological effect | Health effect: Health condition: Schizophrenia Paraquat poisoning Hyperoxalemia Rhabdomyolysis Cancer: Lung cancer |
| Disposition | Route of exposure: Enteral: Ingestion Source: Endogenous Food Synthetic: Personal care product Biological: Animal Plant: Poaceae Coffea Fabaceae Glycine max Cucurbitaceae Theobroma cacao Biological location: Tissue and substructures: All tissues Organ and components: Prostate Biofluid and excreta: Urine Saliva Feces Blood Breast milk Sweat Cerebrospinal fluid Subcellular: Cytoplasm Mitochondria Cell and elements: Extracellular |
| Role | Industrial application: Food and nutrition Personal care products Chelating agent Household products: Antimicrobial agent Pharmaceutical industry: Pharmaceutical |
| Physico-Chemical Properties |
| Physico-Chemical Properties - Experimental | Show entries Search: | Property | Value | Reference | --- | Boiling Point | Not Available | | | Charge | Not Available | | | Density | Not Available | | | Experimental logP | Not Available | | | Experimental pKa | Not Available | | | Experimental Water Solubility | Not Available | | | Isoelectric point | Not Available | | | Mass Composition | Not Available | | | Melting Point | Not Available | | | Optical Rotation | Not Available | | Showing 1 to 10 of 14 entries Previous 1 2 Next |
| Spectra |
| Spectra | |
| EI-MS/GC-MS | | Type | Description | Splash Key | View | --- --- | | EI-MS | Mass Spectrum (Electron Ionization) | splash10-002f-9400000000-fb757b614278bb898b54 | 2014-09-20 | View Spectrum | | GC-MS | Citric acid, 4 TMS, GC-MS Spectrum | splash10-0002-0941000000-c310fda0e6a19bf6ae40 | Spectrum | | GC-MS | Citric acid, 4 TMS, GC-MS Spectrum | splash10-0002-0941000000-f1dfda3e9abc7cfb6b2a | Spectrum | | GC-MS | Citric acid, 4 TMS, GC-MS Spectrum | splash10-006t-0952000000-ba3a1a80815f65afd05c | Spectrum | | GC-MS | Citric acid, non-derivatized, GC-MS Spectrum | splash10-0002-0951000000-48857e9baebe16d9fc8d | Spectrum | | GC-MS | Citric acid, 4 TMS, GC-MS Spectrum | splash10-00di-9531000000-b6fd7d038634694f0873 | Spectrum | | GC-MS | Citric acid, 4 TMS, GC-MS Spectrum | splash10-00di-0593000000-b193bbba8c0cefdfe42a | Spectrum | | GC-MS | Citric acid, non-derivatized, GC-MS Spectrum | splash10-0002-0941000000-c310fda0e6a19bf6ae40 | Spectrum | | GC-MS | Citric acid, non-derivatized, GC-MS Spectrum | splash10-0002-0941000000-f1dfda3e9abc7cfb6b2a | Spectrum | | GC-MS | Citric acid, non-derivatized, GC-MS Spectrum | splash10-006t-0952000000-ba3a1a80815f65afd05c | Spectrum | | GC-MS | Citric acid, non-derivatized, GC-MS Spectrum | splash10-0002-0951000000-48857e9baebe16d9fc8d | Spectrum | | GC-MS | Citric acid, non-derivatized, GC-MS Spectrum | splash10-00di-9531000000-b6fd7d038634694f0873 | Spectrum | | GC-MS | Citric acid, non-derivatized, GC-MS Spectrum | splash10-00di-0593000000-b193bbba8c0cefdfe42a | Spectrum | | GC-MS | Citric acid, non-derivatized, GC-MS Spectrum | splash10-0002-0941000000-573230e47a4efb1e36a5 | Spectrum | | Predicted GC-MS | Citric acid, non-derivatized, Predicted GC-MS Spectrum - 70eV, Positive | splash10-002k-6900000000-d1a61a52f6efdec3101a | Spectrum | | Predicted GC-MS | Citric acid, 4 TMS, Predicted GC-MS Spectrum - 70eV, Positive | splash10-02ib-6019300000-6d5ebea10441c6d4149a | Spectrum | | Predicted GC-MS | Citric acid, non-derivatized, Predicted GC-MS Spectrum - 70eV, Positive | Not Available | Spectrum | | Predicted GC-MS | Citric acid, TMS_1_1, Predicted GC-MS Spectrum - 70eV, Positive | Not Available | Spectrum | | Predicted GC-MS | Citric acid, TMS_1_2, Predicted GC-MS Spectrum - 70eV, Positive | Not Available | Spectrum | | Predicted GC-MS | Citric acid, TMS_1_3, Predicted GC-MS Spectrum - 70eV, Positive | Not Available | Spectrum | | Predicted GC-MS | Citric acid, TMS_2_1, Predicted GC-MS Spectrum - 70eV, Positive | Not Available | Spectrum | | Predicted GC-MS | Citric acid, TMS_2_2, Predicted GC-MS Spectrum - 70eV, Positive | Not Available | Spectrum | | Predicted GC-MS | Citric acid, TMS_2_3, Predicted GC-MS Spectrum - 70eV, Positive | Not Available | Spectrum | | Predicted GC-MS | Citric acid, TMS_2_4, Predicted GC-MS Spectrum - 70eV, Positive | Not Available | Spectrum | | Predicted GC-MS | Citric acid, TMS_3_1, Predicted GC-MS Spectrum - 70eV, Positive | Not Available | Spectrum | | Predicted GC-MS | Citric acid, TMS_3_2, Predicted GC-MS Spectrum - 70eV, Positive | Not Available | Spectrum | |
| MS/MS | | Type | Description | Splash Key | View | --- --- | | MS/MS | LC-MS/MS Spectrum - Quattro_QQQ 10V, Negative (Annotated) | splash10-03di-2900000000-956001b034bbc0b7da96 | 2012-07-24 | View Spectrum | | MS/MS | LC-MS/MS Spectrum - Quattro_QQQ 25V, Negative (Annotated) | splash10-000l-9100000000-55e34e2c1616942fd4f2 | 2012-07-24 | View Spectrum | | MS/MS | LC-MS/MS Spectrum - Quattro_QQQ 40V, Negative (Annotated) | splash10-000f-9000000000-7b8be3e90a4daf85f04a | 2012-07-24 | View Spectrum | | MS/MS | LC-MS/MS Spectrum - LC-ESI-ITFT (LTQ Orbitrap XL, Thermo Scientfic) , Negative | splash10-0006-0911200000-af80d22f720facc4813a | 2012-08-31 | View Spectrum | | MS/MS | LC-MS/MS Spectrum - LC-ESI-ITFT (LTQ Orbitrap XL, Thermo Scientfic) , Negative | splash10-014i-9000000000-6f4fedbb19821898fb22 | 2012-08-31 | View Spectrum | | MS/MS | LC-MS/MS Spectrum - LC-ESI-ITFT (LTQ Orbitrap XL, Thermo Scientfic) , Negative | splash10-03di-0900000000-8eff68ab89f7ce2262be | 2012-08-31 | View Spectrum | | MS/MS | LC-MS/MS Spectrum - LC-ESI-ITFT (LTQ Orbitrap XL, Thermo Scientfic) , Negative | splash10-0fsi-0019300000-b370c4c3dd656c8eb729 | 2012-08-31 | View Spectrum | | MS/MS | LC-MS/MS Spectrum - LC-ESI-ITFT (LTQ Orbitrap XL, Thermo Scientfic) , Negative | splash10-0006-0912100000-05e8ceb6dd97f7b9af98 | 2012-08-31 | View Spectrum | | MS/MS | LC-MS/MS Spectrum - LC-ESI-ITFT (LTQ Orbitrap XL, Thermo Scientfic) , Negative | splash10-014i-9000000000-e4b67745443c48a63bd6 | 2012-08-31 | View Spectrum | | MS/MS | LC-MS/MS Spectrum - LC-ESI-ITFT (LTQ Orbitrap XL, Thermo Scientfic) , Negative | splash10-03di-0900000000-92536abec982e0a653a0 | 2012-08-31 | View Spectrum | | MS/MS | LC-MS/MS Spectrum - LC-ESI-ITFT (LTQ Orbitrap XL, Thermo Scientfic) , Negative | splash10-03di-0190000000-35772df08490f451d828 | 2012-08-31 | View Spectrum | | MS/MS | LC-MS/MS Spectrum - LC-ESI-QQ (API3000, Applied Biosystems) 10V, Negative | splash10-0006-0900000000-35856cc258368d13e1ff | 2012-08-31 | View Spectrum | | MS/MS | LC-MS/MS Spectrum - LC-ESI-QQ (API3000, Applied Biosystems) 20V, Negative | splash10-03di-3900000000-7170fdcb9749e6d6c2fb | 2012-08-31 | View Spectrum | | MS/MS | LC-MS/MS Spectrum - LC-ESI-QQ (API3000, Applied Biosystems) 30V, Negative | splash10-01p9-9500000000-9a502981b0a9e1ea35ad | 2012-08-31 | View Spectrum | | MS/MS | LC-MS/MS Spectrum - LC-ESI-QQ (API3000, Applied Biosystems) 40V, Negative | splash10-000i-9100000000-ed0c86b90e1f4966d025 | 2012-08-31 | View Spectrum | | MS/MS | LC-MS/MS Spectrum - LC-ESI-QQ (API3000, Applied Biosystems) 50V, Negative | splash10-0a4r-9000000000-b8a975c2d639d8c91afb | 2012-08-31 | View Spectrum | | MS/MS | LC-MS/MS Spectrum - LC-ESI-QTOF (UPLC Q-Tof Premier, Waters) , Negative | splash10-03di-2900000000-30970a4250098e148658 | 2012-08-31 | View Spectrum | | MS/MS | LC-MS/MS Spectrum - LC-ESI-QQ , negative | splash10-0006-0900000000-a587dddf188152c5d7b0 | 2017-09-14 | View Spectrum | | MS/MS | LC-MS/MS Spectrum - LC-ESI-QQ , negative | splash10-03di-3900000000-7170fdcb9749e6d6c2fb | 2017-09-14 | View Spectrum | | Predicted MS/MS | Predicted LC-MS/MS Spectrum - 10V, Positive | splash10-0037-0900000000-06dccacb53e75edc8b8d | 2016-09-12 | View Spectrum | | Predicted MS/MS | Predicted LC-MS/MS Spectrum - 20V, Positive | splash10-02cu-5900000000-bfea9d561968b2670b13 | 2016-09-12 | View Spectrum | | Predicted MS/MS | Predicted LC-MS/MS Spectrum - 40V, Positive | splash10-002r-7900000000-b1d32e4661447b09a017 | 2016-09-12 | View Spectrum | | Predicted MS/MS | Predicted LC-MS/MS Spectrum - 10V, Negative | splash10-0005-1900000000-d074598ec2180f8b600c | 2016-09-12 | View Spectrum | | Predicted MS/MS | Predicted LC-MS/MS Spectrum - 20V, Negative | splash10-0k92-3900000000-9917f196271a495d3d1e | 2016-09-12 | View Spectrum | | Predicted MS/MS | Predicted LC-MS/MS Spectrum - 40V, Negative | splash10-0pb9-9600000000-513b016c7c951d1e0b12 | 2016-09-12 | View Spectrum | |
| NMR | | Type | Description | | View | --- --- | | 1D NMR | 1 H NMR Spectrum (1D, 500 MHz, H 2 O, experimental) | | Spectrum | | 1D NMR | 13 C NMR Spectrum (1D, 125 MHz, H 2 O, experimental) | | Spectrum | | 1D NMR | 1 H NMR Spectrum (1D, 400 MHz, DMSO-d6, experimental) | | Spectrum | | 1D NMR | 13 C NMR Spectrum (1D, 22.53 MHz, DMSO-d6, experimental) | | Spectrum | | 1D NMR | 1 H NMR Spectrum (1D, D 2 O, experimental) | | Spectrum | | 1D NMR | 1 H NMR Spectrum (1D, D 2 O, experimental) | | Spectrum | | 1D NMR | 1 H NMR Spectrum (1D, D 2 O, experimental) | | Spectrum | | 1D NMR | 13 C NMR Spectrum (1D, D 2 O, experimental) | | Spectrum | | 1D NMR | 13 C NMR Spectrum (1D, 100 MHz, D 2 O, predicted) | | Spectrum | | 1D NMR | 1 H NMR Spectrum (1D, 100 MHz, D 2 O, predicted) | | Spectrum | | 1D NMR | 13 C NMR Spectrum (1D, 200 MHz, D 2 O, predicted) | | Spectrum | | 1D NMR | 1 H NMR Spectrum (1D, 200 MHz, D 2 O, predicted) | | Spectrum | | 1D NMR | 13 C NMR Spectrum (1D, 300 MHz, D 2 O, predicted) | | Spectrum | | 1D NMR | 1 H NMR Spectrum (1D, 300 MHz, D 2 O, predicted) | | Spectrum | | 1D NMR | 13 C NMR Spectrum (1D, 400 MHz, D 2 O, predicted) | | Spectrum | | 1D NMR | 1 H NMR Spectrum (1D, 400 MHz, D 2 O, predicted) | | Spectrum | | 1D NMR | 13 C NMR Spectrum (1D, 500 MHz, D 2 O, predicted) | | Spectrum | | 1D NMR | 1 H NMR Spectrum (1D, 500 MHz, D 2 O, predicted) | | Spectrum | | 1D NMR | 13 C NMR Spectrum (1D, 600 MHz, D 2 O, predicted) | | Spectrum | | 1D NMR | 1 H NMR Spectrum (1D, 600 MHz, D 2 O, predicted) | | Spectrum | | 1D NMR | 13 C NMR Spectrum (1D, 700 MHz, D 2 O, predicted) | | Spectrum | | 1D NMR | 1 H NMR Spectrum (1D, 700 MHz, D 2 O, predicted) | | Spectrum | | 1D NMR | 13 C NMR Spectrum (1D, 800 MHz, D 2 O, predicted) | | Spectrum | | 1D NMR | 1 H NMR Spectrum (1D, 800 MHz, D 2 O, predicted) | | Spectrum | | 1D NMR | 13 C NMR Spectrum (1D, 900 MHz, D 2 O, predicted) | | Spectrum | | 2D NMR | [1 H, 1 H]-TOCSY. Unexported temporarily by An Chi on Oct 15, 2021 until json or nmrML file is generated. 2D NMR Spectrum (experimental) | | Spectrum | | 2D NMR | [1 H, 13 C]-HSQC NMR Spectrum (2D, 600 MHz, H 2 O, experimental) | | Spectrum | |
| External Links |
| ChemSpider ID | Not Available |
| ChEMBL ID | Not Available |
| KEGG Compound ID | Not Available |
| Pubchem Compound ID | Not Available |
| Pubchem Substance ID | Not Available |
| ChEBI ID | Not Available |
| Phenol-Explorer ID | Not Available |
| DrugBank ID | Not Available |
| HMDB ID | Not Available |
| CRC / DFC (Dictionary of Food Compounds) ID | Not Available |
| EAFUS ID | Not Available |
| Dr. Duke ID | Not Available |
| BIGG ID | Not Available |
| KNApSAcK ID | Not Available |
| HET ID | Not Available |
| Food Biomarker Ontology | Not Available |
| VMH ID | Not Available |
| Flavornet ID | Not Available |
| GoodScent ID | Not Available |
| SuperScent ID | Not Available |
| Wikipedia ID | Not Available |
| Phenol-Explorer Metabolite ID | Not Available |
| Duplicate IDS | Not Available |
| Old DFC IDS | Not Available |
| Associated Foods |
| Show entries Search: | Food | Content Range | Average | Reference | --- --- | | | | | | | Abalone | Expected but not quantified | Not Available | Inferred by geno/metabolomic analysis | | Abiyuch | Expected but not quantified | Not Available | Inferred by geno/metabolomic analysis | | Acerola | Expected but not quantified | Not Available | Inferred by geno/metabolomic analysis | | Acorn | Expected but not quantified | Not Available | Inferred by geno/metabolomic analysis | | Adzuki bean | Expected but not quantified | Not Available | Inferred by geno/metabolomic analysis | | Agar | Expected but not quantified | Not Available | Inferred by geno/metabolomic analysis | | Agave | Expected but not quantified | Not Available | Inferred by geno/metabolomic analysis | | Alaska blueberry | Expected but not quantified | Not Available | Inferred by geno/metabolomic analysis | | Alaska wild rhubarb | Expected but not quantified | Not Available | Inferred by geno/metabolomic analysis | | Alfalfa | Expected but not quantified | Not Available | Inferred by geno/metabolomic analysis | Processing... Showing 1 to 10 of 455 entries Previous 1 2 3 4 5 … 46 Next |
| Biological Effects and Interactions |
| Health Effects / Bioactivities | Not Available |
| Enzymes | Not Available |
| Pathways | Not Available |
| Metabolism | Not Available |
| Biosynthesis | Not Available |
| Organoleptic Properties |
| Flavours | Not Available |
| Files |
| MSDS | Not Available |
| References |
| Synthesis Reference | Not Available |
| General Reference | Not Available |
| Content Reference | |
This project is supported by the Canadian Institutes of Health Research, Canada Foundation for Innovation, and by The Metabolomics Innovation Centre (TMIC), a nationally-funded research and core facility that supports a wide range of cutting-edge metabolomic studies. TMIC is funded by Genome Canada, Genome Alberta, and Genome British Columbia, a not-for-profit organization that is leading Canada's national genomics strategy with $900 million in funding from the federal government.
FooDB Version 1.0 — Contact Us
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. |
13043 | https://dermnetnz.org/imagedetail/14934-ash-leaf-macule-in-tuberous-sclerosis | Are you a healthcare professional
GO TO DERMNET PRO
Main menu
Common skin conditions
NEWS
Join DermNet PRO
Read more
Quick links
Skin checker
Try our skin symptom checker
Ash leaf macule in tuberous sclerosis
Ash leaf macule in tuberous sclerosis
Ash leaf macule
© DermNet
You can use or share this image if you comply with our image licence. Please provide a link back to this page.
For a high resolution, unwatermarked copy contact us here. Fees apply.
Source: dermnetnz.org
Join our newsletter
RESOURCES
Skin checker
PO-PASI scoring
AI image dataset
Quizzes
Glossary
CONTACT
Contact us
Website feedback
Volunteer
Donate
ABOUT
About DermNet
Editorial process
Website terms
Image licence
FAQ
Privacy settings
Privacy policy
© DermNet® 2025
IMPORTANT NOTICE: DermNet does not provide a free online consultation service. If you have any concerns with your skin or its treatment, see a dermatologist for advice. |
13044 | https://artofproblemsolving.com/wiki/index.php/Divisibility_rules?srsltid=AfmBOorXGbB3mzWIAh-UDiYffOovmpq07_x8i3oj7OTC-bj6-wb9ZVFp | Art of Problem Solving
Divisibility rules - AoPS Wiki
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical CampusVisit the Virtual Campus
Sign In
Register
online school
Class ScheduleRecommendationsOlympiad CoursesFree Sessions
books tore
AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates
community
ForumsContestsSearchHelp
resources
math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten
contests on aopsPractice Math ContestsUSABO
newsAoPS BlogWebinars
view all 0
Sign In
Register
AoPS Wiki
ResourcesAops Wiki Divisibility rules
Page
ArticleDiscussionView sourceHistory
Toolbox
Recent changesRandom pageHelpWhat links hereSpecial pages
Search
Divisibility rules
These divisibility rules help determine when positive integers are divisible by particular other integers. All of these rules apply for base-10only -- other bases have their own, different versions of these rules.
Contents
[hide]
1 Divisibility Videos
2 Basics
2.1 Divisibility Rule for 2 and Powers of 2
2.2 Divisibility Rule for 3 and 9
2.3 Divisibility Rule for 5 and Powers of 5
2.4 Divisibility Rule for 7
2.5 Divisibility Rule for 10 and Powers of 10
2.6 Divisibility Rule for 11
2.7 General Rule for Composites
2.7.1 Example
3 Advanced
3.1 General Rule for Primes
3.2 Divisibility Rule for 13
3.3 Divisibility Rule for 17
3.4 Divisibility Rule for 19
3.5 Divisibility Rule for 29
3.6 Divisibility Rule for 49
4 Special
4.1 Mod-preserving tests
4.1.1 Mod-preserving for 7
4.1.2 Mod-preserving for 13
4.2 Block tests
4.2.1 Small blocks -- 101 and 1001
4.2.2 Bigger blocks -- 10001 and 10000001
4.2.3 Type 2 blocks -- 111 and 11111
5 Problems
6 Resources
6.1 Books
6.2 Classes
7 See also
Divisibility Videos
Basics
Divisibility Rule for 2 and Powers of 2
A number is divisible by if and only if the last digits of the number are divisible by . Thus, in particular, a number is divisible by 2 if and only if its units digit is divisible by 2, i.e. if the number ends in 0, 2, 4, 6 or 8.
Proof
Divisibility Rule for 3 and 9
A number is divisible by 3 or 9 if and only if the sum of its digits is divisible by 3 or 9, respectively. Note that this does not work for higher powers of 3. For instance, the sum of the digits of 1899 is divisible by 27, but 1899 is not itself divisible by 27.
Proof
Divisibility Rule for 5 and Powers of 5
A number is divisible by if and only if the last digits are divisible by that power of 5.
Proof
Divisibility Rule for 7
Rule 1: Partition into 3 digit numbers from the right (). The alternating sum () is divisible by 7 if and only if is divisible by 7.
Proof
Rule 2: Truncate the last digit of , double that digit, and subtract it from the rest of the number (or vice-versa). is divisible by 7 if and only if the result is divisible by 7.
Proof
Rule 3: "Tail-End divisibility." Note. This only tells you if it is divisible and NOT the remainder. Take a number say 12345. Look at the last digit and add or subtract a multiple of 7 to make it zero. In this case we get 12380 or 12310 (both are acceptable; I am using the former). Lop off the ending 0's and repeat. 1238 - 28 ==> 1210 ==> 121 - 21 ==> 100 ==> 1 NOPE. Works in general with numbers that are relatively prime to the base (and works GREAT in binary). Here's one that works. 12348 - 28 ==> 12320 ==> 1232 +28 ==> 1260 ==> 126 + 14 ==> 14 YAY!
Tiny extension to tell you the remainder:
Count how many zeroes you lop off and mod 6.
Multiply mod 7 with the corresponding number
Zeroes (mod 6) Number to multiply by
0 1
1 3
2 2
3 6
4 4
5 5
And that's the remainder.
Divisibility Rule for 10 and Powers of 10
If a number is power of 10, define it as a power of 10. The exponent is the number of zeros that should be at the end of a number for it to be divisible by that power of 10.
Example: A number needs to have 6 zeroes at the end of it to be divisible by 1,000,000 because .
Divisibility Rule for 11
A number is divisible by 11 if the alternating sum of the digits is divisible by 11.
Proof
General Rule for Composites
A number is divisible by , where the prime factorization of is , if the number is divisible by each of .
Example
For the example, we will check if 55682168544 is divisible by 36.
The prime factorization of 36 to be . Thus we must check for divisibility by 4 and 9 to see if it's divisible by 36.
Since the last two digits, 44, of the number is divisible by 4, so is the entire number.
To check for divisibility by 9, we look to see if the sum of the digits is divisible by 9. The sum of the digits is 54 which is divisible by 9.
Thus, the number is divisible by both 4 and 9 and must be divisible by 36.
Advanced
General Rule for Primes
For every prime number other than 2 and 5, there exists a rule similar to rule 2 for divisibility by 7. For a general prime , there exists some number such that an integer is divisible by if and only if truncating the last digit, multiplying it by and subtracting it from the remaining number gives us a result divisible by . Divisibility rule 2 for 7 says that for , . The divisibility rule for 11 is equivalent to choosing . The divisibility rule for 3 is equivalent to choosing . These rules can also be found under the appropriate conditions in number bases other than 10. Also note that these rules exist in two forms: if is replaced by then subtraction may be replaced with addition. We see one instance of this in the divisibility rule for 13: we could multiply by 9 and subtract rather than multiplying by 4 and adding.
is any number so that
Divisibility Rule for 13
Rule 1: Truncate the last digit, multiply it by 4 and add it to the rest of the number. The result is divisible by 13 if and only if the original number was divisble by 13. This process can be repeated for large numbers, as with the second divisibility rule for 7.
Proof
Rule 2: Partition into 3 digit numbers from the right (). The alternating sum () is divisible by 13 if and only if is divisible by 13.
Proof
Rule 3: Works for . Let . If is odd add 39 to . Round up to the nearest multiple of 80, call the result . Find . Check: Is .
Proof
Divisibility Rule for 17
Truncate the last digit, multiply it by 5 and subtract from the remaining leading number. The number is divisible if and only if the result is divisible. The process can be repeated for any number.
Proof
Divisibility Rule for 19
Truncate the last digit, multiply it by 2 and add to the remaining leading number. The number is divisible if and only if the result is divisible. This can also be repeated for large numbers.
Proof
Divisibility Rule for 29
Truncate the last digit, multiply it by 3 and add to the remaining leading number. The number is divisible if and only if the result is divisible. This can also be repeated for large numbers.
Proof
Divisibility Rule for 49
Why 49? For taking pesky out of a root.
Useful below 4900. Round up to a multiple of 50, call it , and subtract the original number, call this . If , it is divisible by 49.
Examples:
Round up: . Difference: . ? Yes!
Round up: . Difference: . ? No!
Round up: . Difference: . ? Yes!
Extension to work for all numbers. Floor divide by 4950, multiply by 50, and add to before calculating
Proof
Special
Mod-preserving tests
These tests allow you take the modulo operation easily.
Mod-preserving for 7
Multiply the first digit by 3 and add it to the rest.
Mod-preserving for 13
Multiply the first digit by 3 and subtract it from the rest
Block tests
As a bonus, these are also mod-preserving
Small blocks -- 101 and 1001
The divisibility for 101 test is simple: Alternate adding and subtracting blocks of two digits starting from the end two, which are added.
Ex. 1102314 by 101
01 + 10 - 23 + 14 ← last block is always two digits and positive =0 so 1102314 is divisible by 101
The divisibility for 1001 is the same, but with blocks of three. (Starting with the end three, this time)
The 1001 test also works for all it's divisors. The most useful are 7, 11, and 13.
Bigger blocks -- 10001 and 10000001
10001 has block size length 4, and factors nicely into 73137.
1000001 has block size 6, and factors into 175882353. 5882353 isn't much use, but 17 is, when we're testing a large number.
Type 2 blocks -- 111 and 11111
A different type of test can be yielded from adding all the blocks, but again starting with the end.
111 has a block length of three, and factors into 37 and 3.
11111 has a length of five, and factors to 41 and 271.
1111111, with a length of seven, can provide a test for 239 and 4649, if you ever need it.
Problems
Practice Problems on Alcumus
Divisibility (Prealgebra)
2000 AMC 8 Problems/Problem 11
2006 AMC 10B Problems/Problem 25
Resources
Books
The AoPS Introduction to Number Theory by Mathew Crawford.
The Art of Problem Solving by Sandor Lehoczky and Richard Rusczyk.
Classes
AoPS Introduction to Number Theory Course
See also
Number theory
Modular arithmetic
Math books
Mathematics competitions
Retrieved from "
Category:
Divisibility Rules
Art of Problem Solving is an
ACS WASC Accredited School
aops programs
AoPS Online
Beast Academy
AoPS Academy
About
About AoPS
Our Team
Our History
Jobs
AoPS Blog
Site Info
Terms
Privacy
Contact Us
follow us
Subscribe for news and updates
© 2025 AoPS Incorporated
© 2025 Art of Problem Solving
About Us•Contact Us•Terms•Privacy
Copyright © 2025 Art of Problem Solving
Something appears to not have loaded correctly.
Click to refresh. |
13045 | https://www.quora.com/Why-does-the-function-f-x-x-not-have-a-minimum-in-the-interval-1-2-This-was-explained-to-me-by-saying-that-the-limit-x-2-x-2-but-f-x-is-never-actually-equal-to-2-since-2-is-not-in-the-domain-but-isnt-there-a-minimum | Why does the function f(x) =-x not have a minimum in the interval [1,2)? This was explained to me by saying that the limit x->2(-x) = -2 but f(x) is never actually equal to -2 since 2 is not in the domain but isn't there a minimum really close to 2? - Quora
Something went wrong. Wait a moment and try again.
Try again
Skip to content
Skip to search
Sign In
Mathematics
Minimum
F (x) =-1-1/5x
Interval Notation
Functions (general)
Limits and Continuty
Domain (mathematics)
Calculus 1
F\left (x\right)=1+x+x^2+...
5
Why does the function f(x) =-x not have a minimum in the interval [1,2)? This was explained to me by saying that the limit x->2(-x) = -2 but f(x) is never actually equal to -2 since 2 is not in the domain but isn't there a minimum really close to 2?
All related (37)
Sort
Recommended
Reuven Harmelin
Lecturer at Technion - Israel Institute of Technology (1982–present) · Author has 2.3K answers and 1.9M answer views
·3y
Suppose you have found that f(x) =-x attains its minimum with respect to the interval [1,2) at x=m very, very, very close to 2, and yet 1<=m<2, and the minimum value is then
f(m)=-m>-2
OK?
Well, absolutely not OK!!!. Why? because take for example
and observe that because 1<=m<2, we have
that is,
belongs to the interval [1,2), but guess what happens there? That is what happens:
Hence, what we assumed to be a minimum of f(x) over the given interval [1,2) is NOT really the minimum, as expected. That disproves the existence of a minimum of f(x)=-x in the interval [1,2).
Continue Reading
Suppose you have found that f(x) =-x attains its minimum with respect to the interval [1,2) at x=m very, very, very close to 2, and yet 1<=m<2, and the minimum value is then
f(m)=-m>-2
OK?
Well, absolutely not OK!!!. Why? because take for example
and observe that because 1<=m<2, we have
that is,
belongs to the interval [1,2), but guess what happens there? That is what happens:
Hence, what we assumed to be a minimum of f(x) over the given interval [1,2) is NOT really the minimum, as expected. That disproves the existence of a minimum of f(x)=-x in the interval [1,2).
Upvote ·
9 1
Promoted by The Hartford
The Hartford
We help protect over 1 million small businesses
·Updated Sep 19
What is small business insurance?
Small business insurance is a comprehensive type of coverage designed to help protect small businesses from various risks and liabilities. It encompasses a range of policies based on the different aspects of a business’s operations, allowing owners to focus on growth and success.
The primary purpose of small business insurance is to help safeguard a business’s financial health. It acts as a safety net, helping to mitigate financial losses that could arise from the unexpected, such as property damage, lawsuits, or employee injuries. For small business owners, it’s important for recovering quickl
Continue Reading
Small business insurance is a comprehensive type of coverage designed to help protect small businesses from various risks and liabilities. It encompasses a range of policies based on the different aspects of a business’s operations, allowing owners to focus on growth and success.
The primary purpose of small business insurance is to help safeguard a business’s financial health. It acts as a safety net, helping to mitigate financial losses that could arise from the unexpected, such as property damage, lawsuits, or employee injuries. For small business owners, it’s important for recovering quickly and maintaining operations.
Choosing the right insurance for your small business involves assessing your unique needs and consulting with an advisor to pick from comprehensive policy options. With over 200 years of experience and more than 1 million small business owners served, The Hartford is dedicated to providing personalized solutions that help you focus on growth and success. Learn about our coverage options!
Upvote ·
999 555
9 1
9 3
Related questions
More answers below
If the limit when h tends to 0 of f(x+h) is equal to f(X), what is the function of f?
What is the limit of the function f(x) = (e^(1/x) - 1) / (e^(1/x) + 1) as x approaches 0?
What are the domain and range of f(x) = (x-2) / (-x^2 + x + 2)?
If x^4 <= f(x) <= x^2 for x in [-1,1] and x^2 <= f(x) <= x^4 for x< -1 and x >1, at what point c do you automatically know lim x tends to c f(x) ? What can you say about the value of the limits at these points
F(x) = (x2-1) /x3-1, what is the limit where X=1?
Alireza Shariati
BS from Isfahan University of Technology (Graduated 2025) · Author has 1.4K answers and 972.7K answer views
·3y
No, there isn't. This is also true about other decreasing/increasing functions on an open interval.
f(x)=−x f(x)=−x doesn't have absolute minima on [1,2)[1,2) since if you evaluate the function at a point, no matter how close to x=2 x=2, there is always another point, closer to x=2 x=2 that gives the function a smaller value. More precisely, there is an uncountable number of points between x=2 x=2 and your point. However, if you add x=2 x=2 to your interval, the minimum value will be f(−2)f(−2).
Upvote ·
9 1
Kelly MacArthur
MS in Math & MS in Stats from The University of Utah · Author has 74 answers and 33.8K answer views
·3y
You ask such a great question and your thought process is totally in the right direction. So, let’s say that there is a minimum close to where x=2. Then, how close does x have to be to 2 to declare it the x-value of a minimum point? Is x=1.9999 close enough? Well, no, because x=1.99999999 is closer to 2. Oh, but then we can find another x-value that’s even closer to x=2, but not exactly equal to x=2. The problem is that we can do this infinitely many times and still be super close to 2 but not equal to 2. In such cases, then, we say that there is no minimum point because we can’t find one poin
Continue Reading
You ask such a great question and your thought process is totally in the right direction. So, let’s say that there is a minimum close to where x=2. Then, how close does x have to be to 2 to declare it the x-value of a minimum point? Is x=1.9999 close enough? Well, no, because x=1.99999999 is closer to 2. Oh, but then we can find another x-value that’s even closer to x=2, but not exactly equal to x=2. The problem is that we can do this infinitely many times and still be super close to 2 but not equal to 2. In such cases, then, we say that there is no minimum point because we can’t find one point close enough to the point (2, -2) since every point you give me that’s close to that point, I can give you another one that’s closer.
Upvote ·
Sohel Zibara
Studied at Doctor of Philosophy Degrees (Graduated 2000) · Author has 5.1K answers and 2.6M answer views
·3y
The function f(x) = - x is a strictly decreasing function on the given interval, it takes its maximum value at x = 1, namely - 1 and decreases all the way to the value - 2 at x = 2. But since the given interval is open at x = 2, the function f can get infinitely close to - 2, but can never take the val...
Upvote ·
9 1
Promoted by Coverage.com
Johnny M
Master's Degree from Harvard University (Graduated 2011)
·Updated Sep 9
Does switching car insurance really save you money, or is that just marketing hype?
This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars.
I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend.
Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t.
Continue Reading
This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars.
I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend.
Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. It always sounded like a hassle. Dozens of tabs, endless forms, phone calls I didn’t want to take.
But recently I decided to check so I used this quote tool, which compares everything in one place. It took maybe 2 minutes, tops. I just answered a few questions and it pulled up offers from multiple big-name providers, side by side. Prices, coverage details, even customer reviews—all laid out in a way that made the choice pretty obvious.
They claimed I could save over $1,000 per year. I ended up exceeding that number and I cut my monthly premium by over $100. That’s over $1200 a year. For the exact same coverage.
No phone tag. No junk emails. Just a better deal in less time than it takes to make coffee. Here’s the link to two comparison sites - the one I used and an alternative that I also tested.
If it’s been a while since you’ve checked your rate, do it. You might be surprised at how much you’re overpaying.
Upvote ·
999 485
999 103
99 17
Related questions
More answers below
Suppose that a function f(x) is defined for all x in [-1,1]. Can anything be said about the existence of lim x->0 f(x)?
If f(x)=1/|x-2|, when x≠2, f(2)=1. How?
How can we say that the limit of f(x) as x approaches a is equal to f(a) if they are not defined at x=a?
What is the limit of f(x) =ln (4-x²) when x approaches -2 from the left and when x approach 2 from the right?
Limit x tends to a f(x)=f(a). What does this mean?
Michael Tam
HACCP Coordinator · Author has 2.5K answers and 949.3K answer views
·3y
f ‘ (x) = - 1 < 0 for all x means that
f(x) is a decreasing function.
If the interval considered has been [1, 2],
the minimum would be f(2) = - 2.
However, the interval is [1, 2), which means
1 <= x < 2.
If there is an epsilon in the interval [1, 2), then
I can always define a delta in the interval
(epsilon, 2) such that f(delta) < f(epsilon).
But delta is never equal to 2.
Thus there is no minimum for f(x) = -x in the interval
[1, 2).
Upvote ·
Carlo
Knows Italian · Author has 1.4K answers and 551.2K answer views
·3y
The range R of
f: [1,2) → ℝ
f(x) = - x
is R=[-1,-2)
Upper and lower boundary of the set R.
Upper boundary = -1 ⇒ SupR = -1
Lower boudary = -2 ⇒ InfR = -2
Note. They always exist for each set.
Maximum and minimum of R (of f(x))
-1∈R ⇒ Maximum f(x) = SupR = -1
-2∉R ⇒ mimimum f(x) does not exist, since InfR∉R.
Upvote ·
Sponsored by Amazon Business
Solutions and supplies to support learning.
Save on essentials and reinvest in students and staff.
Sign Up
99 85
Albert Yang
BS in Mathematics, University of California, Davis · Author has 8.7K answers and 1.8M answer views
·3y
If the interval was [1,2] instead of [1,2), the minimum would indeed be -2. However, since the right endpoint is open, no matter what number you pick, you can always find one closer to -2, i.e. is smaller than the number you picked. For example, if you pick -1.9999, then (-1.9999 - 2)/2 = -1.99995, which is smaller than -1.9999 but greater than -2.
Upvote ·
Gordon M. Brown
Math Tutor at San Diego City College (2018-Present) · Author has 6.2K answers and 4.3M answer views
·1y
Related
Could we create a single variable function f(x) that has local minimum at x=-2 and no local maximum?
Yes. The simplest example might be f(x) = (x + 2)^2. In fact, an infinite number of such functions exist, as the graph below suggests:
We can get away with doing this because, technically, an absolute minimum is also a local minimum. Yet some may not be convinced by this technicality. What about a quartic function that contains both a local minimum and a separate absolute minimum? For example, begin with the function f(x) = x^4 - 3x^3 + 2x. Then add a constant a to each value of x in the function, and, using a slider for a, shift the graph to the left until the local minimum is located at x = -
Continue Reading
Yes. The simplest example might be f(x) = (x + 2)^2. In fact, an infinite number of such functions exist, as the graph below suggests:
We can get away with doing this because, technically, an absolute minimum is also a local minimum. Yet some may not be convinced by this technicality. What about a quartic function that contains both a local minimum and a separate absolute minimum? For example, begin with the function f(x) = x^4 - 3x^3 + 2x. Then add a constant a to each value of x in the function, and, using a slider for a, shift the graph to the left until the local minimum is located at x = -2, while the absolute minimum is located elsewhere.
Alas, this maneuver doesn’t seem to get us the counterexample that we want. That’s because the presence of two minima, one local, one absolute, appears to introduce an inevitable local maximum in between—in this case, in the vicinity of x = -1:
Upvote ·
9 7
Promoted by US Auto Insurance Now
US Auto Insurance Now
Helping Drivers Find Great Car Insurance Deals
·Sep 23
What are some of the most effective ways to save money?
Making smart financial decisions doesn't have to be complicated. In 2025, there are several simple yet highly effective money hacks that can make a huge difference in your financial health. These aren't complicated investment strategies; they are practical, everyday habitsthat help you keep more of your hard-earned money.
Here are 5 easy ways to boost your savings and make your income work for you:
Automate Your Savings and Investments Set up an automatic transfer to your savings account or investment portfolio the same day your paycheck hits. Even if it's a small amount like $25 a week, it
Continue Reading
Making smart financial decisions doesn't have to be complicated. In 2025, there are several simple yet highly effective money hacks that can make a huge difference in your financial health. These aren't complicated investment strategies; they are practical, everyday habitsthat help you keep more of your hard-earned money.
Here are 5 easy ways to boost your savings and make your income work for you:
Automate Your Savings and Investments Set up an automatic transfer to your savings account or investment portfolio the same day your paycheck hits. Even if it's a small amount like $25 a week, it adds up quickly and builds wealth with zero effort. The key is "out of sight, out of mind"—you won't be tempted to spend what you never see.
Leverage High-Yield Savings Accounts Don't let your emergency fund or short-term savings sit idle in a traditional bank account earning a fraction of a percent. Look for a high-yield savings account (HYSA). These accounts often have no fees and offer significantly higher interest rates, allowing your money to grow passively and effortlessly.
Cancel Unused Subscriptions Take an hour to review your bank and credit card statements from the last few months. You'll likely find subscriptions for streaming services, apps, or gym memberships you rarely use. Canceling these small, recurring expenses can free up dozens of dollars each month that can be redirected toward savings or debt payment.
Compare Car and Home Insurance Rates with a Free Tool Did you know you could be overpaying on your car or home insurance by hundreds of dollars each year without even knowing it? Insurance rates can change dramatically based on market conditions, your driving history, and even your credit score.
The best part is that instead of calling dozens of companies, there is a tool that does the hard work for you. It lets you compare real-time quotes from a network of over 1000 insurers in minutes. You simply enter your information once and see all your options side-by-side. Current users save, on average, up to $600 per year by finding a better rate. Don't leave money on the table; use this tool to see how much you could be saving today.
Start a Side Hustle In the gig economy, there are countless ways to earn extra income on your own schedule. Whether it's driving for a rideshare service, freelancing your skills (writing, graphic design, social media), or selling crafts online, a side hustle can be a powerful way to accelerate debt payoff or build up your savings. The extra income can be a great boost to your financial goals.
Upvote ·
99 26
9 1
Gordon M. Brown
Math Tutor at San Diego City College (2018-Present) · Author has 6.2K answers and 4.3M answer views
·Apr 6
Related
What is the limit of f(x) =x²-3 if x>1 and -3x+4 if x<1 at x=1?
Very simply:
Textbooks on calculus are quite clear on this matter. The limit as x approaches 1 from the left is not identical to the limit as x approaches 1 from the right. Therefore, the limit simpliciter does not exist.
It’s worth pointing out that f(x) is undefined at x = 1. That turns out not to be germane to the problem. Had the right-hand limit also equaled 1, we would say that the limit does exist, and is equal to 1 notwithstanding that f(1) is undefined.
I would suggest that you read your textbook more assiduously than is evident here.
Continue Reading
Very simply:
Textbooks on calculus are quite clear on this matter. The limit as x approaches 1 from the left is not identical to the limit as x approaches 1 from the right. Therefore, the limit simpliciter does not exist.
It’s worth pointing out that f(x) is undefined at x = 1. That turns out not to be germane to the problem. Had the right-hand limit also equaled 1, we would say that the limit does exist, and is equal to 1 notwithstanding that f(1) is undefined.
I would suggest that you read your textbook more assiduously than is evident here.
Upvote ·
9 8
Jörg Straube
M.Sc. in Computer Science, ETH Zurich (Graduated 1987) · Author has 6.3K answers and 1.7M answer views
·2y
Related
What is the limit of f(x) =ln (4-x²) when x approaches -2 from the left and when x approach 2 from the right?
The domain of f(x) = ln(4 - x^2) is the open interval (-2 .. 2).
So you can not build the lim x→-2- and lim x→2+ as f(x) is not defined.
However, the lim x→-2+ and lim x→2- is defined and is -oo
Continue Reading
The domain of f(x) = ln(4 - x^2) is the open interval (-2 .. 2).
So you can not build the lim x→-2- and lim x→2+ as f(x) is not defined.
However, the lim x→-2+ and lim x→2- is defined and is -oo
Upvote ·
9 4
Philip Lloyd
Specialist Calculus Teacher, Motivator and Baroque Trumpet Soloist. · Author has 6.8K answers and 52.8M answer views
·2y
Related
What is the solution for the question,"find the limit of f(x) as x tends to 2 from the left if f(x) = (x³-2 if x>=2, (1+x²) if x<2?
This seems to be a little “mixed up” and somewhat pointless.
I will draw the graph to explain…
I feel a strong urge to make this question into one with at least some meaning!
Conclusion, we can see that f(x) approaches 6 from both sides but because the actual point (2, 6) is not included, the function is not continuous...
Continue Reading
This seems to be a little “mixed up” and somewhat pointless.
I will draw the graph to explain…
I feel a strong urge to make this question into one with at least some meaning!
Conclusion, we can see that f(x) approaches 6 from both sides but because the actual point (2, 6) is not included, the function is not continuous...
Upvote ·
9 4
9 1
Samuel Arthur
Mathematics is fun and integrals can be tamed. · Author has 515 answers and 195.1K answer views
·1y
Related
Given that f(x) = 3x + 2, what is the limit of f as x approaches 2?
Q: If f(x)=3 x+2, then find lim x→2 f(x).Q: If f(x)=3 x+2, then find lim x→2 f(x).
A:A:
We are going to find the limit lim x→2 3 x+2 lim x→2 3 x+2, then here we solve it as followed by the simple method, direct substitution, and prove why the answer of the limit is true through epsilon-delta definition and helps of numerical and tabular method that approximation.
Direct Substitution and Limit Laws
A direct substitution in limits is the basic property that we used to evaluate the limit analytically by replacing the variable with something that was approached by the given variable.
Case 1: Direct Substitu
Continue Reading
Q: If f(x)=3 x+2, then find lim x→2 f(x).Q: If f(x)=3 x+2, then find lim x→2 f(x).
A:A:
We are going to find the limit lim x→2 3 x+2 lim x→2 3 x+2, then here we solve it as followed by the simple method, direct substitution, and prove why the answer of the limit is true through epsilon-delta definition and helps of numerical and tabular method that approximation.
Direct Substitution and Limit Laws
A direct substitution in limits is the basic property that we used to evaluate the limit analytically by replacing the variable with something that was approached by the given variable.
Case 1: Direct Substitution
Let f(x)f(x) be any function, either polynomials or rational functions for x→a x→a, where a a is in the domain of function f(x)f(x). Then
lim x→a f(x)=f(a)lim x→a f(x)=f(a)
In order to evaluate the limits analytically, you have to know the following properties of limits:
Case 2: Properties of Limits
Constant: lim x→a c=c lim x→a c=c
Constant Multiple: lim x→a c f(x)=c lim x→a f(x)lim x→a c f(x)=c lim x→a f(x)
Powers and Radicals: (3a)lim x→a x n=a n(3a)lim x→a x n=a n For the Case 3b and 3c, when we assume that if n∈N n∈N, then (3b)lim x→a[f(x)]n=[lim x→a f(x)]n(3c)lim x→a n√f(x)=n√lim x→a f(x)(3b)lim x→a[f(x)]n=[lim x→a f(x)]n(3c)lim x→a f(x)n=lim x→a f(x)n
Sum and Difference: lim x→a f(x)±g(x)=lim x→a f(x)±lim x→a g(x)lim x→a f(x)±g(x)=lim x→a f(x)±lim x→a g(x)
Products: lim x→a f(x)⋅g(x)=lim x→a f(x)⋅lim x→a g(x)lim x→a f(x)⋅g(x)=lim x→a f(x)⋅lim x→a g(x)
Quotients: lim x→a f(x)g(x)=lim x→a f(x)lim x→a g(x),if lim x→a g(x)≠0 lim x→a f(x)g(x)=lim x→a f(x)lim x→a g(x),if lim x→a g(x)≠0
Alternative 1:
lim x→2 3 x+2=lim x→2 3 x+lim x→2 2lim x→a f(x)±g(x)=lim x→a f(x)±lim x→a g(x)=3 lim x→2 xlim x→a c f(x)=c lim x→a f(x)+lim x→2 2lim x→a c=c=3(2)+2=6+2=8 lim x→2 3 x+2=lim x→2 3 x+lim x→2 2⏟lim x→a f(x)±g(x)=lim x→a f(x)±lim x→a g(x)=3 lim x→2 x⏟lim x→a c f(x)=c lim x→a f(x)+lim x→2 2⏟lim x→a c=c=3(2)+2=6+2=8
Alternative 2:
Alternative 1 is correct as well as Alternative 2, so you can evaluate the limits either indirectly, using the properties of limits, or using the substitution directly, as in Case 1.
lim x→2 3 x+2=3(2)+2=6+2=8 lim x→2 3 x+2=3(2)+2=6+2=8
Therefore all these alternatives are correct.
Proof lim x→2 3 x+2=8 by ε-δ Definition Proof lim x→2 3 x+2=8 by ε-δ Definition
Here, we are trying to prove our answer previously for the limits by the formal definition of limit. Here, the formal definition is given by the epsilon-delta definition, which was from Augustin-Louis Cauchy's phrases on the mathematically rigorous meaning assigned, which defines that our limit is correct since if the function f(x)f(x) becomes arbitrarily close to a a single integer λ λ as x x approaches a a, then by the mathematical expression we write that as
lim x→a f(x)=λ lim x→a f(x)=λ
Therefore we did this here for lim x→2 3 x+2=8 lim x→2 3 x+2=8. Here, we start by guessing a value chosen to show that for each ε>0 ε>0, there exists a δ>0 δ>0 such that,
|(3 x+2)−8|<ε whenever 0<|x−2|<δ|(3 x+2)−8|<ε whenever 0<|x−2|<δ
or other ways we can also say that
if 0<|x−2|<δ then|(3 x+2)−8|<ε if 0<|x−2|<δ then|(3 x+2)−8|<ε
Because our choice of δ δ depends on ε ε, the connection we established for two absolute values results from this algebraic process.
|(3 x+2)−8|⟹|3 x−6|⟹|3(x−2)|⟹3|x−2||(3 x+2)−8|⟹|3 x−6|⟹|3(x−2)|⟹3|x−2|
Since we had 3|x−2|3|x−2| previously, we can choose δ=ε 3 δ=ε 3 for a given ε>0 ε>0, because it works for this proof. In fact, by making this choice, we imply our statement that,
0<|x−2|<δ=ε 3 0<|x−2|<δ=ε 3
That means,
|(3 x+2)−8|⟹3|x−2|<3⋅ε 3=ε|(3 x+2)−8|⟹3|x−2|<3⋅ε 3=ε
As you can see here in the graph, for x x-values within δ δ of 2 2 (x≠2 x≠2), then values of f(x)f(x) are within ε ε of 8 8. This behavior is evident from both sides as they converge towards the limit. This table approximation reinforce the fact we states before.
x 1.9 1.99 1.999 2 2.001 2.01 2.1 f(x)7.7 7.97 7.997 8 8.003 8.03 8.3 x 1.9 1.99 1.999 2 2.001 2.01 2.1 f(x)7.7 7.97 7.997 8 8.003 8.03 8.3
That means our answer is true for and we give this expression checkmark as we notice that correct.
lim x→2 3 x+2=8✓✓lim x→2 3 x+2=8
Upvote ·
9 2
Mark Gritter
recreational mathematician · Author has 5.7K answers and 11.8M answer views
·7y
Related
What is the limit of f(x) when x approaches 2 from the right? If the function is not defined for x greater than 2.
If f(x) is not defined for x>2 x>2, then the limit “from the right” (from numbers greater than 2) does not exist either. The limit from the left may or may not exist.
For example, the function f(x)=log(2−x)f(x)=log(2−x) is not defined (on the real numbers) for x≥2 x≥2. Its limit from the left is −∞−∞ because as x x starts below 2 and gets closer to 2, log(2−x)log(2−x) gets more and more negative. Its limit from the right is undefined, because there are no values greater than 2 which we can use to construct a limit.
Upvote ·
9 3
Related questions
If the limit when h tends to 0 of f(x+h) is equal to f(X), what is the function of f?
What is the limit of the function f(x) = (e^(1/x) - 1) / (e^(1/x) + 1) as x approaches 0?
What are the domain and range of f(x) = (x-2) / (-x^2 + x + 2)?
If x^4 <= f(x) <= x^2 for x in [-1,1] and x^2 <= f(x) <= x^4 for x< -1 and x >1, at what point c do you automatically know lim x tends to c f(x) ? What can you say about the value of the limits at these points
F(x) = (x2-1) /x3-1, what is the limit where X=1?
Suppose that a function f(x) is defined for all x in [-1,1]. Can anything be said about the existence of lim x->0 f(x)?
If f(x)=1/|x-2|, when x≠2, f(2)=1. How?
How can we say that the limit of f(x) as x approaches a is equal to f(a) if they are not defined at x=a?
What is the limit of f(x) =ln (4-x²) when x approaches -2 from the left and when x approach 2 from the right?
Limit x tends to a f(x)=f(a). What does this mean?
Is the function f(x) =(x-1) (x-2) bijection?
Why is the limit of f(x) as x tends to a=k if f(x)=k?
What is : 2 - 2 x 2 + 2?
What is the limit when x approaches 3 to the left (right)? F(x) = x^2-9, if x is less than or equal to 3 and x+2, if x is greater than 3
If f(x/(x^2-1)) =x/(x^2+1), what is f(x)?
Related questions
If the limit when h tends to 0 of f(x+h) is equal to f(X), what is the function of f?
What is the limit of the function f(x) = (e^(1/x) - 1) / (e^(1/x) + 1) as x approaches 0?
What are the domain and range of f(x) = (x-2) / (-x^2 + x + 2)?
If x^4 <= f(x) <= x^2 for x in [-1,1] and x^2 <= f(x) <= x^4 for x< -1 and x >1, at what point c do you automatically know lim x tends to c f(x) ? What can you say about the value of the limits at these points
F(x) = (x2-1) /x3-1, what is the limit where X=1?
Suppose that a function f(x) is defined for all x in [-1,1]. Can anything be said about the existence of lim x->0 f(x)?
Advertisement
About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press ·
© Quora, Inc. 2025 |
13046 | https://www.cut-the-knot.org/m/Geometry/GeneralizationSimson.shtml | A Generalization of Simson Line
Typesetting math: 100%
Site...
What's new
Content page
Front page
Index page
About
Privacy policy
Help with math
Subjects...
Arithmetic
Algebra
Geometry
Probability
Trigonometry
Visual illusions
Articles...
Cut the knot!
What is what?
Inventor's paradox
Math as language
Problem solving
Collections...
Outline mathematics
Book reviews
Interactive activities
Did you know?
Eye opener
Analogue gadgets
Proofs in mathematics
Things impossible
Index/Glossary
Simple math...
Fast Arithmetic Tips
Stories for young
Word problems
Games and puzzles
Our logo
Make an identity
Elementary geometry
A Generalization of Simson Line
What Might This Be About?
Problem
Let A B C be a triangle, line ℓ pass through the circumcenter O; point P lie on the circumcircle. Let A P,B P,C P meet L at A P,B P,C P, respectively. Denote A 0,B 0,C 0 the projections of A P,B P,C P onto B C,C A,A B, respectively. Then A 0,B 0,C 0 are collinear.
Moreover, the new line passes through the midpoint of O H, where H the orthocenter of Δ A B C. If ℓ passes through P, the line coincides with the simson.
Solution 1
Let X=ℓ∩A C and M be the midpoint of A C. Let Y,Z be the projection of P,B on A C, respectively. Let H A,H B,H C,H P be the projection of A,B,C,P on ℓ, respectively. Let A′,B′,C′,P′ be the orthopole of ℓ with respect to Δ B C P,Δ C A P,Δ A B P,Δ A B C, respectively. Let R be the Poncelet point of {A,B,C,P} (It's well-known that R is the midway between P and the orthocenter H of Δ A B C).
From a lemma at post #2 at the artofproblemsolving forum, we get A′,B′,C′,P′ lie on a line τ. Since ⊙(A P A 0 C) is the pedal circle of A P with respect to Δ P A C, so from Fontene theorem we get B′∈⊙(A P A 0 C), implying A 0,A P,C,H C,B′ are concyclic.
Since H C A′⊥P B,H C B′⊥P A,H P A′⊥B C,H P B′⊥A C, so ∠B′H C A′=∠A P B=∠A C B=∠B′H P A′, from which A′,B′,H C,H P are also concyclic.
From the Reim theorem and H P A′∥A P A 0 we get A 0∈A′B′≡τ (similar discussion for B 0,C 0). It's well-known that R lie on the 9-point circle of Δ A B C, so P′,M,Z,R are concyclic at the 9-point circle of Δ A B C. Similarly B′,M,Y,R are concyclic at the 9-point circle of Δ A B C. Since B,H B,P′,X,Z are concyclic at the pedal circle of X with respect to Δ A B C, so ∠A Z P′=∠X H B P′=90∘−∠(A C,τ) (note that H B P′⊥A C). Similarly we can prove ∠C Y B′=90∘−∠(A C,τ), implying Z P′∥Y B′, thus, from Reim theorem, we get P′,R,B′ are collinear, i.e., R∈τ≡A 0 B 0 C 0¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯.
Solution 2
Fix the line ℓ and animate P. The pencils P A,P B,P C are projective inducing a proyectivity on ℓ, i.e. the series A p,B p,C p are projective ⟹ series A 0,B 0,C 0 are projective.
Let D,F be the antipodes of A,C on the circumcircle (O) and consider the case when A p∈B F. If B C p cuts (O) again at D′, then by Pascal theorem for A P C F B D′, it follows that A p,C p,C F∩A D′ are collinear ⟹D≡D′⟹∠C p B A=90∘⟹B≡A 0≡C 0⟹A 0↦C 0 is a perspectivity ⟹A 0 C 0 goes through a fixed point. When P coincides with {X,Y}≡ℓ∩(O), then A 0 C 0 becomes Simson lines of X,Y meeting at the orthopole T of ℓ⟹T∈A 0 C 0 and similarly T∈B 0 C 0⟹A 0,B 0,C 0 are collinear on a line τ passing through T.
Let H be the orthocenter of △A B C and let X be the midpoint of H P lying on 9-point circle (N). It's known that T∈(N) when O∈ℓ. Now since X∧¯¯¯¯P∧¯¯¯¯A 0 with fixed points at (N)∩B C, then it follows that X↦A 0 is a stereographic projection of (N) onto B C⟹X∈T A 0≡τ.
Solution 3
If there exists a point W on the circumcircle of △A B C, denote by d W the Simson line of W WRT △A B C Now, let Q be the reflection of P WRT the line L and let M be the midpoint of P H (H is the orthocenter of △A B C) We will prove that: M A 0∥d Q.
Indeed: T,D are the projections of P,H on B C M P=M H⇒M D=M T. On the other hand:
(T M,T D)=(d P,d S)=1 2.S P=(A S,A P)=(A O,A P)=−(P O,P A)(mod π)
It follows that
(1) △T M D∼△P O A.
(2) A 0 T A 0 D=A P P A P A (because A D∥A P A 0∥P T)
The two imply that △T M A 0∼△P O A P, so that
(M T,M A 0)=(O A P,O P)=1 2.Q P=(d P,d Q) (mod π)
I.e., M A 0∥d Q.
Similarly: M B 0,M C 0∥d Q, implying M,A 0,B 0,C 0¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ or A 0,B 0,C 0¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ bisects H P.
Solution 4
Lemma 1 (well-known)
The two circles with diameter A C P,C A P intersects at two points X,Y , one of them (say X) lies on (O), the second, (say Y ), lies on the nine-point circle of △P A C.}
Proof of Lemma 1
Let C O cut (O) again at K; K A P cut (O) at X; X C P cut (O) again at L, then by Pascal in hexagon L A P C K X we get A,O,L collinear. Hence X lies on the circle with diameter C A P as well as the circle with diameter A C P. Let the circle with diameter A C P cut P C at A 1, the circle with diameter C A P cut P A at C 1, then A A 1,C C 1 are two altitudes of △P A C. If they intersect at H 1 then H 1 obviously lies on the radical axis of the two circles in question, thus H 1 lies on X Y. Note that
∠C 1 Y A 1=∠C 1 Y X+∠X Y A 1=∠C 1 C X+∠X A H 1=360∘−∠A X C−∠A H 1 C=(180∘−∠A X C)+(180∘−∠A H 1 C)=∠A P C+∠A P C=2∠A P C.
This means that if I is the midpoint of P H 1 then C 1 I Y A 1 cyclic, or Y lies on the nine-point circle of △P A C.
Lemma 2
A 0 C 0 passes through Y.
Proof of Lemma 2
Note that ∠X Y C 0=∠X A C 0=∠X C A 0=∠X Y A 0. This implies that A 0,C 0,Y are collinear.
Let A Q,C T the altitudes of △A B X with H the orthocenter. Easy to get A,H 1,C,H cyclic and the circle they lie on is the mirror of the circle O over A C. If R is the midpoint of A C and M midpoint of P H then it is easy to see that R, M are the two common points of the nine-point circle of △P A C and the nine-point circle of △B A C. In other words, M is the midpoint of P H. So if S is the midpoint of P C 4 then M S∥C 0 C P.
Now we are going to show that M lies on the line Y C 0 A 0. Note that
∠X Y M=∠X Y A 1−∠M Y A 1=∠X A A 1−(180−∠M S A 1)=∠X A A 1−(180∘−∠C 0 C P A 1)=∠X A A 1−∠C 0 A A 1=∠X Y C 0.
This means that Y,M,C 0 are collinear or M lies on the line Y C 0 A 0, as desired.
Next, let the circle with diameter C B P cut the circle with diameter B C P intersect at W,Y . By Lemma 1, Z lies on (O) and W lies on the nine-point circle of P B C. There is no problem to see that the nine-point circle of △P A C goes through S, M. By Lemma 2, we have C 0,B 0,W collinear. Since ∠W M S=∠W C S=∠W C 0 C P and M S∥C 0 C P, then W,M,C 0 are collinear. Hence all the points A 0,C 0,M,Y,B 0,W lie on a line d; this line bisects the line segment joining P and the orthocenter of △B A C (that is P H). Of course then the line l goes through P , then X,C P,A P≡P ; P 1≡Y, d is the Simson line of △A B C.
Acknowledgment
Oai Thanh Dào posted the problem in September 2014 at the Advanced Plane Geometry group (message 1781) and subsequently at the CutTheKnotMath facebook page alnog with Telv Cohl's solution (Solution 1). Solution 2 is by Luis Gonzalez, Solution 3 by Tran Quang Huy, Solution 4 is by Tran Lam.
Related material
Read more...
Simson Line - the simson
Simson Line: Introduction
Simson Line
Three Concurrent Circles
9-point Circle as a locus of concurrency
Miquel's Point
Circumcircle of Three Parabola Tangents
Angle Bisector in Parallelogram
Simsons and 9-Point Circles in Cyclic Quadrilateral
Reflections of a Point on the Circumcircle
Simsons of Diametrically Opposite Points
Simson Line From Isogonal Perspective
Pentagon in a Semicircle
Simson Line in Disguise
Two Simsons in a Triangle
Carnot's Theorem
|Contact||Front page||Contents||Geometry|
Copyright © 1996-2018 Alexander Bogomolny
73255809 |
13047 | https://www.quora.com/How-can-I-mathematically-show-the-relation-between-the-launch-height-and-the-range-covered-by-the-projectile | How to mathematically show the relation between the launch height and the range covered by the projectile - Quora
Something went wrong. Wait a moment and try again.
Try again
Skip to content
Skip to search
Sign In
Mathematics
Oblique Projectiles Motio...
Physics
Kinematic Equations
Range of a Projectile
Launch Angle
Gravitational Projectile ...
Application of Projectile
Motion Projectile Vector
5
How can I mathematically show the relation between the launch height and the range covered by the projectile?
All related (37)
Sort
Recommended
Jafar Mortadha
Mechanical Engineer · Author has 1.4K answers and 3.6M answer views
·8y
You need to use the equations of motion. Assuming the projectile makes an angle θ θ with the horizontal, you need to find the horizontal component of velocity and the total time taken by the projectile to land. The total time can be found by taking the overall vertical displacement of the projectile, which is equal to launch height in the downward direction.
If θ=0 θ=0, R=√2 v 2 h g R=2 v 2 h g
Continue Reading
You need to use the equations of motion. Assuming the projectile makes an angle θ θ with the horizontal, you need to find the horizontal component of velocity and the total time taken by the projectile to land. The total time can be found by taking the overall vertical displacement of the projectile, which is equal to launch height in the downward direction.
If θ=0 θ=0, R=√2 v 2 h g R=2 v 2 h g
Upvote ·
9 6
9 1
9 1
Sponsored by RedHat
Know what your AI knows, with open source models.
Your AI should keep records, not secrets.
Learn More
99 36
Related questions
More answers below
What is the relation between horizontal range and maximum height of projectile?
What is the relation between the max height and the max range of a projectile?
What happens when the height of a projectile is doubled in range?
What is the relationship between projection angle and height and range of the projectile?
Why does a projectile's range increase as the launch height is increased, if everything else is kept constant?
Mary Sanchez
Tallness Specialist: Elevate Stature Expert · Author has 67 answers and 134.3K answer views
·1y
The relation between launch height and projectile range is expressed through projectile motion equations. By understanding the physics behind projectile motion, you can derive mathematical relationships that illustrate how launch height influences range. Explore my Quora Profile for detailed discussions on physics and mathematical modeling.
Upvote ·
9 2
Geoff Mangum
Self-study 1973-2019. · Author has 3.8K answers and 3.1M answer views
·2y
Find the ratio of the two.
range = cos(angle)•v(launch) • sin(angle) • v(launch) • 2/g
range = cos(angle) • sin(angle) • v(launch)^2 • 2/g
peak h = sin(angle) • v(launch)/2 • sin(angle) • v(launch)/g
peak h = sin(angle) • v(launch) • sin(angle) • v(launch) • 1 / (2•g)
peak = sin(angle) • sin(angle) • v(launch)^2 • 1 / (2•g)
h/range = sin(angle) • sin(angle) • v(launch)^2 • 1 / (2•g) / cos(angle) • sin(angle) • v(launch)^2 • 2/g
h/range = sin(angle) / cos(angle) • 1 / 2•g • g / 2
h/range = tan(angle) / 4
range/h = 4 / tan(angle)
Ex: range = 34 m, angle = 30 degrees
34/h = 4 / tan(30) = 6.928
h = 34 m / 6.9
Continue Reading
Find the ratio of the two.
range = cos(angle)•v(launch) • sin(angle) • v(launch) • 2/g
range = cos(angle) • sin(angle) • v(launch)^2 • 2/g
peak h = sin(angle) • v(launch)/2 • sin(angle) • v(launch)/g
peak h = sin(angle) • v(launch) • sin(angle) • v(launch) • 1 / (2•g)
peak = sin(angle) • sin(angle) • v(launch)^2 • 1 / (2•g)
h/range = sin(angle) • sin(angle) • v(launch)^2 • 1 / (2•g) / cos(angle) • sin(angle) • v(launch)^2 • 2/g
h/range = sin(angle) / cos(angle) • 1 / 2•g • g / 2
h/range = tan(angle) / 4
range/h = 4 / tan(angle)
Ex: range = 34 m, angle = 30 degrees
34/h = 4 / tan(30) = 6.928
h = 34 m / 6.928 = 4.91 m
Ex: peak h = 15 m, angle = 48 degrees
range/15 m = 4 / tan(48) = 1.1106
range = 1.1106 • 15 m = 16.66 m
Upvote ·
Les McLean
Ph.D. in Engineering · Author has 4.9K answers and 12.3M answer views
·5y
Originally Answered: What is the equation relating to the range and launch height of a projectile? ·
The notes from my lecture “Projectiles 101” may be useful to you:
At any time t, a projectile's horizontal and vertical displacement are:
x = VtCos θ where V is the initial velocity, θ is the launch angle
y = VtSinθ – ½gt^2
The velocities are the time derivatives of displacement:
Vx = VCosθ (note that Vx does not depend on t, so Vx is constant)
Vy = VSinθ – gt
Velocity = Vx i + Vy j
The magnitude of velocity is √(Vx^2 + Vy^2)
At maximum height, Vy = 0 = VSinθ – gt
So at maximum height, t = (VSinθ)/g [total flight time = 2t]
The range R of a projectile launched at an angle θ with a velocity V is:
R = V^2 Sin
Continue Reading
The notes from my lecture “Projectiles 101” may be useful to you:
At any time t, a projectile's horizontal and vertical displacement are:
x = VtCos θ where V is the initial velocity, θ is the launch angle
y = VtSinθ – ½gt^2
The velocities are the time derivatives of displacement:
Vx = VCosθ (note that Vx does not depend on t, so Vx is constant)
Vy = VSinθ – gt
Velocity = Vx i + Vy j
The magnitude of velocity is √(Vx^2 + Vy^2)
At maximum height, Vy = 0 = VSinθ – gt
So at maximum height, t = (VSinθ)/g [total flight time = 2t]
The range R of a projectile launched at an angle θ with a velocity V is:
R = V^2 Sin2θ / g
The maximum height H is
H = V^2 Sin^2(θ) / 2g
Upvote ·
9 2
Promoted by JH Simon
JH Simon
Author of 'How To Kill A Narcissist'
·Updated Fri
Narcissist abuse has lead me into a very dark depression. How do I overcome this?
Pendulate between the darkness and the light.
Depression is a healing tonic which restores the Self to a point of equilibrium. Remember that while in the narcissistic relationship you were identified with a grandiose construct, i.e. the false Self of the narcissist. Your old identity was demolished, and you were reprogrammed according to the narcissist’s tastes. This false identity is now crumbling, and your ego is undergoing a process of grief. That is what the depression is.
Your ego drew a sense of identity from the narcissist, and it wants that identity back. It does not care what kind of id
Continue Reading
Pendulate between the darkness and the light.
Depression is a healing tonic which restores the Self to a point of equilibrium. Remember that while in the narcissistic relationship you were identified with a grandiose construct, i.e. the false Self of the narcissist. Your old identity was demolished, and you were reprogrammed according to the narcissist’s tastes. This false identity is now crumbling, and your ego is undergoing a process of grief. That is what the depression is.
Your ego drew a sense of identity from the narcissist, and it wants that identity back. It does not care what kind of identity you have; only that you should have one. It does not realise that you can rebuild yourself in a more actualised, empowered way.
Before you can do that, however, you must grieve. Ideally you want to direct all of your awareness into the depression, to expand your consciousness and accept the depression in all of its intensity. However, that might be too overwhelming initially. Instead, take time each day to sit in an upright position and simply direct your consciousness toward the feeling of depression for as long as you can tolerate. Note its intensity. Where in the body does it manifest? In the chest? In your stomach? Let your face droop, let your body soften, let yourself be as sad and depressed as needed. Go with the flow. Do not think about it or analyse it, simply observe it and allow it to happen. This is how you allow the grieving process to complete itself. Just when you think it will never end, it will begin to transform.
But that could be days, weeks or months away. For now, simply take time out each day to do this practice. When you become overwhelmed, and surely you will at the beginning, change up and do something that brings you relaxation and joy. Take a bath, spend time with a good friend, watch your favourite TV show, go for a walk, do exercise. When you are sufficiently filled, go back into the dark and sit there i.e. be conscious with it. You can be sure that when the work is done, the sun will shine again, and the darkness will recede back into the depths of your being. Then the spiritual growth can begin.
Best of luck.
If you have just started your narcissistic abuse recovery journey, check out How To Kill A Narcissist. Or if you wish to immunise yourself against narcissists and move on for good, take a look at How To Bury A Narcissist.
Upvote ·
999 507
99 18
99 12
Related questions
More answers below
What is the mathematical relation between the maximum horizontal range and the maximum height of a projectile?
How do I calculate the relationship between the maximum height and the range of a projectile?
At what angle of projection are the horizontal range and maximum height equal in projectile motion?
For what launch angle is the maximum height of a projectile 1/3 of its range?
Which angle projectile range is maximum?
Assistant
Bot
·
1y
To show the mathematical relationship between the launch height and the range of a projectile, we can start with the basic equations of projectile motion. The range R R of a projectile launched at an angle θ θ with an initial velocity v 0 v 0 from a height h h can be derived using the following steps.
Step 1: Break down the motion
Vertical Motion:
The vertical motion is described by the equation:
y(t)=h+v 0 sin(θ)t−1 2 g t 2 y(t)=h+v 0 sin(θ)t−1 2 g t 2
where:
y(t)y(t) is the vertical position at time t t,
g g is the acceleration due to gravity (approximately 9.81 m/s 2 9.81 m/s 2),
h h is the i
Continue Reading
To show the mathematical relationship between the launch height and the range of a projectile, we can start with the basic equations of projectile motion. The range R R of a projectile launched at an angle θ θ with an initial velocity v 0 v 0 from a height h h can be derived using the following steps.
Step 1: Break down the motion
Vertical Motion:
The vertical motion is described by the equation:
y(t)=h+v 0 sin(θ)t−1 2 g t 2 y(t)=h+v 0 sin(θ)t−1 2 g t 2
where:
y(t)y(t) is the vertical position at time t t,
g g is the acceleration due to gravity (approximately 9.81 m/s 2 9.81 m/s 2),
h h is the initial height.
Horizontal Motion:
The horizontal motion is described by:
x(t)=v 0 cos(θ)t x(t)=v 0 cos(θ)t
Step 2: Determine the time of flight
To find the time of flight T T, we set y(T)=0 y(T)=0 (when the projectile hits the ground):
0=h+v 0 sin(θ)T−1 2 g T 2 0=h+v 0 sin(θ)T−1 2 g T 2
This is a quadratic equation in T T:
−1 2 g T 2+v 0 sin(θ)T+h=0−1 2 g T 2+v 0 sin(θ)T+h=0
Using the quadratic formula T=−b±√b 2−4 a c 2 a T=−b±b 2−4 a c 2 a, where a=−1 2 g a=−1 2 g, b=v 0 sin(θ)b=v 0 sin(θ), and c=h c=h:
T=−v 0 sin(θ)±√(v 0 sin(θ))2+2 g h−g T=−v 0 sin(θ)±(v 0 sin(θ))2+2 g h−g
This simplifies to:
T=v 0 sin(θ)+√(v 0 sin(θ))2+2 g h g T=v 0 sin(θ)+(v 0 sin(θ))2+2 g h g
Step 3: Calculate the range
Now, substitute T T into the horizontal motion equation:
R=x(T)=v 0 cos(θ)T R=x(T)=v 0 cos(θ)T
Substituting for T T:
R=v 0 cos(θ)(v 0 sin(θ)+√(v 0 sin(θ))2+2 g h g)R=v 0 cos(θ)(v 0 sin(θ)+(v 0 sin(θ))2+2 g h g)
This gives the range R R as a function of the launch height h h, the launch angle θ θ, and the initial velocity v 0 v 0.
Conclusion
The range R R is influenced by the launch height h h. As h h increases, the time of flight T T increases, which in turn increases the range R R. The exact relationship is more complex but can be analyzed through the equation derived above. For practical applications, you can analyze specific cases by substituting values for h h, v 0 v 0, and θ θ.
Upvote ·
Allen Ries
Math Major University of Alberta · Author has 25.1K answers and 9.7M answer views
·2y
first you need v to know how long the projectile is in the air. I assume we already know the height ‘s’ above the ground and the initial vertical velocity ‘u’. Then s=ut+1/2gt². Solve this equation for the time ‘t’.
The range ‘r’ is calculated as r= ut where ‘u’ is the horizontal velocity and time ‘t’ is the time found in step 1.
If the initial velocity is not vertical use : r=u²sin2θ/g.
Upvote ·
Nguyễn Hiếu Long
Physics enthusiast
·7y
Related
What is the relation between horizontal range and maximum height of projectile?
First, there are some essential equation regarding one-dimension motion with constant acceleration:
v=v 0+a t v=v 0+a t (1)
x=x 0+v 0 t+1 2 a t 2 x=x 0+v 0 t+1 2 a t 2 (2)
v 2−v 2 0=2 a s v 2−v 0 2=2 a s (3)
(where v is the final velocity, v 0 v 0 is the initial velocity, x is the displacement, s is the distance the object travels along the direction of acceleration)
(All this set of equations can be proved with simple calculus)
Now apply to two-dimension motion, Set up the projectile problem as shown below,
Apply (1) to find velocity components:
On the x-axis: u x=u 0 c o s(θ)u x=u 0 c o s(θ)(since there is no acceleration on x-axis)
On the y_axis
Continue Reading
First, there are some essential equation regarding one-dimension motion with constant acceleration:
v=v 0+a t v=v 0+a t (1)
x=x 0+v 0 t+1 2 a t 2 x=x 0+v 0 t+1 2 a t 2 (2)
v 2−v 2 0=2 a s v 2−v 0 2=2 a s (3)
(where v is the final velocity, v 0 v 0 is the initial velocity, x is the displacement, s is the distance the object travels along the direction of acceleration)
(All this set of equations can be proved with simple calculus)
Now apply to two-dimension motion, Set up the projectile problem as shown below,
Apply (1) to find velocity components:
On the x-axis: u x=u 0 c o s(θ)u x=u 0 c o s(θ)(since there is no acceleration on x-axis)
On the y_axis: u y=u 0 s i n(θ)−g t u y=u 0 s i n(θ)−g t (sine a y=−g a y=−g)
Apply (3) on the y axis, note that when the object reaches maximum height, v y=0 v y=0
=>0 2−(u 0 s i n(θ))2=2(−g)H 0 2−(u 0 s i n(θ))2=2(−g)H
=>H=(u 0 s i n(θ))2 2 g H=(u 0 s i n(θ))2 2 g
Now, calculate the total flight time of the object, at t = 0 u y=u 0 s i n(θ)u y=u 0 s i n(θ) , when the object touches the ground, u y=−u 0 s i n(θ)u y=−u 0 s i n(θ)
Apply (1): −u 0 s i n(θ)=u 0 s i n(θ)−g t 1−u 0 s i n(θ)=u 0 s i n(θ)−g t 1
=>t 1=2 u 0 s i n(θ)g t 1=2 u 0 s i n(θ)g
To find the horizontal range:
R=u x.t 1=u 0 c o s(θ)2 u 0 s i n(θ)g R=u x.t 1=u 0 c o s(θ)2 u 0 s i n(θ)g
which finally gives, R=u 2 0 g 2 s i n(θ)c o s(θ)R=u 0 2 g 2 s i n(θ)c o s(θ)
So the relation between H and R is
R=4 H c o s(θ)s i n(θ)R=4 H c o s(θ)s i n(θ)
Using the same method (set of 3 equations), we can solve a harder problem from this: find the range and maximum height when throw an object at a height h, and see the difference.
Hope this helps!
Upvote ·
99 66
9 2
Sponsored by Digital News Wire
What Everyone Should Know in the Digital Age.
Understanding the Evolving Concept of Financial Literacy in a Technology-Driven Economy.
Learn More
Ali Abdulla
Expert in Education, professor of Nuclear physics,Author · Author has 22.6K answers and 6.2M answer views
·8y
You can read the projectiles subject in any classical mechanical book, such as Griffith or Goldestein,you find the range is X = h cot theta.where h is the height and theta is the angle.
Your response is private
Was this worth your time?
This helps us sort answers on the page.
Absolutely not
Definitely yes
Upvote ·
Sushant Mishra
B.sc in Physics, St. Xavier's College, Ranchi (Graduated 2019) · Upvoted by
Mate Bacha
, Masters Physics & Another Natural Science, Kamla Nehru Public Higher Secondary School (2022)
·7y
Related
At what angle of projection are the horizontal range and maximum height equal in projectile motion?
Upvote ·
999 150
9 8
9 2
Sponsored by VAIZ.com
Tool Like Asana — Try For Free Our Better Alternative!
VAIZ — Better Asana Alternative That Boosts Productivity with Built-in Doc Editor & Task Management.
Sign Up
99 14
Les McLean
Ph.D. in Engineering · Author has 4.9K answers and 12.3M answer views
·6y
Related
What is the relationship between projection angle and height and range of the projectile?
The range R of a projectile launched at an angle θ with a velocity Vo is:
R = Vo^2 Sin2θ / g
The maximum height H is
H = Vo^2 Sin^2(θ) / 2g
EXAMPLE:
A projectile is launched with a velocity Vo = 100m/s at an angle of 30° to the horizontal.
The range R = (100^2)(0.866)/9.81 = 882.77m
The maximum height is H = (100^2)(0.5^2)/19.62 = 127.42
Another way to show the relationship is to use Vo^2 = Rg/Sin2θ from range and Vo^2 = 2Hg/Sin^2(θ) from height,
So Rg/Sin2θ = 2Hg/Sin^2(θ)
Or H/R = Sin^2θ/2Sin2θ
Simplifying using trig identities:
H/R = (Tanθ)/4
Or Tanθ = 4H/R
Upvote ·
9 4
Philip Lloyd
Former Specialist Calculus Teacher and Mentor.. · Author has 6.8K answers and 52.8M answer views
·1y
Related
At what angle does the range of projectile become equal to the height of projectile?
When does R = H
Suppose an object is thrown from O with a velocity V at an angle α.
T...
Upvote ·
9 7
Steven Digby
Physicist/Educator for 4 years · Author has 2.6K answers and 467.9K answer views
·Updated 9mo
Related
How do you calculate the height of a projectile that goes straight up and then back down?
The first thing to note, in this problem, is that at the point of maximum height the velocity is zero, because the motion slows down in the upward direction then speeds up in the downward direction, so it's reasonable to expect there to be a point where it's zero.
v = 0
The next stage is to make an assumption. We're going to assume that the acceleration due to gravity is -9.8m/s² for the whole journey. The negative sign is because I have (arbitrarily) assigned a positive value to the upward velocity so the downward direction of the acceleration is negative. This is because acceleration and veloc
Continue Reading
The first thing to note, in this problem, is that at the point of maximum height the velocity is zero, because the motion slows down in the upward direction then speeds up in the downward direction, so it's reasonable to expect there to be a point where it's zero.
v = 0
The next stage is to make an assumption. We're going to assume that the acceleration due to gravity is -9.8m/s² for the whole journey. The negative sign is because I have (arbitrarily) assigned a positive value to the upward velocity so the downward direction of the acceleration is negative. This is because acceleration and velocity are both vector values with direction built in to them and in this problem they are acting in opposite directions. The maths would still work if I made velocity negative and acceleration positive.
a = -9.8
The third stage requires an extra piece of information. Either the time to reach maximum height (t) or the initial velocity (u) as the object was projected straight up.
If we know the initial velocity (u) then the equation is
v² = u² + 2as
u² = 19.6 x s
s = u² / 19.6
If we know the time taken then the equation is
v = u + at
u = 9.8 x t
s = (9.8 x t)² / 19.6
s = 4.9 x t²
And that's our first model.
If the projectile goes up very high then it might not be safe to assume constant acceleration due to gravity. Instead, we would need to divide the journey into segments and do the above calculations for each segment with a calculated value for acceleration at each height.
a = MG / d²
(d is the distance from the Earth's centre of gravity. It might be that d = R + s, where R is the radius of the Earth)
And that's our second model
The more segments you have the more accurate your answer will be. If you can use integration to sum up over infinitesimal segments, all the better.
Note, that if the initial velocity is greater than escape velocity then the projectile will never reach zero velocity and never come back down.
Upvote ·
9 1
Eduardo Dequilla
B.S. Physics from Ateneo de Davao University (Graduated 2005) · Author has 8.2K answers and 12.4M answer views
·3y
Related
If the max height of a projectile is 15 m, what will be the max range?
If the max height of a projectile is 15 m then its maximum range is 4 max height and it is equal to 60 m. The screenshot of the output of my projectile motion program gives the value of the range as 60.0 m. This happens only when the launched angle is 45 degrees. Shown below has a zenith and maxH of 15.0 m and a range value of 60.0 m. The angle theta is 45 degrees.
Continue Reading
If the max height of a projectile is 15 m then its maximum range is 4 max height and it is equal to 60 m. The screenshot of the output of my projectile motion program gives the value of the range as 60.0 m. This happens only when the launched angle is 45 degrees. Shown below has a zenith and maxH of 15.0 m and a range value of 60.0 m. The angle theta is 45 degrees.
Upvote ·
9 1
Related questions
What is the relation between horizontal range and maximum height of projectile?
What is the relation between the max height and the max range of a projectile?
What happens when the height of a projectile is doubled in range?
What is the relationship between projection angle and height and range of the projectile?
Why does a projectile's range increase as the launch height is increased, if everything else is kept constant?
What is the mathematical relation between the maximum horizontal range and the maximum height of a projectile?
How do I calculate the relationship between the maximum height and the range of a projectile?
At what angle of projection are the horizontal range and maximum height equal in projectile motion?
For what launch angle is the maximum height of a projectile 1/3 of its range?
Which angle projectile range is maximum?
What is the range of the projectile motion launched from some height?
How do you show that range = maximum height?
What is vertical range of projectile?
How do you calculate the time to reach the maximum height for a projectile?
How do you find the launch angle theta that will give the same max height and range for a projectile?
Related questions
What is the relation between horizontal range and maximum height of projectile?
What is the relation between the max height and the max range of a projectile?
What happens when the height of a projectile is doubled in range?
What is the relationship between projection angle and height and range of the projectile?
Why does a projectile's range increase as the launch height is increased, if everything else is kept constant?
What is the mathematical relation between the maximum horizontal range and the maximum height of a projectile?
Advertisement
About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press ·
© Quora, Inc. 2025 |
13048 | https://www.cia.gov/the-world-factbook/field/population/country-comparison/ | Go to CIA.gov
WFBThe World Factbook
Country Comparisons
Population
Compares estimates from the US Bureau of the Census based on statistics from population censuses, vital statistics registration systems, or sample surveys pertaining to the recent past and on assumptions about future trends.
237Results
Filter Rank Type
total
Filter Regions
All
| Rank | Country | Date of Information |
---
| 1 | China | 1,416,043,270 | 2024 est. |
| 2 | India | 1,409,128,296 | 2024 est. |
| 3 | United States | 341,963,408 | 2024 est. |
| 4 | Indonesia | 281,562,465 | 2024 est. |
| 5 | Pakistan | 252,363,571 | 2024 est. |
| 6 | Nigeria | 236,747,130 | 2024 est. |
| 7 | Brazil | 220,051,512 | 2024 est. |
| 8 | Bangladesh | 168,697,184 | 2024 est. |
| 9 | Russia | 140,820,810 | 2024 est. |
| 10 | Mexico | 130,739,927 | 2024 est. |
| 11 | Japan | 123,201,945 | 2024 est. |
| 12 | Ethiopia | 118,550,298 | 2024 est. |
| 13 | Philippines | 118,277,063 | 2024 est. |
| 14 | Congo, Democratic Republic of the | 115,403,027 | 2024 est. |
| 15 | Egypt | 111,247,248 | 2024 est. |
| 16 | Vietnam | 105,758,975 | 2024 est. |
| 17 | Iran | 88,386,937 | 2024 est. |
| 18 | Turkey (Turkiye) | 84,119,531 | 2024 est. |
| 19 | Germany | 84,119,100 | 2024 est. |
| 20 | Thailand | 69,920,998 | 2024 est. |
| 21 | United Kingdom | 68,459,055 | 2024 est. |
| 22 | France | 68,374,591 | 2024 est. |
| 23 | Tanzania | 67,462,121 | 2024 est. |
| 24 | Italy | 60,964,931 | 2024 est. |
| 25 | South Africa | 60,442,647 | 2024 est. |
| 26 | Kenya | 58,246,378 | 2024 est. |
| 27 | Burma | 57,527,139 | 2024 est. |
| 28 | Korea, South | 52,081,799 | 2024 est. |
| 29 | Sudan | 50,467,278 | 2024 est. |
| 30 | Colombia | 49,588,357 | 2024 est. |
| 31 | Uganda | 49,283,041 | 2024 est. |
| 32 | Spain | 47,280,433 | 2024 est. |
| 33 | Algeria | 47,022,473 | 2024 est. |
| 34 | Argentina | 46,994,384 | 2024 est. |
| 35 | Iraq | 42,083,436 | 2024 est. |
| 36 | Afghanistan | 40,121,552 | 2024 est. |
| 37 | Canada | 38,794,813 | 2024 est. |
| 38 | Poland | 38,746,310 | 2024 est. |
| 39 | Morocco | 37,387,585 | 2024 est. |
| 40 | Angola | 37,202,061 | 2024 est. |
| 41 | Saudi Arabia | 36,544,431 | 2024 est. |
| 42 | Uzbekistan | 36,520,593 | 2024 est. |
| 43 | Ukraine | 35,661,826 | 2024 est. |
| 44 | Ghana | 34,589,092 | 2024 est. |
| 45 | Malaysia | 34,564,810 | 2024 est. |
| 46 | Mozambique | 33,350,954 | 2024 est. |
| 47 | Peru | 32,600,249 | 2024 est. |
| 48 | Yemen | 32,140,443 | 2024 est. |
| 49 | Venezuela | 31,250,306 | 2024 est. |
| 50 | Nepal | 31,122,387 | 2024 est. |
| 51 | Cameroon | 30,966,105 | 2024 est. |
| 52 | Cote d'Ivoire | 29,981,758 | 2024 est. |
| 53 | Madagascar | 29,452,714 | 2024 est. |
| 54 | Australia | 26,768,598 | 2024 est. |
| 55 | Niger | 26,342,784 | 2024 est. |
| 56 | Korea, North | 26,298,666 | 2024 est. |
| 57 | Syria | 23,865,423 | 2024 est. |
| 58 | Taiwan | 23,595,274 | 2024 est. |
| 59 | Burkina Faso | 23,042,199 | 2024 est. |
| 60 | Mali | 21,990,607 | 2024 est. |
| 61 | Sri Lanka | 21,982,608 | 2024 est. |
| 62 | Malawi | 21,763,309 | 2024 est. |
| 63 | Zambia | 20,799,116 | 2024 est. |
| 64 | Kazakhstan | 20,260,006 | 2024 est. |
| 65 | Chad | 19,093,595 | 2024 est. |
| 66 | Senegal | 18,847,519 | 2024 est. |
| 67 | Chile | 18,664,652 | 2024 est. |
| 68 | Ecuador | 18,309,984 | 2024 est. |
| 69 | Guatemala | 18,255,216 | 2024 est. |
| 70 | Romania | 18,148,155 | 2024 est. |
| 71 | Netherlands | 17,772,378 | 2024 est. |
| 72 | Zimbabwe | 17,150,352 | 2024 est. |
| 73 | Cambodia | 17,063,669 | 2024 est. |
| 74 | Benin | 14,697,052 | 2024 est. |
| 75 | Guinea | 13,986,179 | 2024 est. |
| 76 | Rwanda | 13,623,302 | 2024 est. |
| 77 | Burundi | 13,590,102 | 2024 est. |
| 78 | Somalia | 13,017,273 | 2024 est. |
| 79 | South Sudan | 12,703,714 | 2024 est. |
| 80 | Bolivia | 12,311,974 | 2024 est. |
| 81 | Tunisia | 12,048,847 | 2024 est. |
| 82 | Belgium | 11,977,634 | 2024 est. |
| 83 | Haiti | 11,753,943 | 2024 est. |
| 84 | Jordan | 11,174,024 | 2024 est. |
| 85 | Cuba | 10,966,038 | 2024 est. |
| 86 | Czechia | 10,837,890 | 2024 est. |
| 87 | Dominican Republic | 10,815,857 | 2024 est. |
| 88 | Azerbaijan | 10,650,239 | 2024 est. |
| 89 | Sweden | 10,589,835 | 2024 est. |
| 90 | Greece | 10,461,091 | 2024 est. |
| 91 | Tajikistan | 10,394,063 | 2024 est. |
| 92 | Portugal | 10,207,177 | 2024 est. |
| 93 | Papua New Guinea | 10,046,233 | 2024 est. |
| 94 | United Arab Emirates | 10,032,213 | 2024 est. |
| 95 | Hungary | 9,855,745 | 2024 est. |
| 96 | Honduras | 9,529,188 | 2024 est. |
| 97 | Belarus | 9,501,451 | 2024 est. |
| 98 | Israel | 9,402,617 | 2024 est. |
| 99 | Sierra Leone | 9,121,049 | 2024 est. |
| 100 | Austria | 8,967,982 | 2024 est. |
| 101 | Togo | 8,917,994 | 2024 est. |
| 102 | Switzerland | 8,860,574 | 2024 est. |
| 103 | Laos | 7,953,556 | 2024 est. |
| 104 | Paraguay | 7,522,549 | 2024 est. |
| 105 | Libya | 7,361,263 | 2024 est. |
| 106 | Hong Kong | 7,297,821 | 2024 est. |
| 107 | Bulgaria | 6,782,659 | 2024 est. |
| 108 | Nicaragua | 6,676,948 | 2024 est. |
| 109 | Serbia | 6,652,212 | 2024 est. |
| 110 | El Salvador | 6,628,702 | 2024 est. |
| 111 | Eritrea | 6,343,956 | 2024 est. |
| 112 | Kyrgyzstan | 6,172,101 | 2024 est. |
| 113 | Congo, Republic of the | 6,097,665 | 2024 est. |
| 114 | Singapore | 6,028,459 | 2024 est. |
| 115 | Denmark | 5,973,136 | 2024 est. |
| 116 | Turkmenistan | 5,744,151 | 2024 est. |
| 117 | Central African Republic | 5,650,957 | 2024 est. |
| 118 | Finland | 5,626,414 | 2024 est. |
| 119 | Slovakia | 5,563,649 | 2024 est. |
| 120 | Norway | 5,509,733 | 2024 est. |
| 121 | Liberia | 5,437,249 | 2024 est. |
| 122 | Lebanon | 5,364,482 | 2024 est. |
| 123 | Costa Rica | 5,265,575 | 2024 est. |
| 124 | Ireland | 5,233,461 | 2024 est. |
| 125 | New Zealand | 5,161,211 | 2024 est. |
| 126 | Georgia | 4,900,961 | 2024 est. |
| 127 | Panama | 4,470,241 | 2024 est. |
| 128 | Mauritania | 4,328,040 | 2024 est. |
| 129 | Croatia | 4,150,116 | 2024 est. |
| 130 | Oman | 3,901,992 | 2024 est. |
| 131 | Bosnia and Herzegovina | 3,798,671 | 2024 est. |
| 132 | Moldova | 3,599,528 | 2024 est. |
| 133 | Uruguay | 3,425,330 | 2024 est. |
| 134 | Mongolia | 3,281,676 | 2024 est. |
| 135 | West Bank | 3,243,369 | 2024 est. |
| 136 | Kuwait | 3,138,355 | 2024 est. |
| 137 | Albania | 3,107,100 | 2024 est. |
| 138 | Puerto Rico | 3,019,450 | 2024 est. |
| 139 | Armenia | 2,976,765 | 2024 est. |
| 140 | Jamaica | 2,823,713 | 2024 est. |
| 141 | Namibia | 2,803,660 | 2024 est. |
| 142 | Lithuania | 2,628,186 | 2024 est. |
| 143 | Qatar | 2,552,088 | 2024 est. |
| 144 | Gambia, The | 2,523,327 | 2024 est. |
| 145 | Gabon | 2,455,105 | 2024 est. |
| 146 | Botswana | 2,450,668 | 2024 est. |
| 147 | Lesotho | 2,227,548 | 2024 est. |
| 148 | Gaza Strip | 2,141,643 | 2024 est. |
| 149 | North Macedonia | 2,135,622 | 2024 est. |
| 150 | Guinea-Bissau | 2,132,325 | 2024 est. |
| 151 | Slovenia | 2,097,893 | 2024 est. |
| 152 | Kosovo | 1,977,093 | 2024 est. |
| 153 | Latvia | 1,801,246 | 2024 est. |
| 154 | Equatorial Guinea | 1,795,834 | 2024 est. |
| 155 | Bahrain | 1,566,888 | 2024 est. |
| 156 | Timor-Leste | 1,506,909 | 2024 est. |
| 157 | Trinidad and Tobago | 1,408,966 | 2024 est. |
| 158 | Cyprus | 1,320,525 | 2024 est. |
| 159 | Mauritius | 1,310,504 | 2024 est. |
| 160 | Estonia | 1,193,791 | 2024 est. |
| 161 | Eswatini | 1,138,089 | 2024 est. |
| 162 | Djibouti | 994,974 | 2024 est. |
| 163 | Fiji | 951,611 | 2024 est. |
| 164 | Comoros | 900,141 | 2024 est. |
| 165 | Bhutan | 884,546 | 2024 est. |
| 166 | Guyana | 794,099 | 2024 est. |
| 167 | Solomon Islands | 726,799 | 2024 est. |
| 168 | Luxembourg | 671,254 | 2024 est. |
| 169 | Suriname | 646,758 | 2024 est. |
| 170 | Macau | 644,426 | 2024 est. |
| 171 | Cabo Verde | 611,014 | 2024 est. |
| 172 | Montenegro | 599,849 | 2024 est. |
| 173 | Brunei | 491,900 | 2024 est. |
| 174 | Malta | 469,730 | 2024 est. |
| 175 | Belize | 415,789 | 2024 est. |
| 176 | Bahamas, The | 410,862 | 2024 est. |
| 177 | Maldives | 388,858 | 2024 est. |
| 178 | Iceland | 364,036 | 2024 est. |
| 179 | Vanuatu | 318,007 | 2024 est. |
| 180 | New Caledonia | 304,167 | 2024 est. |
| 181 | Barbados | 304,139 | 2024 est. |
| 182 | French Polynesia | 303,540 | 2024 est. |
| 183 | Sao Tome and Principe | 223,561 | 2024 est. |
| 184 | Samoa | 208,853 | 2024 est. |
| 185 | Guam | 169,532 | 2024 est. |
| 186 | Saint Lucia | 168,038 | 2024 est. |
| 187 | Curacao | 153,289 | 2024 est. |
| 188 | Aruba | 125,063 | 2024 est. |
| 189 | Kiribati | 116,545 | 2024 est. |
| 190 | Grenada | 114,621 | 2024 est. |
| 191 | Tonga | 104,889 | 2024 est. |
| 192 | Virgin Islands | 104,377 | 2024 est. |
| 193 | Jersey | 103,387 | 2024 est. |
| 194 | Antigua and Barbuda | 102,634 | 2024 est. |
| 195 | Saint Vincent and the Grenadines | 100,647 | 2024 est. |
| 196 | Micronesia, Federated States of | 99,603 | 2024 est. |
| 197 | Seychelles | 98,187 | 2024 est. |
| 198 | Isle of Man | 92,269 | 2024 est. |
| 199 | Andorra | 85,370 | 2024 est. |
| 200 | Marshall Islands | 82,011 | 2024 est. |
| 201 | Dominica | 74,661 | 2024 est. |
| 202 | Bermuda | 72,800 | 2024 est. |
| 203 | Guernsey | 67,787 | 2024 est. |
| 204 | Cayman Islands | 66,653 | 2024 est. |
| 205 | Turks and Caicos Islands | 60,439 | 2024 est. |
| 206 | Greenland | 57,751 | 2024 est. |
| 207 | Saint Kitts and Nevis | 55,133 | 2024 est. |
| 208 | Faroe Islands | 52,933 | 2024 est. |
| 209 | Northern Mariana Islands | 51,118 | 2024 est. |
| 210 | Sint Maarten | 46,215 | 2024 est. |
| 211 | American Samoa | 43,895 | 2024 est. |
| 212 | Liechtenstein | 40,272 | 2024 est. |
| 213 | British Virgin Islands | 40,102 | 2024 est. |
| 214 | San Marino | 35,095 | 2024 est. |
| 215 | Saint Martin | 32,996 | 2024 est. |
| 216 | Monaco | 31,813 | 2024 est. |
| 217 | Gibraltar | 29,683 | 2024 est. |
| 218 | Palau | 21,864 | 2024 est. |
| 219 | Anguilla | 19,416 | 2024 est. |
| 220 | Wallis and Futuna | 15,964 | 2024 est. |
| 221 | Tuvalu | 11,733 | 2024 est. |
| 222 | Nauru | 9,892 | 2024 est. |
| 223 | Saint Helena, Ascension, and Tristan da Cunha | 7,943 | 2024 est. |
| 224 | Cook Islands | 7,761 | 2024 est. |
| 225 | Saint Barthelemy | 7,086 | 2024 est. |
| 226 | Montserrat | 5,468 | 2024 est. |
| 227 | Saint Pierre and Miquelon | 5,132 | 2024 est. |
| 228 | Falkland Islands (Islas Malvinas) | 3,142 | 2021 est. |
| 229 | Svalbard | 2,556 | 2025 est. |
| 230 | Tokelau | 2,453 | 2024 est. |
| 231 | Paracel Islands | 2,230 | 2024 est. |
| 232 | Niue | 1,815 | 2024 est. |
| 233 | Norfolk Island | 1,739 | 2021 est. |
| 234 | Christmas Island | 1,692 | 2021 est. |
| 235 | Holy See (Vatican City) | 1,000 | 2024 est. |
| 236 | Cocos (Keeling) Islands | 593 | 2021 est. |
| 237 | Pitcairn Islands | 50 | 2025 est. | |
13049 | https://math.stackexchange.com/questions/5023227/using-wolfram-alpha-for-intergration | definite integrals - using wolfram-alpha for intergration - Mathematics Stack Exchange
Join Mathematics
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Mathematics helpchat
Mathematics Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
using wolfram-alpha for intergration
Ask Question
Asked 8 months ago
Modified8 months ago
Viewed 106 times
This question shows research effort; it is useful and clear
0
Save this question.
Show activity on this post.
Is it right place for asking for a help on building proper expression
in wolfram alpha?
I would like to find interval [-m,m][-m,m] for an integral whose area is known in a following problem.
The problem:
Find radius r r of the circle whose center lies on the circumference
of another circle of radius R R and common area of both circles is equal to half of area of circle of radius R R.
visualization of the problem
The approach:
My approach to solve this was using the fact that in x,y x,y coordinate system that common area is bounded by two functions:
f(x)=R 2−x 2−−−−−−−√f(x)=R 2−x 2
g(x)=−r 2−x 2−−−−−−√+R g(x)=−r 2−x 2+R
and, according to fundamental theorem of calculus (the use of which I found
in the similar question: link), there is an equation:
∫m−m f(x)−g(x)d x=π R 2 2∫−m m f(x)−g(x)d x=π R 2 2
where m m can be derived from an equation:
f(m)=g(m)f(m)=g(m)
and:
{m=4 r 2 R 2−r 4√2 R,{m=4 r 2 R 2−r 4 2 R,
−m=4 r 2 R 2−r 4√2 R}−m=4 r 2 R 2−r 4 2 R}
And if a point (m, f(m))(m, f(m)) is known then radius r r being looked for could be found easily.
However making WA do its job seems not that simple: link
solve ( Integrate[(Sqrt[(R^2-x^2])-(-Sqrt[r^2-x^2]+R),{x,-Sqrt[(4(rR)^2-r^4]/2R,Sqrt[(4(rR)^2-r^4]/2R}] = (πR^2)/2 ) for r
Wolfram|Alpha doesn't understand your query
If there is a positive answer to my initial question then how to make that
query understandable to WA?
definite-integrals
wolfram-alpha
Share
Share a link to this question
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this question to receive notifications
asked Jan 14 at 17:59
mtthmtth
9 1 1 bronze badge
5
One thing that you could start with is to fix the brackets, e.g. in Sqrt[(R^2-x^2] you never close the open parenthesis. This probably won't help much - WA doesn't necessarily use Mathematica's syntax and some inputs are too much for it. It also might be confused about rR - it doesn't necessarily know that this is r R. You can always try the Mathematica Engine which is free, if you don't have access to the full software.Randy Marsh –Randy Marsh 2025-01-14 18:10:44 +00:00 Commented Jan 14 at 18:10
r R r R is a new variable, and definitely not what you want, namely r⋅R r⋅R.Dietrich Burde –Dietrich Burde 2025-01-14 18:36:50 +00:00 Commented Jan 14 at 18:36
I don’t understand why W o l f r a m W o l f r a m doesn’t implement yet its language in W o l f r a m A l p h a W o l f r a m A l p h a Felix Marin –Felix Marin 2025-01-14 20:57:55 +00:00 Commented Jan 14 at 20:57
You can use M a t h e m a t i c a M a t h e m a t i c a freely with W o l f r a m C l o u d W o l f r a m C l o u d.Felix Marin –Felix Marin 2025-01-14 21:00:40 +00:00 Commented Jan 14 at 21:00
Beside the syntax errors as already pointed by @Randy Marsh, I do not think that you could obtain a closed form solution for the zero of the function. Moroover, as writen, I suppose that R<1 2 R<1 2.Claude Leibovici –Claude Leibovici 2025-01-15 09:56:36 +00:00 Commented Jan 15 at 9:56
Add a comment|
0
Sorted by: Reset to default
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
definite-integrals
wolfram-alpha
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Report this ad
Linked
0How do you find the interval for an integral whose area is a given number?
Related
1Inputting complicated equation into Wolfram
2Is the solution from wolfram alpha for x 4+b x 3+f=0 x 4+b x 3+f=0 only wrong for b=0 b=0?
1How to make Wolfram Alpha give me a range of possible answers for an equation if the subject is assumed to be an integer?
0I found an unusual (at least for me) answer in Wolfram Alpha I don't understand
0Why does Wolfram Alpha give different answers to this algebraic equation depending on how it is entered?
1Finding area of circle exterior to parabola using double integral.
0How to integrate to get sinh? (Wolfram Alpha)
6Strange symbol from Wolfram Alpha when finding closed-form solution of a number
Hot Network Questions
Cannot build the font table of Miama via nfssfont.tex
Passengers on a flight vote on the destination, "It's democracy!"
What's the expectation around asking to be invited to invitation-only workshops?
How to start explorer with C: drive selected and shown in folder list?
What meal can come next?
Childhood book with a girl obsessessed with homonyms who adopts a stray dog but gives it back to its owners
Can peaty/boggy/wet/soggy/marshy ground be solid enough to support several tonnes of foot traffic per minute but NOT support a road?
Explain answers to Scientific American crossword clues "Éclair filling" and "Sneaky Coward"
Does the mind blank spell prevent someone from creating a simulacrum of a creature using wish?
Riffle a list of binary functions into list of arguments to produce a result
Does the Mishna or Gemara ever explicitly mention the second day of Shavuot?
What is the feature between the Attendant Call and Ground Call push buttons on a B737 overhead panel?
My dissertation is wrong, but I already defended. How to remedy?
"Unexpected"-type comic story. Aboard a space ark/colony ship. Everyone's a vampire/werewolf
Storing a session token in localstorage
ICC in Hague not prosecuting an individual brought before them in a questionable manner?
What’s the usual way to apply for a Saudi business visa from the UAE?
Is it ok to place components "inside" the PCB
alignment in a table with custom separator
Can you formalize the definition of infinitely divisible in FOL?
Proof of every Highly Abundant Number greater than 3 is Even
Determine which are P-cores/E-cores (Intel CPU)
Drawing the structure of a matrix
Who is the target audience of Netanyahu's speech at the United Nations?
more hot questions
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Mathematics
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
13050 | https://www.webqc.org/molecular-weight-of-CrCl3%2A6h2o.html | CrCl36H2O molar mass
Printed from
Molar Mass, Molecular Weight and Elemental Composition Calculator
Enter a chemical formula to calculate its molar mass and elemental composition:
Molar mass of _CrCl 36H 2 O_ is 266.4468 g/mol
Convert between CrCl 36H 2 O weight and moles
| Compound | Moles | Weight, g |
---
| CrCl 36H 2 O | | | | |
Elemental composition of CrCl 36H 2 O
| Element | Symbol | Atomic weight | Atoms | Mass percent |
--- ---
| Chromium | Cr | 51.9961 | 1 | 19.5146 |
| Chlorine | Cl | 35.453 | 3 | 39.9175 |
| Hydrogen | H | 1.00794 | 12 | 4.5395 |
| Oxygen | O | 15.9994 | 6 | 36.0284 |
| Computing molar mass step by step |
| First, compute the number of each atom in CrCl 36H 2 O: Cr: 1, Cl: 3, H: 12, O: 6 Then, lookup atomic weights for each element in periodic table: Cr: 51.9961, Cl: 35.453, H: 1.00794, O: 15.9994 Now, compute the sum of products of number of atoms to the atomic weight: Molar mass (CrCl 36H 2 O) = ∑ Count i Weight i = Count(Cr) Weight(Cr) + Count(Cl) Weight(Cl) + Count(H) Weight(H) + Count(O) Weight(O) = 1 51.9961 + 3 35.453 + 12 1.00794 + 6 15.9994 = 266.4468 g/mol |
| Mass Percent Composition | Atomic Percent Composition |
--- |
| Cr Chromium (19.51%) Cl Chlorine (39.92%) H Hydrogen (4.54%) O Oxygen (36.03%) | Cr Chromium (4.55%) Cl Chlorine (13.64%) H Hydrogen (54.55%) O Oxygen (27.27%) |
| Mass Percent Composition |
| Cr Chromium (19.51%) Cl Chlorine (39.92%) H Hydrogen (4.54%) O Oxygen (36.03%) |
| Atomic Percent Composition |
| Cr Chromium (4.55%) Cl Chlorine (13.64%) H Hydrogen (54.55%) O Oxygen (27.27%) |
| Related |
| Oxidation state calculator |
| Compound properties |
Computing molar mass (molar weight)
To calculate molar mass of a chemical compound enter its formula and click 'Compute'. In chemical formula you may use:
Any chemical element. Capitalize the first letter in chemical symbol and use lower case for the remaining letters: Ca, Fe, Mg, Mn, S, O, H, C, N, Na, K, Cl, Al.
Functional groups: D, T, Ph, Me, Et, Bu, AcAc, For, Tos, Bz, TMS, tBu, Bzl, Bn, Dmg
parenthesis () or brackets [].
Common compound names.
Examples of molar mass computations: NaCl, Ca(OH)2, K4[Fe(CN)6], CuSO45H2O, nitric acid, potassium permanganate, ethanol, fructose, caffeine, water.
Molar mass calculator also displays common compound name, Hill formula, elemental composition, mass percent composition, atomic percent compositions and allows to convert from weight to number of moles and vice versa.
Computing molecular weight (molecular mass)
To calculate molecular weight of a chemical compound enter it's formula, specify its isotope mass number after each element in square brackets.
Examples of molecular weight computations: CO2, SO2.
Definitions
Molecular mass (molecular weight) is the mass of one molecule of a substance and is expressed in the unified atomic mass units (u). (1 u is equal to 1/12 the mass of one atom of carbon-12)
Molar mass (molar weight) is the mass of one mole of a substance and is expressed in g/mol.
Mole is a standard scientific unit for measuring large quantities of very small entities such as atoms and molecules. One mole contains exactly 6.022 ×10 23 particles (Avogadro's number)
Steps to calculate molar mass
Identify the compound: write down the chemical formula of the compound. For example, water is H 2 O, meaning it contains two hydrogen atoms and one oxygen atom.
Find atomic masses: look up the atomic masses of each element present in the compound. The atomic mass is usually found on the periodic table and is given in atomic mass units (amu).
Calculate molar mass of each element: multiply the atomic mass of each element by the number of atoms of that element in the compound.
Add them together: add the results from step 3 to get the total molar mass of the compound.
Example: calculating molar mass
Let's calculate the molar mass of carbon dioxide (CO 2):
Carbon (C) has an atomic mass of about 12.01 amu.
Oxygen (O) has an atomic mass of about 16.00 amu.
CO 2 has one carbon atom and two oxygen atoms.
The molar mass of carbon dioxide is 12.01 + (2 × 16.00) = 44.01 g/mol.
Lesson on computing molar mass
Practice what you learned:
Practice calculating molar mass
Weights of atoms and isotopes are from NIST article.
Related: Molecular weights of amino acids
molecular weights calculated today
Please let us know how we can improve this web app.
Chemistry tools
Gas laws
Unit converters
Periodic table
Chemical forum
Constants
Symmetry
Contribute
Contact us
Choose languageDeutschEnglishEspañolFrançaisItalianoNederlandsPolskiPortuguêsРусский中文日本語한국어
How to cite?
MenuBalanceMolar massGas lawsUnitsChemistry toolsPeriodic tableChemical forumSymmetryConstantsContributeContact us
How to cite?
Choose languageDeutschEnglishEspañolFrançaisItalianoNederlandsPolskiPortuguêsРусский中文日本語한국어
WebQC is a web application with a mission to provide best-in-class chemistry tools and information to chemists and students.
By using this website, you signify your acceptance of Terms and Conditions and Privacy Policy.
Do Not Sell My Personal Information
© 2025 webqc.org All rights reserved |
13051 | https://www.sciencedirect.com/topics/mathematics/union-of-event | Union of Event - an overview | ScienceDirect Topics
Skip to Main content
Journals & Books
Union of Event
In subject area:Mathematics
The union of events is defined as the new event that consists of all outcomes that are in either event A, event B, or both. It occurs whenever at least one of the events A or B occurs.
AI generated definition based on:Introductory Statistics (Third Edition), 2010
How useful is this definition?
Press Enter to select rating, 1 out of 3 stars Press Enter to select rating, 2 out of 3 stars Press Enter to select rating, 3 out of 3 stars
About this page
Add to MendeleySet alert
Discover other topics
1. On this page
On this page
Definition
Chapters and Articles
Related Terms
Recommended Publications
On this page
Definition
Chapters and Articles
Related Terms
Recommended Publications
Chapters and Articles
You might find these chapters and articles relevant to this topic.
Chapter
Probability
2010, Introductory Statistics (Third Edition)Sheldon M. Ross
Definition
Any set of outcomes of the experiment is called an event. We designate events by the letters A, B, C, and so on. We say that the event A occurs whenever the outcome is contained in A.
For any two events A and B, we define the new event A ∪ B, called the union of events A and B, to consist of all outcomes that are in A or in B or in both A and B. That is, the event A ∪ B will occur if either A or B occurs.
In Example 4.1(a), if A = {g} is the event that the child is a girl and B = {b} is the event that it is a boy, then A ∪ B = {g, b}. That is, A ∪ B is the whole sample space S.
In Example 4.1(c), let
A={all outcomes starting with 4}
be the event that the number 4 horse wins; and let
B={all outcomes whose second element is 2}
be the event that the number 2 horse comes in second. Then A ∩ B is the event that either the number 4 horse wins or the number 2 horse comes in second or both.
A graphical representation of events that is very useful is the Venn diagram . The sample space S is represented as consisting of all the points in a large rectangle, and events are represented as consisting of all the points in circles within the rectangle. Events of interest are indicated by shading appropriate regions of the diagram. The colored region of Fig. 4.1 represents the union of events A and B.
Sign in to download full-size image
FIGURE 4.1. A Venn diagram: shaded region is A ∪ B.
For any two events A and B, we define the intersection of A and B to consist of all outcomes that are both in A and in B. That is, the intersection will occur if both A and B occur. We denote the intersection of A and B by A ∩ B. The colored region of Fig. 4.2 represents the intersection of events A and B.
Sign in to download full-size image
FIGURE 4.2. Shaded region is A ∩ B.
In Example 4.1(b), if A = {(H, H), (H, T)} is the event that the first coin lands heads and B = {(H, T), (T, T)} is the event that the second coin lands tails, then A ∩ B = {(H, T)} is the event that the first coin lands heads and the second lands tails.
In Example 4.1(c), if A is the event that the number 2 horse wins and B is the event that the number 3 horse wins, then the event A ∪ B does not contain any outcomes and so cannot occur. We call the event without any outcomes the null event, and designate it as Ø. If the intersection of A and B is the null event, then since A and B cannot simultaneously occur, we say that A and B are disjoint, or mutually exclusive. Two disjoint events are pictured in the Venn diagram of Fig. 4.3.
Sign in to download full-size image
FIGURE 4.3. A and B are disjoint events.
For any event A we define the event A c, called the complement of A, to consist of all outcomes in the sample space that are not in A. That is, A c will occur when A does not, and vice versa. For instance, in Example 4.1(a), if A = {g} is the event that the child is a girl, then A c ={b} is the event that it is a boy. Also note that the complement of the sample space is the null set, that is, S c = Ø. Figure 4.4 indicates A c, the complement of eventA.
Sign in to download full-size image
FIGURE 4.4. Shaded region is A c.
We can also define unions and intersections of more than two events. For instance, the union of events A, B, and C, written A ∪ B ∪ C, consists of all the outcomes of the experiment that are in A or in B or in C. Thus, A ∪ B ∪ C will occur if at least one of these events occurs. Similarly, the intersection A ∩ B ∩ C consists of the outcomes that are in all the events A, B, and C. Thus, the intersection will occur only if all the events occur.
We say that events A, B, and C are disjoint if no two of them can simultaneously occur.
Show more
View chapterExplore book
Read full chapter
URL:
Book 2010, Introductory Statistics (Third Edition)Sheldon M. Ross
Chapter
Probability
2017, Introductory Statistics (Fourth Edition)Sheldon M. Ross
Key Terms
Experiment: Any process that produces an observation.
Outcome: The observation produced by an experiment.
Sample space: The set of all possible outcomes of an experiment.
Event: Any set of outcomes of the experiment. An event is a subset of sample space S. The event is said to occur if the outcome of the experiment is contained in it.
Union of events: The union of events A and B, denoted by A∪B, consists of all outcomes that are in A or in B or in both A and B.
Intersection of events: The intersection of events A and B, denoted by A∩B, consists of all outcomes that are in both A and B.
Complement of an event: The complement of event A, denoted by A c, consists of all outcomes that are not in A.
Mutually exclusive or disjoint: Events are mutually exclusive or disjoint if they cannot occur simultaneously.
Null event: The event containing no outcomes. It is the complement of sample space S.
Venn diagram: A graphical representation of events.
Probability of an event: The probability of event A, denoted by P(A), is the probability that the outcome of the experiment is contained in A.
Addition rule of probability: The formula
P(A∪B)=P(A)+P(B)−P(A∩B)
Conditional probability: The probability of one event given the information that a second event has occurred. We denote the conditional probability of B given that A has occurred by P(B|A).
Sign in to download full-size image
Multiplication rule: The formula
P(A∩B)=P(A)P(B|A)
A generalized version of the multiplication rule is given by
P(A 1∩A 2∩⋯∩A n)=P(A 1)P(A 2|A 1)P(A 3|A 1∩A 2)⋯P(A n|A 1∩⋯∩A n−1)
Independent: Two events are said to be independent if knowing whether a specific one has occurred does not change the probability that the other occurs.
Show more
View chapterExplore book
Read full chapter
URL:
Book 2017, Introductory Statistics (Fourth Edition)Sheldon M. Ross
Chapter
Inductive Logic
2011, Handbook of the History of LogicUlrike von Luxburg, Bernhard Schölkopf
5.1 The union bound
The union bound is a simple but convenient tool to transform the standard law of large numbers of individual functions into a uniform law of large numbers over a set of finitely many functions. Suppose the set F consists just of finitely many functions, that is F={f 1,f 2,…,f m}. Each of the functions f i ∈ F satisfies the standard law of large numbers in form of the Chernoff bound, that is
(14)P(|R(f i)−R emp(f i)|≥ε)≤2 exp(−2 n ε 2).
Now we want to transform these statements about the individual functions f i into a uniform law of large numbers. To this end, note that we can rewrite:
(15)P(sup f∈F|R(f)−R emp(f)|≥ε)=P(|R(f 1)−R emp(f 1)|≥ε or|R(f 2)−R emp(f 2)|≥ε or…or|R(f m)−R emp(f m)|≥ε)
(16)≤∑i=1 m P(|R(f i)−R emp(f i)|≥ε)≤2 m exp(−2 n ε 2)
Let us go through these calculations step by step. The first equality comes from the way the supremum is defined. Namely, the supremum over certain expressions is larger than ɛ if at least one of the expressions is larger than ɛ, which leads to the statements with the “or” combinations. The next step uses a standard tool from probability theory, the union bound. The union bound states that the probability of a union of events (that is, events coupled with “or”) is smaller or equal to the sum of the individual probabilities. Finally, the last step is a simple application of the Chernoff bound of Eq. (14) to each of the terms in the sum.
From left to right, the statements in Eq. (16) show us how to convert the Chernoff bound for individual functions f i into a bound which is uniform over a finite numberm of functions. As we can see, the difference between the Chernoff bound for the individual functions and the right hand side of (16) is just a factor m. If the function space F is fixed, this factor can be regarded as a constant, and the term 2 m exp(−2 nɛ 2) still converges to 0 as n → ∞. Hence, the empirical risk converges to 0 uniformly over F as n → ∞. That is, we have proved that empirical risk minimization over a finite set F of functions is consistent with respect to F.
We next describe a trick used by Vapnik and Chervonenkis to reduce the case of an infinite function class to the case of a finite one. It consists of introducing what is sometimes called a ghost sample. It will enable us to replace the factor m in (16) by more general capacity measures that can be computed for infinite function classes.
View chapterExplore book
Read full chapter
URL:
Handbook2011, Handbook of the History of LogicUlrike von Luxburg, Bernhard Schölkopf
Chapter
Basic Concepts in Probability
2013, Markov Processes for Stochastic Modeling (Second Edition)Oliver C. Ibe
1.1 Introduction
The concepts of experiments and events are very important in the study of probability. In probability, an experiment is any process of trial and observation. An experiment whose outcome is uncertain before it is performed is called a random experiment. When we perform a random experiment, the collection of possible elementary outcomes is called the sample space of the experiment, which is usually denoted by Ω. We define these outcomes as elementary outcomes because exactly one of the outcomes occurs when the experiment is performed. The elementary outcomes of an experiment are called the sample points of the sample space and are denoted by w i, i=1, 2, …. If there are n possible outcomes of an experiment, then the sample space is Ω={w 1, w 2, …, w n}.
An event is the occurrence of either a prescribed outcome or any one of a number of possible outcomes of an experiment. Thus, an event is a subset of the sample space. For example, if we toss a die, any number from 1 to 6 can appear. Therefore, in this experiment, the sample space is defined by
Ω={1,2,3,4,5,6}
The event “the outcome of the toss of a die is an even number” is a subset of Ω and is defined by
E={2,4,6}
For a second example, consider a coin-tossing experiment in which each toss can result in either a head (H) or a tail (T). If we toss a coin three times and let the triplet xyz denote the outcome “x on the first toss, y on the second toss, and z on the third toss,” then the sample space of the experiment is
Ω={HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}
The event “one head and two tails” is a subset of Ω and is defined by
E={HTT,THT,TTH}
If a sample point x is contained in event A, we write x∈A. For any two events A and B defined on a sample space Ω, we can define the following new events:
•
A∪B is the event that consists of all sample points that are either in A or in B or in both A and B. The event A∪B is called the union of events A and B.
•
A∩B is the event that consists of all sample points that are in both A and B. The event A∩B is called the intersection of events A and B. Two events are defined to be mutually exclusive if their intersection does not contain a sample point; that is, they have no outcomes in common. Events A 1, A 2, A 3, … are defined to be mutually exclusive if no two of them have any outcomes in common and the events collectively have no outcomes in common.
•
A−B is the event that consists of all sample points that are in A but not in B. That is, A−B={x∈A|x∉B}. The event A−B is called the difference of events A and B. Note that A−B is different from B−A. The difference is sometimes denoted by A_B_, and A−B=A∩B¯, where B¯ is defined as the complement of event B, which is the part of Ω that is not contained in B. That is, B¯=Ω−B.
•
B⊂A is the event where all the sample points in event B are contained in event A, and we say that B is a subset of A.
Consider an abstract space Ω; that is, Ω is a space without any special structure. Let F be a family of subsets of Ω with the following properties:
1.
Ø∈F and Ω∈F, where Ø is the empty space called the null event.
2.
If A∈F, then A¯∈F.
3.
F is closed under countable unions and intersections; that is, if A 1, A 2, … are events in F, then ∪k=1∞A k and ∩k=1∞A k are both in F.
Under these conditions, F is defined to be a σ-algebra (or σ-field). A probability measure defined on a σ-algebra F of Ω is a function P that maps points in F onto the closed interval [0,1]. Thus, for an event A in F, the function P[A] is called the probability of event A. The probability measure P satisfies the following Kolmogorov axioms:
1.
As stated earlier, for any event A∈F, 0≤P[A]≤1.
2.
P[Ω]=1, which means that with probability 1, the outcome will be a sample point in the sample space.
3.
For any set of n disjoint events A 1, A 2, …, A n in F,
P[A 1∪A 2∪⋯∪A n]=P[A 1]+P[A 2]+⋯+P[A n]
That is, for any set of mutually exclusive events defined on the same space, the probability of at least one of these events occurring is the sum of their respective probabilities.
The triple (Ω, F, P) is called a probability space. The following results are additional properties of a probability measure:
1.
P[A¯]=1−P[A], which states that the probability of the complement of A is one minus the probability of A.
2.
P[Ø]=0, which states that the impossible (or null) event has probability zero.
3.
If A⊂B, then P[A]≤P[B]. That is, if A is a subset of B, the probability of A is at most the probability of B (or the probability of A cannot exceed the probability of B).
4.
If A=A 1∪A 2∪⋯∪A n, where A 1, A 2, …, A n are mutually exclusive events, then
P[A]=P[A 1]+P[A 2]+⋯+P[A n]
5.
For any two events A and B, P[A]=P[A∩B]+P[A∩B¯], which follows from the set identity: A=(A∩B)∪(A∩B¯). Since A∩B and A∩B¯ are mutually exclusive events, the result follows.
6.
For any two events A and B, P[A∪B]=P[A]+P[B]−P[A∩B].
7.
We can extend Property 6 to the case of three events. If A 1, A 2, A 3 are three events in F, then
P[A 1∪A 2∪A 3]=P[A 1]+P[A 2]+P[A 3]−P[A 1∩A 2]−P[A 1∩A 3]−P[A 2∩A 3]+P[A 1∩A 2∩A 3]
This can be further generalized to the case of n arbitrary events in F as follows:
P[A 1∪A 2∪⋯∪A n]=∑i=1 n P[A i]−∑1≤i<j≤n P[A i∩A j]+∑1≤i<j<k≤n P[A i∩A j∩A k]−⋯
That is, to find the probability that at least one of the n events occurs, first add the probability of each event, then subtract the probabilities of all possible two-way intersections, then add the probabilities of all possible three-way intersections, and so on.
1.1.1 Conditional Probability
Let A and B denote two events. The conditional probability of event A given event B, denoted by P[A|B], is defined by
P[A|B]=P[A∩B]P[B]P[B]>0
For example, if A denotes the event that the sum of the outcomes of tossing two dice is 7 and B denotes the event that the outcome of the first die is 4, then the conditional probability of event A given event B is defined by
P[A|B]=P[A∩B]P[B]=P[{4,3}]P[{4,1}]+P[{4,2}]+P[{4,3}]+P[{4,4}]+P[{4,5}]+P[{4,6}]=(1/36)(6/36)=1 6
1.1.2 Independence
Two events A and B are defined to be independent if the knowledge that one has occurred does not change or affect the probability that the other will occur. In particular, if events A and B are independent, the conditional probability of event A, given event B, P[A|B], is equal to the probability of event A. That is, events A and B are independent if
P[A|B]=P[A]
Because by definition P[A∩B]=P[A|B]P[B], an alternative definition of independence of events is that events A and B are independent if
P[A∩B]=P[A]P[B]
The definition of independence can be extended to multiple events. The n events A 1, A 2, …, A n are said to be independent if the following conditions are true:
P[A i∩A j]=P[A i]P[A j]P[A i∩A j∩A k]=P[A i]P[A j]P[A k]⋮P[A 1∩A 2∩⋯∩A n]=P[A 1]P[A 2]⋯P[A n]
This is true for all 1≤i<j<k<⋯≤n. That is, these events are pairwise independent, independent in triplets, and so on.
1.1.3 Total Probability and the Bayes’ Theorem
A partition of a set A is a set {A 1, A 2, …, A n} with the following properties:
a.
A i⊆A, i=1, 2, …, n, which means that A is a set of subsets.
b.
A i∩A k=Ø, i=1, 2, …, n; k=1, 2, …, n; i≠k, which means that the subsets are mutually (or pairwise) disjoint; that is, no two subsets have any element in common.
c.
A 1∪A 2∪⋯∪A n=A, which means that the subsets are collectively exhaustive. That is, the subsets together include all possible values of the set A.
Let {A 1, A 2, …, A n} be a partition of the sample space Ω, and suppose each one of the events A 1, A 2, …, A n has nonzero probability of occurrence. Let B be any event. Then
P[B]=P[B|A 1]P[A 1]+P[B|A 2]P[A 2]+L+P[B|A n]P[A n]=∑i=1 n P[B|A i]P[A i]
This result is defined as the total probability of event B.
Suppose event B has occurred, but we do not know which of the mutually exclusive and collectively exhaustive events A 1, A 2, …, A n holds true. The conditional probability that event A k occurred, given that B occurred, is given by
P[A k|B]=P[A k∩B]P[B]=P[A k∩B]∑i=1 n P[B|A i]P[A i]
where the second equality follows from the total probability of event B. Because P[A k∩B]=P[B|A k]P[A k], the preceding equation can be rewritten as follows:
P[A k|B]=P[A k∩B]P[B]=P[B|A k]P[A k]∑i=1 n P[B|A i]P[A i]
This result is called the Bayes’ formula (or Bayes’ rule).
Show more
View chapterExplore book
Read full chapter
URL:
Book 2013, Markov Processes for Stochastic Modeling (Second Edition)Oliver C. Ibe
Chapter
Elements of probability
2021, Introduction to Probability and Statistics for Engineers and Scientists (Sixth Edition)Sheldon M. Ross
3.3 Venn diagrams and the algebra of events
A graphical representation of events that is very useful for illustrating logical relations among them is the Venn diagram. The sample space S is represented as consisting of all the points in a large rectangle, and the events E,F,G,…, are represented as consisting of all the points in given circles within the rectangle. Events of interest can then be indicated by shading appropriate regions of the diagram. For instance, in the three Venn diagrams shown in Figure 3.1, the shaded areas represent, respectively, the events E∪F, EF, and E c. The Venn diagram of Figure 3.2 indicates that E⊂F.
Sign in to download full-size image
Figure 3.1. Venn diagrams. (A) Shaded region: E ∪ F; (B) shaded region: EF; (C) shaded region: E c.
Sign in to download full-size image
Figure 3.2. Venn diagram.
The operations of forming unions, intersections, and complements of events obey certain rules not dissimilar to the rules of algebra. We list a few of these.
Commutative law E ∪ F = F ∪ E EF =FE
Associative law(E ∪ F)∪G = E ∪ (F ∪ G)(EF)G = E(FG)
Distributive law(E∪F)G=EG∪FG EF ∪ G = (E ∪ G)(F ∪ G)
These relations are verified by showing that any outcome that is contained in the event on the left side of the equality is also contained in the event on the right side and vice versa. One way of showing this is by means of Venn diagrams. For instance, the distributive law may be verified by the sequence of diagrams shown in Figure 3.3.
Sign in to download full-size image
Figure 3.3. Proving the distributive law. (A) Shaded region: EG; (B) shaded region: FG; (C) shaded region: (E ∪ F)G, (E ∪ F)G = EG ∪ FG.
The following useful relationship between the three basic operations of forming unions, intersections, and complements of events is known as DeMorgan's laws.
(E∪F)c=E c F c(EF)c=E c∪F c
Show more
View chapterExplore book
Read full chapter
URL:
Book 2021, Introduction to Probability and Statistics for Engineers and Scientists (Sixth Edition)Sheldon M. Ross
Chapter
Basic Probability Concepts
2014, Fundamentals of Applied Probability and Random Processes (Second Edition)Oliver C. Ibe
1.2 Sample Space and Events
The concepts of experiments and events are very important in the study of probability. In probability, an experiment is any process of trial and observation. An experiment whose outcome is uncertain before it is performed is called a random experiment. When we perform a random experiment, the collection of possible elementary outcomes is called the sample space of the experiment, which is usually denoted by Ω. We define these outcomes as elementary outcomes because exactly one of the outcomes occurs when the experiment is performed. The elementary outcomes of an experiment are called the sample points of the sample space and are denoted by w i,i=1,2,…. If there are n possible outcomes of an experiment, then the sample space is Ω={w 1,w 2,…,w n}.
An event is the occurrence of either a prescribed outcome or any one of a number of possible outcomes of an experiment. Thus, an event is a subset of the sample space. For example, if we toss a die, any number from 1 to 6 may appear. Therefore, in this experiment the sample space is defined by
(1.1)Ω=123456
The event “the outcome of the toss of a die is an even number” is the subset of Ω and is defined by
(1.2)E=246
For a second example, consider a coin tossing experiment in which each toss can result in either a head (H) or tail (T). If we toss a coin three times and let the triplet xyz denote the outcome “x on first toss, y on second toss and z on third toss,” then the sample space of the experiment is
(1.3)Ω=HHH HHT HTH HTT THH THT TTH TTT
The event “one head and two tails” is the subset of Ω and is defined by
(1.4)E=HTT THT TTH
Other examples of events are as follows:
▪
In a single coin toss experiment with sample space Ω={H,T}, the event E={H} is the event that a head appears on the toss and E={T} is the event that a tail appears on the toss.
▪
If we toss a coin twice and let xy denote the outcome “x on first toss and y on second toss,” where x is head or tail and y is head or tail, then the sample space is Ω={HH,HT,TH,TT}. The event E={HT,TT} is the event that a tail appears on second toss.
▪
If we measure the lifetime of an electronic component, such as a chip, the sample space consists of all nonnegative real numbers. That is,
Ω=x|0≤x<∞
The event that the lifetime is not more than 7 hours is defined as follows:
E=x|0≤x≤7
▪
If we toss a die twice and let the pair (x,y) denote the outcome “x on first toss and y on second toss,” then the sample space is
(1.5)Ω=11 12 13 14 15 16 21 22 23 24 25 26 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 54 55 56 61 62 63 64 65 66
The event that the sum of the two tosses is 8 is denoted by
E=26 35 44 53 62
For any two events A and B of a sample space Ω, we can define the following new events:
▪
A∪B is event that consists of all sample points that are either in A or in B or in both A and B. The event A∪B is called the union of events A and B.
▪
A∩B is event that consists of all sample points that are in both A and B. The event A∩B is called the intersection of events A and B. Two events are defined to be mutually exclusive if their intersection does not contain a sample point; that is, they have no outcomes in common. Events A 1,A 2,A 3,… are defined to be mutually exclusive if no two of them have any outcomes in common and the events collectively have no outcomes in common.
▪
A−B=A_B_ is event that consists of all sample points that are in A but not in B. The event A−B (also denoted by A_B_) is called the difference of events A and B. Note that A−B is different from B−A.
The algebra of unions, intersections and differences of events will be discussed in greater detail when we study set theory later in this chapter.
Show more
View chapterExplore book
Read full chapter
URL:
Book 2014, Fundamentals of Applied Probability and Random Processes (Second Edition)Oliver C. Ibe
Chapter
The concept of probability and basic results
2015, An Introduction to Probability and Statistical Inference (Second Edition)George G. Roussas
2.1 Definition of Probability and Some Basic Results
When a random experiment is entertained, one of the first questions which arise is, what is the probability that a certain event occurs? For instance, in reference to Example 26 in Chapter 1, one may ask: What is the probability that exactly one head occurs; in other words, what is the probability of the event B = {HTT, THT, TTH}? The answer to this question is almost automatic and is 3/8. The relevant reasoning goes like this: Assuming that the three coins are balanced, the probability of each one of the 8 outcomes, considered as simple events, must be 1/8. Since the event B consists of 3 sample points, it can occur in 3 different ways, and hence its probability must be 3/8.
This is exactly the intuitive reasoning employed in defining the concept of probability when two requirements are met: First, the sample space S has finitely many outcomes, S={s 1,…,s n}, say, and second, each one of these outcomes is “equally likely” to occur, has the same chance of appearing, whenever the relevant random experiment is carried out. This reasoning is based on the underlying symmetry. Thus, one is led to stipulating that each one of the (simple) events {s i}, i = 1, …, n, has probability 1/n. Then the next step, that of defining the probability of a composite event A, is simple; if A consists of m sample points, A={s i 1,…,s i m}, say (1 ≤ m ≤ n) (or none at all, in which case m = 0), then the probability of A must be m/n. The notation used is: P({s 1})=⋯=P({s n})=1 n and P(A)=m n. Actually, this is the so-called classical definition of probability. That is,
Classical Definition of Probability
Let S be a sample space, associated with a certain random experiment and consisting of finitely many sample points n, say, each of which is equally likely to occur whenever the random experiment is carried out. Then the probability of any event A, consisting of m sample points (0 ≤ m ≤ n), is given by P(A)=m n.
In reference to Example 26 in Chapter 1, P(A)=4 8=1 2=0.5. In Example 27 (when the two dice are unbiased), P(X=7)=6 36=1 6≃0.167, where the r.v. X and the event (X = 7) are defined in Section 1.3 of Chapter 1. In Example 29, when the balls in the urn are thoroughly mixed, we may assume that all of the (m + n)(m + n − 1) pairs are equally likely to be selected. Then, since the event A occurs in 20 different ways, P(A)=20(m+n)(m+n−1). For m = 3 and n = 5, this probability is P(A)=20 56=5 14≃0.357.
From the preceding (classical) definition of probability, the following simple properties are immediate: For any event A,P(A)≥0;P(S)=1; if two events A 1 and A 2 are disjoint (A 1 ∩ A 2 = ∅), then P(A 1 ∪ A 2) = P(A 1) + P(A 2). This is so because, if A 1={s i 1,…,s i k},A 2={s j 1,…,s j ℓ}, where all s i 1,…,s i k are distinct from all s j 1,…,s j ℓ, then A 1∪A 2={s i 1,…,s i k s j 1,…,s j ℓ} and P(A 1∪A 2)=k+ℓ n=k n+ℓ n=P(A 1)+P(A 2).
In many cases, the stipulations made in defining the probability as above are not met, either because S has not finitely many points (as is the case in Examples 32, 33–35 Example 33 Example 34 Example 35 (by replacing C and M by ∞), and 36–40 in Chapter 1 Example 36 Example 37 Example 38 Example 39 Example 40), or because the (finitely many) outcomes are not equally likely. This happens, for instance, in Example 26 when the coins are not balanced and in Example 27 when the dice are biased. Strictly speaking, it also happens in Example 30. In situations like this, the way out is provided by the so-called relative frequency definition of probability. Specifically, suppose a random experiment is carried out a large number of times N, and let N(A) be the frequency of an event A, the number of times A occurs (out of N). Then the relative frequency of A is N(A)N. Next, suppose that, as N → ∞, the relative frequencies N(A)N oscillate around some number (necessarily between 0 and 1). More precisely, suppose that N(A)N converges, as N → ∞, to some number. Then this number is called the probability of A and is denoted by P(A). That is, P(A)=lim N→∞N(A)N. (It will be seen later in this book that the assumption of convergence of the relative frequencies N(A)/N is justified subject to some qualifications.) To summarize,
Relative Frequency Definition of Probability
Let N(A) be the number of times an event A occurs in N (identical) repetitions of a random experiment, and assume that the relative frequency of A,N(A)N, converges to a limit as N → ∞. This limit is denoted by P(A) and is called the probability of A.
At this point, it is to be observed that the empirical data show that the relative frequency definition of probability and the classical definition of probability agree in the framework in which the classical definition applies.
From the relative frequency definition of probability and the usual properties of limits, it is immediate that: P(A) ≥ 0 for every event A; P(S)=1; and for A 1, A 2 with A 1 ∩ A 2 = ∅,
P(A 1∪A 2)=lim N→∞N(A 1∪A 2)N=lim N→∞(N(A 1)N+N(A 2)N)=lim N→∞N(A 1)N+lim N→∞N(A 2)N=P(A 1)+P(A 2);
that is, P(A 1 ∪ A 2) = P(A 1) + P(A 2), provided A 1 ∩ A 2 = ∅. These three properties were also seen to be true in the classical definition of probability. Furthermore, it is immediate that under either definition of probability, P(A 1 ∪ … ∪ A k) = P(A 1) + ⋯ + P(A k), provided the events are pairwise disjoint; A i ∩ A j = ∅, i ≠ j.
The above two definitions of probability certainly give substance to the concept of probability in a way consonant with our intuition about what probability should be. However, for the purpose of cultivating the concept and deriving deep probabilistic results, one must define the concept of probability in terms of some basic properties, which would not contradict what we have seen so far. This line of thought leads to the so-called axiomatic definition of probability due to Kolmogorov.
Axiomatic Definition of Probability
Probability is a function, denoted by P, defined for each event of a sample space S, taking on values in the real line ℜ, and satisfying the following three properties:
P(1)
P(A) ≥ 0 for every event A (nonnegativity of P).
P(2)
P(S)=1 (P is normed).
P(3)
For countably infinite many pairwise disjoint events A i, i = 1, 2, …, A i ∩ A j = ∅, i ≠ j, it holds
P(A 1∪A 2∪⋯)=P(A 1)+P(A 2)+⋯;or P(∪i=1∞A i)=∑i=1∞P(A i)(sigma-addivity(σ-additivity)of P).
Comments on the Axiomatic Definition
(1)
Properties (P1) and (P2) are the same as the ones we have seen earlier, whereas property (P3) is new. What we have seen above was its so-called finitely additive version; that is, P(∪i=1 n A i)=∑i=1 n P(A i), provided A i ∩ A j = ∅, i ≠ j. It will be seen below that finite additivity is implied by σ-additivity but not the other way around. Thus, if we are to talk about the probability of the union of countably infinite many pairwise disjoint events, property (P3) must be stipulated. Furthermore, the need for such a union of events is illustrated as follows: In reference to Example 32, calculate the probability that the first head does not occur before the n th tossing. By setting A i={T…T︸i−1 H},i=n,n+1,…, what we are actually after here is P(A n ∪ A n+1 ∪ ⋯) with A i ∩ A j = ∅, i ≠ j, i and j ≥ n.
(2)
Property (P3) is superfluous (reduced to finite-additivity) when the sample space S is finite, which implies that the total number of events is finite.
(3)
Finite-additivity is implied by additivity for two events, P(A 1 ∪ A 2) = P(A 1) + P(A 2), A 1 ∩ A 2 = ∅, by way of mathematical induction.
Here are two examples in calculating probabilities.
Example 1
In reference to Example 1 in Chapter 1, take n = 58, and suppose we have the following configuration:
| Arsenic | Barium |
--- |
| High Mercury | Low Mercury |
| High | Low | High | Low |
| High | 1 | 3 | 5 | 9 |
| Low | 4 | 8 | 10 | 18 |
Calculate the probabilities mentioned in (i) (a)-(d).
Discussion
For simplicity, denote by B h the event that the site selected has a high barium concentration, and likewise for other events figuring below. Then:
(i)(a)
B h = (A h ∩ B h ∩ M h) ∪ (A h ∩ B h ∩ M ℓ) ∪ (A ℓ ∩ B h∩ M h) ∪ (A ℓ ∩ B h ∩ M ℓ) and the events on the right-hand side are pairwise disjoint. Therefore (by the basic property #2 in Section 2.1.1),
P(B h)=P(A h∩B h∩M h)+P(A h∩B h∩M ℓ)+P(A ℓ∩B h∩M h)+P(A ℓ∩B h∩M ℓ)=1 58+3 58+4 58+8 58=16 58=8 29≃0.276.
(i)(b)
Here, P(M h∩A ℓ∩B ℓ)=P(A ℓ∩B ℓ∩M h)=10 58=5 29≃0.172.
(i)(c)
Here, the required probability is as in (a):
P(A h∩B h∩M ℓ)+P(A h∩B ℓ∩M h)+P(A ℓ∩B h∩M h)=12 58=6 29≃0.207.
(i)(d)
As above,
P(A h∩B ℓ∩M ℓ)+P(A ℓ∩B h∩M ℓ)+P(A ℓ∩B ℓ∩M h)=27 58≃0.466.
Example 2
In ranking five horses in a horse race (Example 31 in Chapter 1), calculate the probability that horse #3 terminates at least second.
Discussion
Let A i be the event that horse #3 terminates at the i th position, i = 1, …, 5. Then the required event is A 1 ∪ A 2, where A 1, A 2 are disjoint. Thus,
P(A 1∪A 2)=P(A 1)+P(A 2)=24 120+24 120=2 5=0.4.
Example 3
In tossing a coin repeatedly until H appears for the first time (Example 32 in Chapter 1), suppose that P{T…T︸i−1 H}=P(A i)=q i−1 p for some 0 <p< 1 and q = 1 − p (in anticipation of Definition 3 in Section 2.4). Then
P(∪i=n∞A i)=∑i=n∞P(A i)=∑i=n∞q i−1 p=p∑i=n∞q i−1=p q n−1 1−q=p q n−1 p=q n−1.
For instance, for p = 1/2 and n = 3, this probability is 1 4=0.25. That is, when tossing a fair coin, the probability that the first head does not appear either the first or the second time (and therefore it appears either the third time or the fourth time, etc.) is 0.25. For n = 10, this probability is approximately 0.00195 ≃ 0.002.
Next, we present some basic results following immediately from the defining properties of the probability. First, we proceed with their listing and then with their justification.
2.1.1 Some Basic Properties of a Probability Function
1.
P (∅) = 0.
2.
For any pairwise disjoint events A 1,…,A n,P(∪i=1 n A i)=∑i=1 n P(A i).
3.
For any event A, P(A c) = 1 − P(A).
4.
A 1 ⊆ A 2 implies P(A 1) ≤ P(A 2) and P(A 2 − A 1) = P(A 2) − P(A 1).
5.
0 ≤ P(A) ≤ 1 for every event A.
6.(i)
For any two events A 1 and A 2:
P(A 1∪A 2)=P(A 1)+P(A 2)−P(A 1∩A 2).
(ii)
For any three events A 1, A 2, and A 3:
P(A 1∪A 2∪A 3)=P(A 1)+P(A 2)+P(A 3)−[P(A 1∩A 2)+P(A 1∩A 3)+P(A 2∩A 3)]+P(A 1∩A 2∩A 3).
7.
For any events A 1,A 2,…,P(∪i=1∞A i)≤∑i=1∞P(A i) (σ-sub-additivity), and P(∪i=1 n A i)≤∑i=1 n P(A i) (finite-sub-additivity).
2.1.2 Justification
1.
From the obvious fact that S=S∪∅∪∅∪… and property (P3),
P(S)=P(S∪∅∪∅∪⋯)=P(S)+P(∅)+P(∅)+⋯
or P(∅) + P(∅) +⋯ = 0. By (P1), this can only happen when P(∅) = 0.
Of course, that the impossible event has probability 0 does not come as a surprise. Any reasonable definition of probability should imply it.
2.
Take A i = ∅ for i ≥ n + 1, consider the following obvious relation, and use (P3) and #1 to obtain:
P(∪i=1 n A i)=P(∪i=1∞A i)=∑i=1∞P(A i)=∑i=1 n P(A i).
3.
From (P2) and #2, P(A∪A c)=P(S)=1 or P(A) + P(A c) = 1, so that P(A c) = 1 − P(A).
4.
The relation A 1 ⊆ A 2, clearly, implies A 2 = A 1 ∪ (A 2 − A 1), so that, by #2, P(A 2) = P(A 1) + P(A 2 − A 1). Solving for P(A 2 − A 1), we obtain P(A 2 − A 1) = P(A 2) − P(A 1), so that, by (P1), P(A 1) ≤ P(A 2).
At this point it must be pointed out that P(A 2 − A 1) need not be P(A 2) − P(A 1), if A 1 is not contained in A 2.
5.
Clearly, ∅⊆A⊆S for any event A. Then (P1), #1 and #4 give: 0=P(∅)≤P(A)≤P(S)=1.
6.(i)
It is clear (e.g., by means of a Venn diagram) that
A 1∪A 2=A 1∪(A 2∩A 1 c)=A 1∪(A 2−A 1∩A 2).
Then, by means of #2 and #4:
P(A 1∪A 2)=P(A 1)+P(A 2−A 1∩A 2)=P(A 1)+P(A 2)−P(A 1∩A 2).
(ii)
Apply part (i) to obtain:
P(A 1∪A 2∪A 3)=P[(A 1∪A 2)∪A 3]=P(A 1∪A 2)+P(A 3)−P[(A 1∪A 2)∩A 3]=P(A 1)+P(A 2)−P(A 1∩A 2)+P(A 3)−P[(A 1∩A 3)∪(A 2∩A 3)]=P(A 1)+P(A 2)+P(A 3)−P(A 1∩A 2)−[P(A 1∩A 3)+P(A 2∩A 3)−P(A 1∩A 2∩A 3)]=P(A 1)+P(A 2)+P(A 3)−P(A 1∩A 2)−P(A 1∩A 3)−P(A 2∩A 3)+P(A 1∩A 2∩A 3).
7.
By the identity in Section 1.2 and (P3):
P(∪i=1∞A i)=P[A 1∪(A 1 c∩A 2)∪…∪(A 1 c∩…∩A n−1 c∩A n)∪…]=P(A 1)+P(A 1 c∩A 2)+⋯+P(A 1 c∩…∩A n−1 c∩A n)+⋯≤P(A 1)+P(A 2)+⋯+P(A n)+⋯(by#4).
For the finite case:
P(∪i=1 n A i)=P[A 1∪(A 1 c∩A 2)∪…∪(A 1 c∩…∩A n−1 c∩A n)]=P(A 1)+P(A 1 c∩A 2)+⋯+P(A 1 c∩…∩A n−1 c∩A n)≤P(A 1)+P(A 2)+⋯+P(A n).
Next, some examples are presented to illustrate some of the properties #1–#7.
Example 4(i)
For two events A and B, suppose that P(A) = 0.3, P(B) = 0.5, and P(A ∪ B) = 0.6. Calculate P(A ∩ B).
(ii)
If P(A) = 0.6, P(B) = 0.3, P(A ∩ B c) = 0.4, and B ⊂ C, calculate P(A ∪ B c ∪ C c).
Discussion
(i)
From P(A ∪ B) = P(A) + P(B) − P(A ∩ B), we get P(A ∩ B) = P(A) + P(B) − P(A ∪ B) = 0.3 + 0.5 − 0.6 = 0.2.
(ii)
The relation B ⊂ C implies C c ⊂ B c and hence A ∪ B c ∪ C c = A ∪ B c. Then P(A ∪ B c ∪ C c) = P(A ∪ B c) = P(A) + P(B c) − P(A ∩ B c) = 0.6 + (1 − 0.3) − 0.4 = 0.9.
Example 5
Let A and B be the respective events that two contracts I and II, say, are completed by certain deadlines, and suppose that: P(at least one contract is completed by its deadline) = 0.9 and P(both contracts are completed by their deadlines) = 0.5. Calculate the probability: P(exactly one contract is completed by its deadline).
Discussion
The assumptions made are translated as follows: P(A ∪ B) = 0.9 and P(A ∩ B) = 0.5. What we wish to calculate is: P[(A ∩ B c) ∪ (A c ∩ B)] = P[(A ∪ B) − (A ∩ B)] = P(A ∪ B) − P(A ∩ B) = 0.9 − 0.5 = 0.4.
Example 6(i)
For three events A, B, and C, suppose that P(A ∩ B) = P(A ∩ C) and P(B ∩ C) = 0. Then show that P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − 2 P(A ∩ B).
(ii)
For any two events A and B, show that P(A c ∩ B c) = 1 − P(A) − P(B) + P(A ∩ B).
Discussion
(i)
We have P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(A ∩ C) − P(B ∩ C) + P(A ∩ B ∩ C). But A ∩ B ∩ C ⊂ B ∩ C, so that P(A ∩ B ∩ C) ≤ P(B ∩ C) = 0, and therefore P(A ∪ B ∪ C) = P(A) + P(B)+ P(C) − 2 P(A ∩ B).
(ii)
Indeed, P(A c ∩ B c) = P((A ∪ B)c) = 1 − P(A ∪ B) = 1 − P(A) − P(B) + P(A ∩ B).
Example 7
In ranking five horses in a horse race (Example 31 in Chapter 1), what is the probability that horse #3 will terminate either first or second or third?
Discussion
Denote by B the required event and let A i = “horse #3 terminates in the i th place,” i = 1, 2, 3. Then the events A 1, A 2, A 3 are pairwise disjoint, and therefore
P(B)=P(A 1∪A 2∪A 3)=P(A 1)+P(A 2)+P(A 3).
But P(A 1)=P(A 2)=P(A 3)=24 120=0.2, so that P(B) = 0.6.
Example 8
Consider a well-shuffled deck of 52 playing cards (Example 28 in Chapter 1), and suppose we draw at random three cards. What is the probability that at least one is an ace?
Discussion
Let A be the required event, and let A i be defined by A i = “exactly i cards are aces,” i = 0, 1, 2, 3. Then, clearly, P(A) = P(A 1 ∪ A 2 ∪ A 3). Instead, we may choose to calculate P(A) through P(A c) = 1 − P(A 0), where
P(A 0)=(48 3)(52 3)=48×47×46 52×51×50=4324 5525,so that P(A)=1201 5525≃0.217.
Example 9
Refer to Example 3 in Chapter 1 and let C 1, C 2, C 3 be defined by C 1 = “both S 1 and S 2 work,” C 2 = “ S 5 works,” C 3 = “both S 3 and S 4 work,” and let C = “current is transferred from point A to point B.” Then P(C) = P(C 1 ∪ C 2 ∪ C 3). At this point (in anticipation of Definition 3 in Section 2.4; see also Exercise 4.14 in this chapter), suppose that:
P(C 1)=p 1 p 2,P(C 2)=p 5,P(C 3)=p 3 p 4,P(C 1∩C 2)=p 1 p 2 p 5,P(C 1∩C 3)=p 1 p 2 p 3 p 4,P(C 2∩C 3)=p 3 p 4 p 5,P(C 1∩C 2∩C 3)=p 1 p 2 p 3 p 4 p 5.
Then:
P(C)=p 1 p 2+p 5+p 3 p 4−p 1 p 2 p 5−p 1 p 2 p 3 p 4−p 3 p 4 p 5+p 1 p 2 p 3 p 4 p 5.
For example, for p 1 = p 2 = p 3 = p 4 = p 5 = 0.9, we obtain
P(C)=0.9+2(0.9)2−2(0.9)3−(0.9)4+(0.9)5≃0.996.
This section is concluded with two very useful results stated as theorems. The first is a generalization of property #6 to more than three events, and the second is akin to the concept of continuity of a function as it applies to a probability function.
Theorem 1
The probability of the union of any n events, A 1, …, A n, is given by:
P(∪j=1 n A j)=∑j=1 n P(A j)−∑1≤j 1<j 2≤n P(A j 1∩A j 2)+∑1≤j 1<j 2<j 3≤n P(A j 1∩A j 2∩A j 3)−⋯+(−1)n+1 P(A 1∩⋯∩A n).
Although its proof (which is by mathematical induction) will not be presented, the pattern of the right-hand side above follows that of property #6(i) and it is clear. First, sum up the probabilities of the individual events, then subtract the probabilities of the intersections of the events, taken two at a time (in the ascending order of indices), then add the probabilities of the intersections of the events, taken three at a time as before, and continue like this until you add or subtract (depending on n) the probability of the intersection of all n events.
Recall that, if A 1 ⊆ A 2 ⊆⋯, then lim n→∞A n=∪n=1∞A n, and if A 1 ⊇ A 2 ⊇ ⋯, then lim n→∞A n=∩n=1∞A n.
Theorem 2
For any monotone sequence of events {A n}, n ≥ 1, it holds P(lim n→∞A n) = lim n→∞P(A n).
This theorem will be employed in many instances, and its use will be then pointed out.
Exercises
1.1
If P(A) = 0.4, P(B) = 0.6, and P(A ∪ B) = 0.7, calculate P(A ∩ B).
1.2
If for two events A and B, it so happens that P(A)=3 4 and P(B)=3 8, show that:
P(A∪B)≥3 4 and 1 8≤P(A∩B)≤3 8.
1.3
If for the events A, B, and C, it so happens that P(A) = P(B) = P(C) = 1, then show that:
P(A∩B)=P(A∩C)=P(B∩C)=P(A∩B∩C)=1.
Hint. Use Properties 2.1.1 (4), 6(i), (ii).
1.4
If the events A, B, and C are related as follows: A ⊂ B ⊂ C and P(A)=1 4,P(B)=5 12, and P(C)=7 12, compute the probabilities of the following events:
A c∩B,A c∩C,B c∩C,A∩B c∩C c,A c∩B c∩C c.
Hint. Use Properties 2.1.1 (3), (4) here, and Proposition 4 in Chapter 1.
1.5
Let S be the set of all outcomes when flipping a fair coin four times, so that all 16 outcomes are equally likely. Define the events A and B by:
A={s∈S;s contains more T's than H's},B={s∈S;there are both H's and T's in s and every T in s precedes every H in s}.
Compute the probabilities P(A), P(B).
1.6
Let S={x integer;1≤x≤200}, and define the events A, B, and C as follows:
A={x∈S;x is divisible by 7},B={x∈S;x=3 n+10,for some positive integer n},C={x∈S;x 2+1≤375}.
Calculate the probabilities P(A), P(B), and P(C).
1.7
If two fair dice are rolled once, what is the probability that the total number of spots shown is:
(i)
Equal to 5?
(ii)
Divisible by 3?
1.8
Students in a certain college subscribe to three news magazines A, B, and C according to the following proportions:
A:20%,B:15%,C:10%,
both A and B: 5%, both A and C: 4%, both B and C: 3%, all three A, B, and C: 2%.
If a student is chosen at random, what is the probability he/she subscribes to none of the news magazines?
Hint. Use Proposition 4 in Chapter 1, and Property 2.1.1 (6) (ii) here.
1.9
A high-school senior applies for admissions to two colleges A and B, and suppose that: P(admitted at A) = p 1, P(rejected by B) = p 2, and P(rejected by at least one, A or B) = p 3.
(i)
Calculate the probability that the student is admitted by at least one college.
(ii)
Find the numerical value of the probability in part (i), if p 1 = 0.6, p 2 = 0.2, and p 3 = 0.3.
1.10
An airport limousine service has two vans, the smaller of which can carry six passengers and the larger nine passengers. Let x and y be the respective numbers of passengers carried by the smaller and the larger van in a given trip, so that a suitable sample space S is given by:
S={(x,y);x=0,…,6 and y=0, 1,…,9}.
Also, suppose that, for all values of x and y, the probabilities P({(x, y)}) are equal. Finally, define the events A, B, and C as follows:
A=“the two vans together carry either 4 or 6 or 10 passengers,”B=“the larger van carries twice as many passengers as the smaller van,”C=“the two vans carry different numbers of passengers.”
Calculate the probabilities: P(A), P(B), and P(C).
1.11
In the sample space S=(0,∞), consider the events A n=(0,1−2 n),n=1,2,…,A=(0,1), and suppose that P(A n)=2 n−1 4 n.
(i)
Show that the sequence {A n} is increasing and that lim n→∞A n=∪n=1∞A n=A.
(ii)
Use the appropriate theorem (cite it!) in order to calculate the probability P(A).
1.12
Show that for any n events A 1, …, A n, it holds:
P(∩i=1 n A i c)≥1−∑i=1 n P(A i).
1.13
Show that, if P(A i) = 1, i = 1, 2, …, then P(∩i=1∞A i)=1.
Hint. Use DeMorgan's laws and Theorem 2 in Chapter 2.
1.14
In a small community 15 families have a number of children as indicated below.
Number of family 4 5 3 2 1
Number of children 1 2 3 4 5
Total number of children 4 10 9 8 5 36
(i)
If a family is chosen at random, what is the probability P(C i), where C i = “the family bas i children,” i = 1, 2, 3, 4, 5?
(ii)
If a child is chosen at random (from among the 36 children in the community), what is the probability P(F i), where F i = “child belongs in a family with i children, ” i = 1, 2, 3, 4, 5?
Show more
View chapterExplore book
Read full chapter
URL:
Book 2015, An Introduction to Probability and Statistical Inference (Second Edition)George G. Roussas
Chapter
Inductive Logic
2011, Handbook of the History of LogicUlrike von Luxburg, Bernhard Schölkopf
5 Capacity Concepts and Generalization Bounds
So far, we have argued which property of the function space determines whether the principle of empirical risk minimization is consistent, i.e., whether it will work “in the limit.” This property was referred to as uniform convergence. To make statements about what happens after seeing only finitely many data points — which in reality will always be the case — we need to take a closer look at this convergence. It will turn out that this will provide us with bounds on the risk, and it will also provide insight into which properties of function classes determine whether uniform convergence can take place. To this end, let us take a closer look at the subject of Theorem 3: the probability
(13)P(sup f∈F|R(f)−R emp(f)|>ε).
Two tricks are needed along the way: the union bound and the method of symmetrization by a ghost sample.
5.1 The union bound
The union bound is a simple but convenient tool to transform the standard law of large numbers of individual functions into a uniform law of large numbers over a set of finitely many functions. Suppose the set F consists just of finitely many functions, that is F={f 1,f 2,…,f m}. Each of the functions f i ∈ F satisfies the standard law of large numbers in form of the Chernoff bound, that is
(14)P(|R(f i)−R emp(f i)|≥ε)≤2 exp(−2 n ε 2).
Now we want to transform these statements about the individual functions f i into a uniform law of large numbers. To this end, note that we can rewrite:
(15)P(sup f∈F|R(f)−R emp(f)|≥ε)=P(|R(f 1)−R emp(f 1)|≥ε or|R(f 2)−R emp(f 2)|≥ε or…or|R(f m)−R emp(f m)|≥ε)
(16)≤∑i=1 m P(|R(f i)−R emp(f i)|≥ε)≤2 m exp(−2 n ε 2)
Let us go through these calculations step by step. The first equality comes from the way the supremum is defined. Namely, the supremum over certain expressions is larger than ɛ if at least one of the expressions is larger than ɛ, which leads to the statements with the “or” combinations. The next step uses a standard tool from probability theory, the union bound. The union bound states that the probability of a union of events (that is, events coupled with “or”) is smaller or equal to the sum of the individual probabilities. Finally, the last step is a simple application of the Chernoff bound of Eq. (14) to each of the terms in the sum.
From left to right, the statements in Eq. (16) show us how to convert the Chernoff bound for individual functions f i into a bound which is uniform over a finite numberm of functions. As we can see, the difference between the Chernoff bound for the individual functions and the right hand side of (16) is just a factor m. If the function space F is fixed, this factor can be regarded as a constant, and the term 2 m exp(−2 nɛ 2) still converges to 0 as n → ∞. Hence, the empirical risk converges to 0 uniformly over F as n → ∞. That is, we have proved that empirical risk minimization over a finite set F of functions is consistent with respect to F.
We next describe a trick used by Vapnik and Chervonenkis to reduce the case of an infinite function class to the case of a finite one. It consists of introducing what is sometimes called a ghost sample. It will enable us to replace the factor m in (16) by more general capacity measures that can be computed for infinite function classes.
5.2 Symmetrization
Symmetrization is an important technical step towards using capacity measures of function classes. Its main purpose is to replace the event sup f∈F|R(f)−R emp(f)| by an alternative event which can be solely computed on a given sample. Assume we are given a sample (X i,Y i)i=1,…,n.We now introduce a new sample called the ghost sample. This ghost sample is just another sample (X i′,Y i′)i=1,…,n which is also drawn iid from the underlying distribution and which is independent of the first sample. It is called ghost sample because we do not need to physically draw this sample in practice. It is just a mathematical tool, that is we play “as if we had a second sample”. Of course, given the ghost sample we can also compute the empirical risk of a function with respect to this ghost sample, and we will denote this risk by R emp′(f). With the help of the ghost sample, one can now prove the following simple statement:
LEMMA 4 Vapnik and Chervonenkis. For mε 2 ≥ 2, we have
(17)P(sup f∈F|R(f)−R emp(f)|>ε)≤2 P(sup f∈F|R emp(f)−R emp′(f)|>ε/2).
Here, the first P refers to the distribution of an iid sample of size n, while the second one refers to the distribution of two samples of size n (the original sample and the ghost sample), that is the distribution of iid samples of size 2n. In the latter case, R emp measures the empirical loss on the first half of the sample, and R emp′on the second half.
Although we do not prove this lemma, it should be fairly plausible: if the empirical risks on two independent n-samples are close to each other, then they should also be close to the true risk.
This lemma is called the symmetrization lemma. Its name refers to the fact that we now look at an event which depends in a symmetric way on a sample, now of size 2 n. The main purpose of this lemma is that the quantity R(f), which cannot be computed on a finite sample, has been replaced by the quantity R emp′(f), which can be computed on a finite sample.
Now let us explain what the symmetrization lemma is used for. In the last section we have seen how to bound the probability of uniform convergence (13) in terms of a probability of an event referring to a finite function class. The crucial observation is now that even if F contains infinitely many functions, the different ways it can classify a training set of n sample points is finite. Namely, for any given training point in the training sample, a function can only take the values −1 or +1. On a sample of n points X 1,…,X n, a function can act in at most 2 n different ways: it can choose each Y i as −1 or +1. This has a very important consequence. Even if a function class F contains infinitely many functions, there are at most 2 n different ways those functions can classify the points of a finite sample of n points. This means that if we consider the term
sup f∈F|R emp(f)−R emp′(f)|
then the supremum effectively only runs over a finite function class. To understand this, note that two functions f,g∈F which take the same values on the given sample have the same empirical risk, that is R emp(f) = R emp(g). The analogous statement holds for the ghost sample and R emp′. Hence, all functions f,g which coincide both on the sample and the ghost sample will lead to the same term |R emp(f)−R emp′(f)|. Thus, the only functions we need to consider to compute the supremum are the 2 2 n functions we can obtain on sample and ghost sample together. Hence, we can replace the supremum over f∈F by the supremum over a finite function class with at most 2 2 n functions.
Note that this step is only possible due to the symmetrization lemma. If we had considered the term sup f∈F|R(f)−R emp(f)|, the argument from above would not hold, as the value of R(f) not only depends on the values of f on the sample.
In the following we now want to show how the insights gained in the symmetrization step can be used to derive a first capacity measure of a function class.
5.3 The shattering coefficient
For the purpose of bounding (13), Lemma 4 implies that the function class F is effectively finite: restricted to the 2 n points appearing on the right hand side of (17), it has at most 2 2 n elements. This is because only the values of the functions on the sample points and the ghost sample points count. The number of effectively different functions can be smaller than 2 2 n, however. For example, it could be the case that F does not contain a single function which takes value +1 on the first training point. We now want to formalize this. Let Z n := ((X 1,Y 1),…,(X n,Y n)) be a given sample of size n. Denote by |F Z n| be the cardinality of F when restricted to {X 1,…,X n}, that is, the number of functions from F that can be distinguished from their values on {X 1,…,X n}. Let us, moreover, denote the maximum number of functions that can be distinguished in this way as N(F,n), where the maximum runs over all possible choices of samples, that is
N(F,n)=max{|F Z n||X 1,…,X n∈x}.
The quantity N(F,n) is referred to as the shattering coefficient of the function class F with respect to sample size n. It has a particularly simple interpretation: it is the number of different outputs (Y 1,…,Y n) that the functions in F can achieve on samples of a given size n. In other words, it measures the number of ways that the function space can separate the patterns into two classes. Whenever N(F,n)=2 n, this means that there exists a sample of size n on which all possible separations can be achieved by functions of the class F. If this is the case, the function space is said to shatter n points. Note that because of the maximum in the definition of N(F,n), shattering means that there exists a sample of n patterns which can be separated in all possible ways — it does not mean that this applies to all possible samples of n patterns.
The shattering coefficient is a capacity measure of a function class, that is it measures the “size” of a function class in some particular way. Note that if a function class F contains very many functions, then the shattering coefficient tends to be larger than for a function class which only contains very few functions. However, the shattering coefficient is more subtle than simply counting the number of functions in a class. It only counts the number of functions in relation to the samples we are interested in. The following section will now finally show how to use the shattering coefficient to derive a generalization bound for empirical risk minimization on infinite function classes F.
5.4 Uniform convergence bounds
Given an arbitrary, possibly infinite function class, we now want to take a look at the right hand side of (17). We now consider a sample of 2 n points, that is a set Z 2 n, where we interpret the first n points as the original sample and the second n points as the ghost sample. The idea is now to replace the supremum over F by the supremum over the set F Z 2 n, use that the set F Z 2 n contains at most N(F,n)≤2 2 n different functions, then apply the union bound on this finite function set and then the Chernoff bound. This leads to a bound like (16), with N(F,2 n) playing the role of m. Essentially, those steps can be written down as follows:
P(sup f∈F|R(f)−R emp(f)|>∈)(due to symmetrization)≤2 P(sup f∈F|R emp(f)−R emp′(f)|>∈/2)(only functions in F Z 2 n are important)=2 P(sup f∈F Z 2 n|R emp(f)−R emp′(f)|>∈/2)(F Z 2 n contains at most N(F,2 n)we functions, independently of Z 2 n)(use union bound argument and Chernoff)≤2 N(F,2 n)exp(−n ε 2/4)
So all in all we see that
(18)P(sup f∈F|R(f)−R emp(f)|>∈)≤2 N(F,2 n)exp(−n ε 2/4).
Now we can use this expression to draw conclusions about consistency of empirical risk minimization. Namely, ERM is consistent for function class F if the right hand side of this expression converges to 0 as n → ∞. Let us look at a few examples.
First of all, consider a case where the shattering coefficient N(F,2 n) is considerably smaller than 2 2 n, say N(F,2 n)≤(2 n)k for some constant k (this means that the shattering coefficient grows polynomially in n). Plugging this in the right hand side of (18), we get
2 N(F,2 n)exp(−n ε 2/4)=2⋅(2 n)k⋅exp(−n ε 2/4)=2 exp(k⋅log(2 n)−n ε 2/4).
Here we can see that for all ɛ, if n is large enough the whole expression converges to 0 for n → ∞. From this we can conclude that whenever the shattering coefficient N(F,2 n) only grows polynomially with n, then empirical risk minimization is consistent with respect to F.
On the other hand, consider the case where we use the function class F all. It is clear that this class can classify each sample in every possible way, that is N(F,2 n)=2 2 n for all values of n. Plugging this in the right hand side of (18), we get
2 N(F,2 n)exp(−n ε 2/4)=2⋅2 2 n exp(−n ε 2/4)=2 exp(n(2 log(2)−ε 2/4)).
We can immediately see that this expression does not tend to 0 when n → ∞, that is we cannot conclude consistency for F all. Note that we cannot directly conclude that ERM using F all is inconsistent, either. The reason is that (18) only gives an upper bound on P(sup f∈F|R(f)−R emp(f)|>ε), that is it only provides a sufficient condition for consistency, not a necessary one. However, with more effort one can also prove necessary statements. For example, a necessary and sufficient condition for consistency of ERM is that
log N(F,n)/n→0
(cf. [Mendelson, 2003], for related theorems also see [Vapnik and Chervonenkis, 1971; Vapnik and Chervonenkis, 1981], or Section 12.4. of [Devroye et al., 1996]). The proof that this condition is necessary is more technical, and we omit it. In case of the examples above, this condition immediately gives the desired results: if N(F,n) is polynomial, then log N(F,n)/n→0. On the other hand, for F all we always have N(F,n)=2 n, thus log N(F,n)/n=n/n=1, which does not converge to 0. Thus, ERM using F all is not consistent.
5.5 Generalization bounds
It is sometimes useful to rewrite (18) “the other way round”. That is, instead of fixing ɛ and then computing the probability that the empirical risk deviates from the true risk by more than ɛ, we specify the probability with which we want the bound to hold, and then get a statement which tells us how close we can expect the risk to be to the empirical risk. This can be achieved by setting the right hand side of (18) equal to some δ> 0, and then solving for ε. As a result, we get the statement that with a probability at least 1 − δ, any function f∈F satisfies
(19)R(f)≤R emp(f)+4 n(log(2 N(F,n))−log(δ)).
In the same way as above, we can use this bound to derive consistency statements. For example, it is now obvio√s that empirical risk minimization is consistent for function class F if the term log(2 N(F,2 n))/n converges to 0 as n → ∞. Again, this is for example the case if N(F,2 n) only grows polynomially with n.
Note that the bound (19) holds for all functions f∈F. On the one side, this is a strength of the bound, as it holds in particular for the function f n minimizing the empirical risk, which is what we wanted. Moreover, many learning machines do not truly minimize the empirical risk, and the bound thus holds for them, too. However, one can also interpret it as a weakness, since by taking into account more information about the function we are interested in, one could hope to get more accurate bounds.
Let us try to get an intuitive understanding of this bound. It tells us that if both R emp(f) and the square root term are small simultaneously, then we can guarantee that with high probability, the risk (i.e., the error on future points that we have not seen yet) will be small. This sounds like a surprising statement, however, it does not claim anything impossible. If we use a function class with relatively small N(F,n), i.e., a function class which cannot “explain” many possible functions, and then notice that using a function of this class we can nevertheless explain data sampled from the problem at hand, then it is likely that this is not a coincidence, and we can have captured some essential aspects of the problem. If, on the other hand, the problem is too hard to learn from the given amount of data then we will find that in order to explain the data (i.e., to achieve a small R emp(f)), we need a function class which is so large that it can basically explain anything, and thus the square root term would be large. Note, finally, that whether a problem is hard to learn is entirely determined by whether we can come up with a suitable function class, and thus by our prior knowledge of it. Even if the optimal function is subjectively very complex, if our function class contains that function, and few or no other functions, we are in an excellent position to learn.
There exists a large number of bounds similar to (18) and its alternative form(19). Differences occur in the constants, both in front of the exponential and in its exponent. The bounds also differ in the exponent of ε (see [Devroye et al., 1996; Vapnik, 1998] and references therein) and in the way they measure capacity. We will not elaborate on those issues.
5.6 The VC dimension
Above we have formulated the generalization bounds in terms of the shattering coefficient N(F,n). The downside is that they are usually difficult to evaluate. However, there exists a large number of different capacity concepts, with different advantages and disadvantages. We now want to introduce the most well known one, the so-called VC dimension (named after Vapnik and Chervonenkis). Its main purpose is to characterize the growth behavior of the shattering coefficient using a single number.
We say that a sample Z n of size n is shattered by function class F if the function class can realize any labeling on the given sample, that is |F Z n|=2 n. The VC dimension of F, denoted by VC(F), is now defined as the largest number n such that there exists a sample of size n which is shattered by F. Formally,
VC(F)=max{n∈N||F Z n|=2 n for some Z n}.
If the maximum does not exist, the VC dimension is defined to be infinity. For many examples of function classes and their VC dimensions see for example [Kearns and Vazirani, 1994] or [Anthony and Biggs, 1992].
A beautiful combinatorial result proved simultaneously by several people ([Sauer, 1972], [Shelah, 1972], [Vapnik and Chervonenkis, 1971]) characterizes the growth behavior of the shattering coefficient and relates it to the VC dimension:
LEMMA 5 Vapnik, Chervonenkis, Sauer, Shelah. Let F be a function class with finite VC dimension d. Then
N(F,n)≤∑i=0 d(i n)
for all n ∈ ℕ. In particular, for all n ≥ d we have
N(F,n)≤(e n d)d.
The importance of this statement lies in the last fact. If n ≥ d, the shattering coefficient behaves like a polynomial function of the sample size n. This is a very remarkable result: once we know the VC-dimension of a function class F is finite, we already know that the shattering coefficients grow polynomially with n. By the results of the last section this implies the consistency of ERM. Note that we also have a statement in the other direction. If the VC-dimension is infinite, this means that for each n there exists some sample which can be shattered by F, that is N(F,n)=2 n. For this case we have already seen above that ERM is not consistent. Together, we achieve the following important characterization:
THEOREM 6. Empirical risk minimization is consistent with respect to F if and only if VC(F)is finite.
One important property to note both about the shattering coefficient and the VC dimension is that they do not depend on the underlying distribution P, they only depend on the function class F. One the one hand, this is an advantage, as all generalization bounds derived from those concepts apply to all possible probability distributions. On the other hand, one can also consider this as disadvantage, as the capacity concepts do not take into account particular properties of the distribution at hand. In this sense, those capacity concepts often lead to rather loose bounds.
5.7 Rademacher complexity
A different concept to measure the capacity of a function space is the Rademacher complexity. Compared to the shattering coefficient and the VC dimension, it does depend on the underlying probability distribution, and usually leads to much sharper bounds than both of them. The Rademacher complexity is defined as follows. Let σ 1, σ 2,… independent random variables which attain the two values +1 and −1 with probability 0.5 each (such random variables are sometimes called Rademacher variables, therefore the name “Rademacher complexity”). For example, they could be the results of repeatedly tossing a fair coin. We formally define the Rademacher complexity R(F) of a function space F as
(20)R(F):=E sup f∈F 1 n∑i=1 n σ i f(X i).
This expression looks complicated, but it has a nice interpretation. For the moment, consider the values σ i as fixed, and interpret label σ i as a label of the point X i. As both σ i and f(X i) only take the values +1 or −1, the product σ i f(X i) takes the value +1 if σ i = f(X i), and −1 if σ i ≠ f(X i). As a consequence, the sum on the right hand side of Equation (20) will be large if the labels f(X i) coincide with the labels σ i on many data points. This means that the function f “fits well” to the labels σ i: if the labels σ i were the correct labels, f would have a small training error R emp. Now taking into account the supremum, we not only look at one function f, but at all functions f∈F. We can see that sup f∈F∑i=1 n σ i f(X i) is large if there exists a function in F which fits well to the given sequence σ i of labels. Finally, recall that the labels σ i are supposed to be random variables. We can consider them as random labels on the data points X i. As we take the expectation over both the data points and the random labels, the overall Rademacher complexity is high if the function space F is able to “fit well” to random labels. This intuition makes sense: a function space has to be pretty large to be able to fit to all kinds of random labels on all kinds of data sets. In this sense, the Rademacher complexity measures how “complex” the function space is: the higher R(F), the larger the complexity of F.
From a mathematical point of view, the Rademacher complexity is convenient to work with. One can prove generalization bounds of the following form: with probability at least 1 − δ,
R(f)≤R emp(f)+2 R(F)+log(1/δ)2 n
Rademacher complexities have some advantages over the classical capacity concepts such as the VC dimension. Most notably, the bounds obtained by Rademacher complexities tend to be much sharper than the ones obtained by the classical tools. The proof techniques are different from the ones explained above, but we do not want to go into details here. For literature on Rademacher complexity bounds, see for example [Mendelson, 2003], [Bousquet et al., 2003] or [Boucheron et al., 2005] and references therein.
5.8 Large margin bounds
Finally, we would like to introduce another type of capacity measure of function classes which is more specialized than the general combinatorial quantities introduced above. Consider the special case where the data space consists of points in the two-dimensional space ℝ 2, and where we want to separate classes by a straight line. Given a set of training points and a classifier f n which can perfectly separate them, we define the margin of the classifier f n as the smallest distance of any training point to the separating line f n (cf. Figure 5). Similarly, a margin can be defined for general linear classifiers in the space ℝ d of arbitrary dimensiond. It can be proved that the VC dimension of a class F ρ of linear functions with all have margin at least ρ can essentially be bounded by the ratio of the radius R of the smallest sphere enclosing the data points with the margin ρ, that is
Sign in to download full-size image
Figure 5. Margin of a linear classifier: the crosses depict training points with label +1, the circles training points with label -1. The straight line is the linear classifier f n, and the dashed line shows the margin. The width ρ of the margin is depicted by the arrow.
VC(F ρ)≤min{d,4 R 2 ρ 2}+1
(cf. Theorem 5.1. in [Vapnik, 1995]). That is, the larger the margin ρ of the functions in the class F ρ, the smaller is its VC dimension. Thus, one can use the margin of classifiers as a capacity concept. One of the most well-known classifiers, the support vector machine (SVM) builds on this result. See [Schölkopf and Smola, 2002] for a comprehensive treatment. An example of a generalization bound involving the large margin is as follows (for a more precise statement see for example Theorem 7.3. in [Schölkopf and Smola, 2002]):
THEOREM 7 Large margin bound. Assume the data space lies inside a ball of radius R in ℝ d. Consider the set F ρ of linear classifiers with margin at least ρ. Assume we are given n training examples. By ν(f) denote the fraction of training examples which have margin smaller than ρ or which are wrongly classified by some classifier f∈F ρ. Then, with probability at least 1 − δ, the true error of any f∈F ρ can be bounded by
R(f)≤v(f)+c n(R 2 ρ 2 log(n)2+log(1/δ))
where c is some universal constant.
5.9 Other generalization bounds and capacity concepts
Above we have introduced a few capacity concepts for function classes such as the shattering coefficient, the VC dimension, or the Rademacher complexity. In the literature, there exist many more capacity concepts, and introducing all of them will be beyond the scope of this overview. However, we would like to point out the general form which most generalization bounds take. Usually, those bounds are composed of three different terms and have a form like: with probability at least 1 − δ,
R(f)≤R emp(f)+capacity(F)+confidence(δ).
That is, one can bound the true risk of a classifier by its empirical risk, some capacity term which in the simplest case only depends on the underlying function class, and a confidence term which depends on the probability with which the bound should hold. Note that by nature, all bounds of this form a worst case bounds: as the bound holds for all functions in the class F, the behavior of the bound is governed by the “worst” or “most badly behaved” function in the class. This point is often used to criticize this approach to statistical learning theory, as natural classifiers do not tend to pick the worst function in a class.
Show more
View chapterExplore book
Read full chapter
URL:
Handbook2011, Handbook of the History of LogicUlrike von Luxburg, Bernhard Schölkopf
Chapter
Symbols and Special Numbers
2014, Mathematical Formulas for Industrial and Mechanical EngineeringSeifedine Kadry
1.4 Probability and Statistics Symbols
| Symbols | How to Read It | How to Use It | Examples |
--- --- |
| P(A) | probability function | probability of event A | P(A)=0.5 |
| P(A∩B) | probability of events intersection | probability that of events A and B | P(A∩B)=0.5 |
| P(A∪B) | probability of events union | probability that of events A or B | P(A∪B)=0.5 |
| P(A|B) | conditional probability function | probability of event A given event B occurred | P(A|B)=0.3 |
| f(x) | probability density function (pdf) | P(a≤x≤b)=∫f(x)d x | |
| F(x) | cumulative distribution function (cdf) | F(x)=P(X≤x) | |
| μ | population mean | mean of population values | μ=10 |
| E(X) | expectation value | expected value of random variable X | E(X)=10 |
| E(X|Y) | conditional expectation | expected value of random variable X given Y | E(X|Y=2)=5 |
| var(X) | variance | variance of random variable X | var(X)=4 |
| σ 2 | variance | variance of population values | σ 2=4 |
| std(X) | standard deviation | standard deviation of random variable X | std(X)=2 |
| σ X | standard deviation | standard deviation value of random variable X | σ X=2 |
| x˜ | median | middle value of random variable x | x˜=5 |
| cov(X,Y) | covariance | covariance of random variables X and Y | cov(X,Y)=4 |
| corr(X,Y) | correlation | correlation of random variables X and Y | corr(X,Y)=3 |
| ρ X,Y | correlation | correlation of random variables X and Y | ρ X,Y=3 |
| Mod | mode | value that occurs most frequently in population | |
| MR | mid range | MR=(x max+x min)/2 | |
| Md | sample median | half the population is below this value | |
| Q 1 | lower/first quartile | 25% of population are below this value | |
| Q 2 | median/second quartile | 50% of population are below this value=median of samples | |
| Q 3 | upper/third quartile | 75% of population are below this value | |
| X | sample mean | average/arithmetic mean | x=(2+5+9)/3=5.333 |
| s 2 | sample variance | population samples variance estimator | s 2=4 |
| S | sample standard deviation | population samples standard deviation estimator | s=2 |
| z x | standard score | z x=(x−x)/s x | |
| X~ | distribution of X | distribution of random variable X | X~N(0,3) |
| N(μ,σ 2) | normal distribution | Gaussian distribution | X~N(0,3) |
| U(a,b) | uniform distribution | equal probability in range a, b | X~U(0,3) |
| exp(λ) | exponential distribution | f(x)=λ e −λx, x≥0 | |
| gamma(c, λ) | gamma distribution | f(x)=λcx c−1 e−λx/Γ(c), x≥0 | |
| χ 2(k) | chi-square distribution | f(x)=x k/2−1 e −x/2/(2 k/2 Γ(k/2)) | |
| F(k 1,k 2) | F distribution | | |
| Bin(n,p) | binomial distribution | f(k)=n C k p k(1−p)n−k | |
| Poisson(λ) | Poisson distribution | f(k)=λ k e −λ/k! | |
| Geom(p) | geometric distribution | f(k)=p(1−p)k | |
| HG(N,K,n) | hyper-geometric distribution | | |
| Bern(p) | Bernoulli distribution | | |
| n! | factorial | n!=1·2·3·⋯·n | 5!=1·2·3·4·5=120 |
| n P k | permutation | P k n=n!(n−k)! | 5 P 3=5!/(5−3)!=60 |
| n C k(n k) | combination | C k n=(n k)=n!k!(n−k)! | 5 C 3=5!/[3!(5−3)!]=10 |
Show more
View chapterExplore book
Read full chapter
URL:
Book 2014, Mathematical Formulas for Industrial and Mechanical EngineeringSeifedine Kadry
Chapter
The concept of probability and basic results
2015, An Introduction to Probability and Statistical Inference (Second Edition)George G. Roussas
Classical Definition of Probability
Let S be a sample space, associated with a certain random experiment and consisting of finitely many sample points n, say, each of which is equally likely to occur whenever the random experiment is carried out. Then the probability of any event A, consisting of m sample points (0 ≤ m ≤ n), is given by P(A)=m n.
In reference to Example 26 in Chapter 1, P(A)=4 8=1 2=0.5. In Example 27 (when the two dice are unbiased), P(X=7)=6 36=1 6≃0.167, where the r.v. X and the event (X = 7) are defined in Section 1.3 of Chapter 1. In Example 29, when the balls in the urn are thoroughly mixed, we may assume that all of the (m + n)(m + n − 1) pairs are equally likely to be selected. Then, since the event A occurs in 20 different ways, P(A)=20(m+n)(m+n−1). For m = 3 and n = 5, this probability is P(A)=20 56=5 14≃0.357.
From the preceding (classical) definition of probability, the following simple properties are immediate: For any event A,P(A)≥0;P(S)=1; if two events A 1 and A 2 are disjoint (A 1 ∩ A 2 = ∅), then P(A 1 ∪ A 2) = P(A 1) + P(A 2). This is so because, if A 1={s i 1,…,s i k},A 2={s j 1,…,s j ℓ}, where all s i 1,…,s i k are distinct from all s j 1,…,s j ℓ, then A 1∪A 2={s i 1,…,s i k s j 1,…,s j ℓ} and P(A 1∪A 2)=k+ℓ n=k n+ℓ n=P(A 1)+P(A 2).
In many cases, the stipulations made in defining the probability as above are not met, either because S has not finitely many points (as is the case in Examples 32, 33–35 Example 33 Example 34 Example 35 (by replacing C and M by ∞), and 36–40 in Chapter 1 Example 36 Example 37 Example 38 Example 39 Example 40), or because the (finitely many) outcomes are not equally likely. This happens, for instance, in Example 26 when the coins are not balanced and in Example 27 when the dice are biased. Strictly speaking, it also happens in Example 30. In situations like this, the way out is provided by the so-called relative frequency definition of probability. Specifically, suppose a random experiment is carried out a large number of times N, and let N(A) be the frequency of an event A, the number of times A occurs (out of N). Then the relative frequency of A is N(A)N. Next, suppose that, as N → ∞, the relative frequencies N(A)N oscillate around some number (necessarily between 0 and 1). More precisely, suppose that N(A)N converges, as N → ∞, to some number. Then this number is called the probability of A and is denoted by P(A). That is, P(A)=lim N→∞N(A)N. (It will be seen later in this book that the assumption of convergence of the relative frequencies N(A)/N is justified subject to some qualifications.) To summarize,
Relative Frequency Definition of Probability
Let N(A) be the number of times an event A occurs in N (identical) repetitions of a random experiment, and assume that the relative frequency of A,N(A)N, converges to a limit as N → ∞. This limit is denoted by P(A) and is called the probability of A.
At this point, it is to be observed that the empirical data show that the relative frequency definition of probability and the classical definition of probability agree in the framework in which the classical definition applies.
From the relative frequency definition of probability and the usual properties of limits, it is immediate that: P(A) ≥ 0 for every event A; P(S)=1; and for A 1, A 2 with A 1 ∩ A 2 = ∅,
P(A 1∪A 2)=lim N→∞N(A 1∪A 2)N=lim N→∞(N(A 1)N+N(A 2)N)=lim N→∞N(A 1)N+lim N→∞N(A 2)N=P(A 1)+P(A 2);
that is, P(A 1 ∪ A 2) = P(A 1) + P(A 2), provided A 1 ∩ A 2 = ∅. These three properties were also seen to be true in the classical definition of probability. Furthermore, it is immediate that under either definition of probability, P(A 1 ∪ … ∪ A k) = P(A 1) + ⋯ + P(A k), provided the events are pairwise disjoint; A i ∩ A j = ∅, i ≠ j.
The above two definitions of probability certainly give substance to the concept of probability in a way consonant with our intuition about what probability should be. However, for the purpose of cultivating the concept and deriving deep probabilistic results, one must define the concept of probability in terms of some basic properties, which would not contradict what we have seen so far. This line of thought leads to the so-called axiomatic definition of probability due to Kolmogorov.
Axiomatic Definition of Probability
Probability is a function, denoted by P, defined for each event of a sample space S, taking on values in the real line ℜ, and satisfying the following three properties:
P(1)
P(A) ≥ 0 for every event A (nonnegativity of P).
P(2)
P(S)=1 (P is normed).
P(3)
For countably infinite many pairwise disjoint events A i, i = 1, 2, …, A i ∩ A j = ∅, i ≠ j, it holds
P(A 1∪A 2∪⋯)=P(A 1)+P(A 2)+⋯;or P(∪i=1∞A i)=∑i=1∞P(A i)(sigma-addivity(σ-additivity)of P).
Comments on the Axiomatic Definition
(1)
Properties (P1) and (P2) are the same as the ones we have seen earlier, whereas property (P3) is new. What we have seen above was its so-called finitely additive version; that is, P(∪i=1 n A i)=∑i=1 n P(A i), provided A i ∩ A j = ∅, i ≠ j. It will be seen below that finite additivity is implied by σ-additivity but not the other way around. Thus, if we are to talk about the probability of the union of countably infinite many pairwise disjoint events, property (P3) must be stipulated. Furthermore, the need for such a union of events is illustrated as follows: In reference to Example 32, calculate the probability that the first head does not occur before the n th tossing. By setting A i={T…T︸i−1 H},i=n,n+1,…, what we are actually after here is P(A n ∪ A n+1 ∪ ⋯) with A i ∩ A j = ∅, i ≠ j, i and j ≥ n.
(2)
Property (P3) is superfluous (reduced to finite-additivity) when the sample space S is finite, which implies that the total number of events is finite.
(3)
Finite-additivity is implied by additivity for two events, P(A 1 ∪ A 2) = P(A 1) + P(A 2), A 1 ∩ A 2 = ∅, by way of mathematical induction.
Here are two examples in calculating probabilities.
Example 1
In reference to Example 1 in Chapter 1, take n = 58, and suppose we have the following configuration:
| Arsenic | Barium |
--- |
| High Mercury | Low Mercury |
| High | Low | High | Low |
| High | 1 | 3 | 5 | 9 |
| Low | 4 | 8 | 10 | 18 |
Calculate the probabilities mentioned in (i) (a)-(d).
Discussion
For simplicity, denote by B h the event that the site selected has a high barium concentration, and likewise for other events figuring below. Then:
(i)(a)
B h = (A h ∩ B h ∩ M h) ∪ (A h ∩ B h ∩ M ℓ) ∪ (A ℓ ∩ B h∩ M h) ∪ (A ℓ ∩ B h ∩ M ℓ) and the events on the right-hand side are pairwise disjoint. Therefore (by the basic property #2 in Section 2.1.1),
P(B h)=P(A h∩B h∩M h)+P(A h∩B h∩M ℓ)+P(A ℓ∩B h∩M h)+P(A ℓ∩B h∩M ℓ)=1 58+3 58+4 58+8 58=16 58=8 29≃0.276.
(i)(b)
Here, P(M h∩A ℓ∩B ℓ)=P(A ℓ∩B ℓ∩M h)=10 58=5 29≃0.172.
(i)(c)
Here, the required probability is as in (a):
P(A h∩B h∩M ℓ)+P(A h∩B ℓ∩M h)+P(A ℓ∩B h∩M h)=12 58=6 29≃0.207.
(i)(d)
As above,
P(A h∩B ℓ∩M ℓ)+P(A ℓ∩B h∩M ℓ)+P(A ℓ∩B ℓ∩M h)=27 58≃0.466.
Example 2
In ranking five horses in a horse race (Example 31 in Chapter 1), calculate the probability that horse #3 terminates at least second.
Discussion
Let A i be the event that horse #3 terminates at the i th position, i = 1, …, 5. Then the required event is A 1 ∪ A 2, where A 1, A 2 are disjoint. Thus,
P(A 1∪A 2)=P(A 1)+P(A 2)=24 120+24 120=2 5=0.4.
Example 3
In tossing a coin repeatedly until H appears for the first time (Example 32 in Chapter 1), suppose that P{T…T︸i−1 H}=P(A i)=q i−1 p for some 0 <p< 1 and q = 1 − p (in anticipation of Definition 3 in Section 2.4). Then
P(∪i=n∞A i)=∑i=n∞P(A i)=∑i=n∞q i−1 p=p∑i=n∞q i−1=p q n−1 1−q=p q n−1 p=q n−1.
For instance, for p = 1/2 and n = 3, this probability is 1 4=0.25. That is, when tossing a fair coin, the probability that the first head does not appear either the first or the second time (and therefore it appears either the third time or the fourth time, etc.) is 0.25. For n = 10, this probability is approximately 0.00195 ≃ 0.002.
Next, we present some basic results following immediately from the defining properties of the probability. First, we proceed with their listing and then with their justification.
Show more
View chapterExplore book
Read full chapter
URL:
Book 2015, An Introduction to Probability and Statistical Inference (Second Edition)George G. Roussas
Related terms:
Probability Theory
Complement of Event
Conditional Probability
Random Variable
Lim Inf
Lim Sup
Sample Point
Sample Space
Venn Diagram
Finite Additivity
View all Topics
Recommended publications
Historia MathematicaJournal
Introductory Statistics (Third Edition)Book • 2010
Games and Economic BehaviorJournal
Operations Research LettersJournal
Browse books and journals
About ScienceDirect
Remote access
Advertise
Contact and support
Terms and conditions
Privacy policy
Cookies are used by this site. Cookie Settings
All content on this site: Copyright © 2025 or its licensors and contributors. All rights are reserved, including those for text and data mining, AI training, and similar technologies. For all open access content, the relevant licensing terms apply. |
13052 | https://math.stackexchange.com/questions/2075365/confused-by-integrating-factors-example | ordinary differential equations - Confused by integrating factors example. - Mathematics Stack Exchange
Join Mathematics
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Mathematics helpchat
Mathematics Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Confused by integrating factors example.
Ask Question
Asked 8 years, 9 months ago
Modified8 years, 9 months ago
Viewed 134 times
This question shows research effort; it is useful and clear
0
Save this question.
Show activity on this post.
I'm struggling to understand how this step works when solving a differential equation by integrating factors.
The question is, solve:
x d y d x−3 y=x 4 cos(x),x d y d x−3 y=x 4 cos(x),
for x>0 x>0.
I don't understand how to get from step (a) to step (b).
(a):1 x 3 y′−3 x 4 y=cos(x),(b):d d x 1 x 3 y=cos(x).(a):1 x 3 y′−3 x 4 y=cos(x),(b):d d x 1 x 3 y=cos(x).
In the notes it mentions that the 2nd term is a derivative of the first term but it doesn't really help me understand what's going on.
Thanks.
ordinary-differential-equations
Share
Share a link to this question
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this question to receive notifications
asked Dec 28, 2016 at 22:38
Ryan TandyRyan Tandy
453 6 6 silver badges 13 13 bronze badges
1
All integrating factor questions are the same. A simpler example might be to see that y′+x y=d d x(x y)y′+x y=d d x(x y)Kaynex –Kaynex 2016-12-28 22:47:02 +00:00 Commented Dec 28, 2016 at 22:47
Add a comment|
3 Answers 3
Sorted by: Reset to default
This answer is useful
2
Save this answer.
Show activity on this post.
We have a differential equation
a(x)y′+b(x)y=c(x)a(x)y′+b(x)y=c(x)
Divide through by a(x)a(x) to obtain
y′+p(x)y=q(x)y′+p(x)y=q(x)
Now imagine that we have a convenient term k(x)k(x) such that
k(x)y′+p(x)k(x)y=(k(x)y)′=k′(x)y+y′k(x)k(x)y′+p(x)k(x)y=(k(x)y)′=k′(x)y+y′k(x)
This yields a differential equation for k(x)k(x), i.e.
k′(x)=k(x)p(x)k′(x)=k(x)p(x)
A separable equation with solution k(x)=C e∫p(x)k(x)=C e∫p(x). Now, because of the way that we have constructed k(x)k(x), when we multiply y′+p(x)y=q(x)y′+p(x)y=q(x) through by k(x)k(x), we obtain
k(x)y′+k(x)p(x)y=(y k(x))′=q(x)k(x)y′+k(x)p(x)y=(y k(x))′=q(x)
Which can be solved for y y by integrating both sides and dividing by k(x)k(x). In your specific case p(x)=−3 x p(x)=−3 x and q(x)=x 3 cos x q(x)=x 3 cosx.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
edited Dec 29, 2016 at 1:10
answered Dec 28, 2016 at 22:46
GuPeGuPe
7,530 5 5 gold badges 37 37 silver badges 68 68 bronze badges
2
Well, explained, but I think p(x)p(x) is missing a minus sign.Mike –Mike 2016-12-29 00:29:11 +00:00 Commented Dec 29, 2016 at 0:29
@Mike good catch, had not noticed.GuPe –GuPe 2016-12-29 01:10:22 +00:00 Commented Dec 29, 2016 at 1:10
Add a comment|
This answer is useful
1
Save this answer.
Show activity on this post.
Use the product rule on 1 x 3 y 1 x 3 y and you'll get exactly the terms from part (a).
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
answered Dec 28, 2016 at 22:42
user402287user402287
204 2 2 silver badges 3 3 bronze badges
Add a comment|
This answer is useful
1
Save this answer.
Show activity on this post.
You can just check that by performing the differentiation in (b)(b) (the notation is kind of misleading):
d d x(1 x 3 y(x))=−3 x 4 y(x)+1 x 3 y′(x).d d x(1 x 3 y(x))=−3 x 4 y(x)+1 x 3 y′(x).
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
answered Dec 28, 2016 at 22:45
Hirek KubicaHirek Kubica
196 6 6 bronze badges
Add a comment|
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
ordinary-differential-equations
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Report this ad
Related
12What are integrating factors, really?
2integrating factors
4Separable Solutions vs Integrating Factors
0Find two integrating factors for this Differential Equation
7Why do Integrating Factors Work?
0How to solve this ODE using integrating factors and differential form
0Finding the general solution by integrating factors -- where am I going wrong (example problem included)?
0Methods of solving linear DEs
Hot Network Questions
Is existence always locational?
Identifying a movie where a man relives the same day
Does the mind blank spell prevent someone from creating a simulacrum of a creature using wish?
alignment in a table with custom separator
Any knowledge on biodegradable lubes, greases and degreasers and how they perform long term?
Can a cleric gain the intended benefit from the Extra Spell feat?
Determine which are P-cores/E-cores (Intel CPU)
What’s the usual way to apply for a Saudi business visa from the UAE?
Sign mismatch in overlap integral matrix elements of contracted GTFs between my code and Gaussian16 results
Should I let a player go because of their inability to handle setbacks?
What were "milk bars" in 1920s Japan?
What "real mistakes" exist in the Messier catalog?
Suspicious of theorem 36.2 in Munkres “Analysis on Manifolds”
My dissertation is wrong, but I already defended. How to remedy?
Is it possible that heinous sins result in a hellish life as a person, NOT always animal birth?
How long would it take for me to get all the items in Bongo Cat?
Bypassing C64's PETSCII to screen code mapping
Matthew 24:5 Many will come in my name!
How to use \zcref to get black text Equation?
Why include unadjusted estimates in a study when reporting adjusted estimates?
How many stars is possible to obtain in your savefile?
ICC in Hague not prosecuting an individual brought before them in a questionable manner?
How can the problem of a warlock with two spell slots be solved?
Why do universities push for high impact journal publications?
more hot questions
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Mathematics
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
13053 | https://vemu.org/uploads/lecture_notes/17_02_2023_252595088.pdf | Discrete Mathematics & Graph Theory (20A54304) Lecture Notes II – BTECH Prepared by Dr.K.R.Sekhar Department of Humanities & Sciences VEMU INSTITUTE OF TECHNOLOGY (Approved by AICTE, New Delhi and Affiliated to JNTUA, Ananthapuramu) Accredited by NAAC, NBA ( EEE , ECE & CSE) & ISO 9001-2015 Certified Institution Near Pakala. P. Kothakota, Chittoor-Tirupati Highway Chittoor, Andhra Pradesh -517112 Website: www.vemu.org R20 Regulations JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY ANANTAPUR (Established by Govt. of A.P., ACT No.30 of 2008) ANANTAPUR – 515 002 (A.P) INDIA Course Code Discrete Mathematics & Graph Theory L T P C 20A54304 3 0 0 3 Course Objectives To introduce the concepts of mathematical logic and gain knowledge in sets, relations and functions Solve problems using counting techniques and combinatorics To introduce generating functions and recurrence relations Use Graph Theory for solving real world problems Course outcomes (CO) : After completion of the course, the student can able to CO-1: Apply mathematical logic to solve problems. CO-2: Apply the conceptual background needed and identify structures of algebraic nature CO-3: Apply basic counting techniques to solve combinatorial problems. CO-4: Formulate problems and solve recurrence relations. CO-5: Apply Graph Theory in solving computer science problems Syllabus UNIT – I- Mathematical Logic Introduction, Statements and Notation, Connectives, Well-formed formulas, Tautology, Duality law, Equivalence, Implication, Normal Forms, Functionally complete set of connectives, Inference Theory of Statement Calculus, Predicate Calculus, Inference theory of Predicate Calculus. UNIT – II- Set theory Basic Concepts of Set Theory, Relations and Ordering, The Principle of Inclusion Exclusion, Pigeon hole principle and its application, Functions composition of functions, Inverse Functions, Recursive Functions, Lattices and its properties. Algebraic structures: Algebraic systems-Examples and General Properties, Semi groups and Monoids, groups, sub groups, homomorphism, Isomorphism. UNIT – III- Elementary Combinatorics Basics of Counting, Combinations and Permutations, Enumeration of Combinations and Permutations, Enumerating Combinations and Permutations with Repetitions, Enumerating Permutations with Constrained Repetitions, Binomial Coefficients, The Binomial and Multinomial Theorems. UNIT – IV- Recurrence Relations Generating Functions of Sequences, Calculating Coefficients of Generating Functions, Recurrence relations, Solving Recurrence Relations by Substitution and Generating functions, The Method of Characteristic roots, Solutions of Inhomogeneous Recurrence Relations. UNIT – V- Graphs Basic Concepts, Isomorphism and Subgraphs, Trees and their Properties, Spanning Trees, Directed Trees, Binary Trees, Planar Graphs, Euler’s Formula, Multigraphs and Euler Circuits, Hamiltonian Graphs, Chromatic Numbers, The Four Color Problem Textbooks: 1. Joe L. Mott, Abraham Kandel and Theodore P. Baker, Discrete Mathematics for Computer Scientists & Mathematicians, 2nd Edition, Pearson Education. 2. J.P. Tremblay and R. Manohar, Discrete Mathematical Structures with Applications to Computer Science, Tata McGraw Hill, 2002 Reference Books: 1. Kenneth H. Rosen, Discrete Mathematics and its Applications with Combinatorics and Graph Theory, 7th Edition, McGraw Hill Education (India) Private Limited. 2. Graph Theory with Applications to Engineering and Computer Science by Narsingh Deo. Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner |
13054 | https://web.pa.msu.edu/people/duxbury/courses/phy481/Fall2008/Lecture7.pdf | PHY481 - Lectures 7 and part of 8 Sections 3.5-3.6 of PS A. Finding the electric field - continued (vi) Uniform shell of charge (shell theorems) The shell theorems state that (i) the electric field inside a uniform shell of charge is zero and (ii) that the field outside the uniform shell of charge is the same as that of a point charge with the same total charge as the shell.
These results are easy to derive from Gauss’s law, in the same way as for the cylindrical case. However now we use spherical surfaces of radius r < R and r > R, where R is now the radius of the spherical shell of charge. The shell has charge density σ and the total charge on the shell is Q = 4πR2σ. We first consider a Gaussian surface consisting of a spherical shell, which is concentric with the charged shell, with radius r < R. Noting that by symmetry the electric field is directed radially and only depends on r, then since there is no enclosed charge the electric field for r < R is zero in agreement with the first shell theorem. Considering a spherical shell of radius r > R, we have, Z S ⃗ E · ⃗ dA = 4πR2E(r) = Q ǫ0 so that ⃗ E = 1 4πǫ0 Qˆ r r2 (1) which proves the second shell theorem.
Now lets prove the first shell theorem directly. Consider a general point ⃗ r inside the sphere. If we place a spherical co-ordinate system at that point, then we can draw a cone of infinitesimal solid angle dΩ= sinθdθdφ extending from this origin to the spherical surface.
The cone has two intersections with the spherical shell, on opposite sides of the sphere. The areas of the spherical shell subtended by this solid angle are dA1 = r2 1dΩand dA2 = r2 2dΩ.
The magnitude of the electric field at ⃗ r due to these two areas of charge is given by, dE = kσ(dA1 r2 1 −dA2 r2 2 ) = 0 (2) This construction applies to all parts of the spherical surface, proving the first shell theorem.
Proof of the second shell theorem is more involved.
It is also relatively straightforward to demonstrate the second shell theorem by direct integration, using polar co-ordinates it may be shown that the electric field in the radial direction for r > R where R is the radius of the shell, is given by, Er(r) = σ 4πǫ0 Z π 0 (r −Rcosθ)2πR2sinθdθ (R2sin2θ + (r −Rcosθ)2)3/2 (3) 1 which reduces to the point charge result.
in deriving this expresion we used dA = 2πR2sinθdθ (vii) Uniformly charged sphere of radius R Again we carry out the flux integral to find, for a surface of radius r, φE = E(r)4πr2 (4) The charge enclosed by this surface changes with r. For r < R, the enclosed charge is given by, q(r < R) = 4π 3 r3ρ (5) while for r > R we have, q(r > R) = 4π 3 R3ρ (6) Using Gauss’s law φE = q/ǫ0, we thus have, E(r < R) = rρ 3ǫ0 = Q 4πǫ0 r R3 (7) where Q = 4πρR3/3 is the total charge on the sphere. The electric field outside the sphere of charge is like that of a point charge (shell theorem), E(r > R) = 1 4πǫ0 Q r2 (8) In both cases, the direction of the electric field is ˆ r. (see problem 3.19) B. Electric potential energy and electric potential Physical definition The electric potential energy (U) is the potential energy due to the electrostatic force.
As always only differences in potential energy correspond to physical observables. However we define a reference potential energy and calculate all differences in potential energy with respect to this reference. In electrostatics, the potential energy is defined to be zero when the charges are an infinite distance apart.
The difference in potential energy in moving a charge between two positions a and b is defined in terms of the work done in moving the charge between these two positions, so that, ∆Uab = Z b a ⃗ Fext · ⃗ ds = − Z b a ⃗ F · ⃗ ds (9) 2 where ⃗ F is the Coulomb force on the charge and ⃗ ds is a small element of distance along the path we choose to take from a to b. Note that Eq. (19) does not depend on the path we take. All paths from a to b give the same change in potential energy. Note also that Eq.
(19) is true only for conservative forces, ie. systems in which there is no dissipation (e.g.
friction).
Instead of using the Coulomb force, we defined the electric field which is the force per unit charge. In the same way we define the electric potential, V , to be the potential energy per unit charge, ie. U = qV . The electric potential is so important it is given its own unit, the volt (V ). In terms of the electric potential, equation (19) is, ∆Vab == − Z b a ⃗ E · ⃗ ds (10) From this equation we see that the units of voltage are Nm/C. We have defined the volt to be the unit of voltage, so we have the relation V = Nm/C or V/m = N/C. So far we have used N/C as the units of electric field. This is usual in electrostatics. However in circuit theory it is usual to use volts/meter (V/m).
If we move a charge q from a point a to a point b, then the charge in potential energy, Uab, of the system is ∆Uab = − Z b a ⃗ F · ⃗ ds (11) - ⃗ F is the electric force on charge q due to all of the other charges in the system - ⃗ ds is a small displacement along the path we choose to take from a to b.
If we define the electric potential to be the potential energy per unit charge, so that V = U/q, then the difference in electric potential is given by, ∆Vab = 1 q∆Uab = − Z b a ⃗ E · ⃗ ds (12) We define the potential energy and hence the potential of a charge at infinity to be zero.
With the definition of potential given in Eq. (22) and taking the potential at infinity to be zero, we can calculate the potential at each position from the electric field.
3 |
13055 | https://www.youtube.com/watch?v=tRBB0cOelrY | A line segment DE is drawn parallel to base BC of A ABC which cuts AB at point D and AC at point...
Doubtnut
3940000 subscribers
7 likes
Description
472 views
Posted: 5 Feb 2022
A line segment DE is drawn parallel to base BC of A ABC which cuts AB at point D and AC at point E. If AB = 5 BD and EC = 3.2 cm, find the length of AE.
Class: 10
Subject: MATHS
Chapter: SIMILARITY (WITH APPLICATIONS TO MAPS AND MODELS)
Board:ICSE
You can ask any doubt from class 6-12, JEE, NEET, Teaching, SSC, Defense and Banking exam on Doubtnut App or You can Whatsapp us at - 8400400400
Link -
Join our courses to improve your performance and Clear your concepts from basic for Class 6-12 School and Competitive exams (JEE/NEET) -
Contact Us:
👉 Have Any Query? Ask Us.
🤙 Call: 01247158250
💬 WhatsApp: 8400400400
📧 Email: info@doubtnut.com
🌐 Website:
Welcome to Doubtnut.
Doubtnut is World’s Biggest Platform for Video Solutions of Physics, Chemistry, Maths and Biology Doubts with over 5 million+ Video Solutions. Doubtnut is a Q&A App for Maths, Physics, Chemistry and Biology (up to JEE Advanced and NEET Level), Where You Can Ask Unlimited Questions by Clicking a Picture of Doubt on the Doubtnut App and Get Instant Video Solution.
Subscribe Our YouTube Channels:
✿ Doubtnut:
✿ Class 11-12, JEE & NEET (Hindi):
✿ Class 11-12, JEE & NEET (English)::
✿ Class 6-10 (Hindi):
✿ Class 6-10 (English):
✿ Doubtnut Govt. Exams:
Follow Us:
🔔 Facebook:
🔔 Instagram:
🔔 Telegram:
🔔 Twitter:
🔔 LinkedIn:
Our Telegram Pages:
🔔 Doubtnut Official:
🔔 Doubtnut IIT JEE:
🔔 Doubtnut NEET:
🔔 Doubtnut CBSE Boards:
🔔 Doubtnut UP Boards:
🔔 Doubtnut Bihar Boards:
🔔 Doubtnut Government Exams:
class 10 maths
class 10
class 10 science
class 10 english
cbse class 10
class 10 chemistry
class 10 english grammar
class 10 biology
class 10 physics
class 10 ncert
sst class 10
class 10 geography
class 10 syllabus cbse
class 10 history
class 10 syllabus
cbse class 10 maths
cbse class 10 board exam
class 10 syllabus english
class 10 syllabus of science
class 10 syllabus science
class 10 syllabus maths
cbse class 10 exam date 2021
cbse class 10 sample paper 2021
cbse class 10 science
cbse class 10 english
class 10 board exam 2021
cbse class 10 board
cbse class 10 physics
cbse class 10 chemistry
cbse class 10 biology
cbse class 10 english grammar
cbse class 10 2021
class 10 syllabus 2021
class 10 syllabus ncert
class 10 syllabus of english
class 10 syllabus of maths
class 10 syllabus physics
class 10 syllabus sst 2021
1 comments
Transcript:
without need get instant video solutions to all your maths physics chemistry and biology doubts just click the image of the question crop the question and get instant video solution download doubt and app today the question is a line segment d each drone parallel to base bc of abc which cuts a b at point d and ac at point e if a b is equal to 5 b d and e c is equal to 3.2 centimeter find the length of a in this question we have given a b is equal to 5 b d by subtracting b d from both sides we get a b minus b d is equal to 4 bd or we can write a d is equal to 4 bd because a b minus b d is equal to a d or a d upon a b is equal to 4 upon 5 now in triangle a d e and triangle abc angle dae is equal to angle bac because angle a is common in both the triangle and angle ade is equal to angle abc because they both are corresponding angles and angle aed is equal to angle a c b because they also both are corresponding angle so triangle ade is similar to triangle abc by a a a rule thus a d upon a b is equal to a e upon ac and we can write ac as a e plus e c so a d upon a b is equal to a e upon a e plus ec by putting values we get 4 upon 5 is equal to ae upon ae plus 3.2 by further solving this we get 5 ae is equal to 4 ae plus 3.2 into 4 or by further solving this we get ae is equal to 12.8 centimeter which is our answer for class 6-12 itj and neet level trusted by more than five crore students download doubt and app today |
13056 | https://www.jlrivet.com/blog/how-strong-are-rivets | How Strong are Rivets - JiLiang
Home
Products
Stainless Steel Rivets
Brass Rivet
Aluminum Rivets
Stainless Steel Screws and Nuts
About Us
Custom
Blog
FAQs
Video
Contact Us
Home
Blog
How Strong are Rivets
Rivets are simple, reliable fasteners whose load‐bearing capacity depends chiefly on three factors:
Material (aluminum, steel, stainless steel, Monel, etc.)
Diameter of the rivet shank
Type (solid vs. blind/pop rivet)
Below is a rough guide to the typical shear and tensile capacities of common solid rivets:
| Rivet Material | Diameter (in) | Shear Strength (lbf) | Tensile Strength (lbf) |
--- --- |
| 2024-T4 Aluminum | 1/8″ (3.2 mm) | ~1,200 lbf | ~1,600 lbf |
| 2024-T4 Aluminum | 3/16″ (4.8 mm) | ~2,700 lbf | ~3,200 lbf |
| Mild Steel (1020) | 1/8″ | ~3,000 lbf | ~4,200 lbf |
| Mild Steel (1020) | 3/16″ | ~6,700 lbf | ~9,300 lbf |
| 304 Stainless Steel | 1/8″ | ~3,500 lbf | ~5,000 lbf |
| Monel | 1/8″ | ~4,500 lbf | ~6,500 lbf |
Shear strength is the force required to slide the joined plates apart along the plane of the rivet.
Tensile strength is the force pulling the rivet straight out of the hole.
How to estimate rivet capacity
Calculate cross-sectional area of the shank:
A=π d 2 4 A = \frac{\pi d^2}{4}A=4 π d 2
2. Multiply by the material’s allowable stress.
For aluminum rivets, a safe design shear stress is around 12–14 ksi (8.3–9.6 kg/mm²).
For steel rivets, use about 30 ksi (21 kg/mm²).
Example for a 1/8″ (0.125 in) steel rivet in shear:
A=π×0.125 2 4≈0.0123 i n 2,F s h e a r=0.0123×30,000≈370 l b f A = \frac{\pi \times 0.125^2}{4} \approx 0.0123\ \mathrm{in}^2, \quad F_{\rm shear} = 0.0123 \times 30{,}000 \approx 370\ \mathrm{lbf}A=4 π×0.12 5 2≈0.0123 in 2,F shear=0.0123×30,000≈370 lbf
(Note: this theoretical value is lower than the empirical ~3,000 lbf because design stresses are conservative.)
Factors that reduce real‐world strength
Hole quality: oversize or out-of-round holes reduce grip.
Proximity to edges: rivets too close to sheet edges can tear out.
Installation quality: insufficient bucking, improper stem pull, or high setting speed can induce cracks or incomplete forming.
Fatigue loading: cyclic loads significantly lower allowable working stress.
Blind (pop) rivets vs. solid rivets
Blind rivets (e.g. aluminum body with steel mandrel) tend to have lower capacities. A typical 1/8″ aluminum pop rivet might only manage ~500 lbf in shear, compared to ~1,200 lbf for a solid 2024-T4 rivet of the same diameter.
In Practice:
Always consult the rivet manufacturer’s datasheet for exact “ultimate” and “working” loads.
Use a suitable safety factor (often 2–4×) on the working load for structural applications.
For critical joints, consider proof‐load or pull‐off testing of a sample rivet installation.
By understanding material limits, geometry, and installation quality, you can select rivets that comfortably exceed the loads your assembly will see.
Prev： What is a countersunk rivet
Next： How to Remove a Rivet from Metal:5 Ways
Related Blogs
How Strong are Rivets
Hammer Drive Rivets:What are hammer rivets?
How to Remove a Rivet from Metal:5 Ways
How strong are double cap rivets
What is a blind rivet
Hammer Drive Rivets Sizes
What is a countersunk rivet
China Jiliang Machinery Hardware Co. , Ltd.
GET A QUOTE
Our Products
>Stainless Steel Rivets>Brass Rivet>Aluminum Rivets>Stainless Steel Screws and Nuts>Machine Screw>Semi Tubular Rivets>Round Head Rivet>Flat Head Rivet>Truss Head Rivet
CONTACT US
Address:No.8,Third Road,Houjie Qiaotou Third Industrial Zone,Houjie Town,Dongguan City
+8616676517879whatsapp: +8616676517879sales5@jiliang988.com
All Rights Reserved © 2025 Jiliang Machinery Hardware Co. , Ltd.
GET A QUOTE
GET IN TOUCH NOW
Send Message
× |
13057 | https://www.joeteacher.org/uploads/7/6/3/0/7630382/cmos_part_2_-_grammar_and_usage.pdf | 5 • Grammar and Usage BRYAN A. GARNER GRAMMAR 5.1 Introduction 5.1 Nouns 5.4 Traditional Classifications 5.4 Properties of Nouns 5.8 Plurals 5.13 Case 5.17 Appositives 5.23 Functional Variations 5.24 Pronouns 5.27 Definition and Uses 5.27 Properties of Pronouns 5.31 Classes of Pronouns 5.38 Personal Pronouns 5.39 Possessive Pronouns 5.49 Reflexive and Intensive Pronouns 5.51 Demonstrative Pronouns 5.52 Reciprocal Pronouns 5.53 Interrogative Pronouns 5.54 Relative Pronouns 5.56 Indefinite Pronouns 5.67 Adjectives 5.68 Types of Adjectives 5.68 Articles as Limiting Adjectives 5.70 Position of Adjectives 5.79 Degrees of Adjectives 5.84 Special Types of Adjectives 5.90 Verbs 5.97 Definitions 5.97 Infinitives 5.106 Participles and Gerunds 5.110 Voice, Mood, Tense, Person, and Number 5.117 Auxiliary Verbs 5.144 “Be”-Verbs 5.154 Adverbs 5.156 Definition and Formation 5.156 Simple versus Compound Adverbs 5.160 Adverbial Degrees 5.162 Position of Adverbs 5.167 Prepositions 5.172 Definition and Types 5.172 Prepositional Phrases 5.176 Other Prepositional Issues 5.184 Limiting Prepositional Phrases 5.187 Prepositional Idioms 5.193 Conjunctions 5.196 Interjections 5.206 SYNTAX 5.210 The Four Traditional Types of Sentence Structures 5.217 English Sentence Patterns 5.221 Clauses 5.225 Ellipsis 5.229 Negation 5.230 Expletives 5.239 Parallel Structure 5.242 Cleft Sentences 5.246 WORD USAGE 5.249 Glossary of Problematic Words and Phrases 5.250 Bias-Free Language 5.251 Grammar 5.3 Grammar Introduction 5.1 The field of grammar. In its usual sense, grammar is the set of rules governing how words are put together in sentences to communicate ideas—or the study of these rules. Native speakers of a language learn them unconsciously. The rules govern most constructions in any given language. The small minority of constructions that lie outside these rules fall mostly into the category of idiom and customary usage.
5.2 Schools of grammatical thought. There are many schools of grammat ical thought—and differing vocabularies for describing grammar. Gram matical theories have been in upheaval in recent years. It seems that the more we learn, the less we know. As the illustrious editor in chief of the Oxford English Dictionary wrote in 1991: “An entirely adequate descrip tion of English grammar is still a distant target and at present seemingly an unreachable one, the complications being what they are.”1 In fact, the more detailed the grammar (it can run to many large volumes), the less likely it is to be of any practical use to most writers and speakers.
5.3 Parts of speech. As traditionally understood, grammar is both a science and an art. Often it has focused—as it does here—on parts of speech and their syntax. Each part of speech performs a particular function in a sen tence or phrase. Traditional grammar has held that there are eight parts of speech: nouns, pronouns, adjectives, verbs, adverbs, prepositions, conjunctions, and interjections.2 Somewhat surprisingly, modern gram marians cannot agree on precisely how many parts of speech there are in English. At least one grammarian says there are as few as three.3 Another insists that there are “about fifteen,” noting that “the precise number is still being debated.”4 This section deals with the traditional eight; each part of speech is treated below. The purpose here is to sketch some of the main lines of English grammar using traditional grammatical terms.
1. Robert W. Burchfield, Unlocking the English Language (New York: Hill and Wang, 1991), 22.
2. See Robert L. Allen, English Grammars and English Grammar (New York: Scribner, 1972), 7.
3. Ernest W. Gray, A Brief Grammar of Modern Written English (Cleveland: World, 1967), 70.
4. R. L. Trask, Language: The Basics (London: Routledge, 1995), 37.
5.4 GRAMMAR AND USAGE Nouns TRADITIONAL CLASSIFICATIONS 5.4 Nouns generally. A noun is a word that names something, whether ab stract (intangible) or concrete (tangible). It may be a common noun (the name of a generic class or type of person, place, thing, process, activity, or condition) or a proper noun (the name of a specific person, place, or thing—hence capitalized). A concrete noun may be a count noun (if what it names can be counted—as with horses or cars) or a mass noun (if what it names is uncountable or collective—as with information or salt).
5.5 Common nouns. A common noun is the generic name of one item in a class or group {a chemical} {a river} {a pineapple}. It is not capitalized un less it begins a sentence or appears in a title. A common noun is usually used with a determiner—that is, an article or other word (e.g., some, few) that indicates the number and definiteness of the noun element {a loaf} {the day} {some person}. Common nouns may be analyzed into three subcategories: concrete nouns, abstract nouns, and collective nouns. A concrete noun is solid or real; it indicates something perceptible to the physical senses {a building} {the wind} {honey}. An abstract noun de notes something you cannot physically see, touch, taste, hear, or smell {joy} {expectation} {neurosis}. A collective noun—which can be viewed as a concrete noun but is often separately categorized—refers to a group or collection of people or things {a crowd of people} {a flock of birds} {a herd of rhinos}.
5.6 Proper nouns. A proper noun is the name of a specific person, place, or thing {John Doe} {Moscow} {the Hope Diamond}, or the title of a work {Citizen Kane}. Proper nouns may be singular {Mary} {London} or plural {the Great Lakes} {the Twin Cities}. A proper noun is always capitalized, regardless of how it is used—unless someone is purposely flouting the rules {k.d. lang}. A common noun may become a proper noun {Old Hick ory} {the Big Easy}, and sometimes a proper noun may be used figura tively and informally, as if it were a common noun {like Moriarty, he is a Napoleon of crime [Napoleon here connotes an ingenious mastermind who is ambitious beyond limits]}. Proper nouns may be compounded when used as a unit to name something {the Waldorf-Astoria Hotel} {Sat urday Evening Post}. Over time, some proper nouns (called eponyms) have developed common-noun counterparts, such as sandwich (from the Earl of Sandwich) and china (the porcelain, from the nation China). Articles Grammar 5.9 and other determiners are used with proper nouns only when part of the noun is a common noun or the determiner provides emphasis {the Savoy Hotel} {Sam? I knew a Sam Hill once}.
5.7 Mass nouns. A mass noun (sometimes called a noncount noun) is one that denotes something uncountable, either because it is abstract {cow ardice} {evidence} or because it refers to an aggregation of people or things taken as an indeterminate whole {luggage} {the bourgeoisie}. The key difference between mass nouns and collective nouns is that unlike collective nouns (which are count nouns), mass nouns almost never take indefinite articles and typically do not have plural forms (a team, but not an evidence; two groups, but not two luggages). A mass noun can stand alone {music is more popular than ever} or with a determiner other than an indefinite article (some music or the music but generally not a music). As the subject of a sentence, a mass noun typically takes a singular verb and pronoun {the litigation is so varied that it defies simple explana tion}. Some mass nouns, however, are plural in form but are treated as grammatically singular {politics} {ethics} {physics} {news}. Others are always grammatically plural {manners} {scissors} {clothes}. But just as singular mass nouns don’t take an indefinite article, plural mass nouns don’t combine with numbers: you’d never say three scissors or six man ners. Some that refer to concrete objects, such as scissors or sunglasses, can be enumerated by adding pair of {a pair of scissors} {three pairs of sunglasses}. Likewise, singular concrete mass nouns can usually be enu merated by adding a unit noun such as piece (with of) {a piece of cutlery} {seven pieces of stationery}. Both singular and plural mass nouns can take indefinite adjectives such as any, less, much, and some that express general quantity {what you need is some courage} {he doesn’t have any manners}.
PROPERTIES OF NOUNS 5.8 Properties of nouns. Nouns have properties of case and number. Some traditional grammarians also consider gender and person to be proper ties of nouns.
5.9 Noun case. In English, only nouns and pronouns have case. Case denotes the relationship between a noun (or pronoun) and other words in a sen tence. Grammarians disagree about the number of cases English nouns possess. Those who consider inflection (word form) the defining charac teristic tend to say that there are two: common, which is the uninflected 5.10 GRAMMAR AND USAGE form, and genitive (or possessive), which is formed by adding - s or just an apostrophe. But others argue that it’s useful to distinguish how the common-case noun is being used in the sentence, whether it is playing a nominative role {the doctor is in} or an objective role {go see the doctor}. They also argue that the label we put on nouns according to their func tion should match those we use for who and for personal pronouns, most of which do change form in the nominative and objective cases (who/ whom, she/her, etc.). See also 5.17-22.
5.10 Noun number. Number shows whether one object or more than one ob ject is referred to, as with clock (singular) and clocks (plural).
5.11 Noun gender. English nouns have no true gender, as that property is un derstood in many other languages. For example, whether a noun refers to a masculine or feminine person or thing does not determine the form of the accompanying article as it does in French, German, Spanish, and many other languages. Still, some English words—almost exclusively nouns denoting people or animals—are inherently masculine {uncle} {rooster} {lad} or feminine {aunt} {hen} {lass} and take the gender- appropriate pronouns. But most English nouns are common in gender and may refer to either sex {relative} {chicken} {child}. Many words once considered strictly masculine—especially words associated with jobs and professions—have been accepted as common (or indefinite) in gen der over time {author} {executor} {proprietor}. Similarly, many forms made feminine by the addition of a suffix {aviatrix} {poetess} have been essentially abandoned. See 5.251-60.
5.12 Noun person. A few grammarians attribute the property of person to nouns, distinguishing first person {I, Dan Walls, do swear that . . .}, second person {you, the professor, are key}, and third person {she, the arbiter, decides}. While those examples all use nouns in apposition to pronouns, that’s not closely relevant to the question whether the nouns themselves have the property of person in any grammatical sense. But using that property in analyzing nouns does help to point out three things. First, as with grammatical case, one argument for the property of person is to keep the properties of nouns parallel to those of pronouns, even though English nouns do not change form at all in first, second, or third person as personal pronouns do. Second, person determines what form other words will take—here, the verbs. Third, the examples illus trate why attributing person to nouns requires a stretch of logic—if the pronouns were not present in the first two examples, the verb would be in the third person, even if Dan Walls were talking about himself and even if the speaker were addressing the professor.
Grammar 5.16 PLURALS 5.13 Plurals generally. Because exceptions abound, a good dictionary or us age guide is essential for checking the standard plural form of a noun. But there are some basic rules for forming plurals, some of which are covered in chapter 7 (see 7.5-15). A few matters of agreement are treated in the paragraphs that follow.
5.14 Plural form with singular sense. Some nouns are plural in form but sin gular in use and meaning {good news is always welcome} {economics is a challenging subject} {measles is potentially deadly}. Also, a plu ral word used as a word is treated as a singular {“mice” is the plural of “mouse”} {“sistren” is an archaic plural}. Some traditional plurals, such as data and (to a lesser extent) media, have gradually acquired a mass noun sense and are increasingly treated as singular. Although tradition alists stick to the plural uses {the data are inconclusive} {the media are largely misreporting the event}, the new singular uses—using the terms in a collective sense rather than as count nouns—exist alongside the older ones {the data shows the hypothesis to be correct} {the media isn’t infallible}. (In the sciences, data is always plural.) In formal contexts, the most reliable approach is to retain the plural uses unless doing so makes you feel as if you’re being artificial, stuffy, and pedantic. Consider using alternative words, such as information and journalists. Or simply choose the newer usage. But make your play and be consistent—vacillating will not win the admiration of readers and listeners.
5.15 Plural-form proper nouns. A plural geographical name is often treated as singular when the name refers to a single entity {the United States is a relatively young nation} {Naples is a very beautiful city}. But there are many exceptions {the Alps have never been totally impassable}. Names of companies, institutions, and similar entities are generally treated as collective nouns—and hence singular in American English, even when they are plural in form {General Motors reports that it will earn a profit} {American Airlines has moved its headquarters}. In British En glish, however, singular nouns that refer to individuals who work inde pendently typically take plural verbs {Manchester United have won the FIFA Cup} {England are now leading in World Cup standings}.
5.16 Tricky anomalies of the plural. Not all English nouns show the usual singular-plural dichotomy. For example, mass nouns such as furniture, spaghetti, and wheat have only a singular form, and oats, scissors, and slacks (= pants) exist only as plurals. Some nouns look singular but are invariably plural {the police were just around the corner} {the vermin 5.17 GRAMMAR AND USAGE seem impossible to eradicate}. Others look plural but are invariably sin gular {the news is good} {linguistics is my major}. Strangely enough, person forms two plurals—persons and people—but people also forms the plural peoples {the peoples of the world}.
CASE 5.17 Function of case. Case denotes the relationship between a noun or pro noun and other words in a sentence.
5.18 Common case, nominative function. The nominative (sometimes called the subjective) function denotes the person, place, or thing about which an assertion in a clause is made {the governor delivered a speech [gover nor is the subject]} {the shops are crowded because the holiday season has begun [shops and season are the subjects of their respective clauses]}. A noun serving a nominative function controls the verb and usually precedes it {the troops retreated in winter [troops is the subject]}, but through inversion it can appear almost anywhere in the sentence {high up in the tree sat a leopard [leopard is the subject]}. A noun or pronoun that follows a be-verb and refers to the same thing as the subject is called apredicate nominative {my show dogs are Australian shepherds [Austra lian shepherds is a predicate nominative]}. Generally, a sentence’s pred icate is the part that contains a verb and makes an assertion about the subject.
5.19 Common case, objective function. The objective (sometimes called the accusative) function denotes either (1) the person or thing acted on by a transitive verb in the active voice {the balloon carried a pilot and a passenger [pilot and passenger are objective: the direct objects of the verb carried]} or (2) the person or thing related to another element by a connective, such as a preposition {place the slide under the microscope [microscope is objective: the object of the preposition under]}. A noun in an objective function usually follows the verb {the queen consulted the prime minister [queen is nominative prime minister is objective]}. But with an inverted construction, the object can appear elsewhere in the sentence {everything else was returned; the jewelry the thieves had already sold [jewelry is objective and thieves is nominative]}. A noun serv ing an objective function is never the subject of the following verb and usually does not control the number of the verb {an assembly of strang ers was outside [the plural noun strangers is the object of the preposition oft the singular noun assembly is the subject of the sentence, so the verb was must also be singular]}.
Grammar 5.22 5.20 Genitive case. The genitive case denotes (1) ownership, possession, or occupancy {the architect’s drawing board} {Arnie’s room}; (2) a rela tionship {the philanthropist’s secretary}; (3) agency {the company’s representative}; (4) description {a summer’s day}; (5) the role of a sub ject {the boy’s application [the boy applied]}; (6) the role of an object {the prisoner’s release [someone released the prisoner]}; or (7) an id iomatic shorthand form of an ©/phrase {one hour’s delay [equal to a delay of one hour]}. The genitive case is also called the possessive case, but possessive is a misleadingly narrow term, given the seven different func tions of this case—true possession, as ordinarily understood, being only one. For instance, the fourth function above is often called the descrip tive possessive. This is a misnomer, however, because the form doesn’t express actual possession but instead indicates that the noun is function ing as a descriptive adjective. The genitive is formed in different ways, depending on the noun or nouns and their use in a sentence. The geni tive of a singular noun is formed by adding -’s {driver’s seat} {engineer’s opinion}. The genitive of a plural noun that ends in -s or -es is formed by adding an apostrophe {parents’ house} {foxes’ den}. The genitive of an irregular plural noun is formed by adding -’s {women’s rights} {mice’s cage}. The genitive of a compound noun is formed by adding the appro priate ending to the last word in the compound {parents-in-law’s mes sage}. All these -’$ and -s' endings are called inflected genitives. See also 7.16-29.
5.21 The "of"-genitive. The preposition ©/may precede a noun or proper name to express relationship, agency, or possession. The choice be tween an inflected genitive and an ©/-construction depends mostly on style. Proper nouns and nouns denoting people or things of higher sta tus usually take the inflected genitive {Hilda’s adventures} {the lion’s paw}. Compare the perils of Penelope with the saucer of the chef. Nouns de noting inanimate things can often readily take either the inflected form or the o/genitive {the theater’s name} {the name of the theater}, but some sound right only in the ©/genitive {the end of everything}. The ©/- genitive is also useful when a double genitive is called for—using both of and a possessive form {an idea of Hill’s} {a friend of my grandfather’s} (see also 7.26).
5.22 Joint and separate genitives. If two or more nouns share possession, the last noun takes the genitive ending. (This is called joint or group possession.) For example, Peter and Harriet's correspondence refers to the correspondence between Peter and Harriet. If two or more nouns possess something separately, each noun takes its own genitive end ing. For example, Peter's and Harriet's correspondence refers to Peter’s 5.23 GRAMMAR AND USAGE correspondence and also to Harriet’s correspondence, presumably with all sorts of people. Joint possession is shown by a single apostrophe plus -5 only when two nouns are used. If a noun and a pronoun are used to ex press joint possession, both the noun and the pronoun must show posses sion. For example, Hilda and Eddie’s vacation becomes (when Eddie has already been mentioned) Hilda’s and his vacation or (if Eddie is speaking in first person) Hilda’s and my vacation.
APPOSITIVES 5.23 Appositives—definition and use. An appositive is a noun element that immediately, follows another noun element in order to define or further identify it {George Washington, our first president, was born in Virginia [our first president is an appositive of the proper noun George Washing ton]}. An appositive is said to be “in apposition” with the word or phrase to which it refers. Commas frame an appositive unless it is restrictive {Robert Burns, the poet, wrote many songs about women named Mary [here, poet is a nonrestrictive appositive noun]} {the poet Robert Burns wrote many songs about women named Mary [Robert Burns restricts poet by precisely identifying which poet]}. A restrictive appositive cannot be removed from a sentence without obscuring the identity of the word or phrase that the appositive relates to. See also 6.28.
FUNCTIONAL VARIATIONS 5.24 Nouns as adjectives. Words that are ordinarily nouns sometimes func tion as other parts of speech, such as adjectives or verbs. A noun-to- adjective transition takes place when a noun modifies another noun {the morning newspaper} {a state legislature} {a varsity sport} (morning, state, and varsity function as adjectives). These are also termed attrib utive nouns. Note that they are typically singular in form (car dealership, not cars dealership). Occasionally the use of a noun as an adjective can produce ambiguity. For example, the phrase fast results can be read as meaning either “rapid results” or (less probably but possibly) “the out come of a fast.” Sometimes the noun and its adjectival form can be used interchangeably—e.g., prostate cancer and prostatic cancer both refer to cancer of the prostate gland. But sometimes the use of the noun instead of the adjective may alter the meaning—e.g., a study group is not neces sarily a studious group. A preposition may be needed to indicate a noun’s relationship to other sentence elements. But if the noun functions as an Grammar 5.27 adjective, the preposition must be omitted; at times this can result in a vague phrase—e.g., voter awareness (awareness of voters or by them?). Context might suggest what preposition is implied, but a reader may have to deduce the writer’s meaning.
5.25 Nouns as verbs. English nouns commonly pass into use as verbs; it has always been so. (The resulting verbs are called denominal verbs.) For ex ample, in 1220 the noun husband meant “one who tills and cultivates the earth” {the husband has worked hard to produce this crop}. It became a verb meaning “to till, cultivate, and tend crops” around 1420 {you must husband your land thoughtfully}. New noun-to-verb transitions often occur in dialect or jargon. For example, the noun mainstream is used as a verb in passages such as more school districts are mainstreaming pupils with special needs. In formal prose, such recently transformed words should be used cautiously if at all.
5.26 Adverbial functions. Words that are ordinarily nouns occasionally func tion as adverbs {we rode single file} {Sam walked home}. This shift usually happens when a preposition is omitted {we rode in a single file} {Sam walked to his home}. Traditional grammarians have typically called such nouns-as-adverbs adverbial objectives. An adverbial objective often modifies an adjective rather than a verb {the team is four mem bers strong}.
Pronouns DEFINITION AND USES 5.27 Pronouns defined. A pronoun is a word used as a substitute for a noun or, sometimes, another pronoun. It is used in one of two ways. (1) A pronoun may substitute for an expressed noun or pronoun, especially to avoid needless repetition. For example, most of the nouns in the sentence The father told the father's daughter that the father wanted the father's daughter to do some chores can be replaced with pronouns (his, he, and her): The father told his daughter that he wanted her to do some chores. (2) A pronoun may also stand in the place of an understood noun. For example, if the person addressed has been identified elsewhere, the question Susan, are you bringing your boots? can be more simply stated as Are you bringing your boots? And in the sentence It is too hot, the indefinite it is understood to mean the temperature (of something). There are also a few word pairs, 5.28 GRAMMAR AND USAGE such as each other, one another, and no one, that function as pronouns. These are called phrasal pronouns.
5.28 Antecedents of pronouns. A pronoun typically refers to an antecedent— that is, an earlier noun, pronoun, phrase, or clause in the same or in a previous sentence. Pronouns with antecedents are called anaphoric pro nouns. (Anaphora refers to the use of a word or phrase to refer to or re place one used earlier.) An antecedent may be explicit or implicit, but it should be clear. Miscues and ambiguity commonly arise from (1) a missing antecedent (as in The clowns act with his dog made it a pleasure to watch, where it is intended to refer to the circus, which is not explicitly mentioned in the context); (2) multiple possible antecedents (as in Scott visited Eric after his discharge from the army, where it is unclear who was discharged—Eric or Scott); and (3) multiple pronouns and antecedents in the same sentence (e.g., When the bottle is empty or the baby stops drinking, it must be sterilized with hot water because if it drinks from a dirty bottle, it could become ill—where one hopes that the hot-water sterilization is for the bottle).
5.29 Adjective as antecedent. A pronoun normally requires a noun or an other pronoun as its antecedent. And because possessives function as adjectives, some writers have argued that possessives should not serve as antecedents of pronouns used in the nominative or objective case. But compare Mr. Blain's background qualified him for the job with Mr. Blain had a background that qualified him for the job. Not only is the identity of “him” perfectly clear in either construction, but the possessive in the first—a usage blessed by respected authorities—makes for a more eco nomical sentence; 5.30 Pronouns without antecedents. Some pronouns do not require anteced ents. The first-person pronouns I and we (as well as me and us) stand for the speaker or a group that includes the speaker, so they almost never have an antecedent. Similarly, the second-person pronoun you usually needs no antecedent {are you leaving?}, although one is sometimes sup plied in direct address {Katrina, do you need something?}. Expletives such as there and it (some of which are pronouns) have no antecedents {it is time to go} {this is a fine mess} (see 5.239-41). And the relative pro noun what and the interrogative pronouns (who, which, what) never take an antecedent {who cares what I think?}. In colloquial usage, they often appears without an antecedent {they say she’s a good golfer}, though skeptical listeners and readers may want to know who “they” are.
Grammar 5.33 PROPERTIES OF PRONOUNS 5.31 Four properties of pronouns. A pronoun has four properties: number, person, gender, and case (see 5.17-22). A pronoun must agree with its antecedent in number, person, and gender. (This is called pronoun antecedent agreement.) But only the third-person singular (he, she, it) is capable of indicating all three. Some pronouns can show only number— first-person singular and plural (I, we) and third-person plural. The second-person pronoun (you) indicates person only: it is no longer ca pable of showing singular or plural, since the form is the same for both in Modern English. First- and third-person personal pronouns (except it), who, and whoever can show nominative and objective case (I, me; we, us; he, him; she, her; they, them; who, whom; whoever, whomever); possessive pronouns represent the genitive case.
5.32 Pronoun number and antecedent. A pronoun’s number is guided by that of its antecedent or referent—that is, a singular antecedent takes a singular pronoun of the same person as the antecedent, and a plural an tecedent takes a plural pronoun of the same person as the antecedent {a book and its cover} {the dogs and their owner}. A collective noun takes a singular pronoun if the members are treated as a unit {the audience showed its appreciation} but a plural if they act individually {the audi ence rushed back to their seats}. A singular noun that is modified by two or more adjectives to denote different varieties, uses, or aspects of the object may take a plural pronoun {British and American writing differ in more ways than just their spelling [here, writing may be thought of as an elided noun after British]}. Two or more singular nouns or pronouns that are joined by and are taken jointly and referred to by a plural pro noun {the boy and girl left their bicycles outside}.
5.33 Exceptions regarding pronoun number and antecedent. There are several refinements to the rules stated in 5.32: (1) When two or more singular antecedents denote the same thing and are connected by and, the pronoun referring to the antecedents is singular {a lawyer and role model received her richly deserved recognition today}. (2) When two or more singular antecedents are connected by and and modified by each, every, or no, the pronoun referring to the antecedents is singular {every college and university encourages its students to succeed}. (3) When two or more singular antecedents are connected by or, nor, either-or, or neither-nor, they are treated separately and referred to by a singular pro noun {neither the orange nor the peach smells as sweet as it should}.
(4) When two or more antecedents of different numbers are connected 5.34 GRAMMAR AND USAGE by or or nor, the pronoun’s number agrees with that of the nearest (usu ally the last) antecedent. If possible, cast the sentence so that the plural antecedent comes last {neither the singer nor the dancers have asked for their paychecks}. (5) When two or more antecedents of different numbers are connected by and, they are referred to by a plural pronoun regardless of the antecedents’ order {the horses and the mule kicked over their water trough}.
5.34 Pronoun with multiple antecedents. When a pronoun has two or more antecedents that differ from the pronoun in person, and the antecedents are connected by and, or, or nor, the pronoun must take the person of only one antecedent. The first person is preferred to the second, and the second person to the third. For example, if the antecedents are in the sec ond and first person, the pronoun that follows is in the first person {you or I should get to work on our experiment [our is in the first person, as is the antecedent 1]}. If the antecedents are in the second and third person, the pronoun that follows takes the second person {you and she can settle your dispute}. If the pronoun refers to only one of the connected nouns or pronouns, it takes the person of that noun {you and Marian have dis cussed her trip report}. At times the pronoun may refer to an antecedent that is not expressed in the same sentence; it takes the number of that antecedent, not of any connected noun or pronoun that precedes it {nei ther they nor I could do his work [his is referring to someone named in a preceding sentence]}.
5.35 Pronoun case. Sets of word forms by which a language differentiates the functions that a word performs in a sentence are called the word’s cases. A pronoun that functions as the subject of a finite verb is in the nomina tive case {they went to town}. A personal pronoun in the possessive case is governed by the gender of the possessor {President Barack Obama took his advisers with him to Hawaii}. A pronoun that functions as the object of a verb or preposition is in the objective case {they gave her a farewell party} {they gave it to him}. A pronoun put after an intransitive verb or participle agrees in case with the preceding noun or pronoun re ferring to the same thing {it is 1} (see 5.45). A pronoun used in an absolute construction is in the nominative: its case depends on no other word {she being disqualified, our best hope is gone}.
5.36 Pronouns in apposition. The case of a pronoun used in an appositive con struction is determined by the function (subject or object) of the words with which it is in apposition {we three—Bruce, Felipe, and I—traveled to Augusta} {she asked us—Barbara, Sarah, and me—to move our cars}.
Grammar 5.39 5.37 Nominative case misused for objective. The objective case governs personal pronouns used as direct objects of verbs {call me tomorrow}, indirect objects of verbs {write me a letter}, or objects of prepositions {makes sense to me}. One of the most persistent slips in English is to misuse the nominative case of a personal pronoun in a compound object: po o r : The test would be simple for you or I. b e t t e r : The test would be simple for you or me.
po o r : Read this and tell Laura and I what you think. b e t t e r : Read this and tell Laura and me what you think.
The mistake may arise from overcorrecting a common error that young children are prone to—using the objective case for a personal pronoun in a compound subject, as in Jim and me want to go swimming. Such prob lems arise in compounds so exclusively that the foolproof way to check for them is to read the sentence with the personal pronoun alone: no one would mistake The test would be simple for I or Read this and tell I what you think for correct grammar.
CLASSES OF PRONOUNS 5.38 Seven classes of pronouns. There are seven classes of pronouns (the examples listed here do not include all forms of each): personal (I,you, he, she, it, we, and they); demonstrative (that and this); reciprocal (each other and one another); interrogative (what, which, and who); relative (that, what, which, and who); indefinite (another, any, each, either, and none); and adjective (any, each, that, this, what, and which).
Many pronouns, except personal pronouns, may function as more than one type—e.g., that may be a demonstrative, relative, or adjective pronoun—depending on its use in a particular sentence.
PERSONAL PRONOUNS 5.39 Form of personal pronouns. A personal pronoun shows by its form whether it is referring to the speaker (first person), the person or thing 5.40 GRAMMAR AND USAGE spoken to (second person), or the person or thing spoken of (third per son). Personal pronouns, in other words, convey the source, goal, and topic of an utterance. By their form they also display number, gender, and case.
5.40 Identification of personal pronouns. The first person is the speaker or speakers {I need some tea} {we heard the news}. The second person shows who is spoken to {you should write that essay tonight}. And the third person shows who or what is spoken of {she is at work} {it is in the glove compartment}. The first-person-singular pronoun I is always capitalized no matter where it appears in the sentence {if possible, I will send you an answer today}. All other pronouns are capitalized only at the beginning of a sentence, unless they are part of an honorific title {Her Majesty, the Queen of England}.
5.41 Changes in form of personal pronouns. Personal pronouns change form (or decline) according to person, number, and case. Apart from the sec ond person, all personal pronouns show number by taking a singular or plural form. Although the second-person pronoun you is both singular and plural, it always takes a plural verb, even if only a single person or thing is addressed.
THE FORMS OF PERSONAL PRONOUNS Singular Pronouns Nominative Objective Genitive Reflexive First person I me my, mine myself Second person you you your, yours yourself Third person he him his himself she her her, hers herself it it its itself Plural Pronouns First person we us our, ours ourselves Second person you you your, yours yourselves Third person they them their, theirs themselves There are four essential rules about the nominative and objective cases. (1) If the pronoun is the subject of a clause, it is in the nominative case {he is vice president}. (2) If the pronoun is the object of a verb, it is objec tive {she thanked him}. (3) If a pronoun is the object of a preposition, it is objective {please keep this between you and me}. (4) If the pronoun is the subject of an infinitive, it is objective {Jim wanted her to sing}.
Grammar 5.46 5.42 Agreement of personal pronoun with noun. A personal pronoun agrees with the noun for which it stands in both gender and number {John writes, and he will soon write well} {Sheila was there, but she couldn’t hear what was said}.
5.43 Personal pronouns and gender. Only the third-person-singular pro nouns directly express gender. In the nominative or objective case, the pronoun takes the antecedent noun’s gender {the president is not in her office today; she’s at a seminar}. In the genitive case, the pronoun always takes the gender of the possessor, not of the person or thing possessed {the woman loves her husband} {Thomas is visiting his sister} {the kit ten pounced on its mother}. Some nouns may acquire gender through personification, a figure of speech that refers to a nonliving thing as if it were a person. Pronouns enhance personification when a feminine or masculine pronoun is used as if the antecedent represented a female or male person (as was traditionally done, for example, when a ship or other vessel was referred to with the pronoun she or her).
5.44 Personal pronoun case. Some special rules apply to personal pronouns. (1) If a pronoun is the subject of a clause, or follows a conjunction but precedes the verb, it must be in the nominative case {she owns a tan briefcase} {although Delia would like to travel, she can’t afford to}. (2) If a pronoun is the object of a verb or preposition, it must be in the objective case {the rustic setting helped him relax} {that’s a matter between him and her}. (3) If a prepositional phrase contains more than one object, all the objects must be in the objective case {will you send an invitation to him and me?}. (4) If a pronoun is the subject of an infinitive, it must be in the objective case {does Tina want me to leave?}.
5.45 Personal pronoun after linking verb. Strictly speaking, a pronoun serv ing as the complement of a fo-verb or other linking verb should be in the nominative case {it was she who asked for a meeting}. In that construc tion, she functions as a predicate nominative; when a pronoun does this, it is termed an attribute pronoun. The same construction occurs when someone who answers a telephone call is asked, “May I speak to [an swerer’s name] ?” The refined response is This is he, not This is him.
5.46 Personal pronoun after "than" or "as-as." The case of a pronoun fol lowing a comparative construction, typically at the end of a sentence, depends on who or what is being compared. In My sister looks more like our father than I [or me], for example, the proper pronoun depends on the meaning. If the question is whether the sister or the speaker looks 5.47 GRAMMAR AND USAGE more like their father, the pronoun should be nominative because it is the subject of an understood verb {my sister looks more like our father than I do}. But if the question is whether the father or the speaker looks more like the sister, the pronoun should be objective because it is the object of a preposition in an understood clause {my sister looks more like my father than she looks like me}. Whatever the writer’s intent with the original sentence, and regardless of the pronoun used, the listener or reader can’t be entirely certain about the meaning. It would be better to reword the sentence and avoid the elliptical construction.
5.47 Special uses of personal pronouns. Some personal pronouns have spe cial uses. (1) He, him, and his have traditionally been used as pronouns of indeterminate gender equally applicable to a male or female person {if the finder returns my watch, he will receive a reward}. Because these pronouns are also masculine-specific, they have in recent years been regarded as sexist when used generically, and their indeterminate- gender use is declining (see 5.251-60). (2) It eliminates gender even if the noun’s sex could be identified. Using it does not mean that the noun has no sex—only that the sex is unknown or unimportant {the baby is smiling at its mother} {the mockingbird is building its nest}. (3) We,you, and they can be used indefinitely—that is, without an antecedent—in the sense of “persons,” “one,” or “people in general.” We is sometimes used by an individual who is speaking for a group {the council’s representative declared, “We appreciate your concern”} {the magazine’s editor wrote, “In our last issue, we covered the archaeological survey of Peru”}. This latter use is called “the editorial w.” Some writers also use w to make their prose appear less personal and to draw in the reader or listener {from these results we can draw only one conclusion}. You can apply in definitely to any person or all persons {if you read this book, you will learn how to influence people [you is indefinite—anyone who reads the book will learn]}. The same is true of they {they say that Stonehenge may have been a primitive calendar [those denoted by they are unidentified and perhaps unimportant to the point]}. (4) It also has several uses as an indefinite pronoun: (a) it may refer to a phrase, clause, sentence, or implied thought {he said that the website is down, but I don’t believe it [without the pronoun it, the clause might be rewritten I don’t believe what he said]}; (b) it can be the subject of a verb (usually a be-verb) without an antecedent noun {it was too far}, or an introductory word or expletive for a phrase or clause that follows the verb {it is possible that Jerry Paul is on vacation}; (c) it can be the grammatical subject in an expression about time, weather, distance, or the like {it is almost midnight} {it is beginning to snow}; and (d) it may be an expletive that anticipates the true gram matical subject or object {I find it hard to accept this situation}.
940 Grammar 5.49 5.48 Singular "they." Normally, a singular antecedent requires a singular pronoun. But because he is no longer universally accepted as a generic pronoun referring to a person of unspecified gender, people commonly (in speech and in informal writing) substitute the third-person-plural pronouns they, them, their, and themselves (or the nonstandard singular themself). While this usage is accepted in those spheres, it is only lately showing signs of gaining acceptance in formal writing (see 5.256), where Chicago recommends avoiding its use (see 5.255).5 When referring spe cifically to a person who does not identify with a gender-specific pro noun, however, they and its forms are often preferred. (They used in this sense was the American Dialect Society’s Word of the Year for 2015.) Like singular you, singular they takes a plural verb. So when the con text requires it, they/them/their/theirs, like you/your/yours (long used as both singular and plural forms), can be used to refer to one person {they have a degree in molecular biology} {their favorite color is blue}. And themself (like yourself) may be used to signal the singular antecedent (though some people will prefer themselves) {they blamed themself [or themselves]}. A number of other gender-neutral singular pronouns are in use, invented for that purpose; forms of these are usually singular and take singular verbs. In general, a person’s stated preference for a specific pronoun should be respected. See also 5.251-60.
POSSESSIVE PRONOUNS 5.49 Uses and forms of possessive pronouns. The possessive pronouns, my, our,your, his, her, its, and their, are used as limiting adjectives to qualify nouns {my dictionary} {your cabin} {his diploma}. Despite their name, possessive pronouns function in a much broader series of relationships than mere possession {my professor} {your argument}. Each form has a corresponding absolute possessive pronoun (also called an independent possessive) that can stand alone without a noun: mine, ours, yours, his, hers, its, and theirs. The independent form does not require an explicit object: the thing possessed may be either an antecedent or something understood {this dictionary is mine} {this cabin of yours is nice} {where is hers?}. An independent possessive pronoun can also stand alone and be treated as a noun: it can be the subject or object of a verb {hers is on 5. The generic singular they was endorsed in 2015 by the editors of the Washington Post, though with a caveat to first try avoiding it if possible. Singular they is more likely to be accepted in British than in American English. See Bas Aarts, Oxford Modem English Gram mar (Oxford: Oxford University Press, 2010), 5 (“the use of the plural pronoun they with a non-specific singular antecedent [is] sanctioned by widespread current usage”).
5.50 GRAMMAR AND USAGE the table} {pass me yours}, or the object of a preposition {put your coat with theirs}. When it is used with the preposition of, a double possessive is produced: that letter of Sheila's becomes that letter of hers. Such a con struction is unobjectionable. Note that none of the possessive personal pronouns is spelled with an apostrophe.
5.50 Possessive pronouns versus contractions. The possessive forms of personal pronouns are my, mine, our, ours, your, yours, his, her, hers, its, and their, theirs. Again, none of them takes an apostrophe. Nor does the possessive form of who {whose). Apart from these exceptions, the apos trophe is a universal signal of the possessive in English, so it is a natural tendency (and a common error) to overlook the exceptions and insert an apostrophe in the pronoun forms that end in -s (or the sibilant -se). Aggravating that tendency is the fact that some of the words have ho mophones that are contractions—another form that is also signaled by apostrophes. The pronouns that don’t sound like legitimate contractions seldom present problems, even if they do end in -s {hers, yours, ours). But several do require special attention, specifically its (the possessive of it) and it's (“it is”); your (the possessive otyou) andjwu’re (“you are”); whose (the possessive of who) and who's (“who is”); and the three homophones their (the possessive of they), there (“in that place” or “in that way”), and they're (“they are”).
REFLEXIVE AND INTENSIVE PRONOUNS 5.51 Basic uses of reflexive and intensive pronouns. The words myself, our selves, yourself, yourselves, himself, herself, itself, and themselves are used in two ways, and it’s useful to distinguish between their functions as reflex ive and intensive personal pronouns. Compare the intensive pronoun in I burned the papers myself (in which the object of burned is papers) with the reflexive pronoun in I burned myself (in which the object of burned is myself). Reflexive pronouns serve as objects that usually look back to the subject of a sentence or clause {the cat scared itself} {Gayla took it on herself to make the first move} {Ayoka dressed herself today} {don’t repeat yourself [the subject of this imperative sentence is understood to be you]}. Intensive pronouns repeat the antecedent noun or pronoun to add emphasis {I myself don’t care} {did you speak with the manager herself?} {Kate herself has won several writing awards} {did you knit that yourself?}. An intensive pronoun is used in apposition to its refer ent, so it’s in the nominative case. A common problem occurs when the -self form does not serve either of those functions. For example, the first- person pronoun in a compound might be used as a subject: Grammar 5.53 po o r : The staff and myself thank you for your contribution. b e t t e r : The staff and I thank you for your contribution.
Or it might be used as an object that does not refer to the subject: po o r : Deliver the equipment to my partner or myself. b e t t e r : Deliver the equipment to my partner or me.
DEMONSTRATIVE PRONOUNS 5.52 Demonstrative pronouns defined. A demonstrative pronoun (or, as it is sometimes called, a deictic pronoun) is one that points directly to its antecedent in the text: this or that for a singular antecedent {this is your desk} {that is my office}, and these or those for a plural antecedent {these have just arrived} {those need to be answered}. This and these point to objects that are near in space, time, or thought, while that and those point to objects that are somewhat remote in space, time, or thought. The an tecedent of a demonstrative pronoun can be a noun, phrase, clause, sen tence, or implied thought, as long as the antecedent is clear. Kind and sort, each referring to “one class,” are often used with an adjectival this or that {this kind of magazine} {that sort of school}. The plural forms kinds and sorts are usually preferred with the plural demonstratives {these kinds of magazines} {those sorts of schools}. A demonstrative pronoun standing alone cannot refer to a human antecedent; it must be followed by a word denoting a person. For example: I heard Mike’s son playing. That child is talented. In the second sentence, it would be erroneous to omit child or some such noun after that.
RECIPROCAL PRONOUNS 5.53 Reciprocal pronouns generally. Each other and one another are called reciprocal pronouns because they express a mutual relationship between elements {after much discussion, the two finally understood each other} {it’s true that we love one another}. Compare the nuances of meaning that a reciprocal or plural reflexive pronoun creates in the same sentence: {after our hike, we all checked ourselves for ticks [each person inspected him- or herself]} {after our hike, we checked one another for ticks [each person inspected one or more of the others]}. Reciprocal pronouns can also take the inflected genitive -’s to express possession {we admired each other’s watch}. In traditional usage, each other is reserved for two 5.54 GRAMMAR AND USAGE {she and I protected each other} and one another for more than two {all five of us watched out for one another}.
INTERROGATIVE PRONOUNS 5.54 Interrogative pronouns defined. An interrogative pronoun asks a ques tion. The three interrogatives are w/io, what, and which. Only one, who, declines: who (nominative), whom (objective), whose (possessive) {who starred in Casablanca?} {to whom am I speaking?} {whose cologne smells so nice?}. In the nominative case, who is used in two ways: (1) as the subject of a verb {who washed the dishes today?} and (2) as a predi cate nominative after a linking verb {it was who?}. In the objective case, whom is used in two ways: (1) as the object of a verb {whom did you see?} and (2) as the object of a preposition {for whom is this building named?}.
5.55 Referent of interrogative pronouns. To refer to a person, who, what, or which can be used. But they are not interchangeable. Who is universal or general: it asks for any one or more persons among a universe of people. The answer may potentially include any person, living or dead, present or absent {who wants to see that movie?} {who were your greatest in spirations?}. Who also asks for a particular person’s identity {who is that person standing near the Emerald Buddha?}. Which and what, when fol lowed by a noun denoting a person or persons, are usually selective or limited; they ask for a particular member of a group, and the answer is limited to the group addressed or referred to {which explorers visited China in the sixteenth century?} {what ice-skater is your favorite?}. To refer to a person, animal, or thing, either which or what may be used {which one of you did this?} {what kind of bird is that?}. When applied to a person, what often asks for the person’s character, occupation, qual ities, and the like {what do you think of our governor?}. When applied to a thing, what is broad and asks for any one thing, especially of a set {what is your quest?} {what is your favorite color?}.
RELATIVE PRONOUNS 5.56 Relative pronouns defined. A relative pronoun is one that introduces a dependent (or relative) clause and relates it to the independent clause. Relative pronouns in common use are who, which, what, and that. Who is the only relative pronoun that declines: who (nominative), whom (objec tive), whose (possessive) {the woman who presented the award} {a source whom he declined to name} {the writer whose book was a best seller}.
Grammar 5.59 Who normally refers to a human being, but it can be used in the first, sec ond, or third person. Which refers only to an animal or a thing. What re fers only to a nonliving thing. Which and what are used only in the second and third person. That refers to a human, animal, or thing, and it can be used in the first, second, or third person. When a relative pronoun qual ifies a noun element in the clause it introduces, it is sometimes called a relative adjective. See also 5.64.
5.57 Gender, number, and case with relative pronouns. A relative pronoun agrees with its antecedent in gender, person, and number. If a personal pronoun follows a relative pronoun, and both refer to the same anteced ent in the independent clause, the personal pronoun takes the gender and number of that antecedent {I saw a farmer who was plowing his fields with his mule}. If the personal pronoun refers to a different an tecedent from that of the relative pronoun, it takes the gender and num ber of that antecedent {I saw the boy and also the girl who pushed him down}. A personal pronoun does not govern the case of a relative pro noun. Hence an objective pronoun such as me may be the antecedent of the nominative pronoun who, although a construction formed in this way sounds increasingly archaic or even incorrect {she was referring to me, who never graduated from college} {it was we whom they objected to}. When a construction may be technically correct but sounds awkward or artificial {I, who am wronged, have a grievance}, the best course may be to use preventive grammar and find a different construction {I have been wronged; I have a grievance} {having been wronged, I have a grievance}.
5.58 Positional nuances of relative pronouns. A relative pronoun is in the nominative case when no subject comes between it and the verb {the pro fessor who lectured was brilliant}. When one or more words intervene between the relative pronoun and the verb, the relative is governed by the following verb or by a verb or a preposition within the intervening clause {the person whom I called is no longer there} {it was John who they thought was in the bleachers}. When a relative pronoun is interroga tive, it refers to the word or phrase containing the answer to the question for its consequent, which agrees in case with the interrogative {whose book is that? Joseph’s}.
5.59 Antecedent of relative pronouns. Usually a relative pronoun’s anteced ent is a noun or pronoun in the independent clause on which the relative clause depends. For clarity, it should immediately precede the pronoun {the diadem that I told you about is in this gallery}. The antecedent may also be a noun phrase or a clause, but the result can sometimes be 5.60 GRAMMAR AND USAGE ambiguous: in the bedroom of the villa, which was painted pink, does the which-dause refer to the bedroom or to the villa? See 5.60.
5.60 Remote relative clauses. For clarity, pronouns must have unambiguous antecedents. A common problem with the relative pronouns that, which, and who arises if you separate the relative clause from the noun to which it refers. The longer the separation, the more pronounced the problem— especially when one or more unrelated nouns fall between the true an tecedent and the clause. Consider the guy down the street that runs through our neighborhood: if the intent is for that runs through our neighborhood to refer to the guy rather than the street, the writer should reword the phrase to make that instantly clear to the reader.
po o r : Stress caused her to lose the freedom from fear of the future, which she once enjoyed.
b e t t e r : Stress caused her to lose what she once enjoyed: freedom from fear of the future.
po o r : After the news came out, the CEO fired the aide, a friend of the chairman, who was the target of the investigation.
b e t t e r : After the news came out, the CEO fired the aide, who was the target of the investigation and also a friend of the chairman.
5.61 Omitted antecedent of relative pronoun. If no antecedent noun is ex pressed, what can be used to mean that which {is this what you were looking for?}. But if there is an antecedent, use a different relative pro noun: who {where is the man who spoke?}, that (if the relative clause is restrictive, i.e., essential to the sentence’s basic meaning) {where are the books that Jones, told us about?}, or which (if the relative clause is non- restrictive, i.e., could be deleted without affecting the sentence’s basic meaning) {the sun, which is shining brightly, feels warm on my face}. See also 6.27.
5.62 Relative pronoun and the antecedent "one." A relative pronoun takes its number from its antecedent. That’s easy enough when the antecedent is simply one. But if one is part of a noun phrase with a plural noun such as one of the few or one of those, the relative pronoun following takes the plu ral word as its antecedent—not one. Treat the pronoun as a plural and use a plural verb. For example, in Lily is one of those people who are famous for beingfamous, the plural verb are links a quality belonging to those people.
5.63 Genitive forms for relative pronouns. The forms of whom and of which are genitives {the child, the mother of whom we talked about, is in kin dergarten} {this foal, the sire of which Belle owns, will be trained as a Grammar 5.66 hunter-jumper}. These forms have an old-fashioned sound and can often be rephrased more naturally {the child whose mother we talked about is in kindergarten}. The relative what forms the genitive of what {a list of what we need}. The relative that forms the genitive of that (the preposi tion being placed at the end of the phrase) {no legend that we know of} or of which {no legend of which we know}.
5.64 "Whose" and "of which." The relatives who and which can both take whose as a possessive form (whose substitutes for of which) {a movie the conclusion of which is unforgettable} {a movie whose conclusion is un forgettable}. Some writers object to using whose as a replacement for of which, especially when the subject is not human, but the usage is centu ries old and is widely accepted as preventing unnecessary awkwardness. Compare the company whose stock rose faster with the company the stock of which rose faster. Either form is acceptable, but the possessive whose is far smoother.
5.65 Compound relative pronouns. Who, whom, what, and which form com pound relative pronouns by adding the suffix -ever. The compound rela tives whoever, whomever, whichever, and whatever apply universally to any or all persons or things {whatever you do, let me know} {whoever needs to write a report about this book may borrow it}.
5.66 "Who" versus "whom." Who and whoever are nominative pronouns. Each can be used as a subject {whoever said that?} or as a predicate nomina tive {it was who?}. Whom and whomever are the objective forms, used as the object of a verb {you called whom?} or of a preposition {to whom are you referring?}. Three problems arise with determining the correct case. First, because the words are so often found in the inverted syntax of an interrogative sentence, their true function in the sentence can be hard to see without sorting the words into standard subject-verb-object syntax. In the following example, sorting the incorrect “I should say whom is calling” makes the case easier to determine: po o r : Whom should I say is calling? b e t t e r : Who should I say is calling?
Second, determining the proper case can be confusing when the pronoun serves a function (say, nominative) in a clause that itself serves a differ ent function (say, objective) in the main sentence. The pronoun’s func tion in its clause determines its case.
po o r : I’ll talk to whomever will listen. b e t t e r : I’ll talk to whoever will listen.
5.67 GRAMMAR AND USAGE po o r : Whoever you choose will suit me. b e t t e r : Whomever you choose will suit me.
In the first example, the entire clause whoever will listen is the object of the preposition to. But in the clause itself, whoever serves as the subject, and that function determines its case. Similarly, in the second example whomever is the object of choose in the clause, so it technically ought to be in the objective case even though the clause itself serves as the subject of the sentence. Third, as the second example above shows, a further distraction can arise when the who-clause itself contains a nested clause, typically of attribution or identification (here, you choose).
INDEFINITE PRONOUNS 5.67 Indefinite pronouns generally. An indefinite pronoun is one that gen erally or indefinitely represents an object, usually one that has already been identified or doesn’t need exact identification. The most common examples are another, any, both, each, either, neither, none, one, other, some, and such. There are also compound indefinite pronouns such as anybody, anyone, anything, everybody, everyone, everything, nobody, no one, oneself, somebody, and someone. Each, either, and neither are also called distributive pronouns because they separate the objects referred to from others referred to nearby. Indefinite pronouns have number. When an in definite pronoun is the subject of a verb, it is usually singular {everyone is enjoying the dinner} {everybody takes notes during the first week}. But sometimes an indefinite pronoun carries a plural sense in informal prose {nobody could describe the music; they hadn’t been listening to it} {everyone understood the risk, but they were lured by promises of big returns}. The forms of indefinite pronouns are not affected by gender or person, and the nominative and objective forms are the same. To form the possessive, the indefinite pronoun may take -’s {that is no one’s fault} {is this anyone’s jacket?} or the adverb else plus -’s {don’t interfere with anybody else’s business} {no one else’s cups were broken}.
Adjectives TYPES OF ADJECTIVES 5.68 Adjectives defined. An adjective is a word (more particularly, a type of word sometimes called an adjunct) modifying a noun or pronoun; it is of Grammar 5.71 ten called a describing word. An adjective tells you what sort, how many, how large or small, whose, etc. It may modify an understood as well as an expressed noun {he is a good as well as a wise man [man is under stood aftergood]}. An adjective may add a new idea to a noun or pronoun by describing it more definitely or fully {red wagon} {human error}. Or it may be limiting {three pigs} {this time}. Most adjectives derive from nouns, as plentiful derives from plenty or as stylish derives from style; some derive from verbs, roots, or other adjectives. Often a suffix creates the adjective. Among the suffixes that often distinguish adjectives are -able {manageable}, -al {mystical}, -ary {elementary}, -ed {hammered}, -en {wooden}, -esque {statuesque}, -fid {harmful}, -ible {inaccessible}, -ic {artistic}, -ish {foolish}, -ive {demonstrative}, -less {helpless}, -like {child like}, -ly {ghostly}, -ous {perilous}, -some {lonesome}, and -y {sunny}. But many adjectives do not have distinctive endings and are recognizable only by their function {old} {tall} {brilliant}.
5.69 Proper adjectives. A proper adjective is one that, being or deriving from a proper name, always begins with a capital letter {a New York minute} {a Cuban cigar} {a Canadian dollar}. (But see 8.61.) A proper name used attributively is still capitalized, but it does not cause the noun it modifies to be capitalized. A place-name containing a comma—such as Toronto, Ontario, or New Delhi, India—should generally not be used as an adjec tive because a second comma may be considered obligatory {we met in a Toronto, Ontario, restaurant}. The comma after Ontario in that sen tence is awkward. Compare the readability of a New Delhi, India, market place with a New Delhi marketplace or a marketplace in New Delhi, India (substituting a prepositional phrase for the proper adjective).
ARTICLES AS LIMITING ADJECTIVES 5.70 Articles defined. An article is a limiting adjective that precedes a noun or noun phrase and determines its use to indicate something definite (the) or indefinite (a or an). An article might stand alone or be used with other adjectives {a road} {an elaborate design} {the yellow-brick road}.
5.71 Definite article. A definite article points to a definite object that (1) is so well understood that it does not need description (e.g., the package is here is a shortened form of the package that you expected is here); (2) is a thing that is about to be described {the sights of Chicago}; or (3) is important {the grand prize}. The definite article belongs to nouns in the singular {the star} or the plural number {the stars}.
5.72 GRAMMAR AND USAGE 5.72 Indefinite article. An indefinite article points to a nonspecific object, thing, or person that is not distinguished from the other members of a class. The thing may be singular {a student at Princeton}, or uncountable {a multitude}, or generalized {an idea inspired by Milton’s Paradise Lost}.
5.73 Indefinite article in specific reference. In a few usages, the indefinite article provides a specific reference {I saw a great movie last night} and the definite article a generic reference {the Scots are talking about inde pendence [generalizing by nationality]}.
5.74 Choosing "a" or "an." With the indefinite article, the choice of a or an depends on the sound of the word it precedes. A precedes words with a consonant sound, including /y/, /h/, and /w/, no matter how the word is spelled {a eulogy} {a historic occasion} {a onetime pass}. An comes be fore words with a vowel sound {an insurance agent} {an X-Files episode} {an hour ago}. The same is true for abbreviations. If the first letter or syl lable is sounded as a consonant, use a {a BTU calculation} {a PDF file}. If the first sound is a vowel, use an {an MBA degree} {an ATM}. See also 7.32-33,10.9.
5.75 Articles with coordinate nouns. With a series of coordinate nouns, an article may appear before each noun, but it is not necessary {the rose bush and hedge need trimming}. If the things named make up a single idea, it’s especially unnecessary to repeat the article {in the highest de gree of dressage, the horse and rider appear to be one entity}. And if the named things are covered by one plural noun, the definite article should not be repeated {in the first and second years of college}. But if you want to distinguish concepts or add emphasis, then do repeat the article {the time, the money, and the effort were all wasted}.
5.76 Effect of article on meaning. Because articles have a demonstrative value, the meaning of a phrase may shift depending on the article used. For example, an officer and gentleman escorted Princess Grace to her car suggests (though ambiguously) that the escort was one man with two de scriptive characteristics. But an officer and a friend escorted Princess Grace to her car suggests that two people acted as escorts. Similarly, do you like the red and blue cloth? suggests that the cloth contains both red and blue threads. But do you like the red and the blue cloth? suggests that two dif ferent fabrics are being discussed. The clearest way to express the idea that the cloth contains both red and blue is to hyphenate the phrase as a compound modifier: red-and-blue cloth; and with two kinds of cloth, the clear expression is either to repeat the word cloth (the red cloth and the Grammar 5.80 blue cloth) or to use cloth with the first adjective rather than the second (the red cloth and the blue), 5.77 Omitted article and zero article. The absence of an article may alter a sentence’s meaning—e.g., the meaning of the news brought us little com fort (we weren’t comforted) changes if a is inserted before little: the news brought us a little comfort (we felt somewhat comforted). An article that is implied but omitted is called a zero article, common in idiomatic usage. For example, in the morning you may make the bed, but at night you go to bed (not the bed)—and notice in the morning versus at night. The zero article usually occurs in idiomatic references to time, illness, transpor tation, personal routines, and meals {by sunset} {has cancer} {travel by train} {go to bed} {make breakfast}.
5.78 Article as pronoun substitute. An article may sometimes substitute for a pronoun. For example, the blanks in a patient who develops the described rash on__ hands should informdoctor may be filled in with either a possessive pronoun or the definite article (the).
POSITION OF ADJECTIVES 5.79 Basic rules for position of adjectives. An adjective that modifies a noun element usually precedes it {perfect storm} {spectacular view} {a good bowl of soup}. Such an adjective is called an attributive adjective. An ad jective may follow the noun element if the adjective (1) expresses special emphasis {reasons innumerable} {captains courageous}; (2) occurs in this position in standard usage {court-martial} {notary public}; (3) is a predicate adjective following a linking verb {I am ready}; (4) functions as an appositive set off by commas or dashes {the man, tall and thin, stood in the corner}; or (5) modifies a pronoun of a type usually followed by an adjective {anything good} {everything yellow} {nothing important} {something wicked}. (An adjective that follows its noun is termed a post positive adjective.) Some adjectives are always in the predicate and never appear before what they modify {the city is asleep} {the door was ajar}. Others appear uniformly before the nouns they modify {utter nonsense} {a mere child}. Phrasal adjectives may precede or follow what they mod ify. When a modifying phrase follows the noun element it modifies, it is traditionally called an adjective phrase. See also 5.92.
5.80 Adjective after possessive. When a noun phrase includes a possessive noun, as in childrens shoes or the company’s president, the adjective follows 5.81 GRAMMAR AND USAGE the possessive {children’s athletic shoes} {the company’s former presi dent} (unless the reference is to athletic children or a former company). The same is true of possessive pronouns {her red dress}.
5.81 Adjective modifying pronoun. When modifying a pronoun, an adjective usually follows the pronoun {the searchers found him unconscious} {some like it hot}, sometimes as a predicate adjective {it was insensi tive} {who was so jealous?}. Occasionally, however, an adjective pre cedes the pronoun it modifies {flustered, he sat down} {they offered her the director position but, uncertain, she demurred}.
5.82 Predicate adjective. A predicate adjective is an adjective that follows a linking verb (see 5.101) but modifies the subject {the child is afraid} {the night became colder} {this tastes delicious} {I feel bad}. If an adjective in the predicate modifies a noun or pronoun in the predicate, it is not a predicate adjective. For example, in the train will be late, the adjective late modifies the subject train. But in the train will be here at a late hour, the adjective late modifies the noun hour, not the subject train. So even though it occurs in the predicate, it is not known as a. predicate adjective, which by definition follows a linking verb.
5.83 Date as adjective. Dates are often used as descriptive adjectives, more often today than in years past. If a month-year or month-day date is used as an adjective, no hyphen or comma is needed {October 31 festivities} {December 2014 financial statement}. If a full month-day-year date is used, then a comma is sometimes considered necessary both before and after the year {the May 27, 2016, ceremonies}. But this construction is awkward because the adjective (which is forward looking) contains two commas (which are backward looking); the construction is therefore best avoided {commencement ceremonies on May 27,2016}.
DEGREES OF ADJECTIVES 5.84 Three degrees of adjectives. An adjective is gradable into three degrees: the positive or absolute {hard}, the comparative {harder}, and the super lative {hardest}. A positive adjective simply expresses an object’s quality without reference to any other thing {a big balloon} {bad news}.
5.85 Comparative adjectives. A comparative adjective expresses the relation ship between a specified quality shared by two things, often to determine which has more or less of that quality {a cheaper ticket} {a happier end Grammar 5.87 ing}. The suffix -er usually signals the comparative form of a common ad jective having one or two syllables {light-lighter} {merry-merrier}. These forms are called synthetic comparatives. A positive adjective with three or more syllables typically takes more (or greater, less, fewer, and so forth) instead of a suffix to form the comparative {intelligent-more intelligent} {purposeful-more purposeful}. These forms are called periphrastic com paratives. As noted, some adjectives with two syllables take the -er suffix {lazy-lazier} {narrow-narrower}, but most two-syllable adjectives take more {more hostile} {more careless}. A two-syllable adjective ending in -er, -le, -ow, -ure, or -y can typically use either the -er suffix or more.
5.86 Superlative adjectives. A superlative adjective expresses the relation ship between at least three things and denotes an extreme of inten sity or amount in a particular shared quality {the biggest house on the block} {the bitterest pill of all}. The suffix -est usually signals the super lative form of a common adjective having one or two syllables {lighter- lightest}. These forms are called synthetic superlatives. An adjective with three or more syllables takes most instead of a suffix to form the super lative {quarrelsome-most quarrelsome} {humorous-most humorous}. These forms are called periphrastic superlatives. Some adjectives with two syllables take the -est suffix {holy-holiest} {noble-noblest}, but most two- syllable adjectives take most {most fruitful} {most reckless}.
5.87 Forming comparatives and superlatives. A few rules govern the form ing of a short regular adjective’s comparative and superlative forms. (1) If the adjective is a monosyllable ending in a single vowel followed by a single consonant, the final consonant is doubled before the suffix is attached {red-redder-reddest}. (2) If the adjective ends in a silent -e, the -e is dropped before the suffix is added {polite-politer-politest}. (3) A participle used as an adjective requires more or most before the partici ple; no suffix is added to form the comparative or the superlative {this teleplay is more boring than the first one} {I am most tired on Fridays}. (4) A few one-syllable adjectives—real, right, and wrong—can take only more and most. Even then, these combinations occur only in informal speech. (5) Eager, proper, and somber, unlike many other two-syllable adjectives, also take only more and most; none can take a suffix. (6) A two-syllable adjective to which the negative prefix un- has been added can usually take either a suffix or more or most, even if the total number of syllables is three {unhappiest} {most unhappy}. (7) Many adjectives are irregular—there is no rule that guides their comparative and superlative forms {good-better-best} {less-lesser-least}. A good dictionary will show the forms of an irregular adjective. (8) An adjective can never take both a suffix and more or most (or less, least, etc.). This is a grammatical fault 5.88 GRAMMAR AND USAGE known as a double comparative (e.g., more greener) or a double superlative (e.g., least greenest). It is stigmatized as nonstandard.
5.88 Equal and unequal comparisons. A higher degree of comparison is sig naled by a suffix (-er or -est) or by more or most. (See 5.85, 5.86.) A lower degree is shown by less (comparative) or least (superlative) {cold-less cold} {less cold-least cold}. Equivalence is shown by the use of the as-as construction {this is as old as that} and sometimes by so, but usually in the negative, where it signals a lower degree {that test was not so hard as the last one}.
5.89 Noncomparable adjectives. An adjective that by definition describes an absolute state or condition—e.g., entire, impossible, pregnant, unique—is called noncomparable. It cannot take a comparative suffix and cannot be coupled with a comparative term (more, most, less, least). Nor can it be intensified by a word such as very, largely, or quite. But on the rare oc casion when a particular emphasis is needed, a good writer may depart from this rule and use a phrase such as more perfect, as the framers of the United States Constitution did in composing its preamble {We the People of the United States, in order to form a more perfect Union...}.
SPECIAL TYPES OF ADJECTIVES 5.90 Participial adjectives. A participial adjective is simply a verb’s partici ple that modifies a noun or pronoun. It can be a present participle (verb ending in -ing) {the dining room} {a walking stick} {a rising star} or a past participle (usually a verb ending in -ed) {an endangered species} {a com pleted assignment} {a proven need}. Some past-participial forms have only this adjectival function, the past-participial verb having taken a dif ferent form {a shaven face} {a graven image}. A past participle function ing as an adjective may itself be modified with an adverb such as quite {a quite fatigued traveler}, barely {a barely concealed wince}, little {a little-known fact}, or an adverbial phrase such as very much {a very much distrusted public official}. If the past participle has gained a strong ad jectival quality, very will do the job alone without the quantitative much {very tired} {very drunk}. But if the participial form seems more like a verb, very needs much to help it do the job {very much appreciated} {very much delayed}. A few past participles (such as bored, interested, pleased, satisfied) are in the middle of the spectrum between those having mostly adjectival qualities and those having mostly verbal qualities. With these few, the quantitative much is normally omitted. See also 5.110.
Grammar 5.93 5.91 Coordinate adjectives. A coordinate adjective is one that appears in a sequence with one or more related adjectives to modify the same noun. Coordinate adjectives should be separated by commas or by and {skilled, experienced chess player} {nurturing and loving parent}. If one adjective modifies the noun and another adjective modifies the idea expressed by the combination of the first adjective and the noun, the adjectives are not considered coordinate and should not be separated by a comma. For ex ample, a lethargic soccer player describes a soccer player who is lethargic. Likewise, phrases such as white brick house and wrinkled canvas jacket are unpunctuated because the adjectives are not coordinate: they have no logical connection in sense (a white house could be made of many dif ferent materials; so could a wrinkled jacket). The most useful test is this: if and would fit between the two adjectives, a comma is necessary.
5.92 Phrasal adjectives. A phrasal adjective (also called a compound modifier) is a phrase that functions as a unit to modify a noun. A phrasal adjec tive follows these basic rules: (1) Generally, if placed before a noun, the phrase should be hyphenated to avoid misdirecting the reader {dog-eat- dog competition}. There may be a considerable difference between the hyphenated and the unhyphenated forms: compare small animal hos pital with small-animal hospital. (2) If a compound noun is an element of a phrasal adjective, the entire compound noun must be hyphenated to clarify the relationship among the words {time-clock-punching em ployees}. (3) If more than one phrasal adjective modifies a single noun, hyphenation becomes especially important {nineteenth-century song- and-dance numbers} {state-inspected assisted-living facility}. (4) If two phrasal adjectives end in a common element, the ending element should appear only with the second phrase, and a suspended hyphen should follow the unattached words to show that they are related to the ending element {middle- and upper-class operagoers}. (5) If the phrasal adjec tive denotes an amount or a duration, the plural should be dropped. For instance, pregnancy lasts nine months but is a nine-month pregnancy, and a shop open 24 hours a day has a 24-hour-a-day schedule. The plural is re tained only for fractions {a two-thirds majority}. (6) If a phrasal adjective becomes awkward, the sentence should probably be recast. For example, The news about the lower-than-expected third-quarter earnings disappointed investors could become The news about the third-quarter earnings, which were lower than expected, disappointed investors. Or perhaps this: Investors were disappointed by the third-quarter earnings, which were lower than ex pected. See also 7.81-89.
5.93 Exceptions for hyphenating phrasal adjectives. There are exceptions to hyphenating phrasal adjectives: (1) If a phrasal adjective follows a 5.94 GRAMMAR AND USAGE linking verb, it is often unhyphenated—e.g., compare a well-trained ath lete with an athlete who is well trained. (2) When a proper name begins a phrasal adjective, the name is not hyphenated {the Monty Python school of comedy}. (3) A two-word phrasal adjective that begins with an adverb ending in -ly is not hyphenated {a sharply worded reprimand} (but a not- so-sharply-worded reprimand). See also 7.84,7.85.
5.94 Adjectives as nouns. An adjective-to-noun shift (sometimes called an adnoun) is relatively common in English. Some adjectives are well es tablished as nouns and are perfectly suitable for most contexts. For ex ample, a postmortem examination is often called a postmortem; collectible objects are collectibles; and French people are the French. Any but the most established among such nouns should be used only after careful consid eration. If there’s an alternative, it will almost certainly be better. For ex ample, there is probably no good reason to use the adjective collaborative as a noun (i.e., as a shortened form of collaborative enterprise) when the perfectly good collaboration is available. See also 5.24-26.
5.95 Adjectives as verbs. Adjective-to-verb shifts are uncommon in English but occur once in a while, usually as jargon or slang {the cargo tanks were inerted by introducing carbon dioxide into them} {it would be silly to low-key the credit for this achievement}. They generally don’t fit com fortably into formal prose.
5.96 Other parts of speech functioning as adjectives. Words that ordinarily function as other parts of speech, but sometimes as adjectives, include nouns (see 5.24), pronouns (see 5.49), and verbs (see 5.107).
Verbs DEFINITIONS 5.97 Verbs generally. A verb shows the performance or occurrence of an ac tion or the existence of a condition or a state of being, such as an emo tion. A verb is the most essential part of speech—the only one that can express a full thought by itself (with the subject understood) {Run!} {En joy!} {Think!}. (One-word sentences such as Why? or Yes alone can ex press complete thoughts as well, but these are in fact elliptical sentences omitting a clause implied by context {Why [did she do that] ?} {Yes[, you may borrow that book].}.) 5.98 Transitive and intransitive verbs. Depending on the presence or absence of an object, a verb is classified as transitive or intransitive. A transitive Grammar 5.100 verb requires an object to express a complete thought; the verb indicates what action the subject exerts on the object. For example, the cyclist hit a curb states what the subject cyclist did to the object curb. (A few transi tive verbs have what are called cognate objects, which are closely related etymologically to the verb {drink a drink} {build a building} {see the sights}.) An intransitive verb does not require an object to express a com plete thought {the rescuer jumped}, although it may be followed by a prepositional phrase serving an adverbial function {the rescuer jumped to the ground}. Many verbs may be either transitive or intransitive, the different usages often distinguishing their meanings. For example, when used transitively, as in the king's heir will succeed him, the verb succeed means “to follow and take the place of”; when used intransitively, as in the chemist will succeed in identifying the toxin, it means “to accomplish a task.” With some verbs, no such distinction is possible. For example, in I will walk;you ride, the verb ride is intransitive. In I will walk;you ride your bike, the verb ride is transitive, but its meaning is unchanged. A verb that is normally used transitively may sometimes be used intransitively to emphasize the verb and leave the object undefined or unknown {the patient eats poorly [how well the patient eats is more important than what the patient eats]}. The test for whether a given verb is transitive is to try it with various possible objects. For each sentence in which an object is plausible, the verb is being used transitively. If an object doesn’t work idiomatically, the verb is being used intransitively.
5.99 Ergative verbs. Some verbs, called ergative or ambitransitive verbs, can be used transitively or intransitively {the impact shattered the windshield} {the windshield shattered}. The noun that serves as the object when the verb’s use is transitive becomes the subject when the verb’s use is intransitive. For example, with the noun door and the verb open, one can say I opened the door (transitive) or the door opened (intransitive). Many verbs can undergo ergative shifts {the torpedo sank the boat} {the boat sank}. For example, the verb ship was once exclusively transitive {the company shipped the books on January 16}, but in commercial usage it is now often intransitive {the books shipped on January 16}. Likewise, grow (generally an intransitive verb) was transitive only in horticultural contexts {the family grew several types of crops}, but commercial usage now makes it transitive in many other contexts {how to grow your busi ness}. Careful writers and editors employ such usages cautiously if at all, preferring well-established idioms.
5.100 Regular and irregular verbs. The past-tense and past-participial forms of most English words are formed by appending -ed to the basic form {draft-drafted-drafted}. If the verb ends in -e, only a -d is appended 5.101 GRAMMAR AND USAGE {charge-charged-charged}. (Sometimes a final consonant is doubled.) These verbs are classified as regular, or weak (the latter is a term used in philology to classify forms of conjugation). But a few common verbs have maintained forms derived mostly from Old English roots {begin-began- begun} {bet-bet-bet} {bind-bound-bound} {bite-bit-bitten}. These verbs are called irregular, or strong. The various inflections of strong verbs defy simple classifications, but many past-tense and past-participial forms (1) change the vowel in the base verb (as begin), (2) keep the same form as the base verb (as bet), (3) share an irregular form (as bind), or (4) change endings (as bite). (The vowel change between cognate forms in category 1 is called an ablaut.) The verb be is highly irregular, with eight forms (is, are, was, were, been, being, be, and am). Because no system of useful clas sification is possible for irregular verbs, a reliable memory and a general dictionary are essential tools for using the correct forms consistently. Further complicating the spelling of irregular verbs is the fact that the form may vary according to the sense of the word. When used to mean “to offer a price,” for example, bid keeps the same form in the past tense and past participle, but when it means “to offer a greeting,” it forms bade (traditionally rhyming with glad) and bidden. The form may also depend on whether the verb is being used literally {wove a rug} or figuratively {weaved in traffic}. Finally, a few verbs that are considered regular have an alternative past tense and past participle that is formed by adding -t to the simple verb form {dream-dreamed} {dream-dreamt}. When these alternatives are available, American English tends to prefer the forms ending in -ed (e.g., dreamed, learned, spelled), while British English often prefers the forms ending in -t (dreamt, learnt, spelt).
5.101 Linking verbs. A linking verb (also called a copula or connecting verb) is one that links the subject to a closely related word in the predicate—a subjective complement. The linking verb itself does not take an object, because it expresses a state of being instead of an action {Mr. Block is the chief executive officer} {that snake is venomous} {his heart’s desire is to see his sister again}. There are two kinds of linking verbs: be-verbs and intransitive verbs that are used in a weakened sense, such as appear, become, feel, look, seem, smell, and taste. The weakened intransitive verbs often have a figurative sense akin to that of become, as in He fell heir to a large fortune (he didn’t physically fall on or into anything) or The river ran dry (a waterless river doesn’t run—it has dried up). (See also 5.170.) Some verbs only occasionally function as linking verbs—among them act {act weird}, get {get fat}, go {go bald}, grow {grow weary}, lie {lie fallow}, prove {prove untenable}, remain {remain quiet}, sit {sit still}, stay {stay trim}, turn {turn gray}, and wax {wax eloquent}. Also, some passive-voice constructions contain linking verbs {this band was judged best in the Grammar 5.104 contest} {she was made sales-force manager}. If a verb doesn’t have a subjective complement, then it doesn’t qualify as a linking verb in that particular construction. For instance, when a fo-verb conveys the sense “to be situated” or “to exist,” it is not a linking verb {Kansas City, Kansas, is across the river} {there is an unfilled receptionist position}. Likewise, if a verb such as appear, feel, smell, sound, or taste is followed by an adver bial modifier instead of a subjective complement {he appeared in court} or a direct object {the dog smelled the scent}, it isn’t a linking verb.
5.102 Phrasal verbs. A phrasal verb is usually a verb plus a preposition (or particle), which serves as an adverb {settle down} {act up} {phase out}. A phrasal verb is not hyphenated, even though its equivalent noun or phrasal adjective might be—e.g., compare to flare up with a flare-up, and compare to step up the pace with a stepped-up pace. Three rules apply: (1) if the phrasal verb has a sense distinct from the component words, use the entire phrase—e.g., hold up means “to rob” or “to delay,” and get rid of and do away with mean “to eliminate”; (2) avoid the phrasal verb if the verb alone conveys essentially the same meaning—e.g., rest up is equiva lent to rest; and (3) don’t compress the phrase into a one-word verb, espe cially if it has a corresponding one-word noun form—e.g., one burns out (phrasal verb) and suffers burnout (noun).
5.103 Principal and auxiliary verbs. Depending on its uses, a verb is classified as principal or auxiliary. A principal verb is one that can stand alone to express an act or state {he jogs} {I dreamed about Xanadu}. If com bined with another verb, it expresses the combination’s leading thought {a tiger may roar}. An auxiliary verb is used with a principal verb to form a verb phrase that indicates mood, tense, or voice {you must study for the exam!} {I will go to the store} {the show was interrupted}. The most commonly used auxiliaries are be, can, do, have, may, must, ought, shall, and will. For more on auxiliary verbs, see 5.144-53.
5.104 Verb phrases. The combination of an auxiliary verb with a principal verb is a verb phrase, such as could happen, must go, or will be leaving. When a verb phrase is modified by an adverb, the modifier typically goes directly after the first auxiliary verb, as in could certainly happen, must always go, and will soon be leaving. The idea that verb phrases should not be “split” in this way is quite mistaken (see 5.171). A verb phrase is negated when the negative adverb not is placed after the first auxiliary {we have not called him}. In an interrogative sentence, the first auxiliary begins the sentence and is followed by the subject {must I repeat that?} {do you want more?}. An interrogative can be negated by placement of not af ter the subject {do you not want more?}, but a contraction is often more 5.105 GRAMMAR AND USAGE natural {don’t you want more?}. Most negative forms can be contracted {we do not-we don’t} {I will not-I won’t} {he has not-he hasn’t} {she does not-she doesn’t}, but I am not is contracted to I'm not (never I amn't). The corresponding interrogative form is aren't I? Sometimes the negative is emphasized if the auxiliary is contracted with the pronoun and the neg ative is left standing alone {he is not-he isn’t-he’s not} {we are not-we aren’t-we’re not} {they have not-they haven’t-they’ve not}.
5.105 Contractions. Most types of writing benefit from the use of contractions. If used thoughtfully, contractions in prose sound natural and relaxed and make reading more enjoyable. Be-verbs and most of the auxiliary verbs are contracted when followed by not: are not-aren't, was not-wasn't, cannot-can't, could not-couldn't, do not-don't, and so on. A few, such as ought not-oughtn't, look or sound awkward and are best avoided. Pro nouns can be contracted with auxiliaries, with forms of have, and with some fo-verbs. Think before using one of the less common contractions, which often don’t work well in prose, except perhaps in dialogue or quo tations. Some examples are I’d’ve (I would have), she’d’ve (she would have), it'd (it would), should've (should have), there're (there are), who're (who are), and would've (would have). Also, some contracted forms can have more than one meaning. For instance, there's may be there is or there has, and I'd may be I had or I would. The particular meaning may not always be clear from the context.
INFINITIVES 5.106 Infinitives defined. An infinitive verb, also called the verb’s root or stem, is a verb that in its principal uninflected form may be preceded by to {to dance} {to dive}. It is the basic form of the verb, the one listed in dictio nary entries. The preposition to is sometimes called the sign of the infini tive {he tried to open the door}, and it is sometimes classed as an adverb. In the active voice, to is generally dropped when the infinitive follows an auxiliary verb {you must flee} and can be dropped after several verbs, such as bid, dare, feel, hear, help, let, make, need, and see {you dare say that to me?}. But when the infinitive follows one of these verbs in the passive voice, to should be retained {he cannot be heard to deny it} {they cannot be made to listen}. The to should also be retained after ought and ought not (see 5.149).
5.107 Uses of the infinitive. The infinitive has great versatility. It is sometimes called a verbal noun because it can function as part of a verb phrase {someone has to tell her} or a noun {to walk away now seems rash}. The Grammar 5.111 infinitive also has limited uses as an adjective or an adverb. As a verb, it can take (1) a subject {we wanted the lesson to end}, (2) an object {try to throw the javelin higher}, (3) a predicate complement {want to race home?}, or (4) an adverbial modifier {you need to think quickly in chess}. As a noun, the infinitive can perform as (1) the subject of a finite verb {to fly is a lofty goal} or (2) the object of a transitive verb or parti ciple {I want to hire a new assistant}. An infinitive may be governed by a verb {cease to do evil}, a noun {we all have talents to be improved}, an adjective {she is eager to learn}, a participle {they are preparing to go}, or a pronoun {let him do it}.
5.108 Split infinitive. Although from about 1850 to 1925 many grammarians stated otherwise, it is now widely acknowledged that adverbs sometimes justifiably separate the to from the principal verb {they expect to more than double their income next year}. See also 5.171.
5.109 Dangling infinitive. An infinitive phrase can be used, often loosely, to modify a verb—in which case the sentence must have a grammatical subject (or an unexpressed subject of an imperative) that could logically perform the action of the infinitive. If there is none, then the sentence may be confusing. For example, in To repair your car properly, it must be sent to a mechanic, the infinitive repair does not have a logical subject; the infinitive phrase to repair your car is left dangling. But if the sentence is rewritten as To repair your car properly, you must take it to a mechanic, the logical subject is you.
PARTICIPLES AND GERUNDS 5.110 Participles generally. A participle is a nonfinite verb that is not limited by person, number, or mood but does have tense. Two participles are formed from the verb stem: the present participle invariably ends in -ing, and the past participle usually ends in -ed. The present participle denotes the verb’s action as being in progress or incomplete at the time expressed by the sentence’s principal verb {watching intently for a mouse, the cat settled in to wait} {hearing his name, Jon turned to answer}. The past participle denotes the verb’s action as being completed {planted in the spring} {written last year}.
5.111 Participial phrases. A participial phrase is made up of a participle plus any closely associated word or words, such as modifiers or complements. It can be used (1) as an adjective to modify a noun or pronoun {nailed to the roof, the slate stopped the leaks} {she pointed to the clerk drooping 5.112 GRAMMAR AND USAGE behind the counter} or (2) as an absolute phrase {generally speaking, I prefer spicy dishes} {they having arrived, we went out on the lawn for our picnic}. For more on participial adjectives, see 5.90, 5.115.
5.112 Gerunds. A gerund is a present participle used as a noun. It is not limited by person, number, or mood. Being a noun, the gerund can be used as (1) the subject of a verb {complaining about it won’t help}; (2) the object of a verb {I don’t like your cooking}; (3) a predicate nominative or com plement {his favorite pastime is sleeping}; or (4) the object of a preposi tion {reduce erosion by terracing the fields}. In some sentences, a ger und may substitute for an infinitive. Compare the use of the infinitive to lie as a noun {to lie is wrong} with the gerund lying {lying is wrong}.
5.113 Distinguishing between participles and gerunds. Because participles and gerunds both derive from verbs, the difference between them de pends on their function. A participle is used as a modifier {the running water} or as part of a verb phrase {the meter is running}; it can be modi fied only by an adverb {the swiftly running water}. A gerund is used as a noun {running is great exercise}; it can be modified only by an adjective {sporadic running and walking makes for a great workout}.
5.114 Fused participles. As nouns, gerunds are modified by adjectives {double-parking is prohibited}, including possessive nouns and pro nouns {Critt’s parking can be hazardous to pedestrians}. By contrast, a present participle is always modified (if at all) by an adverb, whether the participle serves as a verb {she’s parking the car now}, an adjective {I’ll be looking for a parking place}, or an adverb {finally parking, we saw that the store had already closed}. It is traditionally considered a linguis tic fault (a fused participle) to use a nonpossessive noun or pronoun with a gerund: po o r : Me painting your fence depends on you paying me first. b e t t e r : My painting your fence depends on your paying me first.
In the poor example, me looks like the subject of the sentence, but it doesn’t agree with the verb depends. Instead, the subject is painting—a gerund, here seeming to be “modified” by me, a pronoun. In the pred icate, you looks like the object of the preposition on, but the true object is the gerund paying. There are times, however, when the possessive is unidiomatic. You usually have no choice but to use a fused participle with a nonpersonal noun {we’re not responsible for the jewelry having been mislaid}, a nonpersonal pronoun {we all insisted on something being Grammar 5.116 done}, or a group of pronouns {the settlement depends on some of them agreeing to compromise}.
5.115 Dangling participles. Both participles and gerunds are subject to dan gling. A participle that has no syntactic relationship with the nearest sub ject is called a dangling participle or just a dangler. In effect, the participle ceases to function as a modifier and functions as a kind of preposition. Often the sentence is illogical, ambiguous, or even incoherent, as in Fre quently used in early America, experts suggest that shaming is an effective punishment (used does not modify the closest noun, experts] it modifies shaming), or Being a thoughtful mother, I believe Meg gives her children good advice (the writer at first seems to be attesting to his or her own thought fulness rather than Meg’s). Recasting the sentence so that the misplaced modifier is associated with the correct noun is the only effective cure {experts suggest that shaming, often used in early America, is an ef fective punishment}. But rewording to avoid the participle or gerund may be preferable {I believe that because Meg is a thoughtful mother, she gives her children good advice}. Using passive voice in an independent clause can also produce a dangler. In Finding that the questions were not ambiguous, the exam grades were not changed, the participle finding “dan gles” because there is no logical subject to do the finding. The sentence can be corrected by using active voice instead of passive, so that the par ticiple precedes the noun it modifies {finding that the questions were not ambiguous, the teacher did not change the exam grades}. Quite often writers will use it or there as the subject of the independent clause after a participial phrase, thereby producing a dangler without a logical subject, as in Reviewing the suggestions, it is clear that no consensus exists. (A possible revision: Our review of the suggestions shows that no consensus exists.) Compare 5.116. See also 5.175.
5.116 Dangling gerunds. A dangling gerund can occur when a participle is the object of a preposition, where it functions as a noun rather than as a modifier. For example, After finishing the research, the screenplay was easy to write (who did the research and who wrote the screenplay?). The best way to correct a dangling gerund is to revise the sentence. The example above could be revised as After Gero finished the research, the screenplay was easy to write, or After finishing the research, Gero found the screenplay easy to write. Dangling gerunds can result in improbable statements. Consider While driving to San Antonio, my phone ran out of power. The phone wasn’t at the wheel, so driving is a dangling gerund that shouldn’t refer to my phone. Clarifying the subject of the gerund improves the sen tence {while I was driving to San Antonio, my phone ran out of power}. Compare 5.115.
5.117 GRAMMAR AND USAGE VOICE, MOOD, TENSE, PERSON, AND NUMBER 5.117 Five properties of verbs. A verb has five properties: voice, mood, tense, person, and number. Verbs are conjugated (inflected) to show these properties.
5.118 Active and passive voice. Voice shows whether the subject acts (active voice) or is acted on (passive voice)—that is, whether the subject per forms or receives the action of the verb. Only transitive verbs are said to have voice. The clause the judge levied a $50 fine is in the active voice because the subject judge is acting. But the tree's branch was broken by the storm is in the passive voice because the subject branch does not break itself—it is acted on by the prepositional object storm. The passive voice is always formed by joining an inflected form of to be (or, in colloquial usage, get) with the verb’s past participle. Compare the ox pulls the cart (active voice) with the cart is pulled by the ox (passive voice). As a mat ter of style, passive voice {the matter will be given careful consider ation} is typically, though not always, inferior to active voice {we will consider the matter carefully}. The choice between active and passive voice may depend on which point of view is desired. For instance, the mouse was caught by the cat describes the mouse’s experience, whereas the cat caught the mouse describes the cat’s. What is important is to be able to identify passive voice reliably. Remember that the mere presence of a fo-verb does not necessarily signal passive voice. For example, he is thinking about his finances isn’t in the passive voice; it’s just a fo-verb plus a present participle.
5.119 Progressive conjugation and voice. If an inflected form of be is joined with a verb’s present participle, a progressive conjugation results {the ox is pulling the cart}. If the verb is transitive, the progressive conjugation is in active voice because the subject is performing the action, not being acted on. But if both the principal verb and the auxiliary are be-verbs fol lowed by a past participle {the cart is being pulled}, the result is a passive voice construction.
5.120 Verb mood. Mood (or mode) indicates the manner in which the verb ex presses an action or state of being. The three moods are indicative, im perative, and subjunctive.
5.121 Indicative mood. The indicative mood is the most common in English. It is used to express facts and opinions and to ask questions {amethysts cost very little} {the botanist lives in a garden cottage} {does that bush produce yellow roses?}.
Grammar 5.124 5.122 Imperative mood. The imperative mood expresses commands {go away!}, direct requests {bring the tray in here}, and, sometimes, permis sion {come in!}. It is simply the verb’s stem used to make a command, a request, an exclamation, or the like {put it here!} {give me a clue} {help!}. The subject of the verb, you, is understood even though the sentence might include a direct address {give me the magazine} {Cindy, take good care of yourself [Cindy is a direct address, not the subject]}. Use the imperative mood cautiously: in some contexts it could be too blunt or un intentionally rude. You can soften the imperative by using a word such as please {please stop at the store}. If that isn’t satisfactory, you might recast the sentence in the indicative {will you stop at the store, please?}.
5.123 Subjunctive mood. Although the subjunctive mood no longer appears with much frequency, it is useful when you want to express an action or a state not as a reality but as a mental conception. Typically, the subjunc tive expresses an action or state as doubtful, imagined, desired, condi tional, hypothetical, or otherwise contrary to fact. Despite its decline, the subjunctive mood persists in stock expressions such as perish the thought, heaven help us, and be that as it may, 5.124 Subjunctive versus indicative mood. The subjunctive mood signals a statement contrary to fact {if I were you}, including wishes {if I were a rich man}, conjectures {oh, were it so}, demands {the landlord insists that the dog go}, and suggestions {I recommend that she take a vaca tion}. Three errors often crop up with these constructions. First, writers sometimes use an indicative verb form when the subjunctive form is needed: po o r : If it wasn’t for your help, I never would have found the place. b e t t e r : If it weren’t for your help, I never would have found the place.
Second, indicative-mood sentences sometimes resemble these subjunc tive constructions but aren’t statements contrary to fact: p o o R: I called to see whether she were in. b e t t e r : I called to see whether she was in.
Third, one often sees If I would have gone, I would..., with two condi tionals, instead of If I had gone, I would... (the better choice). Although the subjunctive mood is often signaled by if, not every i/takes a subjunc tive verb. When the action or state might be true but the writer does not know, the indicative is called for instead of the subjunctive {if I am right 2fi5 5.125 GRAMMAR AND USAGE about this, please call} {if Napoleon was in fact poisoned with arsenic, historians will need to reevaluate his associates}.
5.125 Present subjunctive mood. The present-tense subjunctive mood is formed by using the base form of the verb, such as be. This form of sub junctive often appears in suggestions or requirements {he recommended that we be ready at a moment’s notice} {we insist that he retain control of the accounting department}. The present-tense subjunctive is also ex pressed by using either be plus the simple-past form of the verb or a past form auxiliary plus an infinitive {the chair proposed that the company be acquired by the employees through a stock-ownership plan} {today would be convenient for me to search for that missing file} {might he take down the decorations this afternoon?}. See also 5.123.
5.126 Past subjunctive mood. Despite its label, the past-tense subjunctive mood refers to something in the present or future but contrary to fact. It is formed using the verb’s simple-past tense, except in the case of be, which becomes were regardless of the subject’s number. For example, the declaration if only I had a chance expresses that the speaker has little or no chance. Similarly, I wish I were safe at home almost certainly means that the speaker is not at home and perhaps not safe—though it could also mean that the speaker is at home but quite unsafe. This past-tense- but-present-sense subjunctive typically appears in the form if I (he, she, it) were {if I were king} {if she were any different}. That is, the subjunc tive mood ordinarily uses a past-tense verb (e.g., were) to connote uncer tainty, impossibility, or unreality where the present or future indicative would otherwise be used. Compare If I am threatened, I will quit (indica tive) with If I were threatened, I would quit (subjunctive), or If the canary sings, I smile (indicative) with If the canary sang (or should sing, or were to sing), I would smile (subjunctive).
5.127 Past-perfect subjunctive mood. Just as the past subjunctive uses a verb’s simple-past-tense form to refer to the present or future, the past-perfect subjunctive uses a verb’s past-perfect form to refer to the past. The past perfect subjunctive typically appears in the form if I (he, she, it) had been {if he had been there} {if I had gone}. That is, the subjunctive mood ordi narily uses a past-perfect verb (e.g., had been) to connote uncertainty or impossibility where the past or past-perfect indicative would otherwise be used. Compare If it arrived, it was not properly filed (indicative) with If it had arrived, it could have changed the course of history (subjunctive).
5.128 Verb tense. Tense shows the time in which an act, state, or condition occurs or occurred. The three major divisions of time are present, past, Grammar 5.130 and future (but see 5.131). Each division of time breaks down further into a perfect tense denoting a comparatively more remote time by indicating that the action has been completed: present perfect, past perfect, and future perfect. And all six of these tenses can be further divided to in clude a progressive tense (also called imperfect or continuous), in which the action continues.
5.129 Present tense. The present tense is the infinitive verb’s stem, also called the present indicative {walk} {drink}. It primarily denotes acts, conditions, or states that occur in the present {the dog howls} {the air is cold} {the water runs}. It is also used (1) to express a habitual action or general truth {cats prowl nightly} {polluted water is a health threat}; (2) to refer to timeless facts, such as memorable persons and works of the past that are still extant or enduring {Julius Caesar describes his strategies in The Gallic War} {the Pompeiian mosaics are exquisite}; and (3) to narrate a fictional work’s plot {the scene takes place aboard the Titanic}. The latter two uses are collectively referred to as the historical-present tense, and the third is especially important for those who write about literature. Char acters in books, plays, and films do things—not did them. If you want to distinguish between present action and past action in literature, the present-perfect tense is helpful {Hamlet, who has spoken with his fa ther’s ghost, reveals what he has learned to no one but Horatio}. See also 5.132.
5.130 Past indicative tense. The past indicative denotes an act, state, or con dition that occurred or existed at some explicit or implicit point in the past {the auction ended yesterday} {we returned the shawl}. For a regular verb, it is formed by adding -ed to its base form {jump-jumped} {spill- spilled}. If the verb ends in a silent -e, only a -d is added to form both the past tense and the past participle {bounce-bounced-bounced}. If it ends in -y preceded by a consonant, the -y changes to an -i before forming the past tense and past participle with -ed {hurry-hurried-hurried}. If it ends in a double consonant {block}, two vowels and a consonant {cook}, or a vowel other than -e {veto}, a regular verb forms the past tense and past participle by adding -ed to its simple form {block-blocked-blocked} {cook-cooked-cooked} {veto-vetoed-vetoed}. If the verb ends in a single vowel before a consonant, several rules apply in determining whether the consonant is doubled. It is always doubled in one-syllable words {pat-patted-patted}. In words of more than one syllable, the final con sonant is doubled if it is part of the syllable that is stressed both before and after the inflection {prefer-preferred-preferred}, but not otherwise {travel-traveled-traveled}. In British English there is no such distinction: all such consonants are doubled. Irregular verbs form the past tense 5.131 GRAMMAR AND USAGE and past participle in various ways {give-gave-given} {stride-strode- stridden} {read-read-read}. See also 5.100.
5.131 Future tense. What is traditionally known as the future tense is formed by using will with a verb’s stem form {will walk} {will drink}. It refers to an expected act, state, or condition {the artist will design a wall mural} {the restaurant will open soon}. Shall may be used instead of will, but in American English it typically appears only in first-person questions {shall we go?} and in statements of legal requirements {the debtor shall pay within 30 days}. In most contexts, will is preferred—or must with legal requirements. Most linguists are now convinced that, technically speaking, English has no future tense at all—that will is simply a modal verb that should be treated with all the others.6 Yet the future tense re mains a part of traditional grammar and is discussed here in the famil iar way.
5.132 Present-perfect tense. The present-perfect tense is formed by using have or has with the principal verb’s past participle {have walked} {has drunk}. It denotes an act, state, or condition that is now completed or continues up to the present {I have put away the clothes} {it has been a long day} {I will apologize, even if I have done nothing wrong}. The pres ent perfect is distinguished from the past tense because it refers to (1) a time in the indefinite past {I have played golf there before} or (2) a past action that comes up to and touches the present {I have played cards for the last eighteen hours}. The past tense, by contrast, indicates a more specific or a more remote time in the past.
5.133 Past-perfect tense. The past-perfect (or pluperfect) tense is formed by using had with the principal verb’s past participle {had walked} {had drunk}. It refers to an act, state, or condition that was completed before another specified or implicit past time or past action {the engineer had driven the train to the roundhouse before we arrived} {by the time we stopped to check the map, the rain had begun falling} {the movie had already ended}.
5.134 Future-perfect tense. The future-perfect tense is formed by using will have with the verb’s past participle {will have walked} {will have drunk}. It refers to an act, state, or condition that is expected to be completed before some other future act or time {the entomologist will have col lected sixty more specimens before the semester ends} {the court will have adjourned by five o’clock}.
6. See, e.g., R. L. Trask, Language: The Basics (London: Routledge, 1995), 58.
Grammar 5.138 5.135 Progressive tenses. The progressive tenses, also known as continuous tenses, show action that progresses or continues. With active-voice verbs, all six basic tenses can be made progressive by using the appropriate be- verb and the present participle of the main verb, as so: present progressive (he is playing tennis); present-perfect progressive (he has been playing tennis); past progressive (he was playing tennis); past-perfect progressive (he had been playing tennis); future progressive (he will be playing tennis); and future-perfect progressive (he will have been playing tennis).
With the passive voice, the present- and past-progressive tenses are made by using the appropriate fo-verb with the present participle being, plus the past participle of the main verb, as so: present (I am being dealt the cards); and past (I was being dealt the cards).
5.136 Verb person. A verb’s person shows whether the act, state, or condition is that of (1) the person speaking (first person), (2) the person spoken to (second person), or (3) the person or thing spoken of (third person).
5.137 Verb number. The number of a verb must agree with the number of the noun or pronoun used with it. In other words, the verb must be singular or plural. Only the third-person present-indicative singular changes form to indicate number and person {I sketch} {you sketch} {she sketches} {they sketch}. The second-person verb is always plural in form, whether one person or more than one person is spoken to {you are a wonderful person} {you are wonderful people}.
5.138 Agreement in person and number. A finite verb agrees with its subject in person and number—which is to say that a singular subject takes a singular verb {the solution works}, while a plural subject takes a plural verb {the solutions work}. When a verb has two or more subjects con nected by and, it agrees with them jointly and is plural {Socrates and Plato were wise}. When a verb has two or more subjects connected by or or nor, the verb agrees with the last-named subject {Bob or his friends have your key} {neither the twins nor Jon is prepared to leave}. When the subject is a collective noun conveying the idea of unity or multitude, the verb is singular {the nation is powerful}. When the subject is a collec tive noun conveying the idea of plurality, the verb is plural {the faculty were divided in their sentiments}. See also 5.15.
5.139 GRAMMAR AND USAGE 5.139 Agreement of indefinite pronouns. An indefinite pronoun such as any body^ anyone, everybody, everyone, nobody, no one, somebody, or some one routinely takes a singular verb {everyone receives credits for this course} {somebody knows where the car is}.
5.140 Relative pronouns as subjects. A relative pronoun used as the subject of a clause can be either singular or plural, depending on the pronoun’s antecedent {a woman who likes skydiving} {people who collect books}. One of the trickiest constructions involves one of those who or one of those that: po o r : She is one of those employees who works tirelessly. b e t t e r : She is one of those employees who work tirelessly.
In this construction, the subject of the verb work is who, and the anteced ent of who is employees, not one. You can see this easily if you reorder the syntax (without adding or subtracting a word): Of those employees who work tirelessly, she is one. See also 5.62.
5.141 False attraction to predicate noun. When the subject and a predicate noun differ in number, the subject governs the number of the verb {me diocrity and complacency are the source of his ire} {the source of his ire is mediocrity and complacency}. A plural predicate noun after a sin gular subject may mislead a writer into error by suggesting a plural verb. When this occurs, the simple correction of changing the number of the verb may make the sentence awkward, and the better approach then is to rework the sentence: po o r : My downfall are sweets. b e t t e r : My downfall is sweets. b e s t : Sweets are my downfall.
5.142 Misleading connectives—"as well as," "along with," "together with," and the like. Adding to a singular subject by using a phrasal connective such as along with, as well as, in addition to, together with, and the like does not make the subject plural. This type of distraction can be doubly mis leading because the intervening material seems to create a compound subject, and the modifying prepositional phrase may itself contain one or more plural objects. If the singular verb sounds awkward in such a sen tence, try the conjunction and instead: po o r : The bride as well as her bridesmaids were dressed in mauve. b e t t e r : The bride as well as her bridesmaids was dressed in mauve. b e s t : The bride and her bridesmaids were dressed in mauve.
Grammar 5.146 5.143 Agreement in first and second person. A personal pronoun used as a subject requires the appropriate verb form according to the person of the pronoun: I am he is Igo he goes you are she is you go she goes we are it is we go it goes they are they go Here comes the tricky point: pronouns joined by or, either-or, or neither- nor are traditionally said to take the verb form that agrees with the nearer subject {either he or I am in for a surprise} {either you or he is right} {neither you nor I am a plumber}. Because these constructions are ad mittedly awkward, speakers and writers typically find another way to express the thought {one of us is in for a surprise} {one of you is right} {neither of us is a plumber}.
AUXILIARY VERBS 5.144 Auxiliary verbs generally. An auxiliary verb (sometimes termed a helping verb) is a highly irregular verb used with one or more other verbs to form voice, tense, and mood. It always precedes the principal verb. The most common auxiliary verbs are explained in the following sections. See also 5.103.
5.145 Modal auxiliaries. A subset of auxiliary verbs, called modal auxiliaries or modals, are used to express ability, necessity, possibility, willingness, obligation, and the like {they might be there} {she could be leaving at this very moment}. They are so called because they indicate the principal verb’s mood. All the verbs described below are modal auxiliaries except the last two: do and have.
5.146 "Can" and "could." Can uses only its stem form in the present indica tive {I can} {it can} {they can}. In the past indicative, can becomes could for all persons {he could see better with glasses}. Can does not have an infinitive form (to be able to is substituted) or a present or past partici ple. (Such words lacking one or more inflected forms normal for their word class are traditionally called defective. Most modal auxiliaries are defective verbs.) When it denotes ability, capacity, or permission, can is always followed by an explicit or implicit bare infinitive as the principal verb {you can carry this trunk}. When used in the sense of permission, can is colloquial for may {can I go to the movies?}. Can also connotes 5.147 GRAMMAR AND USAGE actual possibility or common experience {storms can be severe in spring} {days can pass before a decision is announced}. Could is often used to talk about the past {she could hum a tune at six months of age} or to discuss someone’s general ability at a given time {when he was eleven, he could drive a golfball 250 yards}. But could is also used as a softer, less definite equivalent of can in reference to future events {we could travel to Cancun if you wanted to}. In this use, the meaning is close to “would be able to” {you could be promoted within six months if you’d just apply yourself!}.
5.147 "May" and "might." May denotes either permission {you may go to the movies} or possibility {I may go to the movies}. In negating permission, may not is sometimes displaced by the more intensive must not. Com pare You may not climb that tree with You must not climb that tree. May most commonly connotes an uncertain possibility {you may find that assignment too difficult}, and it often becomes might {you might find that assignment too difficult}. Is there a connotative difference? Yes: may tends to express likelihood {we may get there on time}, while might ex presses a stronger sense of doubt {we might get there on time—if the traffic clears}. Might can also express a contrary-to-fact hypothetical {we might have been able to make it if the traffic had been better}.
5.148 "Must." Must denotes a necessity that arises from someone’s will {we must obey the rules}, from circumstances {you must ask what the next step is}, or from rule or obligation {all applications must be received by May 31 to be valid}. Must also connotes a logical conclusion {that must be the right answer} {that must be the house we’re looking for} {it must have been Donna who phoned}. This auxiliary verb does not vary its form in either the present or past indicative. It does not have an infinitive form (to have to is substituted) or a present or past participle. Denoting obligation, necessity, or inference, must is always used with a bare infinitive {we must finish this design} {everyone must eat} {the movie must be over by now}.
5.149 "Ought." Ought denotes either what is reasonably expected of a person as a matter of duty {they ought to fix the fence} or what we guess or con clude is probable {they left at dawn, so they ought to be here soon}. It is more emphatic than should but less strong than must. This verb does not vary its form in either the present or past indicative. It has no infinitive form, or present or past participle. Denoting a duty or obligation, ought is always used with an infinitive, even in the negative {we ought to invite some friends} {the driver ought not to have ignored the signal}. To is oc Grammar 5.153 casionally omitted after not {you ought not worry}, but the better usage is to include it {you ought not to worry}. See also 5.106.
5.150 "Should." Should, the past-indicative form of shall, is used for all per sons, and always with a principal verb {they should be at home} {should you read that newspaper?}. Should does not have an infinitive form or a present or past participle. Should often carries a sense of duty, compul sion, or expectation {I should review those financial-planning tips} {you should clean the garage today} {it should be ready by now}. Sometimes it carries a sense of inference {the package should have been delivered today}. And sometimes it conveys the speaker’s attitude {how should I know?} {you shouldn’t have to deal with that}. Should and ought are quite similar and often interchangeable in discussions of what is re quired, what is advisable, or what we think it is right for people either to do or to have done. Should is slightly less emphatic than ought, but it appears with greater frequency.
5.151 "Will" and "would." In its auxiliary uses, will uses only its stem form in the present indicative {she will} {they will}. In the past indicative, the only form for all persons is would {we would go fishing on Saturdays} {she would say that!}. Will often carries a sense of the future {she will be at her desk tomorrow} or, in the past form would, expresses a conditional statement {I would recognize the house if I saw it again}. It can also ex press certainty {I’m sure you will understand}; decisions and other types of volition {I really will work out more}; requests, orders, and offers {will you stop that!} {will you take $5 for it?}; or typical behavior {she will read for hours on end}.
5.152 "Do." The auxiliary verb do (sometimes called a dummy auxiliary) fre quently creates emphatic verbs. It has two forms in the present indica tive: does for the third-person singular and do for all other persons. In the past indicative, the only form for all persons is did. The past participle is done. As an auxiliary verb, do is used only in the present indicative {we do plan some charity work} and past indicative {did you speak?}. When the verb in an imperative statement is coupled with not, do also appears {do not touch!} {don’t be an idiot!}. When denoting performance, do can also act as a principal verb {he does well in school} {they do good work}. Do can sometimes substitute for a verb, thereby avoiding repe tition {Marion dances well, and so do you} {he caught fewer mistakes than you did}.
5.153 "Have." This verb has two forms in the present indicative: has for the third-person singular and have for all other persons. In the past indicative, 5.154 GRAMMAR AND USAGE the only form for all persons is had; the past participle is also had. When have functions as an auxiliary verb, the present or past indicative of have precedes the past participle of a verb to form that verb’s present-perfect or past-perfect indicative mood {I have looked everywhere} {he had looked for a better rate}. When preceding an infinitive, have denotes ob ligation or necessity {I have to finish this paper tonight!}. Had plus to and an infinitive expresses the past form of must {I had to leave yester day afternoon}. When denoting possession, action, or experience, have functions as a sentence’s principal verb {she has a car and a boat} {you have a mosquito on your neck} {we’ll have a party next week}. Have may also be used with do to express actual or figurative possession {do you have the time?} {do we have room?} {Vicky did not have her coat}.
“be ”-v e r bs 5.154 Forms of "be"-verbs. The verb be has eight forms (be, is, are, was, were, been, being, and am) and has several special uses. First, it is sometimes a sentence’s principal verb meaning “exist” {I think, therefore I am}. Sec ond, it is more often used as an auxiliary verb {I was born in Lubbock}. When joined with a verb’s present participle, it denotes continuing or progressive action {the train is coming} {the passenger was waiting}. When joined with a past participle, the verb becomes passive {a signal was given} {an earring was dropped} (see 5.118). Often this type of con struction can be advantageously changed to active voice {he gave the signal} {she dropped her earring}. Third, be is the most common linking verb that connects the subject with something affirmed of the subject {truth is beauty} {we are the champions}. Occasionally a fo-verb is used as part of an adjective {a rock star wannabe [want to be]} {a would-be hero} or noun {a has-been}.
5.155 Conjugation of "be"-verbs. Be is conjugated differently from other verbs. (1) The stem is not used in the present indicative form. Instead, be has three forms: for the first-person singular, am; for the third-person singu lar, is; and for all other persons, are. (2) The present participle is formed by adding -mg to the root be {being}. It is the same for all persons, but the present perfect requires also using am, is, or are {I am being stalked} {it is being reviewed} {you are being photographed}. (3) The past indica tive has two forms: the first- and third-person singular use was; all other persons use were {she was} {we were}. (4) The past participle for all per sons is been {I have been} {they have been}. (5) The imperative is the verb’s stem {be yourself!}.
Grammar 5.158 Adverbs DEFINITION AND FORMATION 5.156 Adverbs generally. An adverb is a word (more particularly, an adjunct) that qualifies, limits, describes, or modifies a verb, an adjective, or an other adverb {she studied constantly [constantly qualifies the verb stud ied]} {the juggler’s act was really unusual [really qualifies the adjective unusual]} {the cyclist pedaled very swiftly [very qualifies the adverb swiftly]}. An adverb may also qualify a preposition, a conjunction, or an entire independent clause {the birds flew right over the lake [right qualifies the preposition over]} {this is exactly where I found it [exactly qualifies the conjunction where]} {apparently you forgot to check your references [apparently qualifies the rest of the clause]}. Some adverbs may modify an adjective {the bids differ by a very small amount} or an adverb {he moved along very quickly} but not a verb. (You can’t say He spoke very or She played very.) Other adverbs of this sort—often called intensifiers—are more, most, much, quite, rather, really, somewhat, and too (see also 5.89). Grammarians have also traditionally used the term ad verb as a catchall category to sweep in words that aren’t readily put into other categories (such as not, please, and the infinitival to and the particle in a phrasal verb).
5.157 Sentence adverbs. An adverb that modifies an entire sentence is called a sentence adverb {fortunately, we’ve had rain this week} {undoubtedly he drove his car to the depot}. Sentence adverbs most commonly indicate doubt or emphasize a statement’s certainty. Some common examples are maybe, possibly, and however.
5.158 Adverbial suffixes. Many adjectives have corresponding adverbs distin guished by the suffix -ly or, after most words ending in -ic, -ally {slow- slowly} {careful-carefully} {public-publicly} {pedantic-pedantically}. Most adjectives ending in -y preceded by a consonant change the -y to -i when the suffix is added, but some don’t {happy-happily} {shy-shyly}. A few adjectives ending in -e drop the vowel {true-truly} {whole-wholly}. If an adjective ends in an -le that is sounded as part of a syllable, it is replaced with -ly {terrible-terribly} {simple-simply}. An adjective that ends in a double -/ takes only a -y suffix {dull-dully}. Many adjectives ending in -le or -ly do not make appealing adverbs {juvenile-juvenilely} {silly—sillily}. If an -ly adverb looks clumsy (e.g., ghastlily, uglily), either rephrase the sentence or use a phrase {in a ghastly manner} {in an ugly 5.159 GRAMMAR AND USAGE way}. A few other suffixes are used for adverbs, especially in informal speech {he rides cowboy-style} {park your cars curbside}. A few nouns form adverbs by taking the ending -ways {side-sideways}, -ward {sky skyward}, or -wise {clock-clockwise}. And adverbial suffixes are some times added to phrases {she replied matter-of-factly}. Finally, not every word ending in -ly is an adverb—some are adjectives {lovely} {curly}.
5.159 Adverbs without suffixes. Many common adverbs don’t have an iden tifying suffix {almost} {never} {here} {now} {just} {seldom} {late} {near} {too}.
SIMPLE VERSUS COMPOUND ADVERBS 5.160 Simple and flat adverbs. A simple adverb is a single word that qualifies a single part of speech {hardly} {now} {deep}. A flat or bare adverb is one that has an -ly form but whose adjectival form may work equally well or even better, especially when used with an imperative in an informal context {drive slow} {hold on tight} {tell me quick}. Some flat adverbs are always used in their adjectival form {work fast} because the -ly has be come obsolete (although it may linger in related words—e.g., steadfast and steadfastly). And the flat adverb may have a different meaning from the -ly adverb. Compare I am working hard with I am hardly working.
5.161 Phrasal and compound adverbs. A phrasal adverb consists of two or more words that function together as an adverb {in the meantime} {for a while} {here and there}. A compound adverb appears to be a single word but is a compound of several words {notwithstanding} {heretofore} {thereupon}. Compound adverbs should be used cautiously and spar ingly because they tend to make the tone stuffy.
ADVERBIAL DEGREES 5.162 Positive adverbs. Like adjectives (see 5.84), adverbs have three degrees: the positive, the comparative, and the superlative. A positive adverb simply expresses a quality without reference to any other thing {the nurse spoke softly} {the choir sang merrily}.
5.163 Comparative adverbs. A comparative adverb compares the quality of a specified action done by two persons, groups, or things {Bitey worked longer than Arachne} {Rachel studied more industriously than Edith}. Most one-syllable adverbs that do not end in -ly form the com Grammar 5.167 parative by taking the suffix -er {sooner} {harder}. These forms are called synthetic comparatives. Multisyllable adverbs usually form the compar ative with more or less {the Shakespearean villain fenced more ineptly than the hero} {the patient is walking less painfully today}. These forms are called periphrastic comparatives. But there are exceptions for adverbs that end in -ly if the -ly is not a suffix {early-earlier}.
5.164 Superlative adverbs. A superlative adverb compares the quality of a specified action done by at least three persons, groups, or things {Sullie bowled fastest of all the cricketers} {of the three doctoral candidates, Dunya defended her dissertation the most adamantly}. In a loose sense, the superlative is sometimes used for emphasis rather than com parison {the pianist played most skillfully}. Most one-syllable adverbs that do not end in -ly form the superlative by taking the suffix -est {soon est} {hardest}. These forms are called synthetic superlatives. Multisyllable adverbs usually form the superlative with most or least {everyone’s eye sight was acute, but I could see most acutely} {of all the people making choices, he chose least wisely}. These forms are called periphrastic su perlatives. There are exceptions for adverbs that end in -ly if the -ly is not a suffix {early-earliest}.
5.165 Irregular adverbs. A few adverbs have irregular comparative and super lative forms {badly-worse-worst} {little-less-least}. A good dictionary is the best resource for finding an irregular adverb’s forms of comparison.
5.166 Noncomparable adverbs. Many adverbs are noncomparable. Some, by their definitions, are absolute and cannot be compared {eternally} {never} {singly} {uniquely} {universally}. Most adverbs indicating time {now} {then}, position {on}, number {first} {finally}, or place {here} are also noncomparable.
POSITION OF ADVERBS 5.167 Placement of adverbs. To avoid miscues, an adverb should generally be placed as near as possible to the word it is intended to modify. For ex ample, in the marathoners submitted their applications to compete imme- diately, what does immediately modify—compete or submitted? Placing the adverb with the word it modifies makes the meaning clear—e.g., the marathoners immediately submitted their applications to compete. A misplaced adverb can completely change a sentence’s meaning. For ex ample, we nearly lost all our camping equipment states that the equipment was saved; we lost nearly all our camping equipment states that almost 5.168 GRAMMAR AND USAGE everything was lost. An adverb’s placement is also important because adverbs show time {we’ll meet again}, place or source {put the flowers here} {where did you get that idea?}, manner {speak softly}, degree or extent {sales are very good} {how far is it to the British pub?}, reason {I don’t know why Pat couldn’t find the right answer}, consequence {we should therefore hasten to support her candidacy}, and number {first, we need to get our facts straight}. Adverbs can also express comments or observations {Vic was undoubtedly late} {Imani clearly recalled every thing}.
5.168 Adverbs that modify words other than verbs. If an adverb qualifies an adjective, an adverb, a preposition, or a conjunction, it should immedi ately precede the word qualified {our vacation was very short} {the flight took too long} {your fence is partly over the property line} {leave only when the bell rings}. The adverb or adverbs modifying a single adjective, grouped with that adjective, are called an adjective cluster {a classically trained pianist}.
5.169 Adverbs that modify intransitive verbs. If an adverb qualifies an intran sitive verb, it generally follows the verb {the students sighed gloomily when homework was assigned} {the owl perched precariously on a thin branch}. Some exceptions are always, never, often, generally, rarely, and seldom, which may precede the verb {mountaineers seldom succeed in climbing K2}.
5.170 Adverbs and linking verbs. Adverbs do not generally follow linking verbs (see 5.101), such as fo-verbs, appear, become, feel, hear, look, seem, smell, and taste. These verbs connect a descriptive word with the clause’s subject; the descriptive word after the verb applies to the subject, not the verb {he seems honest}. To determine whether a verb is a linking verb, consider whether the descriptive word describes the action or condition, or the subject. For example, the sculptor feels badly literally describes an impaired tactile sense (though that couldn’t conceivably be the intended meaning). But the sculptor feels bad describes the sculptor as unwell or perhaps experiencing guilt (bad being not an adverb but a predicate adjective). Those adverbs that typically precede intransitive verbs (see 5.169), however, may modify linking verbs {she quickly became uncom fortable} {he sometimes seems dishonest}.
5.171 Adverb within a verb phrase. When an adverb qualifies a verb phrase, the normal place for the adverb is between the auxiliary verb and the prin cipal verb {the administration has consistently repudiated this view} {the reports will soon generate controversy} {public opinion is sharply Grammar 5.173 divided}. (See 5.104.) Some adverbs may follow the principal verb {you must go quietly} {are you asking rhetorically?}. There has never been a rule against placing an adverbial modifier between the auxiliary verb and the principal verb in a verb phrase. In fact, it’s typically preferable to put the adverb there {the heckler was abruptly expelled} {the bus had been seriously damaged in the crash}. Sometimes it is perfectly appropriate to split an infinitive with an adverb to add emphasis, clarify meaning, or produce a natural sound. (See 5.108.) A verb’s infinitive or to form is split when an intervening word immediately follows to {to bravely assert}. If the adverb bears the emphasis in a phrase {to boldly go} {to strongly favor}, the split infinitive is justified and often even necessary. But if moving the adverb to the end of the phrase doesn’t suggest a dif ferent meaning or impair the sound, then you have an acceptable way to avoid splitting the infinitive. Recasting a sentence just to eliminate a split infinitive or to avoid splitting the infinitive can alter the nuance or meaning of the sentence. For example, ifs best to always get up early (al ways modifies get up) is not quite the same as ifs always best to get up early (always modifies best). It can also make the phrasing sound unnatural— e.g., ifs best to get up early always.
Prepositions DEFINITION AND TYPES 5.172 Prepositions generally. A preposition is an uninflected function word or phrase linking a noun element (the preposition’s object) with another part of the sentence to show the relationship between them. Prepositions express such notions as position (about, above, below, on, under), direc tion (in, into, to, toward), time (after, before, during, until), and source (from, of, out of). A preposition’s object (sometimes termed an oblique object) is usually a noun, or else a pronoun in the objective case {between me and them}. Usually a preposition comes before its object, but there are exceptions. For example, a preposition used with the relative pro noun that (or with that understood) always follows the object {this is the moment (that) I’ve been waiting for}. It also frequently, but not always, follows the relative pronouns which {which alternative is your decision based on?} {this is the alternative on which my decision is based} and whom {there is a banker (whom) I must speak with} {I can’t tell you to whom you should apply}. See also 5.180.
5.173 Simple and compound prepositions. Many prepositions are relatively straightforward. A simple preposition consists of a single monosyllabic 5.174 GRAMMAR AND USAGE word {as} {at} {by}. A compound preposition has two or more syllables; it may be made up of two or more words {into} {outside} {upon}.
5.174 Phrasal prepositions. A phrasal preposition, sometimes called a complex preposition, is two or more separate words used as a prepositional unit. These include according to, because of, by means of, by reason of, by way of, contrary to, for the sake of, in accordance with, in addition to, in apposition with, in case of, in consideration of, in front of, in regard to, in respect to, in spite of, instead of, on account of, out of, with reference to, with regard to, and with respect to. Many of these phrasal prepositions are symptoms of offi cialese, bureaucratese, or other types of verbose style. If a single-word preposition will do in context, use it. For example, if about will replace with regard to or in connection with, a judicious editor will inevitably pre fer to use the simpler expression.
5.175 Participial prepositions. A participial preposition is a participial form that functions as a preposition (or sometimes as a subordinating con junction). Examples are assuming, barring, concerning, considering, during, notwithstanding, owing to, provided, regarding, respecting, and speaking of Unlike other participles, these words do not create danglers when they have no subject {considering the road conditions, the trip went quickly} {regarding Watergate, he had nothing to say}. See 5.115.
PREPOSITIONAL PHRASES 5.176 Prepositional phrases generally. A prepositional phrase consists of a preposition, its object, and any words that modify the object. A prepo sitional phrase can be used as a noun {for James to change his mind would be a miracle}, an adverb (also called an adverbial phrase) {we strolled through the glade}, or an adjective (also called an adjectival phrase) {we’d love to see the cathedrals of Paris}.
5.177 Prepositional function. Prepositions signal many kinds of relationships. For example, a preposition may express a spatial relationship {to} {from} {out of} {into}, time {at} {for} {throughout} {until}, cause {because of} {on account of}, means {like} {with} {by}, possession {without} {of}, exceptions {but for} {besides} {except}, support {with} {for}, opposition {against}, or concession {despite} {for all} {notwithstanding}.
5.178 Placement of prepositional phrases. A prepositional phrase with an ad verbial or adjectival function should be as close as possible to the word it modifies to avoid awkwardness, ambiguity, or unintended meanings.
Grammar 5.183 Compare Is there a person with a small dog named Sandy here? (is the per son or the dog named Sandy?) with Is there a person named Sandy here with a small dog? Or compare The woman with the Popular Front circulates petitions with The woman circulates petitions with the Popular Front.
5.179 Refinements on placement. If a prepositional phrase equally modifies all the elements of a compound construction, the phrase follows the last element in the compound {the date, the place, and the budget for the wedding have been decided}. If the subject is singular and followed by a plural prepositional phrase, the predicate is singular—e.g., compare the predicate in the man and his two daughters have arrived with that in the man with two daughters has arrived and in the man has arrived with his two daughters.
5.180 Ending a sentence with a preposition. The traditional caveat of yester year against ending sentences or clauses with prepositions is an unnec essary and pedantic restriction. And it is wrong. As Winston Churchill is said to have put it sarcastically, “That is the type of arrant pedantry up with which I shall not put.” A sentence that ends in a preposition may sound more natural than a sentence carefully constructed to avoid a fi nal preposition. Compare, for example, This is the case I told you about with This is the case about which I told you. The “rule” prohibiting terminal prepositions was an ill-founded superstition based on a false analogy to Latin grammar. Today many grammarians use the dismissive term pied- piping for this phenomenon.
5.181 Clashing prepositions. If a phrasal verb {give in} precedes a preposi tional phrase {in every argument}, the back-to-back prepositions, if they are the same, will clash {he gives in in every argument}. Recast the sen tence when possible to avoid such juxtaposed prepositions—e.g., rather than continue arguing, he always gives in, or in every argument, he gives in. See also 6.55. For more on phrasal verbs, see 5.102.
5.182 Elliptical prepositional phrases. Sometimes a prepositional phrase is el liptical, being an independent expression without an antecedent. It often starts a clause and is normally detachable from the statement without af fecting the meaning. Elliptical prepositional phrases include for example, for instance, in any event, in a word, in the last analysis, and in the long run {in any event, call me when you arrive}.
5.183 Pronoun case in prepositional phrase. If a pronoun appears in a prep ositional phrase, the pronoun is usually in the objective case {with me} 5.184 GRAMMAR AND USAGE {alongside her} {between them} (see also 5.19, 5.35). But note that than may function as either a conjunction or a preposition {he’s taller than I [am]} {he’s taller than me}. In edited English, taller than I has predom inated over taller than me in American English from its very beginnings, and in British English it predominated until the 1990s. Throughout the literary history of Modern English, than me, than her, etc. have been re garded as less polished (to say the least) than than I, than she, etc. That is to say, in formal registers than (like as) is considered a conjunction, not a preposition. But in spoken English, than and as are often treated as prep ositions that take a pronoun in the objective case {you’re better than me} {you’re as well known as me}. A possessive pronoun may be used before the preposition’s object {to my house}.
OTHER PREPOSITIONAL ISSUES 5.184 Prepositions and functional variation. Some words that function as prepositions may also function as other parts of speech. The distinguish ing feature of a preposition is that it always has an object. A word such as above, behind, below, by, down, in, off, on, or up can be used as either an adverb or a preposition. When used as a preposition, it takes an ob ject {let’s slide down the hill}. When used as an adverb, it does not {we sat down}. Some conjunctions may serve as prepositions (e.g., than and but). Compare the prepositional but in everyone but Fuzzy traveled abroad last summer (but is used to mean “except”) with the conjunctive but in I like the cut but not the color (but introduces a clause containing an implied separate action: I don't like the color).
5.185 Use and misuse of "like." Like is probably the least understood prep osition. Its traditional function is adjectival, not adverbial, so that like governs nouns and noun phrases {teens often see themselves as star- crossed lovers like Romeo and Juliet}. As a preposition, like is followed by a noun or by a pronoun in the objective case {the person in that old portrait looks like me}. Increasingly today in ordinary speech, like dis places as or as if as a conjunction to connect clauses. For example, in it happened just like I said it would happen, traditional grammarians would want to replace like with as; and in youre looking around like you've mis placed something, like with as if. Because as and as if are conjunctions, they are followed by pronouns in the nominative case {do you work too hard, as I do?}. Although like as a conjunction has been considered non standard since the seventeenth century, today it is common in dialectal and colloquial usage {he ran like he was really scared}.
Grammar 5.190 5.186 Use and misuse of "only." Only functions as an adjective, an adverb, and a conjunction, and it can modify any part of speech. It is probably poorly placed in sentences more often than any other word. Only emphasizes the word or phrase that immediately follows it. When only appears too early in the sentence, it has a deemphasizing effect; it can also alter the meaning of the sentence or produce ambiguity. Compare I bought only tomatoes at the market (I bought nothing else) with I bought tomatoes only at the market (I bought nothing other than tomatoes or I didn’t buy toma toes from any other place?). In idiomatic spoken English, only is placed before the verb, regardless of what it modifies: I only bought tomatoes at the market. This may be acceptable in speech because the speaker can use intonation to make the meaning clear. But since in writing there is no guidance from intonation, rigorous placement of only aids reader com prehension.
LIMITING PREPOSITIONAL PHRASES 5.187 Avoiding overuse of prepositions. Prepositions can easily be overused. Stylistically, a good ratio to strive for is one preposition for every ten to fifteen words. Five editorial methods can reduce the number of preposi tions in a sentence.
5.188 Cutting prepositional phrases. If the surrounding prose’s context per mits, a prepositional phrase can be eliminated—e.g., the most important ingredient in this recipe could be reduced to the most important ingredient when it appears within a passage focused on a particular recipe.
5.189 Cutting unnecessary prepositions. A noun ending in -ance, -ence, -ity, -ment, -sion, or -tion is often formed from a verb {qualification-qualify} {performance-perform}. These nouns are sometimes called nominaliza- tions or zombie nouns, and they often require additional words, especially prepositions (that is, during her performance of the concerto is essentially equivalent to while she performed the concerto, but it is somewhat more abstract and requires the preposition of). Using a verb instead of a nomi- nalization often eliminates one or two prepositions. For example, toward maximization of becomes simply to maximize, so that our efforts toward maximization of profits failed might be edited down to our efforts to maxi mize profits failed.
5.190 Replacing prepositional phrases with adverbs. A strong adverb may re place a weaker prepositional phrase. For example, the president spoke with force is weak compared with the president spoke forcefully.
5.191 GRAMMAR AND USAGE 5.191 Replacing prepositional phrases with genitives. A genitive may replace a prepositional phrase, especially an o/-genitive. For example, I was dismayed by the complexity of the street map essentially equals The street map’s complexity dismayed me. See 5.20.
5.192 Using active voice to eliminate prepositions. Changing from the long passive voice (with by after the verb) to an active-voice construction al ways eliminates a preposition. For example, the ship was sailed by an expe rienced crew equals an experienced crew sailed the ship.
PREPOSITIONAL IDIOMS 5.193 Idiomatic uses of prepositions. Among the most persistent word-choice issues are those concerning prepositions. Which prepositions go with which words? You fill A with B but instill B into A; you replace A with B but substitute B for A; you prefix A to B preface B with A; you force A into B but enforce B on A; finally, A implies B, so you infer B from A. And that’s only the beginning of it.
5.194 Shifts in prepositional idiom. While prepositional idioms often give non native speakers of English nightmares, even native speakers of English may need to double-check them from time to time. Often the language undergoes some shifting. There may be a difference between traditional literary usage (oblivious of) and prevailing contemporary usage (oblivious to). Sometimes the writer may choose one or the other preposition for reasons of euphony. (Is it better, in a given context, to ruminate on, about, or over a specified,problem?) Sometimes, too, the denotative and conno- tative differences can be striking: it’s one thing to be smitten with another and quite a different thing to be smitten by another.
5.195 List of words and the prepositions construed with them. The list be low contains the words that most often give writers trouble. Note that some of the words included here—such as verbs that can be used transi tively {the tire abutted the curb} or words that can be used without fur ther qualification {she refused to acquiesce} {his words were considered blasphemy}—do not always take prepositions.
abide (vb.): with (“stay”); by (“obey”); none (transitive) abound (vb.): in, with [resources] absolve (vb.): from [guilt]; of [obligation] abut (vb.): on, against [land]; none (transitive) Grammar 5.195 accompanied (adj.): by (not with) [something or someone else] accord (vb.): in, with [an opinion]; to [aperson] acquiesce (vb.): in [a decision]; to [pressure] acquit (vb.): of (not from) [a charge]; none (transitive) adept (adj.): at [an activity]; in [an art] admit (vb.) (“acknowledge”): none (not to) (transitive) admit (vb.) (“let in”): to, into admit (vb.) (“allow”): of anxious (adj.): about, over (preferably not to) [a concern] badger (vb.): into [doing something]; about [a situation] ban (vb.): from [a place] ban (n.): on [a thing; an activity]; from [a place] based (adj.): on (preferably not upon) [a premise]; in [a place; a field of study]; at [a place] becoming (adj.): on, to [a person]; of [an office or position] bestow (vb.): on (preferably not upon) [an honoree] binding (adj.): on (preferably not upon) [a person] blasphemy (n.): against [a religious tenet] center (vb.): on, upon (not around) [a primary issue] chafe (vb.): at [doing something]; under [an irritating authority] coerce (vb.): into [doing something] cohesion (n.): between, among [things; groups] collude (vb.): with [a person to defraud another] commiserate (vb.): with [a person] compare (vb.): with (literal comparison); to (poetic or metaphorical comparison) comply (vb.): with (not to) [a rule; an order] confide (vb.): to, in [a person] congruence (n.): with [a standard] connive (vb.): at [a bad act]; with [another person] consider (vb.): none (transitive); as [one of several possible aspects (not as a sub stitute for “to be”)]; for [a position] consist (vb.): of [components (said of concrete things)]; in [qualities (said of ab stract things)] contemporary (adj.): with [another event] contemporary (n.): of [another person] contiguous (adj.): with, to [another place] contingent (adj.): on (preferably not upon) contrast (vb.): to, with [a person or thing] conversant (adj.): with, in [a field of study] convict (vb.): of, for (not in) depend (vb.): on (preferably not upon) differ (vb.): from [a thing or quality]; with [a person]; about, over, on [an issue] 5.195 GRAMMAR AND USAGE different (adj.): from (but when a dependent clause follows different, the conjunc tion than is a defensible substitute for from what: “movies today are different than they were in the fifties”) dissent (n. & vb.): from, against (preferably not to or with) dissimilar (adj.): to (not from) dissociate (vb.): from enamored (adj.): of (not with) equivalent (adj.): to, in (preferably not with) excerpt (n.): from (not of) forbid (vb.): to (formal); from (informal) foreclose (vb.): on [mortgaged property] hale (vb.): to, into [a place]; before [a magistrate] hegemony (n.): over [rivals]; in [a region] identical (adj.): with (preferred by purists), to [something else] impatience (n.): with [a person]; with, at, about [a situation] impose (vb.): on (preferably not upon) [a person] inaugurate (vb.): as [an officer]; into [an office] inculcate (vb.): into, in [a person] independent (adj.): of (not from) [something else] infringe (vb.): none (transitive); on (preferably not upon) [a right] inhere (vb.): in (not within) [a person; a thing] inquire (vb.): into [situations]; of [people]; after [people] instill (vb.): in, into (not with) [a person] juxtapose (vb.): to (not with) mastery (n.): of [a skill or knowledge]; over [people] militate (vb.): against [a harsher outcome] mitigate (vb.): none (transitive) oblivious (adj.): of (preferred), to [a danger; an opportunity] off (prep. & adv.): none (not of) predilection (n.): for [a preferred thing] predominate (vb.) (not transitive): in, on, over [a field; rivals] preferable (adj.): to (not than), over [an alternative] pretext (n.): for [a true intention] reconcile (vb.): with [aperson]; to [a situation] reticent (adj.): about [speaking; a topic] sanction (n.): for [misbehavior]; of [a sponsoring body]; to [a person; an event] shiver (vb.): from [cold]; at [something frightening] stigmatize (vb.): none (transitive); as [dishonorable] subscribe (vb.): to [a periodical or an opinion]; for [stock] trade (vb.): for (“swap”); in (“sell”); with (“do business with”); at (“patronize”); in [certain goods]; on (“buy and sell at”) trust (n.): in [faith]; for (“beneficial trust”) undaunted (adj.): in [a task]; by [obstacles] Grammar 5.199 unequal (adj.): to [a challenge]; in [attributes] used (adj.): to (“accustomed”); for (“applied to”) vexed (adj.): with [someone]; about, at [something] Conjunctions 5.196 Conjunctions defined. A conjunction is a function word that connects sentences, clauses, or words within a clause {my daughter graduated from college in December, and my son will graduate from high school in May [and connects two sentences]} {I said hello, but no one answered [but connects two clauses]} {we’re making progress slowly but surely [but joins two adverbs within a clause]}. In Standard English, conjunc tions connect pronouns in the same case {he and she are colleagues} {the teacher encouraged her and me}. A pronoun following the conjunction than or as is normally in the nominative case even when the clause that follows is understood {you are wiser than I [am]} {you seem as pleased as she [does]}—except in informal or colloquial English {you are wiser than me}. In the latter instance, than can be read as a preposition (see 5.183).
5.197 Simple versus compound conjunctions. A conjunction may be simple, a single word such as and, but, if, or, or though. Most are derived from prep ositions. Compound conjunctions are single words formed by combining two or more words. Most are relatively modern formations; they include words such as although, because, nevertheless, notwithstanding, and unless. Phrasal conjunctions are connectives made up of two or more separate words. Examples are as though, inasmuch as, in case, provided that, so that, and supposing that. The two main classes of conjunctions are coordinat ing and subordinating.
5.198 Coordinating conjunctions. Coordinating conjunctions join words or groups of words of equal grammatical rank, such as two nouns, two verbs, two phrases, or two clauses {are you speaking to him or to me?} {the re sults are disappointing but not discouraging}. Coordinating conjunc tions are further broken down into copulative, adversative, disjunctive, and final. A coordinating conjunction may be either a single word or a correlative conjunction.
5.199 Correlative conjunctions. Correlative conjunctions are conjunctions used in pairs, often to join successive clauses that depend on each other to form a complete thought. Some examples of correlative conjunctions are as-as, if-then, either-or, neither-nor, both-and, where-there, so-as, and 5.200 GRAMMAR AND USAGE not only-but also. Correlative conjunctions must frame structurally iden tical or matching sentence parts {she wanted both to win the gold medal and to set a new record}; in other words, each member of the pair should immediately precede the same part of speech {they not only read the book but also saw the movie} {if the first claim is true, then the second claim must be false}.
5.200 Subordinating conjunctions. A subordinating conjunction connects clauses of unequal grammatical rank. The conjunction introduces a clause that is dependent on the independent clause {follow this road un til you reach the highway} {that squirrel is friendly because people feed it} {Marcus promised that he would help}. A pure subordinating conjunc tion has no antecedent and is not a pronoun or an adverb {take a message if someone calls}.
5.201 Special uses of subordinating conjunctions. Subordinating conjunc tions or conjunctive phrases often denote the following relationships: (1) Comparison or degree— e.g., than (if it follows comparative adverbs or adjectives, or if it follows else, rather, other, or otherwise), as, else, other wise, rather, as much as, as far as, and as well as {is a raven less clever than a magpie?} {these amateur musicians play as well as professionals} {it’s not true as far as I can discover}. (2) Time—e.g., since, until, as long as, as soon as, before, after, when, as, and while {while we waited, it began to snow} {the tire went flat as we were turning the corner} {we’ll start the game as soon as everyone understands the rules} {the audience returned to the auditorium after the concert’s resumption was announced}. (3) Condition or assumption—e.g., if, though, unless, except, without, and once {once you sign the agreement, we can begin remodeling the house} {your thesis must’be presented next week unless you have a good rea son to postpone it} {I’ll go on this business trip if I can fly first class}. (4) Reason or concession—e.g., as, inasmuch as, why, because, for, since, though, although, and albeit {since you won’t share the information, I can’t help you} {Sir John decided to purchase the painting although it was very expensive} {she deserves credit because it was her idea}. (5) Purpose or result—e.g., that, so that, in order that, and such that {we dug up the yard so that a new water garden could be laid out} {he sang so loudly that he became hoarse}. (6) Place—e.g., where {I found a great restaurant where I didn’t expect one to be}. (7) Manner—e.g., as if and as though {he swag gers around the office as if he were an executive}. (8) Appositions—e.g., and, or, what, and that {the buffalo, or American bison, was once nearly extinct}. (9) Indirect questions—e.g., whether, why, and when {he could not say whether we were going the right way}.
Grammar 5.203 5.202 Adverbial conjunctions. An adverbial conjunction connects two clauses and also qualifies a verb {the valet has forgotten where Alvaro’s car is parked [where qualifies the verb is parked]}. There are two types of ad verbial conjunctions: relative and interrogative. A relative adverbial con junction does the same job as any other adverbial conjunction, but it has an antecedent {do you recall that cafe where we first met? [cafe is the antecedent of where]}. An interrogative adverbial conjunction indirectly states a question {Barbara asked when we are supposed to leave [when poses the indirect question]}. Some common examples of conjunctive relative adverbs are after, as, before, now, since, so, until, when, and where. Interrogative adverbs are used to ask direct and indirect questions; the most common are why, how, when, where, and what {I don’t see how you reached that conclusion}.
5.203 Beginning a sentence with a conjunction. There is a widespread belief- one with no historical or grammatical foundation—that it is an error to begin a sentence with a conjunction such as and, but, or so. In fact, a substantial percentage (often as many as 10 percent) of the sentences in first-rate writing begin with conjunctions. It has been so for centuries, and even the most conservative grammarians have followed this prac tice. Charles Allen Lloyd’s words from 1938 fairly sum up the situation as it stands even today: Next to the groundless notion that it is incorrect to end an English sentence with a preposition, perhaps the most widespread of the many false beliefs about the use of our language is the equally groundless notion that it is incorrect to begin one with “but” or “and.” As in the case of the superstition about the prepositional ending, no textbook supports it, but apparently about half of our teachers of En glish go out of their way to handicap their pupils by inculcating it. One cannot help wondering whether those who teach such a monstrous doctrine ever read any English themselves.7 Still, but as an adversative conjunction can occasionally be unclear at the beginning of a sentence. Evaluate the contrasting force of the but in question, and see whether the needed word is really and', if and can be substituted, then but is almost certainly the wrong word. Consider this example: He went to school this morning. But he left his lunch box on the kitchen table. Between those sentences is an elliptical idea, since the two actions are in no way contradictory. What is implied is something like 7. Charles Allen Lloyd, We Who Speak English: And Our Ignorance of Our Mother Tongue (New York: Thomas Y. Crowell, 1938), 19.
5.204 GRAMMAR AND USAGE this: He went to school, intending to have lunch there, but he left his lunch behind. Because and would have made sense in the passage as origi nally stated, but is not the right word—the idea for the contrastive but should be explicit. To sum up, then, but is a perfectly proper way to open a sentence, but only if the idea it introduces truly contrasts with what precedes. For that matter, but is often an effective way of introducing a paragraph that develops an idea contrary to the one preceding it.
5.204 Beginning a sentence with "however." However has been used as a con junctive adverb since the fourteenth century. Like other adverbs, it can be used at the beginning of a sentence. But however is more ponderous and has less impact than the simple but. As a matter of style, however is more effectively used within a sentence to emphasize the word or phrase that precedes it {The job seemed exciting at first. Soon, however, it turned out to be exceedingly dull.}. For purposes of euphony and flow, not of grammar, many highly accomplished writers shun the sentence starting however as a contrasting word. Yet the word is fine in that posi tion in the sense “in whatever way” (not followed by a comma) {however that may be, we’ve now made our decision}.
5.205 Conjunctions and the number of a verb. Coordinating and disjunctive conjunctions affect whether a verb should be plural or singular. Con junctions such as and and through indicate that grouped sentence ele ments impart plurality, so a plural verb is correct {the best vacation and the worst vacation of my life were on cruises} {the first through seventh innings were scoreless}. But conjunctions such as or and either-or dis tinguish the elements and do not impart plurality, so the singular verb is used if the elements are singular {a squirrel or a chipmunk raids the bird feeder every day} {either William or Henry dances with Lady Hill}. Other types of conjunctions have no effect on the verb’s number; for ex ample, if and is used as a copulative conjunction, the verb that follows may be singular {Andres’s bicycle was new, and so was his helmet}. See also 7.8.
Interjections 5.206 Interjections defined. An interjection or exclamation is a word, phrase, or clause that denotes strong feeling {never again!} {you don’t say!}. An interjection has little or no grammatical function in a sentence; it is used absolutely {really, I can’t understand why you put up with the situation} {oh no, how am I going to fix the damage?} {hey, it’s my turn next!}. It is frequently allowed to stand as a sentence by itself {Oh! I’ve lost my Syntax 5.211 wallet!} {Ouch! I think my ankle is sprained!} {Get out!} {Whoa!}. Intro ductory words like well and why may also act as interjections when they are meaningless utterances {well, I tried my best} {why, I would never do that}. The punctuation offsetting the interjections distinguishes them. Compare the different meanings of Well, I didn't know him with I didn't know him well, and Why, here you are! with I have no idea why you are here and Why? I have no idea. See also 6.34, 6.35.
5.207 Use of interjections. Interjections are natural in speech {your order should be shipped, oh, in eight to ten days} and frequently used in dia logue (and formerly in poetry). As a midsentence interrupter, an inter jection may direct attention to one’s phrasing or reflect the writer’s or speaker’s attitude toward the subject, especially if the tone is informal or colloquial {because our business proposal was, ahem, poorly presented, our budget will not be increased this year}.
5.208 Interjections and functional variation. Because interjections are usually grammatically independent of the rest of the sentence, all other parts of speech may be used as interjections. A word that is classified as some other part of speech but used with the force of an interjection is called an exclamatory noun, exclamatory adjective, etc. Some examples are good! (adjective); idiot! (noun); help! (verb); indeed! (adverb); me! (pronoun); and! (conjunction); quickly! (adverb).
5.209 Words that are exclusively interjections. Some words are used only as interjections—for example, ouch, whew, ugh,psst, and oops.
Syntax 5.210 Syntax defined. Syntax is the collective term we use to denote all the rules governing how words are arranged into sentences. In an ana lytic language like English—one that, unlike a synthetic language, uses word order to show word relations (as opposed to inflections of various kinds)—syntax is particularly important in expressing meaning.
5.211 Statements. Most sentences are statements having a declarative struc ture in which (1) the clause contains a subject and (2) the subject pre cedes the verb. Sometimes in speech and informal writing, the subject is merely implied {[he] missed the ball} {[I] think I’ll go to the store}. In a few negative idioms, the subject may follow part of the verb phrase {scarcely had we arrived when we had to return}.
5.212 GRAMMAR AND USAGE 5.212 Questions. Sentences that seek to elicit information are known as ques tions. They have an interrogative structure, which typically begins with a question word. There are three main types: (1) yes-no questions, which are intended to prompt an affirmative or negative response {will we be gone long?}; (2) w/i- questions, so called because they characteristi cally start with w/zo, what, when, where, why, which, or how (not quite a wh- word, but it counts) {which apples do you want?}; and (3) alternative questions, which prompt a response relating to options mentioned in the sentence {would you rather play golf or tennis?}.
5.213 Some exceptional types of questions. Four types of interrogative utter ances aren’t classifiable under the three categories given in 5.212. Two are yes-no questions. The first is the spoken sentence in which one’s pitch rises at the end, in a questioning way—but the structure is that of a declarative sentence {he’s going to Corpus Christi?}. To show vexation in such a question, the question mark may be paired with an exclamation point {she’s going to Padre Island?!}. The second special type of yes-no question is the tag question, in which the interrogative inversion appears at the end of a statement {he has arrived, hasn’t he?} {it’s good, isn’t it?}. A few tag questions are signaled by particular words without the inter rogative inversion {it’s raining, right?} {you’re tired, eh?} {you want to go, yes?}. A third special type is the exclamatory question, in which the interrogative structure appears but when the statement is spoken, one’s tone normally falls at the end {isn’t it nice out here!} {how great is this!}. Finally, a rhetorical question is phrased in the interrogative structure but is meant as an emphatic or evocative statement, without the expectation of an answer {why should I care?} {who knows how long it might take?}.
5.214 Directives. A directive or imperative is a sentence that instructs some body to do or not to do something. The word command is sometimes used as a synonym, but most grammarians consider the term command more appropriate for one of the eight main types of directives, all of which are in the imperative mood of the verb: (1) command {come here now!}; (2) prohibition {don’t do that!}; (3) invitation {join us for dinner!}; (4) warning {watch out for rattlesnakes!}; (5) plea {stay here} {help!}; (6) request {put your book away}; (7) well-wishing {play well} {have a good time!}; and (8) advice {put on some insect repellent}.
5.215 Exceptional directives. Several directives depart from these common patterns, as when the subject is expressed {sit you down} {you stay there}; when they begin with let {let’s have a picnic} {let us wait}; or when they begin with do {do help yourself}.
Syntax 5.220 5.216 Exclamations. An exclamation expresses the extent to which a speaker is moved, aroused, impressed, or disgusted by something. It can take the form of a simple interjection {by golly!} {pishposh!}. Or it can follow a sentence structure consisting of what or how followed by a subject and verb {what an extraordinary novel this is!} {how well she writes!}. Excla mations are sometimes elliptically expressed {what finery!} {how pretty!} {how ugly!}. In formal, literary English, exclamations can be signaled by inverted word order {little did I expect such unfair treatment}.
The Four Traditional Types of Sentence Structures 5.217 Simple sentence. A simple sentence consists of a single independent clause with no dependent clause {no man is an island}. A sentence can be simple despite having internal compound constructions serving as subjects, main verbs, objects of prepositions, and others {time and tide wait for no man}.
5.218 Compound sentence. A compound sentence contains two independent clauses (called coordinate clauses) with no dependent clause {the rain was heavy, and my umbrella was not much help}. Grammarians are divided on the question whether one type of sentence should be labeled com pound or simple: She arrived early and stayed late. Traditional grammar ians have tended to call this a simple sentence with a compound predi cate (where arrived and stayed are coordinate verbs). Transformational grammarians have tended to call it a compound sentence with an elided subject in the second clause {she arrived early [,] and [she] stayed late}.
5.219 Complex sentence. A complex sentence contains a single independent clause with one or more dependent clauses {I’ll be home after I finish work}. Such a sentence may have only one dependent clause {she won because she practiced so hard}, or it may contain a variety of depen dent clauses {the books that were nominated argued that most be havioral differences among people aren’t genetic in origin [that were nominated is an adjective clause; that most behavioral differences among people aren't genetic in origin is a noun clause]}.
5.220 Compound-complex sentence. A compound-complex sentence con tains multiple independent clauses and at least one dependent clause {it was a beautiful evening, so after we left work we went for a walk [after we left work is a dependent clause between two independent clauses]}. It 5.221 GRAMMAR AND USAGE differs from a complex sentence only in containing more than one inde pendent clause. Like the independent clauses of a compound sentence, those of a compound-complex sentence are called coordinate clauses.
English Sentence Patterns 5.221 Importance of word order. English is known as an analytic language- one that depends largely on word order. (A synthetic language, such as Latin, depends largely on inflectional forms of words.) In the transition from Old English (AD 450-1100) to Middle English (1100-1500), the language lost most of its inflected forms—except those for pronouns (I-me-mine etc.). Nouns no longer have nominative and accusative cases. Instead, word order governs meaning. Consider this example: Michael likes crystal. Michael is the subject, likes the verb, and crystal the object. It’s the basic subject-verb-object (SVO) pattern. We deduce the mean ing from the position of the words: Michael is an admirer and perhaps a collector of fine glass. If we change it to Crystal likes Michael, the mean ing is transformed because of the SVO order. We now infer that some one named Crystal thinks fondly about someone named Michael. The SVO pattern is highly significant: it governs the meaning of most English statements. Departures from it typically signal either unusual emphasis or the posing of a question (as opposed to the making of a statement).
5.222 The basic SVO pattern. Despite the seeming potential for monotony in having sentence after sentence using the same fundamental word order, English offers enough variety in vocabulary and in sentence elements that can function as subjects, verbs, and objects to keep things interest ing. Consider these examples, all of which use the pattern but with inter esting levels of sophistication: Mary S likes pomegranates.
V O O The umpire we were talking about rejected our arguments.
S V O O The woman down the street is selling loaves of bread.
S o V O The obstacles that we face create opportunities.
S V O Syntax 5.225 How you think of yourself affects S V the problems of everyday life and 6 perceived as being well adjusted.
O both the way you approach 6 the degree to which you’re O 5.223 5.224 5.225 All seven syntactic patterns. Syntactic patterns other than the SVO pat tern are available, but they are limited to specific types that include two to four of these elements: subject (S), verb (V), [direct] object (O), indi rect object (IO), complement (C), adverbial (A). Here are all seven basic clause patterns: S + V: Sandy smiled.
S + V + O: Sandy hit the ball.
S + V + C: Sandy is eager.
S + V + A: Sandy plays well.
S + V + IO + O: Sandy gave Jerry the ball.
S + V + O + C: Sandy got her bag wet.
S + V + O + A: Sandy wrote her score on the card.
Variations on syntactic order. When clause elements appear in a differ ent order, the inversion may indicate either a question {is Sandy all right? [V-S-C]} or a special kind of emphasis: Yoda my name is!
""c- S~ V Bully you say!
o V v Inversions of this type achieve a special emphasis precisely because they depart from the normal sequence of sentence elements.
Clauses Clauses. A clause is a grammatical unit that contains a subject, a finite verb, and any complements that the verb requires. An independent clause can stand alone as a sentence {Jose saw a squirrel}, while a dependent clause cannot stand alone because of the presence of a word by which 5.226 GRAMMAR AND USAGE it would normally be linked to an independent clause {because he was hungry, he sat down for a meal}. A dependent clause is usually intro duced either by a relative pronoun (making it a relative clause) or by a subordinating conjunction, which establishes the semantic relationship between the independent clause and the dependent one. Combining re lated ideas by linking one or more dependent clauses to an independent one is called subordination, and the result is a complex sentence. Because a dependent clause is always subordinate to an independent clause for contextual meaning, it is also called a subordinate clause. A dependent clause commonly serves one of several functions: the direct object of a verb {everyone believed that the note was genuine [the t/^t-clause is the direct object of believed]}; an adjectival clause modifying a noun element {he who hesitates is lost [who hesitates adjectivally modifies he]}; an ad verbial clause modifying a verb or verb phrase {I bought the car despite my father’s warning not to [the Aspite-clause modifies the verb bought]}.
5.226 Relative clauses. A relative clause is a subordinate clause that is intro duced by a relative pronoun and modifies the noun element (or sen tence or clause) it follows {the car that you own} {those who follow his progress} {they were ten minutes late to the opera, which meant they couldn’t enter until the end of the first act}. In some relative clauses, called contact clauses, the relative pronoun is merely implied {all the people you mention have already registered [the relative pronoun who is implied inpeople [whom]you mention]}. Because the necessary connec tive is omitted, contact clauses are a type of elliptical clause—one often involving what is known as a whiz-deletion (so called because it so often amounts to the omission of who is).
5.227 Appositive clauses. A clause used in apposition to a noun element in the sentence is called an appositive clause. Though these are often (but not always) introduced with the same words that introduce relative clauses (that, which, who), the two differ in that a relative clause functions only within the sentence, while an appositive clause is self-contained: with its introductory relative pronoun removed, it could stand on its own as a grammatical sentence {we all heard the report that the beloved broad caster had died [without that, the remaining appositive clause is gram matically complete: the beloved broadcaster had died]}.
5.228 Conditional clauses. A conditional clause (also called a. protasis) is an adverbial clause, typically introduced by if or unless (or should, although, though, despite, or another subordinating conjunction), establishing the condition in a conditional sentence. Usually this is a direct condition, in dicating that the main clause (also called the apodosis) is dependent on Syntax 5.230 the condition being fulfilled. Sometimes, however, the clause may ex press an indirect condition {if I recall correctly, his assistant’s name is Miljana}, alternative conditions {the party will be a success whether or not it rains}, or an open range of possibilities {whatever you’re doing, it’s working}. Most often, though, a conditional clause expresses a direct condition, which may be open (real or factual) or hypothetical (closed or unreal). An open condition leaves unanswered the question whether the condition will be fulfilled {if you don’t finish the work on time, we’ll have to reevaluate our arrangement}. A hypothetical condition, on the other hand, assumes that the condition has not been, is not, or is unlikely to be fulfilled {if he had only remembered to wear a raincoat, he wouldn’t have ruined his new suit} {if I had a hammer, I could fix this creaky stair} {the transition would be much harder if she left without giving notice}.
Ellipsis 5.229 Ellipsis generally. A grammatical ellipsis (sometimes called an omission) occurs when part of a clause is left understood and the reader or listener is able to supply the missing words. (For the use of three dots to indi cate text omitted from a direct quotation or for faltering or interrupted speech, see 13.50-58.) This “recovery” of omitted words is possible because of shared idiomatic knowledge, context, and what’s called the principle of recoverability {he preferred chocolate, she vanilla [preferred is understood in the second clause]}. A sentence containing such an ellipsis is called an elliptical sentence. In colloquial speech, an ellipsis is useful to avoid repetition, shorten the message, and make it easier to under stand. It’s particularly appropriate for commands and exclamations, and especially when asking or answering a question whose complete answer would essentially repeat the question. For example: Thank you. (I thank you.) One lump or two? (Would you like one lump of sugar or two?) Glad you like it! (I’m glad that you like it!) Which is better? And why? (Which choice is better, and why is it better?) [Can you tell me who built this house?] The Tucker family. (Yes, I can tell you. The Tucker family built this house.) Negation 5.230 Negation generally. A statement may be expressed in positive or neg ative terms. Negation is the grammatical process of reversing the ex 5.231 GRAMMAR AND USAGE pression in a sentence. There are four common types: (1) using the neg ative particle not or no; (2) using negating pronouns such as nobody, none, no one, nothing, or negating adverbs such as nowhere, never, neither; (3) using the coordinating conjunctions neither and nor (or both of them as correlative conjunctions); (4) using words that are negative in mean ing and function, such as hardly (= almost not), scarcely (= almost not), barely (= almost not),/hv (= not many; not much), little (= not much), rarely (= almost never), and seldom (= almost never)—or words having negative affixes such as a- {atypical}, dis- {disrobe}, in- {inimitable} (to gether with the assimilated forms il-, im-, and ir-), non- {nonemployee}, un- {untidy}, -less {careless}, and -free {hassle-free}.
5.231 The word "not." The simplest and most common form of negation in volves using the particle not. Used with ordinary verbs and with auxiliary verbs, not typically negates a verb, an object, a phrase, or a clause. Not typically precedes whatever sentence element is being negated. To ne gate an ordinary verb in the present- or past-tense indicative mood, the verb is replaced by a compound of do or did plus not and a bare infinitive.
They sell newspapers in the hotel.
They do not sell newspapers in the hotel.
Kerri sings at the opera today.
Kerri does not sing at the opera today.
The waiter returned with our order.
The waiter did not return with our order.
Not usually immediately follows the principal verb or an auxiliary. If there are two or more verbs in the negative expression, not always fol lows the first of them.
I am happy.
I am not happy.
I should leave for work.
I should not leave for work.
I should leave for work, but I cannot find my glasses.
With participles, not precedes the participle {not given any warning, Josue nonchalantly opened the door} {not coming to any conclusions, the Syntax 5.233 jury decided to suspend deliberations} {not having heard the news, Brett innocently asked how Tara was doing in school}. The subject is normally elided from the participial phrase. Not doesn’t have to negate everything that follows it. It may be limited to the element immediately following {I discovered not a scientific breakthrough but a monstrous develop ment}. A sentence containing not may be qualified by another element that limits the extent of the negation. The word’s or clause’s placement may significantly alter the scope of negation. For example: He definitely did not accept the job offer.
[It is final: he rejected the job offer.] He did not definitely accept the job offer.
[It is uncertain: he might still reject the offer.] We have not eaten yet.
[We have not eaten, but we expect to eat at some time.] Not can be contracted to -n't and appended to most auxiliary verbs with out changing the form of the verb (e.g., are not arent, would not wouldn't, has not hasn't). The exceptions, involving am, can, do, will, and shall, are well known to native speakers of the language: am am not -> [no contraction with negative: use l’m not etc.] can cannot can’t will - will not -> won’t shall -> shall not shan’t Shan't isn’t used in American English except in jest; it still sometimes appears in British English.
5.232 The word "no/' Unlike not, which can negate any element of a sentence, no negates only adjectives and nouns. When used with an adjective phrase, it might produce ambiguity. For example, in we found no eggs, it’s clear that the speaker found nothing. But in we found no fresh eggs for sale, does the speaker mean they found no eggs at all, only eggs that weren’t fresh, or eggs that were fresh but not for sale?
5.233 Using pronouns and adverbs for negation. Pronouns such as nobody, none, no one, and nothing and adverbs such as nowhere and never also re sult in negation. These words make it unnecessary to use not. They can help reduce the number of words and improve the flow of a sentence.
5.234 GRAMMAR AND USAGE We did not see anyone in the audience.
We saw no one in the audience.
The children do not have anything to do.
The children have nothing to do.
You do not ever listen!
You never listen!
I cannot put the groceries anywhere. There’s nowhere to put the groceries.
5.234 Using "neither" and "nor." The correlative conjunctions neither and nor negate alternatives simultaneously. Traditionally, only pairs are framed by neither-nor, but writers and speakers sometimes use a neither-nor-nor construction, as in the last example below.
The dog and the cat are not friendly.
Neither pet is friendly.
Neither the dog nor the cat is friendly.
The radiator does not leak, and the water pump also does not leak.
Neither the radiator nor the water pump leaks.
Neither John nor Sally nor Brenda can attend the meeting.
In that last example, some writers include only the last nor. But again, a simple neither-nor construction isn’t recommended with three or more elements, the sequence neither-nor-nor being preferable.
5.235 Negative interrogative and imperative statements. In a negative inter rogative statement, the first auxiliary verb may be contracted with not: Aren't you doing your homework tonight? If it is not contracted, then not or no precedes the negated element {are you not doing your homework tonight?} {is there no satisfying you?}. Questions phrased with a negating word are called (unsurprisingly) negative questions; those without nega tion are positive questions. In an imperative statement, the negative par ticle always follows the imperative verb or is contracted with it {come no closer!} {don’t talk back!}.
5.236 Double negatives. When a sentence contains two negatives, in Standard English they are usually thought to cancel each other out to make a mild positive {he didn’t not say anything [he did say something]} {this isn’t an Syntax 5.239 uncommon problem [it’s more or less common]}. In dialect, by contrast, the sentence is often meant to express an emphatic negative {he didn’t say nothing [he said nothing at all]} {we’re not going nowhere special [we’re going somewhere, but it isn’t special]}. Multiple negatives often lead to ambiguity. For example, in I wouldn’t be surprised if Dan doesn’t find the hammer, does the speaker expect Dan to find the hammer or not to find it? In general, though, multiple negation results in a cancellation of other negatives {we didn’t say the children couldn’t come along [we didn’t forbid the children’s coming]}.
5.237 Other forms of negation. A sentence can express negation even though it doesn’t contain any plainly negative elements. Two common means of achieving this effect are using but in the sense “if not” and using except in the sense “but not” {what is a pampered dog but [= if not] a child in a fur suit?} {you may borrow the car except [= but not] when it is raining}.
5.238 "Any" and "some" in negative statements. When the negating particle is not, then any- words must be used with it, not some- words. Any- words include any, anyone, anybody, anything, and anywhere. Some- words in clude some, someone, somebody, something, and somewhere.
n o t t h is : I don’t want to see somebody. [Unless the meaning is one particular person I’m not naming.] b u t t h is : I don’t want to see anybody.
n o t t h is : There aren’t some seats left. b u t t h is : There aren’t any seats left.
Expletives 5.239 Expletives generally. Though expletive commonly denotes a swearword {expletive deleted}, in grammar expletive signifies a word that has no lex ical meaning but serves a merely structural role in a sentence—as a noun element. The two most common expletives are it {it is true!} and there {there must be an answer}. An expletive it or there may be in the subject position, especially when the subject of a sentence is a clause {it is a rule that children must raise their hands to speak during class [the rule is that children must raise their hands to speak during class]} {it is better to stay here than to go there [to stay here is better than to go there]}. In this position, the expletive shifts the emphasis to the predicate containing the true subject. The sentence implies a “who” or “what” question that is answered by the subject. For example, It is foolish to ignore facts tells the 5.240 GRAMMAR AND USAGE reader “what” it is foolish to ignore and emphasizes “facts.” An expletive it may also take the position of a direct object, especially when the real object is a clause or noun phrase {some people don’t like it that stores are open for business on Thanksgiving [some people don’t like stores being open for business on Thanksgiving]}. Using an expletive in this way can tighten a verb phrase and emphasize the object. Compare it was taken for granted that our team would win with we took for granted that our team would win.
5.240 Expletive "it." Whereas the pronoun it adds meaning to a sentence be cause it has an antecedent or else is the formal subject of a fo-verb in the sense of “a person” or “a thing,” an expletive it adds no meaning and takes the subject’s or object’s place when the subject or object shifts to the predicate: It is not known what happened can be restated as What happened is not known. Usually readers have no difficulty intuitively un derstanding whether they’re encountering a pronoun it or an expletive it. But when the expletive and the pronoun appear close together, they may cause the reader to stumble {The much-anticipated feast was a disap pointment; it was poorly cooked and presented. It is hard to believe that such a famous chef thought it would be edible, let alone delight gour mands.}. Avoid having several its in a passage clash in this way. Some other names for the expletive it are ambient it, anticipatory it, dummy it, empty it, introductory it, nonreferential it, and prop it.
5.241 Expletive "there." The word there is also frequently used as an expletive with be or an intransitive verb (especially a linking verb) followed by the subject {there are many different viewpoints presented in the students’ essays} {there were several hundred members present at the confer ence}. An expletive there shouldn’t be confused with there as an adverb of place. Compare There seemed to be someone with Someone seemed to be there.
Parallel Structure 5.242 Parallel structure generally. Parallel constructions—series of like sen tence elements—are common in good writing. Compound structures may link words {win, lose, or draw}, phrases {government of the people, by the people, for the people}, dependent clauses {that all men are created equal; that they are endowed by their Creator with certain unalienable rights; that among these are life, liberty, and the pursuit of happiness}, or sentences {I came; I saw; I conquered}. Every element of a parallel series must be a functional match (word, phrase, clause, sentence) and Syntax 5.244 serve the same grammatical function in the sentence (e.g., noun, verb, adjective, adverb). This syntactic linking of matching elements is called coordination. When linked items do not match, the syntax of the uncoor dinated sentence breaks down: po o r : She did volunteer work in the community kitchen, the homeless shelter, and taught free ESL classes offered by her church.
b e t t e r : She did volunteer work in the community kitchen, the homeless shel ter, and her church, where she taught free ESL classes.
po o r : The candidate is a former county judge, state senator, and served two terms as attorney general.
b e t t e r : The candidate is a former county judge, state senator, and two-term attorney general.
In the second example, for instance, the subject, verb, and modifier (the candidate is a former) fit with the noun phrases county judge and state sen ator , but the third item in the series renders nonsense: The candidate is a former served two terms as attorney general. The first two elements in the series are nouns, while the third is a separate predicate. The corrected version makes each item in the series a noun element.
5.243 Prepositions and parallel structure. In a parallel series of prepositional phrases, repeat the preposition with every element unless they all use the same preposition. A common error occurs when a writer lets two or more of the phrases share a single preposition but inserts a different one with another element: p o o R: I looked for my lost keys in the sock drawer, the laundry hamper, the bath room, and under the bed.
b e t t e r : I looked for my lost keys in the sock drawer, in the laundry hamper, in the bathroom, and under the bed.
If the series had not included under the bed, the preposition could have been used once to apply to all the objects: I looked for my lost keys in the sock drawer, the laundry hamper, and the bathroom, 5.244 Paired joining terms and parallel structure. Correlative conjunctions such as either-or, neither-nor, both-and, and not only-but also and some adverb pairs such as where-there, as-so, and if-then must join grammati cally parallel sentence elements. It is a common error to mismatch ele ments framed by correlatives.
5.245 GRAMMAR AND USAGE p O o R: I’d like to either go into business for myself or else to write freelance travel articles.
b e t t e r : I’d like either to go into business for myself or else to write freelance travel articles.
po o r : Our guests not only ate all the turkey and dressing but both pumpkin pies as well.
b e t t e r : Our guests ate not only all the turkey and dressing but both pumpkin pies as well.
In the second example, the verb ate, when placed after the first correla tive, attaches grammatically to all the turkey but not to both pumpkin pies as well. When moved outside the two phrases containing its direct ob jects, it attaches to both—and the phrasing becomes parallel.
5.245 Auxiliary verbs and parallel structure. If an auxiliary verb appears be fore a series of verb phrases, it must apply to all of them. A common error is to include one phrase that takes a different auxiliary verb: po o r : The proposal would streamline the application process, speed up admis sion decisions, and has proved to save money when implemented by other schools.
b e t t e r : The proposal would streamline the application process, speed up ad mission decisions, and save money.
b e t t e r : The proposal would streamline the application process and speed up admission decisions. It has proved to save money when implemented by other schools.
The auxiliary verb would in that example renders the nonsensical would has proved when parsed with the third element of the predicate series. The first solution resolves that grammatical conflict, while the second breaks out the third into a separate sentence—which also avoids shifting from future tense to past tense in midsentence.
Cleft Sentences 5.246 Cleft sentences defined. A cleft sentence opens with a special type of subject clause (an it-clause, a w/iat-clause, or a similar clause) that changes the focus by adding two or three words (such as it, was, and who; there, are, and that; or what and was) {it was the manager who handled the customer’s complaint} {there are still some missing items that have to be accounted for} {what the campaign lacked was a vibrant slogan}.
Syntax 5.248 Most often the sentence begins with an expletive it and a be-verb (the it- clause). The subject clause emphasizes new information that identifies a person, a place, a time, an object, a cause, etc. For example: It was Manuel who met Adam in college. (The focus is on the actor.) It was Adam whom Manuel met in college. (The focus is on the person that the actor met.) It was in college that Manuel met Adam. (The focus is on the time or place when they met.) The part of a cleft sentence beginning with the relative pronoun usually refers to information already given. Hence it may be reduced when the information that would be in the final clause is understood: When did Manuel and Adam become acquainted?
It was in college (that they met).
5.247 Types of cleft sentences. A cleft sentence may be declarative {it is the quality of the work that concerns me} {there was an incident that led to the concert’s postponement}. Or it may be interrogative {is it the quality of the work that you’re concerned about?} {what was the incident that led to the concert’s postponement?}. It may also be positive or negative. A positive cleft sentence states a truth. A negative cleft sentence uses simple negation (not, no) to state the contrary. Often the relative pronoun is elided in a whiz-deletion (as with the bracketed words below): po s it iv e : There are many movies [that are] worth seeing this weekend. n e g a t iv e : There are not many movies [that are] worth seeing this weekend. n e g a t iv e : There are no movies [that are] worth seeing this weekend.
See also 5.226.
5.248 Use of cleft sentences. A cleft sentence is sometimes used for dramatic effect to signal a shift or a beginning, especially to create an interesting lead-in for a topic: It was hours later that Burns discovered he’d left his wallet on the counter.
It was in 1912 that shipbuilders and legislators learned the cost of not providing ocean liners with adequate numbers of lifeboats.
In some contexts, a cleft sentence may imply a contrast or mistake. For instance, It’s not Joan who wants to be a social activist implies that Joan is 5.249 5.250 5.249 GRAMMAR AND USAGE being distinguished from another person or that someone else has been mistakenly identified.
Word Usage Grammar versus usage. The great mass of linguistic issues that writers and editors wrestle with don’t really concern grammar at all—they con cern usage: the collective habits of a language’s native speakers. It’s an arbitrary fact, but ultimately an important one, that corollary means one thing and correlation something else. Yet there seems to be an irresistible law of language that two words so similar in sound will inevitably be con fused by otherwise literate users of language—a type of mistake called catachresis. Some confusions, such as the one just cited, are relatively new. Others, such as lay versus lie and infer versus imply, are much older.
Good usage versus common usage. The best dictionaries are signaled by the imprints of Merriam-Webster, Webster’s New World, American Heritage, Oxford University Press, and Random House. But one must use care and judgment in consulting any dictionary. The mere presence of a word in the dictionary’s pages does not mean that the word is in all respects fit for print as Standard Written English. The dictionary merely describes how speakers of English have used the language; despite occa sional usage notes, lexicographers generally disclaim any intent to guide writers and editors on the thorny points of English usage—apart from collecting evidence of what others do. So infer is recorded as meaning, in one of its senses, imply; irregardless as meaning regardless; restauranteur as meaning restaurateur; and on and on. That is why, in the publishing world, it is generally necessary to consult a style or usage guide, such as Fowler's Modern English Usage or my own Garner's Modern English Usage.
a; an. Use the indefinite article a before any word beginning with a consonant sound {a euphonious phrase} {a utopian dream}. Use an before any word be ginning with a vowel sound {an officer} {an honorary degree}. The word histor ical and its variations cause missteps, but if the h in these words is pronounced, it takes an a {an hour-long talk at a historical society}. Likewise, an initialism (whose letters are sounded out) may be paired with one article, while an acro nym (which is pronounced as a word) beginning with the same letter is paired with the other {an HTML website for a HUD program}.
ability; capability; capacity. Ability refers to a person’s physical or mental power Word Usage 5.250 or skill to do something {the ability to ride a bicycle}. Capability refers more generally to power or ability to do something challenging {she has the capa bility to play soccer professionally} or to the quality of being able to use or be used in a certain way {a jet with long-distance-flight capability}. Capacity refers especially to a vessel’s ability to hold or contain something {a high- capacity fuel tank}. Used figuratively, capacity refers especially to a person’s physical or mental power to learn {an astounding capacity for mathematics}. It can also be used as a synonym for ability {capacity for love}, as a formal word for someone’s job, position, or role {in an advisory capacity}, as a word denoting an amount that can be produced or dealt with {full capacity}, or as a means of denoting size or power {engine capacity}.
abjure; adjure. To abjure is to deny or renounce publicly, especially under oath {the defendant abjured the charge of murder} or to declare one’s permanent abandonment of a place {abjure the realm}. To adjure is to charge someone to do something as if under oath {I adjure you to keep this secret} or to try earnestly to persuade {the executive committee adjured all the members to approve the plan}. Some writers misuse adjure for either abhor (= to detest) or require (= to mandate).
about; approximately. When idiomatically possible, use the adverb about in stead of approximately. In the sciences, however, approximately is preferred {approximately 32 coding-sequence differences were identified}. Avoid cou pling either word with another word of approximation, such as guess or esti mate.
abstruse. See obtuse.
accept; except. To accept something is to receive it {accept this gift} or regard it as proper {accept the idea}. To except something is to exclude it or leave it out {club members will be excepted from the admission charge}, and to except to something is to object to it.
access, vb. The use of nouns as verbs has long been one of the most common ways that word-usage changes happen in English. Today, few people quibble with using contact, debut, or host, for example, as a verb. Access can be safely used as a verb when referring to computing {access a computer} {access the internet} {access a database}. Outside the digital world, though, it can be jar ring and is best avoided.
accord; accordance. The first word means “agreement” {we are in accord on the treaty’s meaning} {we have reached an accord}. The second word means “conformity” {the book was printed in accordance with modern industry stan dards}.
acquiesce. To acquiesce is to do what someone else wants or to passively allow something to happen. The connotation is usually acceptance without enthusi asm or even with opposition that is not acted on. The word traditionally takes the preposition in {the minority party acquiesced in the nomination}, although to is also accepted. With is not standard.
307 5.250 GRAMMAR AND USAGE actual fact, in. Redundant. Try actually instead, or simply omit.
acuity; acumen. What is acute is sharp, and these two words apply to mental sharpness. Acuity most often refers to sharpness of perception—the ability to think, see, or hear clearly {visual acuity}. Acumen always refers to mental prowess, especially the ability to think quickly and make good judgments.
adduce; deduce; induce. To adduce is to give as a reason, offer as a proof, or cite as an example in order to prove that something is true {as evidence of reliability, she adduced her four years of steady volunteer work as a nurse’s aide}. Deduce and induce are opposite processes. To deduce is to reason from general principles to specific conclusions, or to draw a specific conclusion from general knowledge {from these clues about who committed the crime, one deduces that the butler did it}. In a related logical sense, to induce is to form a general principle based on specific observations {after years of study ing ravens, the researchers induced a few of their social habits}. In its more common uses, however, to induce is (1) to persuade someone to do something, especially something unwise {nothing could induce me to try that again}, or (2) to cause a particular physical response {induce labor} {induce vomiting}.
adequate; sufficient; enough. Adequate refers to the suitability of something in a particular circumstance {an adequate explanation} {adequate provisions}. Sufficient refers to an amount that is enough to meet a particular need (always with an abstract concept, a mass noun, or a plural) {sufficient water} {sufficient information} {sufficient cause} {sufficient resources}. Enough, the best word for everyday purposes, meaning “as much or as many as are needed or wanted,” modifies both count nouns {enough people} and mass nouns {enough oil}.
adherence; adhesion. With a few exceptions, the first term is figurative, the sec ond literal. Your adherence to the transportation code requires the adhesion of an inspection sticker to your windshield.
adjure. See abjure, administrator. See executor.
admission; admittance. Admission is generally figurative, suggesting particu larly the rights and privileges granted with permission to enter {the student won admission to a first-rate university} or the price paid for entry {admission is $10}. Admittance is more limited and more a matter of physical entry, but it too is tinged with the idea of permission {no admittance beyond this point}.
adopted; adoptive. Adopted applies to a child or dependent {adopted son}. It is incorrect when applied to the ones who do the adopting; instead, use adoptive, the more general adjective corresponding to adopt {adoptive parents}.
adverse; averse. Though etymologically related, these words have undergone differentiation. Adverse means either “strongly opposed” or “unfavorable” and typically refers to things (not people) {adverse relations between the nations complicated matters} {an adverse wind blew the ship off course}. Averse means “feeling negatively about” or “having a strong dislike or unwillingness,” and it refers to people {he’s averse to asking for directions}.
Word Usage 5.250 affect; effect. Affect, almost always a verb, means “to influence or do something that produces a change; to have an effect on” {the adverse publicity affected the election}. To affect can also mean “to pretend to have a particular feeling or manner” {affecting a Scottish accent}. (The noun affect has a specialized meaning in psychology: emotional expressiveness. Consult your dictionary.) Effect, usually a noun, means “an outcome, result” {the candidate’s attempted explanations had no effect} or “a change caused by an event, action, occur rence, etc.” {harmful effects of smoking}. But it may also be a verb meaning “to make happen, produce” {the goal had been to effect a major change in campus politics}.
affirmative, in the; in the negative. These are slightly pompous ways of saying yes and no. They result in part because people are unsure how to punctuate yes and no. The ordinary way is this: he said yes (without quotation marks around yes, and without a capital); she said no (ditto).
afflict. See inffict.
affront. See effrontery.
after having [+ past participle]. Though common, this phrasing is redundant. Try instead after [+ present participle]: change after having passed the audition, she... to after passing the audition, she... Or this: having passed the audition, she... See 5.110.
afterward, adv.; afterword, n. The first means “later”; the second means “an epilogue.” On afterward(s), see toward.
aggravate. Traditionally, aggravate most properly means “to intensify (some thing bad)” {aggravate an injury} {an aggravated crime}. If the sense is “to bother,” try annoy or irritate or exasperate instead.
aid; aide. Aid can be a verb (= to help) or a noun (= assistance). Aide is a noun (= helper), as in teacher’s aide; in military parlance, it denotes someone as signed to help a superior officer {general’s aide}.
ain’t. This contraction is famously dialectal—a word not to be used except either in the dialogue of a nonstandard speaker or in jest.
alibi. Avoid this as a synonym for excuse. The traditional sense is “the defense of having been elsewhere when a crime was committed.” all (of). Delete the of whenever possible {all the houses} {all my children}. The most common exception occurs when all of precedes a nonpossessive pronoun {all of us} {all of them}.
alleged. Traditional usage applies this participial adjective to things, especially acts {alleged burglary}, not to the actors accused of doing them {alleged bur glar}. That distinction is still observed by some publications, but it has largely been abandoned. Although allegedly /a-lej-ad-lee/ has four syllables, alleged has only two: /a-lejd/.
all ready. See already.
all right. Two words. Avoid alright, which has long been regarded as nonstan dard.
30A 5.250 GRAMMAR AND USAGE all together. See altogether.
allude; elude; illude. To allude is to hint at something indirectly {he alluded to the war by mentioning “our recent national unpleasantness”}. It’s often loosely used where refer or quote would be better—that is, where there is a direct men tion or quotation. To elude is to avoid capture {the fox eluded the hunters}. To illude (quite rare) is to deceive {your imagination might illude you}.
allusion; reference. An allusion is an indirect or casual mention or suggestion of something {the cockroach in this story is an allusion to Kafka}. A reference is a direct or formal mention {the references in this scholarly article have been meticulously documented}. See reference.
alongside. This term, meaning “at the side of,” should not be followed by of a lot. Two words, not one.
already; all ready. The first refers to time {the movie has already started}; the second refers to preparation {are the actors all ready?}.
alright. See all right.
altar, n.; alter, vb. An altar is a table or similar object used for sacramental pur poses. To alter is to change.
alternate, adj. & n.\ alternative, adj. & n. Alternate implies (1) a substitute for another {we took the alternate route} or (2) every other or every second {al ternate Saturdays}. Alternative implies availability as another, usually sounder choice or possibility {alternative fuel sources}. The noun uses are analogous {the awards committee named her as alternate} {we have no alternative}.
altogether; all together. Altogether means “wholly” or “entirely” {that story is altogether false}. All together refers to a unity of time or place {the family will be all together at Thanksgiving}.
amend; emend. The first is the general term, meaning “to change or add to something written or spoken” {the city amended its charter to abolish at-large council districts} or “to make better” {amend your behavior!}. The second means “to remove one or more mistakes from” (as of a text) {for the second printing, the author emended several typos that had reached print in the first}. The noun corresponding to amend is amendment} the one corresponding to emend is emendation.
amiable; amicable. Both mean “friendly,” but amiable refers to people who are easy to like {an amiable waiter} and amicable to relationships that involve goodwill and a lack of quarreling {an amicable divorce}.
amid. See between.
among. See between.
amount; number. Amount is used with mass nouns {a decrease in the amount of pollution} {a small amount of money}. Number is used with count nouns {a growing number of dissidents} {the number of coins in your pocket}.
an. See a.
and. Popular belief to the contrary, this conjunction usefully begins sentences, typically outperforming moreover, additionally, in addition, further, and fur Word Usage 5.250 thermore. Yet it does not occur as a sentence-starter as often as but. See but; see also 5.203.
and/or. Avoid this Janus-faced term. It can often be replaced by and or or with no loss in meaning. Where it seems needed {take a sleeping pill and/or a warm drink}, try... or..., or both {take a sleeping pill or a warm drink, or both}. But think of other possibilities {take a sleeping pill, perhaps with a warm drink}.
anecdotal. This adjective corresponds to anecdote, but in one sense the words have opposite connotations. An anecdote is a story that is thought (but not known) to be true. But anecdotal evidence refers to accounts that are suspect because they are not rigorously verified.
angry. See mad.
anxious. Avoid it as a synonym for eager. The standard sense is “worried, ner vous, distressed.” anyone; any one. The one-word anyone is a singular indefinite pronoun used in reference to no one in particular {anyone would know that}. The two-word phrase any one is a more emphatic form of any, referring to a single person or thing in a group {do you recognize any one of those boys?} {I don’t know any one of those stories}.
anyplace. See anywhere.
anyway; anyways. The former is standard; the latter, traditionally considered di alectal, has made inroads into the speech of many otherwise educated people born since about 1980. But it remains nonstandard.
anywhere; any place. The first is preferred for an indefinite location {my keys could be anywhere}. But any place (two words) is narrower when you mean “any location” {they couldn’t find any place to sit down and rest}. Avoid the informal one-word anyplace.
appertain. See pertain.
appraise; apprise. To appraise is to assess or put a value on something {the jew eler appraised the necklace}. To apprise is to inform or notify someone about something {keep me apprised of any developments}.
appreciate. Three senses: (1) to understand fully; (2) to increase in value; (3) to be grateful for (something). Sense 3 often results in verbose constructions; in stead of I would appreciate it if you would let me know, try I would appreciate your letting me know or, more simply, please let me know.
apprise. See appraise.
approve; endorse. Approve implies positive thought or a positive attitude rather than action apart from consent. Endorse implies both a positive attitude and active support.
approve (of). Approve alone connotes official sanction or acceptance {the finance committee approved the proposed budget}. Approve of suggests thinking favor ably about {she approved of her sister’s new hairstyle}.
approximately. See about.
apt; likely. Both mean “fit, suitable,” but apt is used for general tendencies or 5.250 GRAMMAR AND USAGE habits {the quarterback is apt to drop the football}. Likely expresses probability {because he didn’t study, it’s likely that he’ll do poorly on the exam}. Although likely is traditional as a synonym of probable, many writers and editors object to its use as a synonym of probably. Apt has two other senses: (1) “exactly right for a given situation or purpose” {an apt remark} and (2) “quick to learn” {an apt pupil}.
area. Often a nearly meaningless filler word, as in the area of partnering skills. Try deleting the area of. In the sciences, however, its more literal meaning is often important and should be retained. See also space.
as far as. Almost always wordy. Avoid the nonstandard phrasing that uses as far as in place of as for—that is, avoid using as far as without the completing verb is concerned or goes. Even with the verb, though, this is usually a wordy con struction. Compare as far as change is concerned, ifs welcome with as for change, ifs welcome.
as is. In reference to an acquisition, as is is framed in quotation marks and refers to the acceptance of something without guarantees or representations of qual ity {purchased “as is”}. The phrase on an “as is” basis is verbose.
as of yet. See as yet.
as per. This phrase, though common in the commercial world, has long been con sidered nonstandard. Instead of as per your request, write as you requested or (less good) per your request. The recent innovation asper usual for as usual is an illiteracy.
assault; battery. These are popularly given the same meaning. But in law assault refers to a threat that causes someone to reasonably fear physical violence, and battery refers to a violent or repugnant intentional physical contact with another person. In the strict legal sense, an assault doesn’t involve touching; a battery does.
assemblage; assembly. An assemblage is an informal collection of people or things. An assembly is a group of people, especially decision makers, organized for a purpose {a national assembly}; a meeting {regular public assemblies}; or the process of putting together the parts of something {instructions for as sembly}.
assent; consent. The meanings are similar, but assent connotes a more affirma tive agreement after careful consideration; consent connotes mere allowance, or sometimes grudging acquiescence.
as such. This pronominal phrase always requires an antecedent for such {science is the organized search for truth and, as such, must be looked upon as an end in itself}. The phrase is now often loosely used as a synonym for therefore. Avoid this misusage {science seeks out truth in an organized way and, as such, must be looked upon as an end in itself}.
assumption; presumption. An assumption is not drawn from strong evidence; typically, it is a hypothesis that one accepts as true without definite proof {your assumption can be tested by looking at the public records}. A presumption im Word Usage 5.250 plies a basis in evidence or at least experience; if uncontradicted, a presump tion may support a decision {the legal presumption of innocence}.
assure. See ensure.
as to. This two-word preposition is best used only to begin a sentence that could begin with on the question of or with regard to {as to those checks, she didn’t know where they came from}. Otherwise, use about or some other preposition.
as yet; as of yet. Stilted and redundant. Usenet, still, so far, or some other equiv alent.
attain; obtain. To attain something is either to accomplish it through effort (e.g., a goal) {she soon attained a position of power} or to reach a particular age, size, level, etc. {the stock market attained a new high this morning}. To obtain something is to get it or gain possession of it {obtaining information}. In best usage, you attain a degree and obtain a diploma. It can be a fine distinction, and in common usage the words are often treated as synonyms.
at the present time; at this time; at present. These are turgid substitutes for now, today, currently, or even nowadays (a word of perfectly good literary standing). Of the three phrases, at present is least suggestive of bureaucratese.
at the time that; at the time when. Use the plain and simple when instead.
auger; augur. The spellings of these words can be tricky because they are pro nounced the same /aw-gr/. The tool for boring is an auger. Augur means “a clairvoyant or seer” (noun) or “to foretell” (verb). Augurs well is an idiomatic equivalent of bodes well. The related noun augury refers to an indication of what will happen in the future.
avenge, vb.; revenge, vb. & n. Avenge connotes an exaction for a wrong {his torically, family grudges were privately avenged}. The corresponding noun is vengeance. Revenge connotes the infliction of harm on another out of anger or resentment {the team is determined to revenge its humiliating loss in last year’s championship game}. Revenge is much more commonly a noun {they didn’t want justice—they wanted revenge}.
averse. See adverse.
avocation; vocation. An avocation is a hobby or pleasant pastime {stamp collect ing is my weekend avocation}. A vocation is one’s profession or, especially in a religious sense, one’s calling {she had a true vocation and became a nun}.
awhile; a while. The one-word version is adverbial; it means “for a short time” {let’s stop here awhile}. The two-word version is a noun phrase that follows the preposition for or in {she worked for a while before beginning graduate studies}.
backward(s). See toward.
bale; bail. The somewhat less common term is bale (= a bundle or to form into a bundle, as of hay or cotton). Bail is most often a verb (= to drain by scooping, as of getting water out of a boat using a pail); it is also a noun and verb regard ing the posting of security to get out of jail pending further proceedings. Bail is also used informally to denote leaving quickly or escaping {the couple bailed 5.250 GRAMMAR AND USAGE from the party}. To bail out someone (a phrasal verb) is to get the person out of trouble.
based on. This phrase has two legitimate and two illegitimate uses. It may unim peachably have verbal force (base being a transitive verb, as in they based their position on military precedent) or, in a passive sense, adjectival force (based be ing read as a past-participial adjective, as in a sophisticated thriller based on a John le Carre novel). Two uses, however, are traditionally considered slipshod. Based on should not have adverbial force (as in Rates are adjusted annually, based on the 91-day Treasury bill) or prepositional force (as a dangling partici ple, as in Based on this information, we decided to stay). Try other constructions {rates are adjusted annually on the basis of the 91-day Treasury bill} {with this information, we decided to stay}.
basis. Much overworked, this word most properly means “foundation; the facts, things, or ideas from which something can be developed.” It often appears in the phrase on a... basis or some similar construction. When possible, substi tute adverbs (personally, not on a personal basis) or simply state the time (daily, not on a daily basis). The plural is bases {the legislative bases are complicated}.
bated breath. So spelled—not baited breath. Someone who waits with bated breath is anxious or excited (literally “holding [abating] one’s breath”).
battery. See assault.
begging the question. This phrase traditionally denotes a logical fallacy of as suming as true what has yet to be proved—or adducing as proof for some prop osition something that’s every bit as much in need of proof as the first proposi tion. For example, someone might try to “prove” the validity of a certain religion by quoting from that religion’s holy text. But the phrase gets misused in many ways—as (erroneously) meaning “prompting a question,” “inviting an obvious question,” “evading a question,” and “ignoring a question.” behalf. In behalf ofmeans “in the interest or for the benefit of” {the decision is in behalf of the patient}. On behalfofmeans “acting as agent or representative of” {on behalf of Mr. Scott, I would like to express heartfelt thanks}.
bemused. This word means “bewildered, distracted, or confused.” It is not a syn onym of amused.
benevolence; beneficence. Benevolence is the attribute of being disposed to kind ness or capable of doing good {the priest’s benevolence was plainly evident}. It applies most often to people but may also apply to things that are beneficial. Beneficence is a major act of kindness or the performance of good deeds gen erally {the villagers thanked him for his beneficence}. The first term denotes a quality, the second conduct.
beside; besides. Beside is a preposition of position, whether literal {beside the road} or figurative {beside the point}. Besides may be a preposition meaning “other than” {who’s going besides us?} or an adverb meaning “also” or “any way” {besides, who wants to know?}.
between; among; amid. Between indicates one-to-one relationships {between Word Usage 5.250 you and me}. Among indicates undefined or collective relationships {honor among thieves}. Between has long been recognized as being perfectly appro priate for more than two objects if multiple one-to-one relationships are un derstood from the context {trade between members of the European Union}. Amid is often used with mass nouns {amid talk of war}—though it can often be used with abstract nouns in the plural {resigned amid rumors of misconduct} {the investigation comes amid growing concerns}. Among is invariably used with plurals of count nouns {among the children}. Avoid amidst and amongst, especially in American English.
between you and me. This is the correct phrasing—not between you and I, which is a classic example of hypercorrection. Both pronouns function as objects of the preposition between. True, Shakespeare put the phrase ’tween you and I in a character’s mouth, but that was at a time when English grammar was much less settled than it came to be in the eighteenth century—and that usage was an outlier even in the Elizabethan era. Further, the sociolinguistic point that Shakespeare might have been making by having a character speak that phrase may well be lost in the mists of time.
bi-; semi-. Generally, bi- means “two” (biweekly means “every two weeks”), while semi- means “half” (semiweekly means “twice a week”). Because these pre fixes are often confused with each other, writers should be explicit about the meaning.
biannual; semiannual; biennial. Biannual and semiannual both mean “twice a year” {these roses bloom biannually}. But biennial means “once every two years” or “every other year” {our legislature meets biennially}. To avoid con fusion, write semiannual instead of biannual, and consider writing once every two years instead of biennial.
billion; trillion. The meanings can vary in different countries. In the United States, a billion is 1,000,000,000. In Great Britain, Canada, and Germany, a billion is traditionally a thousand times more than that (a million millions, or what Americans call a trillion)—though the American English sense now predominates even in British English. Further, in Great Britain a trillion is traditionally a million million millions, what Americans would call a quintil lion (1,000,000,000,000,000,000). Although the American definitions are gaining acceptance, writers need to remember the historical geographic dis tinctions. See also 9.8.
blatant; flagrant. An act that is blatant is both bad and plain for all to see {a bla tant error}. One that flagrant is done brazenly as well as openly, often with a stronger suggestion of shocking illegality or immorality {a flagrant violation of the law}.
bombastic. A bombastic speech or essay is pompously long-winded and self- important but essentially empty of substance. The word has nothing to do with temper.
born; borne. Born is used only as an adjective {a born ruler} or in the fixed 5.250 GRAMMAR AND USAGE passive-voice verb to be born {the child was born into poverty}. Borne is the has borne three children}. It is also used to form compound terms {foodborne} {vectorborne}.
both-and. These correlative conjunctions should frame matching syntactic parts. Hence don’t write She is both a writer and she skis professionally, but instead She is both a writer and a professional skier.
breach, n. & vb.; breech, n. A breach is a gap in or violation of something {a breach of contract} or a serious disagreement {healing the breach between the nations}. To breach is to break, break open, or break through {breach the castle walls}. Breech refers to the lower or back part of something, especially the buttocks {a breech birth} or the part of a modern firearm where bullets are inserted {the rifle’s breech}.
bring; take. The distinction may seem obvious, but the error is common. The simple question is, Where is the action directed? If it’s toward you, use bring {bring home the bacon}. If it’s away from you, use take {take out the trash}. You take (not bring) your car to the mechanic.
but. Popular belief to the contrary, this conjunction usefully begins contrasting sentences, typically with greater strength and speed than however. Avoid put ting a comma after it. Cf. and; see also 5.203.
by means of. Often verbose. Use by or with if either one suffices.
by reason of. Use because or because of unless by reason ofis part of an established phrase {by reason of insanity}.
cache; cachet. Cache, a count noun, refers either to a quantity of goods or valu ables that have been stashed away or to a storage buffer within a computer. Cachet, generally a mass noun, refers most commonly to prestige or fetching appeal—or else a seal on a document or a commemorative design. Cachet sometimes appears, incorrectly, as cache.
can; could. Can means “to be able to” and expresses certainty {I can be there in five minutes}. Could is better for a sense of uncertainty or a conditional state ment {could you stop at the cleaners today?} {if you send a deposit, we could hold your reservation}. See 5.146.
can; may. Can most traditionally applies to physical or mental ability {she can do calculations in her head} {the dog can leap over a six-foot fence}. In col loquial English, can also expresses a request for permission {can I go to the movies?}, but this usage is not recommended in formal contexts. May suggests possibility {the class may have a pop quiz tomorrow} or permission {you may borrow my car}. A denial of permission is properly phrased formally with may not {you may not borrow my credit card} or, less formally, with cannot or can’t {you can’t use the computer tonight}. See 5.146,5.147.
cannon; canon. A cannon is an artillery weapon that fires metal balls or other missiles. A canon is (1) a general rule or principle, (2) an established criterion, Word Usage 5.250 (3) the sum of a writer or composer’s work, (4) the collective literature ac cepted by a scholastic discipline, (5) a piece of music in which a tune is started by one performer and is mimicked by each of the others, or (6) a Christian priest having special duties within a church or cathedral.
capability. See ability, capacity. See ability.
capital; capitol. A capital is a seat of government (usually a city) {Jefferson City is the capital of Missouri}. A capitol is a building in which a legislature meets {the legislature opened its new session in the capitol today}.
carat; karat; caret. Carat measures the weight of a gemstone; karat measures the purity of gold. To remember the difference, think of 24Kgold. (In British En glish, the spelling carat serves in both senses.) Caret is a mark on a manuscript indicating where matter is to be inserted; borrowed from Latin in the seven teenth century, it literally means “(something) is lacking.” career; careen. The word career’s career as a verb meaning “to go full speed” may be about over, except in British English (in which the two verbs contend in what is still a tight race). In American English, its duties have been assumed by careen (traditionally, “to tip to one side while moving”), even though nothing in that verb’s time-honored definition denotes high speed. So today in Amer ican English it’s typically careened down the hill but in British English careered down the hill.
caret. See carat.
case. This multifaceted word is often a sign of verbal inflation, especially in its uses as a near-synonym of situation. For example, in case means “if”; in most cases means “usually”; in every case means “always.” The word is justifiably used in law (in which a case is a lawsuit or judicial opinion) and in medicine (in which the word refers to an instance of a disease or disorder). By extension, it has analogous senses in social work, criminal detection, etc. Of course, the word can also denote a box or container {briefcase}, an argument or set of reasons {state your case}, or a grammatical word form.
cause celebre. This word most strictly denotes a legal case, especially a prosecu tion, that draws great public interest. By extension, it refers to a notorious ep isode, event, or even person. It does not properly denote a person’s pet cause. Though it retains its acute and grave accents, the phrase is now considered naturalized enough not to be italicized (except when called out as a phrase, as in the next sentence). Yet the plural retains its French form: causes celebres.
censer; censor, n.; sensor. The correct spellings can be elusive. A censer is either a person who carries a container of burning incense or the container itself. A censor is a person who suppresses objectionable subject matter. A sensor is a mechanical or electronic device for discovering light, heat, movement, etc.
censor, vb.; censure, vb. To censor is to review books, films, letters, and the like to remove objectionable material—that is, to suppress {soldiers’ letters are 5.250 GRAMMAR AND USAGE often censored in wartime}. To censure is to criticize strongly or disapprove, or to officially reprimand {the House of Representatives censured the president for the invasion} {in some countries the government censors the press; in the United States the press often censures the government}.
center around. Although this illogical phrasing does have apologists, stylists tend to use either center on or revolve around.
certainty; certitude. If you are absolutely sure about something, you display both certainty (firm conviction) and certitude (assurance of being certain). That fact you are sure about, however, is a certainty but not a certitude—the latter is a trait applied to people only.
chair; chairman; chairwoman; chairperson. Chair is widely regarded as the best gender-neutral choice. Since the mid-seventeenth century, chair has re ferred to an office of authority. See also 5.251-60.
childish; childlike. Childlike is used positively to connote innocence, eagerness, and freshness {a childlike smile}. Childish is pejorative; it connotes immatu rity, silliness, and unreasonableness {childish ranting}.
chord; cord. Chord denotes (1) a group of harmonically consonant notes {major chords} {.minor chords} or (2) a straight line joining the ends of an arc (sense 2 being a technical term in mathematics and engineering). Cord is the word denoting a thick string or rope {spinal cord} {umbilical cord} {vocal cord}, an enclosed wire that supplies electricity to an appliance or other equipment, or a quantity of firewood.
circumstances. Both in the circumstances and under the circumstances are accept able, but under is now much more common in American English. In predomi nates in British English.
cite, h .; site. As a noun, cite is colloquial for citation, which refers to a source of information {a cite to Encyclopaedia Britannica}. A site is a place or location used for a particular purpose {building site} {website}. Cf. sight.
citizen; subject. In a governmental sense, these are near-synonyms that should be distinguished. A citizen owes allegiance to a nation whose sovereignty is a collective function of the people {a citizen of Germany}. A subject owes alle giance to an individual sovereign because the form of government is monar chical {a subject of the queen}.
class. This word denotes a category or group of things that are considered to gether because of their similarities {the class of woodwind instruments}. Prop erly, a class is never one type {the oboe is a type of woodwind} or one kind of thing {a drum is one kind of percussion instrument}.
classic; classical. Classic means “important, authoritative, outstanding” {The Naked Night is one of Ingmar Bergman’s classic films}. Classical applies to a traditional set of values in literature, music, design, and other fields {classi cal Greek} {a classical composer} or to the definitive or earliest-characterized form {classical EEC syndrome}.
clean; cleanse. Although various cleaning agents are called “cleansers,” clean Word Usage 5.250 displaced cleanse long ago in most of the word’s literal senses. Cleanse retains the Old English root meaning “pure”: its use today usually refers to spiritual or moral (or gastrointestinal) purification.
cleave. This verb was originally two different words, and that difference is re flected in the opposite meanings that cleave has: (1) to cut apart {to cleave meat} and (2) to cling together {standing in the rain, his clothes cleaving to his body}. (When a term is its own antonym, it is known as a contronym.) The con jugations are (1) cleave, cleft (or clove), cleft (or cloven); and (2) cleave, cleaved, cleaved.
clench; clinch. Clench, which connotes a physical action, normally involves a per son’s hands, teeth, jaw, or stomach {he clenched his hand into a fist}. Clinch, the more common term, has mostly figurative uses about finally achieving something after a struggle {clinched the title} {clinched the victory}. But there is an exception to the nonphysical uses of clinch: if two people clinch, they hold each other’s arms tightly, as in boxing.
climactic; climatic. Climactic is the adjective corresponding to climax {during the movie’s climactic scene, the projector broke}. Climatic corresponds to cli mate {the climatic conditions of northern New Mexico}.
clinch. See clench.
close proximity. Redundant. Write either close or in proximity.
closure; cloture. Closure denotes the temporary or permanent closing or final res olution of something. Cloture denotes the parliamentary procedure of closing debate and taking a vote on a legislative bill or other measure.
cohabit; cohabitate. Cohabit is the traditional verb for living with another person in a sexual relationship without being married. Cohabitate, a back-formation from cohabitation, is best avoided.
collaborate; corroborate. To collaborate is to cooperate on some undertaking, especially in the arts or sciences {the participants are collaborators}. To corrob orate something is to back up its reliability with proof or evidence {the expert corroborated the witness’s testimony}.
collegial; collegiate. Collegial answers to colleague {a healthy collegial work envi ronment}; collegiate answers to college {collegiate sports}.
commendable; commendatory. What is done for a worthy cause is commend able {commendable dedication to helping the poor}. What expresses praise is commendatory {commendatory plaque}.
common; mutual. What is common is shared by two or more people {borne by different mothers but having a common father}. What is mutual is reciprocal or directly exchanged by and toward each other {mutual obligations}. Strictly, friend in common is better than mutual friend in reference to a third person who is a friend of two others.
commonweal; commonwealth. The commonweal is the public welfare. Tradi tionally, a commonwealth was a state established by public compact or by the consent of the people to promote the general good (commonweal), and where 5.250 GRAMMAR AND USAGE the people reserved supreme authority. In the United States, the word is syn onymous with state, four of which are still called commonwealths: Kentucky, Massachusetts, Pennsylvania, and Virginia. The Commonwealth of Puerto Rico is also a US territory.
compare. To compare with is to discern both similarities and differences between things. To compare to is to liken things or to note primarily similarities be tween them, especially in the active voice {Are you comparing me to him? I hope not!}.
compelled; impelled. If you are compelled to do something, you have no choice in the matter {Nixon was compelled by the unanimous Supreme Court decision to turn over the tapes}. If you are impelled to do something, you still may not like it, but you are convinced that it must be done {the voter disliked some candidates but was impelled by the income-tax issue to vote a straight-party ticket}. Whereas compel connotes an outside force, impel connotes an inner drive.
compendious; voluminous. These are not synonyms, as many apparently be lieve. Compendious means “concise, abridged.” Voluminous, literally “occupy ing many volumes,” most commonly means “vast” or “extremely lengthy.” complacent; complaisant; compliant. To be complacent is to be content with oneself and one’s life—with the suggestion that one may be smugly unwilling to improve or unprepared for future trouble. To be complaisant is to be easygo ing and eager to please others. To be compliant is to be amenable to orders or to a regimen imposed by others.
compliment; complement. A compliment is a flattering or praising remark {a compliment on your skill}. A complement is something that completes or brings to perfection {the lace tablecloth was a complement to the antique silver}. The words are also verbs: to compliment is to praise, while to complement is to sup plement adequately or to complete. In the grammatical sense, a complement is a word or phrase that follows the verb to complete the predicate. The cor responding adjectives are complimentary, meaning (1) “expressing praise” or (2) “given to someone free of charge”; and complementary, meaning (1) “going well together, despite differences,” or (2) “consisting of two geometric angles that, added together, take up 90 degrees.” comprise; compose. Use with care. To comprise is “to consist of, to include” {the whole comprises the parts}. To compose is “to make up, to form the substance of something” {the parts compose the whole}. The phrase is comprised of, though increasingly common, remains nonstandard. Instead, try is composed of or consists of See include.
concept; conception. Both words may refer to an abstract thought, but concep tion also means “the act of forming an abstract thought.” Avoid using either word as a high-sounding equivalent of idea, design, thought, or program.
condole, vb.; console, vb. These are closely related but not identical. To condole Word Usage 5.250 with is to express sympathy to {community leaders condoled with the victims’ families}. The corresponding noun is condolence {they expressed their condo lences at the funeral}. To console is to comfort in a time of distress or disap pointment {the players consoled their humiliated coach}. The corresponding noun is consolation {their kind words were small consolation}.
confidant; confidante; confident. A confidant is a close companion, someone (male or female) you confide in. Confidante, a feminine form, is a fading alter native spelling of confidant (used only in reference to a female confidant). It reflects French gender spellings. Confident is the adjective meaning “sure that something will happen in the way one wants or expects” or “sure that some thing is true.” congruous; congruent. Both terms mean “in harmony, in agreement.” The first is seen most often in its negative form, incongruous, meaning “strange, unex pected, or unsuitable in a particular situation” {the modern house looks in congruous in this old neighborhood}. The second is used in math to describe triangles that are identical in their angles as well as in the length of their sides {congruent angles}.
connote; denote. To connote (in reference to language) is to convey a meaning beyond the basic one, especially through emotive nuance {the new gerund parenting and all that it connotes}. To denote (again in reference to language) is to specify the literal meaning of something {the phrase freezing point de notes 32 degrees Fahrenheit or 0 degrees Celsius}. Both words have figurative uses {all the joy that parenthood connotes} {a smile may not denote happi ness}.
consent. See assent.
consequent; subsequent. The first denotes causation; the second does not. A consequent event always happens after the event that caused it, as does a sub sequent event. But a subsequent event does not necessarily occur as a result of the first: it could be wholly unrelated but merely later in time.
consider. Add as only when you mean “to examine or discuss for a particular pur pose” {handshaking considered as a means of spreading disease}. Otherwise, omit as {we consider him qualified}.
consist. There are two distinct phrases: consist of and consist in. The first, by far the more common one, applies to the physical components that make up a tan gible thing {the computer-system package consists of software, the CPU, the monitor, and a printer}. The second refers to the essence of a thing, especially in abstract terms {moral government consists in rewarding the righteous and punishing the wicked}.
console. See condole.
contact, vb. If you mean write or call or email, say so. But contact is undeniably a brief way of referring to communication without specifying the means.
contagious; infectious. Both broadly describe a disease that is communicable.
5.250 GRAMMAR AND USAGE But a contagious disease spreads by direct contact with an infected person or animal {rabies is a contagious disease}. An infectious disease is spread by germs on a contaminated object or element, such as earth or water {tetanus is infectious but not contagious}.
contemporary; contemporaneous. Both express coinciding time, but contem porary usually applies to people, and contemporaneous applies to things or actions. Because contemporary has the additional sense “modern,” it is un suitable for contexts involving multiple times. That is, a reference to Roman, Byzantine, and contemporary belief systems is ambiguous; change contemporary to modern.
contemptuous; contemptible. If you are contemptuous, you are feeling and showing that you think someone or something deserves no respect. If you are contemptible, others will have that attitude toward you.
content; contents. Content applies to the ideas, facts, or opinions in a written or oral presentation {the lecture’s content was offensive to some who were pres ent}. Contents usually denotes physical ingredients: the things that are inside a box, bag, room, or other container {the package’s contents were difficult to discern by x-ray}. If the usage suggests many items, material or nonmaterial, contents is correct {table of contents} {the investigative report’s contents}.
continual; continuous. What is continual may go on for a long time, but always there are brief interruptions, so that it can be characterized as intermittent or frequently repeated {continual nagging}. What is continuous never stops—it remains constant or uninterrupted {continuous flow of water}. A line that is continuous has no gaps or holes in it.
contravene; controvert. To contravene is to conflict with or violate (the law, a rule, etc.) {the higher speed limit contravenes our policy of encouraging fuel conservation}. To controvert is to challenge or contradict {the testimony con troverts the witness’s prior statement}.
convince. See persuade.
copyright, vb. This verb, meaning “to obtain the legal right to be the only pro ducer or seller of a book, play, film, or other creative work for a specific length of time,” is conjugated copyright-copyrighted-copyrighted. Note the spelling, which has nothing to do with write.
cord. See chord.
corollary; correlation. A corollary is either (1) a subsidiary proposition that fol lows from a proven mathematical proposition, often without requiring ad ditional evidence to support it, or (2) a natural or incidental result of some action or occurrence. A correlation is a positive connection between things or phenomena. If used in the context of physics or statistics, it denotes the de gree to which the observed interactions and variances are not attributable to chance alone.
corporal; corporeal. What is corporal relates in some way to the body {corporal Word Usage 5.250 punishment}; what is corporeal has a physical form that can be touched {not our spiritual but our corporeal existence}.
corps; core. A corps is a body of like workers, as in an army, with special duties and responsibilities {Marine Corps} {press corps}. It is often misspelled like its homophone, core, which denotes the central or most important part of some thing {the core of the problem} {the earth’s core}.
correlation. See corollary.
corroborate. See collaborate.
could. See can.
couldn’t care less. This is the standard phrasing. Avoid the illogical form could care less.
councillor; counselor. A councillor is one who sits on a council {city councillor}. A counselor is one whose job is to help and advise people with problems {per sonal counselor}. In British English, the spelling is counsellor.
couple. Using couple as an adjective has traditionally been regarded as nonstan dard phrasing—though it is increasingly common as a casualism. Add of {we watched a couple of movies}. When referring to two people as a unit {married couple}, the noun couple takes either a singular or a plural verb {the couple is happy} {the couple are honeymooning in Ravello}. When the pronoun they follows couple—if a pronoun is used at all, it is normally plural—the plural verb is preferable {the couple were delighted by their friends’ responses}.
court-martial. Two words joined by a hyphen, whether the phrase functions as a noun or as a verb. Because martial acts as an adjective meaning “military,” the plural of the noun is courts-martial. The third-person-singular verb is court- martials {if the general court-martials him, he’ll have much to answer for}. In American English, the inflected spellings of the verb are court-martialed, court-martialing; in British English, the spellings are court-martialled, court- martialling.
credible; creditable; credulous. Credible means “believable; deserving trust”; creditable means “praiseworthy; deserving approval”; credulous means “gull ible; tending to believe whatever one is told—and therefore easily deceived.” The most common error involving cognate forms of these words is in the mal- apropism strains credulity. If some form of that cliche must be used, it should read strains credibility.
crevice; crevasse. Size matters. A crack in the sidewalk is a crevice (accent on the first syllable) because it’s narrow and typically not very deep; a fissure in a glacier or a dam is a crevasse (accent on the second syllable) because it’s a deep open crack.
criminal. See unlawful.
criteria. This is the plural form of criterion (= a standard for judging): one crite rion, two criteria. The double plural criterias is a solecism.
damp, vb.; dampen. Both words convey the sense “to moisten.” Damp also 5.250 GRAMMAR AND USAGE means “to reduce with moisture” {damp the fire} or “to diminish vibration or oscillation of [a wire or voltage] ” {damp the voltage}. In a figurative sense, dampen means “to make [a feeling, mood, activity, etc.] less intense or enjoy able” {dampen one’s hopes}.
data. Though originally this word was a plural of datum, it is now commonly treated as a mass noun and coupled with a singular verb. In formal writing (and always in the sciences), use data as a plural. Whatever you do, though, use the term consistently within a single writing—either singular or (more for mally) plural.
deadly; deathly. Deadly means “capable of causing death” {deadly snake venom} or “likely to cause as much harm as possible” {deadly enemies}. Deathly means “arousing thoughts of death or a dead body” {deathly silence}.
decide whether; decide if. See determine whether.
decimate. This word literally means “to kill every tenth person,” a means of re pression that goes back to Roman times. But the word has come to mean “to inflict heavy damage or destroy a large part of something,” and this use has long been predominant. Avoid decimate when you are referring to complete destruction. That is, don’t say that a city was completely decimated.
deduce. See adduce.
defamation; libel; slander. Defamation is the communication of a falsehood that damages someone’s reputation. If it is recorded, especially in writing, it is libel; otherwise, it is slander.
definite; definitive. Definite means “clear, exact” {a definite yes}. Definitive means either “not subject to further revision in the near future” {we have a definitive agreement} or “of such high quality as to be unimprovable for a long time” {the definitive guide}.
delegate. See relegate.
deliberate, adj.; deliberative. As an adjective, deliberate means either “planned; carefully thought out” {a deliberate response} or “slow and steady” {deliber ate progress}. Deliberative means “of, characterized by, or involving debate”; the word most often applies to an assembly {deliberative body} or a process {deliberative meetings}.
denote. See connote.
denounce; renounce. To denounce is either to criticize harshly, especially in public {they denounced the prisoner swap}, or to accuse, as by giving incrim inating information about someone’s illegal political activities to the authori ties {denounced him to the police}. To renounce is either to relinquish or reject {renounced her citizenship} or to declare publicly that one no longer believes something or will no longer behave in some way {renounce violence}.
dependant, n.; dependent, adj. &n. In British English, the first is the preferable noun {he claimed three dependants on his tax return}; the second is the adjec tive {the family has become dependent on welfare}. But in American English, dependent is the usual form as both noun and adjective.
Word Usage 5.250 depend on. Although upon is best reduced to on in this phrase, no further reduc tion is idiomatic: depend demands an on. Hence don’t write That depends how we approach the problem but rather That depends on how we approach the problem.
deprecate. In general, to deprecate is to strongly disapprove or criticize. But in the phrase self-deprecating—which began as a mistaken form of self-depreciating but is now standard—the sense of deprecate is “to belittle.” In computing, dep recate serves as a warning: a deprecated feature or function is one that may be phased out of a future release of software, so users should begin looking for alternatives.
derisive; derisory. What is derisive ridicules as stupid or silly {derisive laughter}. What is derisory invites or deserves ridicule {that derisory “banana” hat}, es pecially when a laughably small amount of money is offered or given {my de risory paychecks}.
deserts; desserts. The first are deserved {he got his just deserts}, the second eaten {the many desserts on the menu}. Just desserts is a common misspelling (unless the meaning is “only postprandial sweets”).
despite; in spite of. For brevity, prefer despite.
determine whether; determine if. The first phrasing is irreproachable style; the second is acceptable as a colloquialism. The same is true of decide whether ver sus decide if.
different. The phrasing different from is generally considered preferable to differ ent than {this company is different from that one}, but sometimes the adver bial phrase differently than is all but required {she described the scene differ ently than he did}. In British English, different to is not uncommon—but it is distinctively British English, whereas different from is standard everywhere.
differ from; differ with. Differ from is the usual construction denoting a contrast {the two species differ from each other in subtle ways}. Differ with regards dif ferences of opinion {the state’s senators differ with each other on many issues}, disburse; disperse. To disburse is to distribute money, especially from a large sum available for some specific purpose. To disperse is (1) to spread in various directions over a wide area {the clouds dispersed} or (2) to cause to go away in different directions {police dispersed the unruly crowd}.
disc. See disk.
discomfort; discomfit. Discomfort is a noun meaning “ill at ease.” It can also be used as a verb meaning “to put ill at ease.” But doing so often invites confusion with discomfit, which originally meant “to defeat utterly.” Today it means “to thwart, confuse, annoy, or embarrass” {the ploy discomfited the opponent}. The distinction has become a fine one, since a discomfited person is also un comfortable. Discomfiture is the corresponding noun.
discreet; discrete. Discreet means either “careful about not divulging secrets or upsetting others” {a discreet silence} or “showing modest taste; nonostenta- tious” {discreet jewelry}. Discrete means “separate, distinct, unconnected” {six discrete parts}.
5.250 GRAMMAR AND USAGE discriminating, adj.; discriminatory. The word discrimination can be used in either a negative or a positive sense, and these adjectives reflect that ambiva lence. Discriminatory means “reflecting a biased, unfair treatment” {discrim inatory employment policy}. Discriminating means “analytically refined, dis cerning, tasteful” {a discriminating palate}.
disinterested. This word should be reserved for the sense “not having a financial personal interest at stake and therefore able to judge a situation fairly; impar tial.” Avoid it as a replacement for uninterested (which means “unconcerned, bored”).
disk; disc. Disk is the usual spelling {hard disk} {disk drive}. But disc is preferred in a few specialized applications {compact disc} {disc brakes} {disc harrow}— particularly where the object in question is circular and flat.
disorganized; unorganized. Both mean “not organized,” but disorganized sug gests (1) a group in disarray, either thrown into confusion or inherently unable to work together {the disorganized 1968 Democratic National Convention in Chicago}, or (2) a person who is exceedingly bad at arranging or planning things {disorganized students}.
disperse. See disburse.
distinctive; distinguished; distinguishable. A distinctive feature is something that makes a person (or place or thing) easy to recognize {U2’s distinctive sound}. But it does not necessarily make that person distinguished (respected and admired) {the distinguished professor wears a distinctive red bow tie}. It does, however, make the person distinguishable (easy to see as being different from something else)—a term that does not carry the positive connotation of distinguished.
dive, vb. The preferred conjugation has traditionally been dive-dived-dived. The irregular form dove, though, has become the slightly predominant past-tense form in American English and should be accepted as standard: dive-dove- dived. Traditionalists will stick to the older inflection.
doctrinal; doctrinaire. Doctrinal means “of, relating to, or constituting a doc trine”; it is neutral in connotation {doctrinal differences}. Doctrinaire means “dogmatic,” suggesting that the person described is stubborn and narrow minded {a doctrinaire ideologue}.
doubtfully, adv. In recent years, this term has come into use as a sentence ad verb functioning as a correlative of hopefully and as an antonym of undoubtedly {Will you be attending the party? Hopefully—but doubtfully. [That is, I hope I’ll be able to go, but I doubt it.]}. Should you abstain from this usage in Stan dard Written English? No doubt.
doubtless, adv. Use this form (it’s called a flat adverb)—not doubtlessly. See also 5.160.
doubt that; doubt whether; doubt if. Doubt that conveys a negative sense of strong skepticism or questioning {I doubt that you’ll ever get your money back}. Doubt whether also conveys a sense of skepticism, though less strong Word Usage 5.250 {the official says that he doubts whether the company could survive}. Doubt if is a casual phrasing for doubt that.
drag. Conjugated drag-dragged-dragged. The past form drug is dialectal.
dream. Either dreamed (more typical in both American English and British En glish) or dreamt is acceptable for the past-tense and past-participial forms.
drink, vb. Correctly conjugated drink-drank-drunk {they had not drunk any fruit juice that day}.
drown, vb. Conjugated drown-drowned-drowned.
drunk, adj.; drunken. Drunk describes a current state of intoxication {drunk driver}. (By contrast, a drunk—like a drunkard—is someone who is habitually intoxicated.) Drunken describes either a trait of habitual intoxication {drunken sot} or intoxicated people’s behavior {a drunken brawl}.
dual; duel. Dual is an adjective meaning “having two parts or two of something” {dual exhaust}. A duel is a fight between two people, especially a formal and often deadly combat with pistols or swords.
due to. In strict traditional usage, due to should be interchangeable with attribut able to {the erratic driving was due to some prescription drugs that the driver had taken} or owed to {thanks are due to all who helped}. When used adverbi ally, due to is often considered inferior to because of or owing to. So in the sen tence Due to the parents' negligence, the entire family suffered, the better phras ing would be Because o/[or Owing to] the parents' negligence, the entire family suffered.
due to the fact that. Use because instead.
dumb. This word means either “stupid” or “unable to speak.” In the second sense, the adjective mute is clearer (and less offensive) for most modern read ers. But on the noun use of mute, see moot.
dying; dyeing. Dying is the present participle of die (= to cease living); dyeing is the present participle of dye (= to color with a liquid).
each. As a noun serving as the subject of a clause, each takes a singular verb {each of them was present that day}. But when it serves as an emphatic appositive for a plural noun, the verb is plural {they each have their virtues} {the newspapers each sell for $3}.
each other; one another. Traditionalists use each other when two things or people are involved, one another when more than two are involved.
eatable. See edible.
economic; economical. Economic means “of, relating to, or involving large-scale finances” {federal economic policy}. Economical means “thrifty; financially efficient; cheap and not wasteful” {an economical purchase}.
edible; eatable. What is edible is fit for human consumption {edible flowers}. What is eatable is at least minimally palatable {the cake is slightly burned but still eatable}.
effect. See affect.
effete. Traditionally, it has meant “worn out, sterile” or “lacking power, character, 5.250 GRAMMAR AND USAGE or vitality.” Today it is often used to mean “snobbish,” “effeminate,” or “un duly pampered.” Because of its ambiguity, the word is best avoided altogether, effrontery; affront. Effrontery is an act of shameless impudence or shocking au dacity. An affront is a deliberate insult.
e.g. See i.e.
either. Like neither, this word takes a singular verb when it functions as subject {is either of the spouses present today?}.
elemental; elementary. Something that is elemental is an essential constituent {elemental ingredients} or a power of nature {elemental force}. Something that is elementary is basic, introductory, or easy {an elementary math prob lem}.
elicit; illicit. To elicit information or a reaction is to get it from someone, espe cially in challenging circumstances {to elicit responses}. Something illicit is disallowed by law or rule and usually also condemned generally by society {an illicit scheme}. Writers often mistakenly use the adjective illicit when they need the verb elicit.
elude. See allude.
embarrass. See harass.
emend. See amend.
emigrate. See immigrate.
eminent; imminent. What is eminent is famous, important, and respected {the eminent professor} or derives from high standing or authority {eminent do main}. What is imminent is looming, likely to happen soon, and almost always bad {imminent disaster}.
emoji; emoticon. An emoji (from the Japanese; pl. emoji) is a pictorial representa tion, or ideogram, that consists of a face, a hand gesture, or an object or symbol intended to express or suggest an emotion or attitude—or any number of ideas or things. An emoticon is a representation of a smiley face or other expressive gesture rendered as a combination of common keyboard characters—e.g.,;-).
emotive; emotional. The first means “arousing intense feeling” {emotive lan guage calculated to persuade the jury}; the second means “of, relating to, or involving intense feelings” {an emotional response}.
empathy; sympathy. Empathy is the ability to understand other people’s feel ings and problems {tremendous empathy with others}. Sympathy is generally compassion and sorrow one feels for another’s misfortunes, especially on a particular occasion {our sympathies are with you}—but it can also be support for a plan or idea {right-wing sympathies} or a mutual understanding and warmth arising from compatibility {there was no personal sympathy between them}.
endemic. See epidemic, endorse. See approve.
enervate; innervate. These words are antonyms. To enervate is to weaken or drain of energy. To innervate is to stimulate or provide with energy.
Word Usage 5.250 enormity; enormousness. Enormity means “monstrousness, moral outrageous ness, atrociousness” {the enormity of the Khmer Rouge’s killings}. Enormous ness means “hugeness” or “immensity” {the enormousness of Alaska}.
enough. See adequate, enquire. See inquire.
ensure; insure; assure. Ensure is the general term meaning “to make sure that something will (or won’t) happen.” In best usage, insure is reserved for under writing financial risk. So we ensure that we can get time off for a vacation, and we insure our car against an accident on the trip. We ensure events and insure things. But we assure people of things by telling them what’s what, so that they won’t worry. The important thing to remember is that we ensure occurrences and assure people.
enthused, adj. Use enthusiastic instead.
enumerable; innumerable. What’s enumerable is countable and listable {the enumerable issues that we need on the agenda}. What’s innumerable can’t be counted, at least not practically {innumerable stars in the sky}. The second word is far more common. Because the two are pronounced so similarly, be wary of using them in speech.
envy. See jealousy.
epidemic; endemic; pandemic. An epidemic disease breaks out, spreads through a limited area (such as a state), and then subsides {an epidemic outbreak of measles}. (The word is frequently used as a noun {a measles epidemic}.) An endemic disease is perennially present within a region or population {malaria is endemic in parts of Africa}. (Note that endemic describes a disease and not a region: it is incorrect to say this region is endemic for [a disease].) A pandemic disease is prevalent over a large area, such as a nation or continent, or the entire world {the 1918-19 flu pandemic}.
equally as. This is typically faulty phrasing. Delete as.
et al. This is the abbreviated form of et alii (“and others”)—the others being people, not things. Since al. is an abbreviation, the period is required—but note that no period follows the et (Latin for “and”). Cf. etc.
etc. This is the abbreviated form of et cetera (“and other things”); it should never be used in reference to people. Etc. implies that a list of things is too extensive to recite. But often writers seem to run out of thoughts and tack on etc. for no real purpose. Also, two redundancies often appear with this abbreviation: (1) and etc., which is poor style because et means “and,” and (2) etc. at the end of a list that begins with for example, such as, e.g., and the like. Those terms properly introduce a short list of examples. Cf. et al.; see also 6.20.
event. The phrase in the event that is a verbose and formal way of saying if. eventuality. This term often needlessly displaces more specific everyday words such as event, result, and possibility.
every day, adv.; everyday, adj. The first is adverbial, the second adjectival. You may wear your everyday clothes every day.
5.250 GRAMMAR AND USAGE every one; everyone. The two-word version is an emphatic way of saying “each” {every one of them was there}; the second is a pronoun equivalent to everybody {everyone was there}.
everywhere. This is the preferable word—not everyplace.
evoke; invoke. To evoke something is to bring it out {evoke laughter} or bring it to mind {evoke childhood memories}. Invoke has a number of senses, including to assert (something) as authority {invoke the Monroe Doctrine}, to appeal (to someone or a higher power) for help {invoke an ally to intervene}, and to conjure up {invoke spirits of the past}.
exceptional; exceptionable. What is exceptional is uncommon, superior, rare, or extraordinary {an exceptional talent}. What is exceptionable is objectionable or offensive {an exceptionable slur}.
executor; administrator. In a will, a person designates an executor to distribute the estate after death. When a person dies without a will or without specify ing an executor, the court will appoint an administrator to do the same. The feminine forms administratrix and executrix are unnecessary and should be avoided.
explicit; implicit. If something is explicit, it is deliberately and clearly spelled out, as in the text of a well-drafted statute. If it is implicit, it is not specifically stated but is either suggested in the wording or necessary to effectuate the purpose. Avoid implicit to mean “complete, unmitigated.” fact that, the. This much-maligned phrase is not always avoidable. But hunt for a substitute before deciding to use it. Sometimes that alone suffices.
farther; further. The traditional distinction is to use farther for a physical dis tance {we drove farther north to see the autumn foliage} and further for a fig urative distance {let’s examine this further} {look no further}. Although it’s a refinement of slight importance, connoisseurs will appreciate it.
faze; phase, vb. To faze is to disturb or disconcert {Jones isn’t fazed by insults}. To phase (usually phase in or phase out) is to schedule or perform a plan, task, or the like in stages {phase in new procedures} {phase out the product lines that don’t sell}. The negative adjective for “unaffected” is unfazed, not unphased.
feel. This verb is weak when used as a substitute for think or believe.
feel bad. Invariably, the needed phrase is feel bad (notfeel badly). See 5.170. fewer. See less.
fictional; fictitious; Active. Fictional (from fiction as a literary genre) means “of, relating to, or involving imagination” {a fictional story}. Fictitious means “imaginary; counterfeit; false” {a fictitious name}., Fictive means “possessing the talent for imaginative creation” {Active gift}; although it can also be a syn onym for fictional, in that sense it is a needless variant. Also, anthropologists use fictive to describe relationships in which people are treated as family mem bers despite having no bond of blood or marriage {Active kin}.
finalize. Meaning “to bring to an end or finish the last part of,” this word has Word Usage 5.250 often been associated with inflated jargon. Although its compactness may rec ommend it in some contexts, use finish when possible.
first. In enumerations, use first, second, third, and so on. Avoid the -ly forms.
fit. This verb is undergoing a shift. It has traditionally been conjugated/it-/itted- fitted, but today fit-fit-fit is prevalent in American English {when she tried on the dress, it fit quite well}. In the passive voice, however, fitted is still normal {the horse was fitted with a new harness}.
flagrant. See blatant.
flair. See flare.
flammable; inflammable. Flammable was invented in the early twentieth cen tury as an alternative to the synonymous word inflammable, which some people misunderstood—dangerously—as meaning “not combustible.” Today flammable is the standard term. Its antonym is nonflammable.
flare; flair. A flare is an unsteady and glaring light {an emergency flare} or a sud den outburst {a flare-up of fighting}. A flair is an outstanding talent {a flair for mathematics} or originality and stylishness {performed with flair}.
flaunt; flout. Flaunt means “to show off ostentatiously” {they flaunted their wealth}. Flout means “to openly disobey” {they flouted the rules}.
flounder; founder. Although the figurative sense of both verbs is “to go wrong,” the literal senses evoke different images. To flounder is to struggle awkwardly, as though walking through deep mud {the professor glared while the unprepared student floundered around for an answer}. To founder (usually in reference to a boat or ship) is to sink or run aground {the ship foundered on the rocks}.
flout. See flaunt.
following. Avoid this word as an equivalent of after. Consider the possible mis cue in Following the presentation, there was a question-and-answer session. After is both simpler and clearer.
forbear, vb.; forebear, n. The terms are unrelated, but the spellings are fre quently confused. To forbear is to refrain {he wanted to speak but decided to forbear [the conjugation is forbear-forbore-forborne]}. A forebear is an ancestor {the house was built by Murray’s distant forebears}.
forego; forgo. To forego is to go before {the foregoing paragraph}. The word ap pears most commonly in the phrase foregone conclusion. To forgo, by contrast, is to do without or renounce {they decided to forgo that opportunity}.
foreword; preface. A book’s foreword (notforward) is an introductory essay writ ten by someone other than the book’s author. An introductory essay written by the book’s author is called a. preface. See 1.40,1.41,1.43,1.47.
forgo. See forego.
former; latter. In the best usage, these words apply only to pairs. The former is the first of two, the latter the second of two.
fortuitous; fortunate. Fortuitous means “happening by chance,” usually (but not always) with a good result {the rotten tree could have fallen at any time; it 5.250 GRAMMAR AND USAGE was just fortuitous that the victims drove by when they did}. Fortunate means “lucky” {we were fortunate to win the raffle}. Today, unfortunately, fortuitous is poaching on the semantic turf of fortunate.
forward(s). See toward, founder. See flounder.
free rein. So written—not free reign.
fulsome, adj. This word does not preferably mean “very full” but “too much, excessive to the point of being repulsive.” Traditionally, a “fulsome speech” is one that is so overpacked with thanks or hyperbole as to sound insincere. The word’s slipshod use arises most often in the cliche fulsome praise, which can suggest the opposite of what the writer probably intends.
further. See farther.
future, in the near. Use soon or shortly instead.
gauntlet; gantlet. Lexicographers and usage critics—especially American ones— have sought since the nineteenth century to make a distinction. Etymologi cally, the two words have different histories: throwing down the gauntlet (= glove) and running the gantlope (= ordeal). But gauntlet has taken over both meanings. The standard phrases have been run the gauntlet and throw down the gauntlet since about 1800—the former phrase by a 10-to-l margin over the competing form run the gantlet. Efforts to separate the terms have run their grueling course.
gentleman. This word is a vulgarism when used as a synonym for man. When used in reference to a cultured, refined man, it is susceptible to some of the same objections as those leveled against lady. Use it cautiously. Cf. lady.
get. Though shunned by many writers as too casual, get often sounds more nat ural than obtain or procure {get a divorce}. It can also substitute for a stuffy become {get hurt}. The verb is conjugated get-got-gotten in American English andget-got-got in British English. Get is the only verb apart from fo-verbs that, when coupled with a past participle, can create a passive-voice construction {get stolen} {get waylaid}.
gibe; jibe; jive. A gibe is a biting insult or taunt: gibes are figuratively thrown at their target {the angry crowd hurled gibes at the miscreant}. To jibe is to be in accord or to agree {the verdict didn’t jibe with the judge’s own view of the facts}. Jive can be either a noun (referring to swing music or to misleading talk that is transparently untrue) or a verb (meaning “to dance to such music” or “to try to mislead with lies”).
gild. See guild.
go. This verb is conjugated go-went-gone. Went appears as a past participle only in dialect.
gourmet; gourmand. Both are aficionados of good food and drink. But a gourmet knows and appreciates the fine points of food and drink, whereas a gourmand tends toward gluttony.
graduate, vb. Whereas graduate means “to grant a diploma to or confer a degree Word Usage 5.250 onf graduate from means “to receive a diploma or degree from (a school, uni versity, or other institution).” A school can graduate a student or a student can graduate from a school, but a student does not graduate a school—at least not in good usage.
grateful; gratified. To be grateful is to be thankful or appreciative. To be gratified is to be pleased, satisfied, or indulged.
grisly; grizzly. What is grisly is gruesome or horrible {grisly details}. What is grizzly is grayish {grizzly hair} or bearish {the North American grizzly bear}.
guild, n.; gild, vb. A guild is an organization of persons with a common interest or profession {a guild of goldsmiths}. To gild is to put a thin layer of gold on some thing {gild a picture frame}, sometimes in a figurative sense {gilding the lily}.
hail; hale. To hail is to salute or greet {hail, Caesar!}, to acclaim enthusiastically {hailed as the greatest novelist of her time}, or to shout as an attention-getter {hail a taxi}. To hale is to compel to go {haled into court}. Hail is also a noun denoting ice-pellet precipitation, or something like it {a hail of insults}. Hale is also an adjective describing someone who is physically sound and free from infirmities.
half (of). Delete the of whenever possible {half the furniture}. When half is fol lowed by a singular noun, the verb is singular {half the state is solidly Demo cratic}; when it is followed by a plural noun, the verb is plural {half the people are Republicans}.
handful. If handful applies to a mass noun, use a singular verb {a handful of trouble is ahead}. But if handful applies to a plural count noun, use a plural verb {only a handful of walnut trees still line Main Street}.
hangar; hanger. One finds hangars (large buildings where aircraft are kept) at an airport {airplane hangars}. Everywhere else, one finds hangers {clothes hang ers} {picture hangers}.
hanged; hung. Hanged is used as the past participle of hang only in its transi tive form when referring to the killing (just or unjust) of a human being by suspending the person by the neck {criminals were hanged at Tyburn Hill}. But if death is not intended or likely, or if the person is suspended by a body part other than the neck, hung is correct {he was hung upside down as a cruel prank}. In most senses, of course, hung is the past form of hang {Abdul hung up his clothes}.
hanger. See hangar.
harass; embarrass. The first word has one r; the second has two. The pronunci ation of harass also causes confusion. The dominant American pronunciation stresses the second syllable, while British English stresses the first.
harebrained. So spelled (after the timid, easily startled animal)—not hairbrained. hark back. So written—preferably not harken back or hearken back.
healthy; healthful. Traditionally, a living thing that is healthy enjoys good health; something that is healthful promotes health {a healthful diet will keep you healthy}. But gradually healthy is taking over both senses.
5.250 GRAMMAR AND USAGE help (to). Omit the to when possible {talking will help resolve the problem}.
he or she. To avoid sexist language, many writers use this alternative phrasing (in place of the generic he). Use it sparingly—preferably after exhausting all other, less obtrusive methods of achieving gender neutrality. In any event, he or she is much preferable to he/she, s/he, (s)he, and the like. See also 5.48,5.255.
historic; historical. The shorter word refers to what is momentous in history {January 16,1991, was a historic day in Kuwait}. Historical, meanwhile, refers simply to anything that pertains to or occurred in history {the historical re cord}. On the question whether to use a or an before historic and historical, see a.
hoard; horde. A hoard is a supply, usually secret and sometimes valuable. Hoard is also a verb meaning “to amass such a supply,” especially when there is no need to do so. A horde was originally a tribe of Asian nomads; today a horde is a large crowd, especially one that moves in a noisy, uncontrolled way.
hoi polloi. This is a mildly disparaging phrase for “common people.” It does not refer to elites, though some writers and speakers misuse it in this way (perhaps from false association with hoity-toity). It is a plural. Although hoi is Greek for “the,” the phrase is commonly rendered the hoi polloi and has been at least since it was used by John Dryden in 1668.
holocaust. When capitalized, this word refers to the Nazi genocide of European Jews in World War II. When not capitalized, it refers (literally or figuratively) to extensive devastation caused by fire or to the systematic and malicious kill ings of human beings on a vast scale. Avoid any light or hyperbolic use of this word.
home in. This phrase is frequently misrendered hone in. (Hone means “to sharpen.”) Home in refers to what homing pigeons and aerial bombs do; the meaning is “to come closer and closer to a target.” homicide. See murder.
hopefully. The old meaning of the word (“in a hopeful manner”) seems unsus tainable; the newer meaning (“I hope” or “it is to be hoped that”), as a sen tence adverb, spread in the 1960s and 1970s and seems here to stay. But many careful writers still deplore the new meaning.
horde. See hoard.
humanitarian. This word means “involving the promotion of human welfare” {humanitarian philanthropy}. Avoid using it in a phrase such as the worst hu manitarian disaster in decades, where it really means just “human.” hung. See hanged.
I; me. When you need a first-person pronoun, use one. It’s not immodest to do so; it’s superstitious not to. But be sure you get the right one {Sally and I are planning to go} {give John or me a call} {keep this between you and me}. See between you and me.
idyllic. An idyll is a short pastoral poem, and by extension idyllic means charming or picturesque. It is not synonymous with ideal (perfect).
Word Usage 5.250 i.e.; e.g. The first is the abbreviation for id est (“that is”); the second is the ab breviation for exempli gratia (“for example”). The English equivalents are preferable in formal prose, though sometimes the compactness of these two- character abbreviations makes them desirable. Always put a comma after ei ther one. See also 6.20, 6.51.
if; whether. While if is conditional, whether introduces an alternative, often in the context of an indirect question. Use whether in two circumstances: (1) to introduce a noun clause: he asked whether his tie was straight (the alternatives are jes and no), and (2) when using i/produces ambiguity. In the sentence he asked if his tie was straight, the literal meaning is “whenever his tie was straight, he asked”; the popular meaning “he wanted someone to tell him whether his tie did or didn’t need straightening” may not be understood by all readers. More tellingly, Call to let me know if you can come means that you should call only if you’re coming. Call to let me know whether you can come means that you should call regardless of whether the answer is yes or no. Avoid substituting if for whether unless your tone is intentionally informal or you are quoting some one. See determine whether; whether.
ilk. This noun commonly means “type” or “sort” in modern usage, and unobjec- tionably so today {of his ilk} {of that ilk}. The Scottish phrase of that ilk means “of the same name or place.” illegal. See unlawful.
illegible; unreadable. Handwriting or printing that is illegible is not clear enough to be read {illegible scrawlings}. Writing that is unreadable is so poorly com posed as to be either incomprehensible or intolerably dull.
illicit. See elicit; unlawful.
illude. See allude.
immigrate; emigrate. To immigrate is to enter a country to live permanently, leaving a past home. To emigrate is to leave one country to live in another one. The cognate forms also demand attention. Someone who moves from Ireland to the United States is an immigrant here and an emigrant there. An emigre is also an emigrant, but especially one in political exile.
imminent. See eminent.
impact; impactful. Resist using impact as a verb. Try affect or influence instead. Besides being hyperbolic, impact is still considered a solecism by traditional ists (though it is gaining ground). Avoid impactful, which is jargon (replace ments include influential and powerfill). Cf. access.
impeachment. Impeachment is the legislative equivalent of an indictment, not a conviction. In the US federal system, the House of Representatives votes on impeachment, and the Senate votes on removal from office.
impelled. See compelled.
implicit. See explicit.
imply; infer. The writer or speaker implies (hints, suggests); the reader or listener infers (deduces). Writers and speakers often use infer as if it were synonymous 5.250 GRAMMAR AND USAGE with imply, but careful writers always distinguish between the two words. See inference.
important; importantly. In the phrase more important(ly)—usually at the outset of a sentence—traditionalists prefer the shorter form as an ellipsis of what is more important, normally with a comma following. But more importantly is now established as a sentence adverb—and it’s unobjectionable.
impractical; impracticable. The first is the more general adjective, meaning “not sensible” or “unrealistic” {impractical planning that doesn’t account for travel expenses}. The second means “impossible to carry out” {landing air craft on that hole-ridden runway proved impracticable}. See also practicable.
in actual fact. See actual fact, in.
inasmuch as. Because or since is almost always a better choice. See since, in behalf of. See behalf.
incidence; incident; instance. Be careful with the first of these words: it has to do with relative rates and ranges {the incidence of albinism within a given society}. Perhaps leave it to scientists and actuaries. An incident (= an event, occurrence, or happening) should be distinguished from an instance (= a case, example).
include; comprise. The basic difference between these near-synonyms is that in clude implies nonexclusivity {the collection includes 126 portraits [suggesting that there is much else in the collection]}, while comprise implies exclusivity {the collection comprises 126 silver spoons [suggesting that nothing else is part of the collection]}. Oddly, in patent law—and there alone—comprise car ries a nonexclusive sense. See comprise.
in connection with. This is a vague, fuzzy phrase {she explained the financial consequences in connection with the transaction} {Ray liked everything in connection with golf} {Phipson was compensated in connection with its re port}. Try replacing the phrase with of, related to, or associated with {she ex plained the financial consequences of the transaction}, about {Ray liked every thing about golf}, or for {Phipson was compensated for its report}.
incredible; incredulous. Incredible properly means “too strange to be believed; difficult to believe.” Colloquially, it is used to mean “astonishingly good” {it was an incredible trip}. Incredulous means “disbelieving, skeptical” {people are incredulous about the rising gas costs}.
inculcate; indoctrinate. One inculcates values into a child but indoctrinates the child with values. That is, inculcate always takes the preposition into and a value or values as its object {inculcate courage into soldiers}. Indoctrinate takes a person as its object {indoctrinate children with the habit of telling the truth}.
individual. Use this word to distinguish a single person from a group. When pos sible, use a more specific term, such as person, adult, child, man, or woman.
indoctrinate. See inculcate.
Word Usage 5.250 induce. See adduce.
in excess of. Try replacing this verbose phrase with more than or over. See over, infectious. See contagious.
infer. See imply.
inference. Use the verb draw, not make, with inference {they drew the wrong infer ences}. Otherwise readers may confuse inference with implication. See imply.
inflammable. See flammable.
inflict; afflict. Events, illnesses, punishments, etc. are inflicted on living things or entities {an abuser inflicts cruelty}. The sufferers are afflicted with or by dis ease, troubles, etc. {agricultural communities are afflicted with drought}.
ingenious; ingenuous. These words are similar in form but not in meaning. In genious describes what is intelligent, clever, and original {an ingenious inven tion}. Ingenuous describes a person who is candid, naive, and without dissimu lation, or an action or statement with those qualities {a hurtful but ingenuous observation}.
innate; inherent. An innate characteristic is one that a living thing has from birth; it should be distinguished, then, from a talent or disposition that one acquires from training or experience. An inherent characteristic is also part of a thing’s nature, but life is not implied. A rock, for example, has an inherent hardness.
innervate. See enervate.
innocent; not guilty. If you are innocent, you are without blame. If you are not guilty, you have been exonerated by a jury. Newspapers avoid the not guilty phrase, though, because the consequences of accidentally leaving off the not could be serious. See pleaded.
innumerable. See enumerable.
in order to; in order for. Often these expressions can be reduced to to and for. When that is so, and rhythm and euphony are preserved or even heightened, use to or for.
in proximity. See close proximity.
inquire. The normal spellings in American English and British English alike are inquire and inquiry. Enquire and enquiry are primarily British English variants.
in regard to. This is the phrase, not the nonstandard in regards to. But try a single word substitute instead: about, regarding, concerning, etc.
insidious; invidious. What is insidious spreads gradually to cause damage—at first without being noticed {an insidious conspiracy}; what is invidious involves moral offensiveness and serious unpleasantness {invidious discrimination}.
in spite of. See despite.
instance. See incidence.
insure. See ensure.
intense; intensive. Intense means (1) “having a strong effect” {intense pressures}, (2) “involving a great deal of effort during a very short time” {intense concen tration}, or (3) “having unduly strong feelings or a demeanor of exaggerated 5.250 GRAMMAR AND USAGE seriousness” {he’s a bit too intense}. Intense is always preferred outside phil osophical and scientific usages. But intensive should be retained in customary terms such as labor-intensive and intensive care.
intently; intensely. An act done intently is done purposefully and with concen tration and determination. One that is done intensely is done with great power, passion, or emotion but not necessarily with deliberate intent.
inter; intern. Inter is a verb meaning “to bury (a dead person)”; the correspond ing noun is interment. An intern is a student working temporarily to gain expe rience, especially in a profession. Intern is also a verb with two senses. As an intransitive verb, it means “to work as an intern” {interning at the US Senate}; the corresponding noun is internship. As a transitive verb, it means “to confine (a civilian) to a certain place or district without a criminal charge, especially in wartime or for political reasons”; the corresponding noun is internment.
in the affirmative. See affirmative, in the.
in the event that. See event.
in the near future. See future, in the near.
in the negative. See affirmative, in the.
inveigh; inveigle. To inveigh is to protest, usually against something {picketers inveighed against annexation}. To inveigle is to cajole or ensnare, especially by misleading {inveigling a friend to attend the party}.
invidious. See insidious.
invoke. See evoke.
irregardless. An error. Use regardless (or possibly irrespective).
it is I; it is me. Both are correct and acceptable. The first phrase, using the first- person predicate nominative, is strictly grammatical (and a little stuffy); the second is idiomatic (and relaxed), and it is often contracted to ifs me. In third- person constructions, however, a greater stringency holds sway in good En glish {this is he} {it isn’t she who has caused such misery}.
its; it’s. Its is the possessive form of it; ifs is the contraction for it is {it’s a sad dog that scratches its fleas}.
jealousy; envy. Jealousy connotes feelings of resentment toward another, partic ularly in matters relating to an intimate relationship {sexual jealousy}. Envy re fers to coveting another’s advantages, possessions, or abilities {his transparent envy of others’ successes}.
jibe; jive. See gibe.
karat. See carat.
kudos. Preferably pronounced /k[y]oo-dos/ (not /-dohz/), this word means “praise and admiration.” It is singular, not plural. Hence avoid kudo is or kudos are.
lady. When used as a synonym for woman—indeed, when used anywhere but in the phrase ladies and gentlemen—this word will be considered objectionable by some readers who think that it refers to a patronizing stereotype. This is especially true when it is used for unprestigious jobs {cleaning lady} or as a Word Usage 5.250 condescending adjective {lady lawyer}. Some will insist on using it to describe a refined woman. If they’ve consulted this entry, they’ve been forewarned. Cf. gentleman.
last; lastly. As with first, second, etc., prefer last when introducing a final point of discussion—or (of course) finally.
latter. See former.
laudable; laudatory. Laudable means “praiseworthy, even if not fully successful” {a laudable effort}. Laudatory means “expressing praise” {laudatory phone calls}.
lay; lie. Admittedly, the traditional conjugations are more blurred than ever. Mas tering them has proved difficult for people. Nevertheless, here goes. Lay is a transitive verb—that is, it demands a direct object {lay your pencils down}. It is inflected lay-laid-laid {I laid the book there yesterday} {these rumors have been laid to rest}. (The children’s prayer Now I lay me down to sleep is a good mnemonic device for the transitive lay.) Lie is an intransitive verb—that is, it never takes a direct object {lie down and rest}. It is inflected lie-lay-lain {she lay down and rested} {he hasn’t yet lain down in twenty-three hours}. In a doc tor’s office, you should be asked to lie back or lie down.
leach; leech. To leach is to percolate or to separate out solids in solution by per colation. A leech is a bloodsucker (whether literal or figurative). By extension of that noun, to leech is either to attach oneself to another as a leech does or to drain the resources of something.
lead. See led.
lease; let. Many Americans seem to think that let is colloquial and of modern ori gin. In fact, the word is three hundred years older than lease and just as proper. One distinction between the two words is that either the owner or the tenant can be said to lease property, but only the owner can be said to let it.
led. This is the correct spelling of the past tense and past participle of the verb lead. It is often misspelled lead, maybe in part because of the pronunciation of the metal lead and the past tense and past participle read, both of which rhyme with led.
leech.See leach.
lend, vb., loan, vb. &n. Lend is the correct term for letting someone use some thing with the understanding that it (or its equivalent) will be returned. The verb loan is standard especially when money is the subject of the transaction— but even then, lend appears somewhat more frequently in edited English. Loan is the noun corresponding to both lend and loan, vb. The past-tense and past- participial form of lend is lent.
less; fewer. Reserve less for singular mass nouns or amounts {less salt} {less soil} {less water}. Reserve fewer for plural count nouns {fewer calories} {fewer people} {fewer suggestions}.
lest. This is one of the few English words that invariably call for a verb in the subjunctive mood {he didn’t want to drive lest he take a wrong turn} {he has 5.250 GRAMMAR AND USAGE turned down the volume lest he disturb his roommates}. The conjunction is somewhat more common in British English than in American English.
let. See lease.
libel. See defamation.
lie. See lay.
life-and-death; life-or-death. The problem of logic aside (life and death being mutually exclusive), the first phrase is the standard idiom {a life-and-death decision}.
light, vb. This verb can be inflected either light-lit-lit or light-lighted-lighted—and irreproachably so. The past-participial adjective tends to be lighted when not modified by an -ly adverb {a lighted building} {a well-lighted hall} but lit if an -ly adverb precedes {brightly lit sconces} {a nicely lit walkway}.
like; as. The use of like as a conjunction (as in the old jingle “tastes good like a cigarette should”) has long been a contentious issue. Traditionally speaking, like is a preposition, not a conjunction equivalent to as {you’re much like me [me is the object of the preposition like]} {do as I say [the conjunction as con nects the imperative do with the independent clause I say]}. As a casualism, however, the conjunctive like has become especially common since the mid twentieth century {nobody cares like I do} {it tastes good like a fine choco late should}. In Standard Written English, a conjunctive like will still provoke frowns among some readers. But the objections are slowly dwindling. If you want your prose to be unimpeachable and heightened, stick to as and as if for conjunctive senses {as we’ve observed, man is a social animal} {it looks as if it might rain}. See also 5.185.
likely. See apt.
literally. This word means “actually; without exaggeration.” It should not be used loosely in figurative senses, as in they were literally glued to their seats (unless glue had in fact been applied). Wherever guides have accepted this usage, they should be disregarded.
loan. See lend.
loathe, vb.; loath, adj. To loathe (the th pronounced as in that) something is to de test it intensely or to regard it with disgust {I loathe tabloid television}. Some one who is loath (the th pronounced as in thing) is reluctant or unwilling {Tracy seems loath to admit mistakes}.
lose; loose, vb.; loosen. To lose something is to be deprived of it. To loose some thing is to release it from fastenings or restraints. To loosen is to make less tight or to ease a restraint. Loose conveys the idea of complete release, whereas loosen refers to only a partial release.
lot. See a lot.
luxuriant; luxurious. The two terms are fairly often confused (an example of catachresis). What is luxuriant is lush and grows abundantly {a luxuriant head of hair}. What is luxurious is lavish, extravagant, and comfortable {a luxurious resort}.
Word Usage 5.250 mad; angry. Some people object to using mad to mean “angry” and would re serve it to mean “insane.” But the first sense dates back seven hundred years and isn’t likely to disappear. As common as it is in everyday use, though, it has been so stigmatized that most people avoid it in formal writing.
majority. This noun preferably denotes countable things {a majority of votes cast}, not uncountable ones {the majority of the time}. Use most whenever it fits. When referring to a preponderance of votes cast, majority takes a singu lar verb {her majority was 7 percent}. But referring to a predominant group of people or things, it can take either a singular verb {the majority in the House was soon swept away} or a plural one {the majority of the voters were against the proposal}. Typically, if a genitive with a plural object follows majority, the verb should be plural {a majority of music teachers prefer using the metro nome}.
malevolent; maleficent. Malevolent describes an evil mind that wishes to harm others {with malevolent intent}. Malejicent is similar but describes desire by the miscreant for accomplishing evil {maleficent bullying}.
malodorous. See odious.
maltreatment. See mistreatment.
mankind. Consider humankind instead.
manslaughter. See murder.
mantle; mantel. A mantle is a long, loose garment like a cloak—almost always today being used in a metaphorical sense {assuming the mantle of a martyr}. A mantel is a wood or stone structure around a fireplace {family pictures on the mantel}.
masterful; masterly. Masterful describes a person who is dominating and im perious. Masterly describes a person who has mastered a craft, trade, or pro fession, or a product of such mastery; the word often means “authoritative” {a masterly analysis}. Because masterly does not readily make an adverb (mas terlily being extremely awkward), try in a masterly way. See also 5.158.
may; can. See can.
may; might. May expresses what is possible, is factual, or could be factual {I may have turned off the stove, but I can’t recall doing it}. Might suggests something that is uncertain, hypothetical, or contrary to fact {I might have won the mar athon if I had entered}. See 5.147.
me. See I.
medal; meddle; metal; mettle. A medal is an award for merit; a metal is a type of substance, usually hard and heavy. To meddle is to interfere. And mettle is a person’s character, courage, and determination to do something no matter how difficult.
media; mediums. In scientific contexts and in reference to mass communica tions, the plural of medium is predominantly media {some bacteria flourish in several types of media} {the media are reporting more medical news}. Although one frequently sees media is, the plural use is recommended. If 5.250 GRAMMAR AND USAGE medium refers to a spiritualist, the plural is mediums {several mediums have held seances here}.
memoranda; memorandums. Although both plural forms are correct, mem oranda has predominated since the early nineteenth century. Don’t use memoranda as if it were singular—the word is memorandum {this memoran dum is} {these memoranda are}.
metal; mettle. See medal.
mete out. The phrase meaning “to distribute” or “to assign” is so spelled {mete out punishment}. Meet out is a common error, especially in the erroneous past tense meeted out.
might. See may.
militate. See mitigate.
minuscule. Something that is minuscule is “very small.” Probably because of the spelling of the modern word mini (and the prefix of the same spelling, which is recorded only from 1936), it is often misspelled miniscule. In printing, minus cules are lowercase letters and majuscules are capital (uppercase) letters.
mistreatment; maltreatment. Mistreatment is the more general term. Maltreat ment denotes a harsh form of mistreatment, involving abuse by rough or cruel handling.
mitigate; militate. To mitigate is to lessen or soften the effects of something un pleasant, harmful, or serious; mitigating circumstances lessen the seriousness of a crime. To militate, by contrast, is to have a marked effect on; the word is usually followed by against {his nearsightedness militated against his ambi tion to become a commercial pilot}. Avoid the mistaken phrase mitigate against for the correct militate against.
moot; mute. Moot (/moot/) means (traditionally) “debatable” {a moot point worth our attention} or (by modern extension) “having no practical signifi cance” {a moot question that is of no account}. Mute (/m[y]oot/) means “si lent, speechless”—and is often considered offensive when used as a noun {deaf-mute}.
more than. See over.
much; very. Much generally intensifies past-participial adjectives {much obliged} {much encouraged} and some comparatives {much more} {much worse} {much too soon}. Very intensifies adverbs and most adjectives {very carefully} {very bad}, including past-participial adjectives that have more adjectival than verbal force {very bored}. See 5.90.
murder; manslaughter; homicide. All three words denote the killing of one person by another. Murder and manslaughter are both unlawful killings, but murder is done maliciously and intentionally. Homicide includes killings that are not unlawful, such as by a police officer acting properly in the line of duty. Homicide also refers to a person who kills another.
mute. See moot.
mutual. See common.
Word Usage 5.250 myself. Avoid using myself as a pronoun in place of I or me—a quirk that arises most often after an and or or. Instead, use it reflexively {I did myself a favor} or emphatically {I myself have tried to get through that book!}. See also 5.51.
naturalist; naturist. Naturalist most often denotes a person who studies natural history, especially a field biologist or an amateur who observes and usually photographs, sketches, or writes about nature. Naturist denotes a nature wor shipper or a nudist.
nauseous; nauseated. Whatever is nauseous, traditionally speaking, induces a feeling of nausea—it makes us feel sick to our stomachs. To feel sick is to be nauseated. Although the use of nauseous to mean nauseated may be too com mon to be called an error anymore, strictly speaking it is poor usage. Because of the ambiguity in nauseous, the wisest course may be to stick to the particip ial adjectives nauseated and nauseating.
necessary; necessitous. Necessary means “required under the circumstances” {the necessary arrangements}. Necessitous means “impoverished” {living in necessitous circumstances}.
neither. Four points. First, like either, this word when functioning as the subject of a clause takes a singular verb {neither of the subjects was given that med icine}. Second, a neither-nor construction should frame grammatically paral lel expressions {neither the room’s being too cold nor the heater’s malfunc tion could justify his boorish reaction} (both noun elements). Third, a simple neither-nor construction should have only two elements {neither bricks nor stones}—though it’s perfectly permissible to multiply nors for emphasis {nei ther snow nor rain nor heat nor gloom of night}. Fourth, the word is acceptably pronounced either /nee-thar/ or /ni-thor/.
no. See affirmative, in the.
noisome. This word has nothing to do with noise. It means noxious, offensive, or foul-smelling {a noisome landfill}.
none. This word may take either a singular or a plural verb. A guideline: if it is followed by a singular noun, treat it as a singular {none of the building was painted}; if by a plural noun, treat it as a plural {none of the guests were here when I arrived}. But for special emphasis, it is quite proper (though possibly stilted) to use a singular verb when a plural noun follows {none of my sugges tions was accepted}.
nonplussed. Traditionally meaning “surprised and confused” {she was non plussed when he took off the mask}, this word is now frequently misused to mean “unfazed”—almost the opposite of its literary sense. Avoid this newer usage, and avoid the variant spelling nonplused. See faze.
notable; noticeable; noteworthy. Notable (= readily noticed) applies both to physical things and to qualities {notable sense of humor}. Noticeable means “detectable with the physical senses” {a noticeable limp}. Noteworthy means “remarkable; deserving attention” {a noteworthy act of kindness}.
not guilty. See innocent.
5.250 GRAMMAR AND USAGE notwithstanding. One word. Less formal alternatives include despite, although, and in spite of. The word notwithstanding may precede or follow a noun {not withstanding her bad health, she decided to run for office} {her bad health notwithstanding, she decided to run for office}.
number. See amount.
observance; observation. Observance means “obedience to a rule or custom” {the family’s observance of Passover}. Observation means either “the watch ing of something” or “a remark based on watching or studying something” {a keen observation about the defense strategy}. Each term is sometimes used when the other would be the better word.
obtain. See attain.
obtuse; abstruse. Obtuse describes a person who can’t understand; abstruse de scribes an idea that is hard to understand. A person who is obtuse is dull and, by extension, dull-witted. What is abstruse is incomprehensible or nearly so.
odious; odorous; odoriferous; malodorous. Odious means “hateful” or “ex tremely unpleasant” {odious Jim Crow laws}. It is not related to the other terms, but it is sometimes misused as if it were. Odorous means “detectable by smell (for better or worse) ” {odorous gases}. Odoriferous means essentially the same thing: it has meant “fragrant” as often as it has meant “foul.” Malodor ous means “smelling quite bad.” The mistaken form odiferous is often used as a jocular equivalent of smelly—but most dictionaries don’t record it.
of. Avoid using this word needlessly after all, off, inside, and outside. Also, pre fer June 2015 over June of 2015. To improve your style, try removing every of- phrase that you reasonably can.
off. Never put of after this word {we got off the bus}.
officious. A person who is officious is aggressively nosy and meddlesome—and overeager to tell people what to do. The word has nothing to do with officer and should not be confused with official.
on; upon. Prefer on to upon unless introducing an event or condition {put that on the shelf, please} {upon the job’s completion, you’ll get paid}. For more about on, see onto.
on behalf of. See behalf.
one another. See each other.
oneself. One word—not one’s self.
onto; on to; on. When is on a preposition, and when an adverb? The sense of the sentence should tell, but the distinction can be subtle. Onto implies a move ment, so it has an adverbial flavor even though it is a preposition {the gymnast jumped onto the bars}. When on is part of the verb phrase, it is an adverb and to is the preposition {the gymnast held on to the bars}. One trick is to mentally say “up” before on: if the sentence still makes sense, then onto is probably the right choice {she leaped onto the capstone}. Alone, on does not imply motion {the gymnast is good on the parallel bars}.
Word Usage 5.250 oppress; repress. Oppress, meaning “to persecute or tyrannize,” is more negative than repress, meaning “to restrain or subordinate.” or. If this conjunction joins singular nouns functioning as subjects, the verb should be singular {cash or online payment is acceptable}.
oral. See verbal.
oration. See peroration.
ordinance; ordnance. An ordinance is a municipal regulation or an authoritative decree. Ordnance is military armament, especially artillery but also weapons and ammunition generally.
orient; orientate. To orient is to get one’s bearings (literally, “to find east”) {it took the new employee a few days to get oriented to the firm’s suite}. Unless used in the sense “to face or turn to the east,” orientate is a poor variation to be avoided.
ought; should. Both express a sense of duty, but ought is stronger. Unlike should, ought requires a fully expressed infinitive, even in the negative {you ought not to see the movie}. Don’t omit the to—as many otherwise well-educated speak ers and writers have begun doing in recent years. See 5.149,5.150.
outside. In spatial references, no of is necessary—or desirable—after this word unless it is used as a noun {outside the shop} {the outside of the building}. But outside of is acceptable as a colloquialism meaning “except for” or “aside from.” over. As an equivalent of more than, this word is perfectly good idiomatic English, overly. Avoid this word, which is not considered the best usage. Try over- as a prefix {overprotective} or unduly {unduly protective}.
pair. This is a singular form, the plural being pairs {three pairs of shoes}. Yet pair may take either a singular verb {this pair of sunglasses was on the table} or a plural one {the pair were inseparable from the moment they met}.
palette; palate; pallet. An artist’s palette is either the board that an artist uses for mixing colors or (collectively) the colors used by a particular artist or avail able in a computer program. Your palate is the roof of your mouth specifically or your taste in food generally. A pallet is a low, usually wooden platform for storing and transporting goods in commerce, or a crude bed consisting of a bag filled with straw.
pandemic. See epidemic.
parameters. Though it may sound elegant or scientific, this word is usually just pretentious when it is used in nontechnical contexts. Stick to boundaries, lim its, guidelines, grounds, elements, or some other word.
partake in; partake of. To partake in is to participate in {the new student refused to partake in class discussions}. To partake of is either to get a part of {partake of the banquet} or to have a quality, at least to some extent {this assault par takes of revenge}.
partly; partially. Both words convey the sense “to some extent; in part” {partly 5.250 GRAMMAR AND USAGE responsible}. Partly is preferred in that sense. But partially has the additional senses of “incompletely” {partially cooked} and “unfairly; in a way that shows bias toward one side” {he treats his friends partially}.
past; passed. Past can be an adjective {past events} (often postpositive {times past}), a noun {remember the past}, a preposition {go past the school}, or an adverb {time flew past}. Passed is the past tense and past participle of the verb pass {we passed the school} {as time passed}.
pastime. This word combines pass (not past) and time. It is spelled with a single 5 and a single t.
peaceable; peaceful. A peaceable person or nation is inclined to avoid strife {peaceable kingdom}. A peaceful person, place, or event is serene, tranquil, and calm {a peaceful day free from demands}.
peak; peek; pique. These three sometimes get switched through writerly blun ders. A peak is an apex, a peek is a quick or illicit glance, and a fit of pique is an episode of peevishness and wounded vanity. To pique is to annoy or arouse: an article piques (not peaks) one’s interest.
pedal; peddle. Pedal is a noun, verb, or adjective relating to the pedal extremity, or foot. As a noun, it denotes a device that is operated by the foot and does some work, such as powering a bicycle or changing the sound of a piano. As a verb, it means to use such a device. As an adjective, it means “of or concerning such a device or its use.” Peddle is a verb meaning either “to try to sell goods to people by traveling from place to place” or “to sell questionable goods to people”—questionable because they may be illegal, harmful, or low quality {peddling magazine subscriptions door to door}.
peek. See peak.
pendant,«.; pendent, adj. A pendant is an item of dangling jewelry, especially one worn around the neck. What is pendent is hanging or suspended from something.
penultimate. This adjective means “next to last” {the penultimate paragraph in the precis}. Many people have started misusing it as a fancy equivalent of ulti mate. The word antepenultimate means “the next to the next-to-last.” people; persons. The traditional view is that persons is used for smaller numbers {three persons} and people with larger ones {millions of people}. But today most people use people even for small groups {only three people were there}.
peroration; oration. A peroration, strictly speaking, is the conclusion of an oration (speech). Careful writers avoid using peroration to refer to a rousing speech or piece of writing.
perpetuate; perpetrate. To perpetuate something is to sustain it or prolong it in definitely {perpetuate the species}. Toperpetrate is to commit or perform an act, especially one that is illegal or morally wrong {perpetrate a crime}.
personally. Three points. First, use this word only when someone does some thing that would normally be done through an agent {the president personally signed this invitation} or to limit other considerations {Jean was affected by Word Usage 5.250 the decision but was not personally involved in it}. Second, personally is redun dant when it modifies a verb that necessarily requires the person’s presence, as in The senator personally shook hands with the constituents. Third, personally shouldn’t appear with I when one is stating an opinion; it weakens the state ment and doesn’t reduce the speaker’s liability for the opinion. The only ex ception arises if a person is required to advance someone else’s view but holds a different personal opinion {in the chamber I voted to lower taxes because of the constituencies I represented; but I personally believed that taxes should have been increased}.
persons. See people.
persuade; convince. Persuade is associated with actions {persuade him to buy a suit}. Convince is associated with beliefs or understandings {she convinced the auditor of her honesty}. The phrase persuade to (do) has traditionally been considered better than convince to (do)—the latter having become common in American English in the 1950s. But either verb will take a that-clause {the committee was persuaded that an all-night session was necessary} {my three- year-old is convinced that Santa Claus exists}.
pertain; appertain. Pertain to, the more common term, means “to relate directly to” {the clause pertains to assignment of risk}. Appertain to means “to belong to or concern something as a matter of form or function” {the defendant’s rights appertain to the Fifth Amendment}.
peruse. This term, which means “to read with great care” (not “to read quickly” or “to scan”) should not be used as a fancy substitute for read.
phase. See faze.
phenomenon. This is the singular {the phenomenon of texting}, the plural being phenomena {cultural phenomena}.
pique. See peak.
pitiable; pitiful. To be pitiable is to be worthy of pity. To be pitiful is either to be very poor in quality or to be so sad or unfortunate as to make people feel sympathy.
pleaded; pled. The first is the standard past-tense and past-participial form {he pleaded guilty} {they have pleaded with their families}. Avoid pled.
plethora. This noun denotes an excess, surfeit, or overabundance. Avoid it as a mere equivalent of “abundance.” populace; populous. The populace is the population of a country as a whole. A populous place is densely populated.
pore. To pore over something written is to read it intently {they pored over every word in the report}. Some writers confuse this word with pour.
practicable; possible; practical. These terms differ in shading. What is practica ble is capable of being done; it’s feasible. What is possible might be capable of happening or being done, but there is some doubt. What is practical is fit for actual use or in a particular situation. See also impractical.
precede; proceed. To precede is to happen before or to go before in some se 5.250 GRAMMAR AND USAGE quence, usually time. It also means “to outrank” or “to surpass” in some mea sure such as importance, but this sense is usually conveyed with the noun pre cedence {the board’s vote takes precedence over the staff’s recommendation}. The word is often misspelled preceed. To proceed is to go on, whether begin ning, continuing, or resuming.
precipitate, adj.; precipitous. What is precipitate occurs suddenly or rashly, with out proper consideration; it describes demands, actions, or movements. What is precipitous is dangerously steep; it describes cliffs and inclines.
precondition. Try condition or prerequisite instead.
predominant, adj.; predominate, vb. Like dominant, predominant is an adjec tive {a predominant point of view}. Like dominate, predominate is a verb {a point of view that predominates throughout the state}. Using predominate as an adjective is nonstandard.
preface. See foreword.
prejudice, vb. Although prejudice is a perfectly normal English noun to denote an all-too-common trait, the corresponding verb is a legalism. For a plain-English equivalent, use harm or hurt. But the past-participial adjective prejudiced is perfectly normal in the sense “biased; harboring strong and often unfair feel ings against.” preliminary to. Make it before, in preparing for, or some other natural phrasing, prescribe. See proscribe.
presently. This word is ambiguous. Write now or soon, whichever you really mean, presumption. See assumption.
preventive. Although the corrupt form preventative is fairly common, the strictly correct form is preventive.
previous to. Make it before.
principle; principal. A principle is a natural, moral, or legal rule {the principle of free speech}. The corresponding adjective is principled {a principled decision}. A principal is a person of high authority or prominence {a school principal} or an initial deposit of money {principal and interest}. Principal is also an adjec tive meaning “most important.” Hence a principal role is a primary one.
prior to. Make it before or until.
proceed. See precede.
process of, in the. You can almost always delete this phrase without affecting the meaning.
propaganda. This is a singular noun denoting information that, being false or misleading, is used by a government or political group to influence people {propaganda was everywhere}. The plural is propagandas.
prophesy; prophecy. Prophesy is the verb meaning “to say what will happen in the future, especially by using supernatural or magical knowledge” {the doomsayers prophesied a market boom despite the bad news}. Prophecy is the noun denoting a prediction, especially one made by someone claiming to have Word Usage 5.250 supernatural or magical powers {their prophecies did not materialize}. Prophe- size is an erroneous form.
proscribe; prescribe. To proscribe something is to prohibit it {legislation that pro scribes drinking while driving}. To prescribe is to say officially what must be done in a particular situation {Henry VIII prescribed the order of succession to include three of his children} or to specify a medical remedy {the doctor prescribed anti-inflammatory pills and certain exercises}.
protuberance. So spelled. Perhaps because protrude means “to stick out,” writers want to spell protuberance (= something that bulges out) with an extra r (after the t). But the words are from different roots.
proved; proven. Proved is the preferred past participle for the verbprove {it was proved to be true}. Use proven only as an adjective {a proven success}.
proximity. See close proximity.
purposely; purposefully. What is done purposely is done deliberately or inten tionally—“on purpose.” What is done purposefully is done with a certain goal or a clear aim in mind. An action may be done purposely without any particular interest in a specific result—that is, not purposefully.
question whether; question of whether; question as to whether. The first phrasing is traditionally considered best. The others are phraseologically in ferior. See as to.
quick(ly). Quickly is the general adverb. But quick is properly used as an adverb in the idiomatic phrases get rich quick and come quick. See also 5.160.
quote; quotation. Traditionally a verb, quote is often used as an equivalent of quotation in speech and informal writing. Also, there is a tendency for writers (especially journalists) to think of quotes as contemporary remarks usable in their writing and of quotations as being wisdom of the ages expressed pithily, rack; wrack. The spelling rack is complex: it accounts for nine different nouns and seven different verbs. Indeed, it is standard in all familiar senses {rack ing his brain} {racked with guilt} {nerve-racking} {rack and ruin}. Wrack is the standard spelling only for the noun meaning “seaweed, kelp.” raise; raze. To raise is to elevate, move upward, enhance, bring up, etc. {we raised some money}. To raze is to demolish, level to the ground, remove, etc. {they razed the building}.
reason. Two points. First, as to reason why, although some object to the supposed redundancy of this phrase, it is centuries old and perfectly acceptable English. Reason that is not always an adequate substitute {can you give reasons why that is preferable to which as a restrictive relative pronoun?}. Second, reason... is because is not good usage—reason.. .is that being preferred {the reason we returned on July 2 is that we wanted to avoid hordes of tourists}.
recur; reoccur. To recur is to happen again and again {his knee problems recurred throughout the rest of the year}, to return to in one’s attention or memory {she recurred to her war experiences throughout our visit}, or to come back to one’s 5.250 GRAMMAR AND USAGE attention or memory {the idea recurred to him throughout the night}. To re occur is merely to happen again {the leak reoccurred during the second big rain}.
reek. See wreak.
reference; referral. A reference is a source of information, a person to provide in formation, an authority for some assertion, or a strong allusion to something. It’s also an attributive adjective {reference book}. It’s not universally accepted as a transitive verb. Referral is a narrower term denoting the practice or an instance of (1) directing someone to another person who can help, especially a professional or a specialist, or (2) relegating some matter to another body for a recommendation or resolution.
refrain; restrain. To refrain is to restrain yourself or to keep from doing some thing; it is typically an act of self-discipline. Other people restrain you by stop ping you from doing something, especially by using physical force {if you don’t refrain from the disorderly conduct, the police will restrain you}. Yet it is pos sible to restrain oneself by controlling one’s own emotions or behavior—and doing so is known as self-restraint.
refute, vb. To refute is to prove that a statement or an idea is wrong—not merely to deny or rebut.
regardless. See irregardless.
regrettable; regretful. What is regrettable is unfortunate or unpleasant enough to make one wish that things were otherwise. A person who is regretfill feels sorry or disappointed about something done or lost. The adverb regrettably, not regretfully, is the synonym of unfortunately.
rein; reign. A rein (usually plural) controls a horse; it is the right word in idioms such as take the reins, give free rein, and, as a verb, rein in. A reign is a state of or term of dominion, especially that of a monarch but by extension also domi nance in some field. This is the right word in idioms such as reign of terror and, as a verb, reign supreme.
relegate; delegate. To relegate is to assign a lesser position than before {the officer was relegated to desk duty pending an investigation}. To delegate is to autho rize a subordinate to act in one’s behalf {Congress delegated environmental regulation to the EPA} or to choose someone to do a particular job or to rep resent an organization or group {she was delegated to find a suitable hotel for the event}.
reluctant. See reticent.
renounce. See denounce.
reoccur. See recur.
repellent; repulsive. Repellent and repulsive both denote the character of driving others away. But repulsive has strong connotations of being so disgusting as to make one feel sick.
repetitive; repetitious. Both mean “occurring over and over.” But whereas re Word Usage 5.250 petitive is fairly neutral in connotation, repetitious has taken on the nuance of tediousness that induces boredom.
repress. See oppress, repulsive. See repellent.
restive; restful. Restive means “so dissatisfied or bored with a situation as to be impatient for change.” Restful means “peaceful, quiet, and conducive to re laxation.” restrain. See refrain.
reticent. Avoid using this word as a synonym for reluctant. It means “unwilling to talk about what one feels or knows; taciturn” {when asked about the incident, the congressional representative became uncharacteristically reticent}.
revenge. See avenge.
rob; steal. Both verbs mean “to wrongfully take [something from another per son].” But rob also includes a threat or act of harming, usually but not always to the person being robbed.
role; roll. A role is an acting part {the role of Hamlet} or the way in which someone or something is involved in an activity or situation, especially in reference to influence {the role that money plays as an incentive}. Roll has many meanings, including a roster {guest roll}; something made or done by rolling {roll of the dice}; and something in the shape of a cylinder or sphere, whether literally {dinner roll} or figuratively {bankroll}. Roll can also be a verb meaning to rotate {roll over!}, to wrap [something] {roll up the leftovers}, or to move forward {the cart rolled down the hill}.
run the gauntlet. See gauntlet.
sacrilegious. This is the correct spelling. There is a tendency by some to switch the i and e on either side of the /, but in fact the word is related to sacrilege, not religion or religious.
seasonal; seasonable. Seasonal means either “happening as expected or needed during a particular time of year” {snow skiing is a seasonal hobby} or “relating to the seasons or a season” {the seasonal aisle stays stocked most of the year, starting with Valentine’s Day gifts in January}. Seasonable means “timely” {seasonable motions for continuance} or “fitting the time of year” {it was un seasonably cold for July}.
self-deprecating. See deprecate.
semi-. See bi-.
semiannual. See biannual.
sensor. See censer.
sensual; sensuous. What is sensual involves indulgence of the physical senses— especially sexual gratification. What is sensuous usually applies to aesthetic enjoyment; it is primarily hack writers who imbue the word with salacious connotations.
sewer; sewage; sewerage. Sewer denotes a wastewater pipe or passage. Sewage 5.250 GRAMMAR AND USAGE denotes the waste carried through such a pipe or passage. Sewerage denotes the sewer system as a whole, including treatment plants and other facilities, and the function of the disposal of sewage and wastewater in general.
shall. This word is complicated. The reality is that shall is little used in everyday contexts outside British English—not in North America but also not in Aus tralia, Ireland, or Scotland. In legal contexts, it frequently appears in statutes, rules, and contracts, supposedly in a mandatory sense but actually quite am biguously. It is perhaps the most widely litigated word in the law—with wildly varying results in its multifarious interpretations. Legal drafters are therefore often advised to avoid it altogether in favor of must, is, will, may, and other words or phrases among which shall’s various meanings can be allocated.8 See also 5.131.
shear; sheer. Shear is the noun or verb relating to (1) the cutting tool or (2) a force af fecting movement, such as a crosswind or the slipping of plates in an earthquake. Sheeris most often an adjective meaning (1) “semitransparent” {a sheer curtain}, (2) “nothing but” {sheer madness}, or (3) “almost vertical” {a sheer cliff}.
shine. When this verb is intransitive, it means “to give or make light”; the past tense is shone {the stars shone dimly}. When it is transitive, it means “to cause to shine”; the past tense is shined {the caterer shined the silver}.
should. See ought.
sight; site. A sight may be something worth seeing {the sights of London} or a device to aid the eye {the sight of a gun}, among other things. A site is a place, whether physical {a mall will be built on this site} or electronic {website}. The figurative expression meaning “to focus on a goal” is to set one's sights. Cf. cite, simplistic. This word, meaning “oversimplified,” has derogatory connotations. Don’t confuse it with simple.
since. This word may relate either to time {since last winter} or to causation {since I’m a golfer, I know what “double bogey” means}. Some writers erroneously believe that the word relates exclusively to time. But the causal since was a part of the English language before Chaucer wrote in the fourteenth century, and it is useful as a slightly milder way of expressing causation than because. Still, if there is any possibility of confusion with the temporal sense, use because.
sink. Inflected sink-sank-sunk. Avoid using sunk as a simple past, as in the ship sunk.
site. See cite; sight.
slander. See defamation.
slew; slough; slue. As a noun, slew (/sloo/) is an informal word equivalent to many or lots {you have a slew of cattle}. It is sometimes misspelled slough (a legitimate noun meaning “a grimy swamp,” pronounced either /sloo/ or 8. See Garner, Legal Writing in Plain English, 2nd ed. (Chicago: University of Chicago Press, 2013), 125-28; Garner’s Dictionary of Legal Usage, 3rd ed. (New York: Oxford University Press, 2011), 952-55 (collecting many authorities).
Word Usage 5.250 /slow/). The phrase slough of despond (from Bunyan’s Pilgrim's Progress ) means “a state of depression or sadness from which one cannot easily lift one self.” This term is etymologically different from slough (/slaf/), meaning “to discard” {slough off dry skin}. As a present-tense verb, to slew is to turn or slide violently or suddenly in a different direction—or to make a vehicle do so {the car keeps slewing sideways}. In American English, a variant spelling of this verb is slue. As a past-tense verb, slew corresponds to the present-tense slay {Cain slew Abel}.
slow. This word, like slowly, may be an adverb. Generally, prefer slowly {go slowly}. But in colloquial usage slow is often used after the verb in a pithy statement, especially an injunction {go slow!} {take it slow}. See also 5.160.
slue. See slew.
sneak. This verb is conjugated as a regular verb: sneak-sneaked-sneaked. Reserve snuck for dialect and tongue-in-cheek usages.
space. As a figurative noun, this word has become a voguish equivalent of area {though not initially interested in journalism, he has decided to move into that space}. Although (or perhaps because) this usage is au courant, avoid it. See also area.
spit. If used to mean “to expectorate,” the verb is inflected spit-spat-spit {he spat a curse} {he has spit many a curse}. But if used to mean “to skewer,” it’s spit- spitted-spitted {the hens have been spitted for broiling}.
stanch. See staunch.
stationary; stationery. Stationary describes a state of immobility or of staying in one place {if it’s stationary, paint it}. Stationery denotes writing materials, es pecially paper for writing letters, usually with matching envelopes {love letters written on perfumed stationery}. To remember the two, try associating the -er in stationery with the -er in paper, or remember that a stationer is someone who sells the stuff.
staunch; stanch. Staunch is an adjective meaning “ardent and faithful” {a staunch Red Sox supporter}. Stanch is the American English verb meaning “to stop the flow”; it is almost always used in regard to bleeding, literally or metaphorically {after New Hampshire the campaign hemorrhaged; only a big win in South Carolina could stanch the bleeding}. In British English, however, staunching the flow is the standard wording.
steal. See rob.
strait; straight. A strait (often pl.) is (1), literally, a narrow channel connecting two large bodies of water separated by two areas of land {Strait of Magellan} or (2), figuratively, a difficult position {dire straits}. This is the word used in com pound terms with the sense of constriction {straitlaced} {straitjacket}. Straight is most often an adjective meaning unbent, steady, sober, candid, honest, or heterosexual.
strata, n. This is the plural for stratum. Keep it plural {Fussell identified nine dis crete strata in American society}. Avoid the double plural stratas.
5.250 GRAMMAR AND USAGE strategy; tactics. A strategy is a long-term plan for achieving a goal. A tactic is a shorter-term method for achieving an immediate but limited success. A strat egy might involve several tactics. By the way, although strategy is so spelled, stratagem has an a in the middle syllable.
subject. See citizen.
subsequent. See consequent.
such. This word, when used to replace this or that—as in “such building was later condemned”—is symptomatic of legalese. Such is actually no more precise than the, this, that, these, or those. It’s perfectly acceptable, however, to use such with a mass noun or plural noun when the meaning is “of that type” or “of this kind” {such impudence galled the rest of the family} {such vitriolic exchanges became commonplace in the following years}. See also as such.
sufficient. See adequate.
supersede. The root of this word derives from sedeo, the Latin word for “to sit, to be established,” not cedo, meaning “to yield.” Hence the spelling varies from the root in words such as concede, recede, and secede. Avoid the variant super cede.
sympathy. See empathy.
systematic; systemic. Systematic means “according to a plan or system, orga nized methodically, or arranged in a system.” Systemic, meaning “affecting the whole of something,” is limited in use to physiological systems {a systemic disease affecting several organs} or, by extension, other systems that may be likened to the body {systemic problems within the corporate hierarchy}.
tactics. See strategy.
take. See bring.
tantalizing; titillating. A tantalizing thing torments us because we want it badly yet it is always just out of reach. A titillating thing tickles us pleasantly, literally or figuratively, and the word often carries sexual connotations.
text, vb. Inflected text-texted-texted, as a regular verb. Avoid using the uninflected text for the past-tense forms.
that; which. These are both relative pronouns (see 5.56-66). In polished Ameri can prose, that is used restrictively to narrow a category or identify a particular item being talked about {any building that is taller must be outside the state}; which is used nonrestrictively—not to narrow a class or identify a particular item but to add something about an item already identified {alongside the of ficer trotted a toy poodle, which is hardly a typical police dog}. Which is best used restrictively only when it is preceded by a preposition {the situation in which we find ourselves}. Nonrestrictively, it is almost always preceded by a comma, a parenthesis, or a dash. (In British English, writers and editors sel dom observe the distinction between the two words.) Is it a useful distinction? Yes. The language inarguably benefits from having a terminological as well as a punctuational means of telling a restrictive from a nonrestrictive relative pronoun, punctuation often being ill-heeded. Is it acceptable to use that in ref Word Usage 5.250 erence to people? Is friends that arrive early an acceptable alternative to friends who arrive early? The answer is yes. Person that has long been considered good idiomatic English. Even so, person who is nearly three times as common as per son that in edited English. See also 6.27.
there; their; they’re. There denotes a place or direction {stay there}. Their is the possessive pronoun {all their good wishes}. They’re is a contraction of they are {they’re calling now}.
therefore; therefor. The words have different senses. Therefore, the common word, means “as a result; for that reason” {the evidence of guilt was slight; therefore, the jury acquitted the defendant}. Therefor, a legalism, means “for it” or “for them” {he took the unworn shirt back to the store and received a refund therefor}.
thus. This is the adverb—not thusly. Use thus (it’s called a flat adverb). See 5.160. till. This is a perfectly good preposition and conjunction {open till 10 p.m.}. It is not a contraction of until and should not be written ’til.
timbre; timber. Timbre is a musical term meaning “tonal quality of the sound made by a particular musical instrument or voice.” Timber is the correct spell ing in all other uses, which relate to trees or wood.
titillating. See tantalizing.
tolerance; toleration. Tolerance is the habitual quality of being tolerant—that is, willing to allow people to say, believe, or do what they want without criticism or punishment. Toleration is a particular instance of being tolerant.
torpid. See turbid.
tortious; tortuous; torturous. What is tortious relates to torts (civil wrongs) or to acts that give rise to legal claims for torts {tortious interference with a con tract}. What is tortuous is full of twists and turns and therefore makes travel difficult {a tortuous path through the woods}. What is torturous involves severe physical and mental suffering {a torturous exam}.
toward; towards. The preferred form in American English is toward: this has been so since about 1900. In British English, towards predominates. The same is true for other directional words, such as upward, downward, forward, and backward, as well as afterward. The use of afterwards and backwards as adverbs is neither rare nor incorrect (and is preferred in British English). For the sake of consistency, many American editors prefer the shorter forms without the final s.
transcript; transcription. A transcript is either a written record, as of a trial or a radio program, or an official record of a student’s classes and grades. Tran scription is the act or process of creating a transcript.
transpire, vb. Although its traditional sense is “to come to be known” {it trans pired that he had paid bribes}, transpire more commonly today means “hap pen” or “occur” {what transpired when I was away?}. In that newer sense, transpire still carries a vague odor of jargon and pretentiousness. But that is disappearing.
5.250 GRAMMAR AND USAGE trillion. See billion.
triumphal; triumphant. Things are triumphal (done or made to celebrate a vic tory) {a triumphal arch}. But only people feel triumphant (displaying pleasure and pride as a result of a victory or success) {a triumphant Caesar returned to Rome}.
turbid; turgid; torpid. Turbid water or liquid is thick and opaque from churned-up mud or detritus {a turbid pond}; by extension, turbid means “unclear, con fused, or disturbed” {a turbid argument}. Turgid means “swollen,” and by ex tension “pompous and bombastic” {turgid prose}. Torpid means “idle, lazy, and sleepy” {a torpid economy}.
ultimate. See penultimate.
unexceptional; unexceptionable. The first means “not very good; no better than average.” The second means “not open to objection.” uninterested. See disinterested.
unique. Reserve this word for the sense “one of a kind.” Avoid it in the sense “special, unusual.” Phrases such as very unique, more unique, somewhat unique, and so on—in which a degree is attributed to unique—aren’t the best usage. See also 5.89.
unlawful; illegal; illicit; criminal. This list is in ascending order of negative con notation. An unlawful act may even be morally innocent (for example, letting a parking meter expire). But an illegal act is something that society formally condemns, and an illicit act calls to mind moral degeneracy {illicit drug use}. Unlike criminal, the first three terms can apply to civil wrongs.
unorganized. See disorganized.
unreadable. See illegible.
upon. See on.
upward(s). See toward.
use; utilize. Use is usually the best choice for simplicity. Utilize is most often an overblown alternative of use, but it is occasionally the better choice when the distinct sense is “to use to best effect” {how to utilize our staff most effec tively}.
venal; venial. A person who is venal is mercenary or open to bribery—willing to use power and influence dishonestly in return for money {a venal government official}; a thing that is venal is purchasable {venal livestock}. A venial fault or sin is trivial enough to be pardonable or excusable {a venial offense} {a venial error}.
verbal; oral. If something is put into words, it is verbal. Technically, verbal covers both written and spoken utterance. If you wish to specify that something was conveyed through speech, use oral.
very. See much.
vocation. See avocation.
voluminous. See compendious.
WordUsage 5.250 waive; wave. To waive is to relinquish claim to or not to insist on enforcing. To wave is to move to and fro.
wangle. See wrangle.
whether. Generally, use whether alone—not with the words or not tacked on {they didn’t know whether to go}. The or not is necessary only when you mean to convey the idea “regardless of whether” {we’ll finish on time whether or not it rains}. On the distinction between whether and if, see if.
which. See that.
while. While may substitute for although or whereas, especially if a conversational tone is desired {while many readers may disagree, the scientific community has overwhelmingly adopted the conclusions here presented}. Yet because while can denote either time or contrast, the word is occasionally ambiguous; when a real ambiguity exists, although or whereas is the better choice.
who; whom. Here are the traditional rules: who is a nominative pronoun used as (1) the subject of a finite verb {it was Jim who bought the coffee today} or (2) a predicate nominative when it follows a linking verb {that’s who}. Whom is an objective pronoun that may appear as (1) the object of a verb {I learned nothing about the man whom I saw} or (2) the object of a preposition {the woman to whom I owe my life}. Today there are two countervailing trends: first, there’s a decided tendency to use who colloquially in most contexts; second, among those insecure about their grammar, there’s a tendency to overcorrect oneself and use whom when who would be correct. Writers and editors of formal prose often resist the first of these; everyone should resist the second. See also 5.66. whoever; whomever. Avoid the second unless you are certain of your grammar {give this book to whoever wants it} {I cook for whomever I love}. If you are uncertain why these examples are correct, use anyone who or (as in the second example) anyone.
who’s; whose. The first is a contraction {who’s on first?}, the second a possessive {whose life is it, anyway?}. Unlike who and whom, whose may refer to things as well as people {the Commerce Department, whose bailiwick includes intellec tual property}. See 5.64.
workers’ compensation. This is the preferred name for workplace accident insurance plans, not workmen’s compensation. Notice that workers is al ways plural. When used as a phrasal adjective, it is hyphenated {workers’- compensation system}.
wrack. See rack.
wrangle; wangle. To wrangle is to argue, especially angrily over a long period {still wrangling over their parents’ estate}. To wangle is to get something or arrange for something to happen by cleverness, manipulation, or trickery {wangle a couple of last-minute tickets}.
wreak; reek. Wreak means (1) “to cause a great deal of harm or many problems” {to wreak havoc on the administration} or (2) “to punish someone in revenge” 5.251 GRAMMAR AND USAGE {to wreak vengeance on his erstwhile friends}. The past tense is wreaked, not wrought. (The latter is an archaic form of the past tense and past participle of work) Reek can be a verb meaning “to stink” or a noun meaning “stench.” wrong; wrongful. These terms are not interchangeable. Wrong has two senses: (1) “immoral, unlawful” {it’s wrong to bully smaller children} and (2) “im proper, incorrect, unsatisfactory” {many of the math answers are wrong}. Wrongful likewise has two senses: (1) “unjust, unfair” {wrongful conduct} and (2) “unsanctioned by law; having no legal right” {it was a wrongful demand on the estate}.
yes. See affirmative, in the.
your; you’re. Your is the possessive form ofyou {your class}. You’re is the contrac tion for you are {you’re welcome}.
Bias-Free Language 5.251 Maintaining credibility. Discussions of bias-free language—language that is neither sexist nor suggestive of other conscious or subconscious prejudices—have a way of descending quickly into politics. But there is a way to avoid the political quagmire: if we focus solely on maintaining credibility with a wide readership, the argument for eliminating bias from our writing becomes much simpler. Biased language that is not central to the meaning of the work distracts many readers and makes the work less credible to them. Few texts warrant a deliberate display of lin guistic biases. Nor is it ideal, however, to call attention to a supposed ab sence of linguistic biases, since this will also distract readers and weaken credibility.
5.252 Gender bias. Consider the issue of gender-neutral language. On the one hand, many reasonable readers find it unacceptable to use the generic masculine pronoun (he in reference to no one in particular). On the other hand, it is unacceptable to many readers (often different readers) either to resort to nontraditional gimmicks to avoid the generic masculine (by using he/she or s/he, for example) or to use they as a kind of singular pronoun (but see 5.48). Either approach sacrifices credibility with some readers.
5.253 Other biases. The same is true of other types of biases, such as slight ing allusions or stereotypes based on characteristics such as race, eth nicity, disability, religion, sexual orientation, transgender status, or birth or family status. Careful writers avoid language that reasonable readers might find offensive or distracting—unless the biased language is central to the meaning of the writing.
Word Usage 5.255 5.254 Bias and the editor's responsibility. A careful editor points out to au thors any biased terms or approaches in the work (knowing, of course, that the bias may have been unintentional), suggests alternatives, and ensures that any biased language that is retained is retained by choice. Although some publishers prefer to avoid certain terms or specific us ages in all cases, Chicago’s editors do not maintain a list of words or usages considered unacceptable. Rather, they adhere to the reasoning presented here and apply it to individual cases. What you should strive for—if you want readers to focus on your ideas and not on the political subtext—is a style that doesn’t even hint at the issue. So unless you’re involved in a debate about, for example, sexism, you’ll probably want a style, on the one hand, that no reasonable person could call sexist and, on the other hand, that never contorts language to be nonsexist.
5.255 Techniques for achieving gender neutrality. Achieving gender neu trality for generic references to people often involves rewording. Nine methods are suggested below because no single method will work for ev ery writer or in every context. Choose the combination of methods that works best in the context you’ve created.
1. Omit the pronoun. Sometimes a personal pronoun is not really necessary. For instance, in the programmer should update the records when data is transferred to her by the head office, if there is only one programmer, the pronoun phrase to her can be omitted: the programmer should update the records when data is transferred by the head office. Note that the shorter sentence is tighter as well as gender-free.
2. Repeat the noun. If a noun and its pronoun are separated by many words, try repeating the noun. For instance, a writer should be careful not to needlessly an tagonize readers, because her credibility would otherwise suffer becomes a writer should be careful not to needlessly antagonize readers, because the writer’s credibil ity would otherwise suffer. Take care not to overuse this technique. Repeating a noun too frequently will irritate readers. If you have to repeat a noun more than twice in a sentence or repeat it too soon, you should probably rewrite the sentence.
3. Use a plural antecedent. By using a plural antecedent, you eliminate the need for a singular pronoun. For instance, a contestant must conduct himself with dig nity at all times becomes contestants must conduct themselves with dignity at all times. The method may cause a slight change in connotation. In the example, a duty becomes a collective responsibility rather than an individual one.
4. Use an article instead of a pronoun. Try replacing the singular personal pro noun with a definite or indefinite article. Quite often you’ll find that the ef fect on the sentence’s meaning is negligible. For instance, A student accused of cheating must actively waive his right to have his guidance counselor present 5.256 GRAMMAR AND USAGE becomes A student accused ofcheating must actively waive the right to have a guidance counselor present.
5. Use the neutral singular pronoun one. Try replacing the gender-specific per sonal pronoun with the gender-neutral singular pronoun one. For instance, an actor in New York is likely to earn more than he is in Paducah becomes an actor in New York is likely to earn more than one in Paducah.
6. Use the relative pronoun who. This technique works best when it replaces a personal pronoun that follows if. It also requires revising the sentence slightly. For instance, employers presume that if an applicant can't write well, he wont he a good employee becomes employers presume that an applicant who can't write well won't be a good employee.
7. Use the imperative mood. The imperative eliminates the need for an explicit pronoun. Although its usefulness is limited in some types of writing, you may find that it avoids prolixity and more forcefully addresses the target audience. For instance, a lifeguard must keep a close watch over children while he is monitor ing the pool becomes keep a close watch over children while monitoring the pool.
8. In moderation, use he or she. Although it is an easy fix, the phrase he or she should be used sparingly, preferably only when no other technique is satis factory. For instance, “abstractitis” is Ernest Gowers's term for writing that is so abstract and obtuse (hence abstruse) that the writer does not even know what he is trying to say becomes “abstractitis” is Ernest Gowers's term for writing that is so abstract and obtuse (hence abstruse) that the writer does not even know what he or she is trying to say. If you find you need to repeat the pronouns in the same sentence, don’t. Revise the sentence instead.
9. Revise the sentence. If no other technique produces a sentence that reads well, rewrite the sentence so that personal pronouns aren’t needed. The amount of revision will vary. For instance, if a boy or girl misbehaves, his or her privileges will be revoked might become if someone misbehaves, that person's privileges will be revoked. And a person who decides not to admit he lied will be considered honest until someone exposes his lie might become a person who denies lying will be con sidered honest until the lie is exposed.
See also 5.256.
5.256 Gender-neutral singular pronouns. Traditionally, the only gender neutral third-person singular personal pronoun in English is it, which doesn’t refer to humans (with very limited exceptions). Clumsy artifices such as s/he and (wo)man or artificial genderless pronouns have been tried—for many years—with no success. They won’t succeed. And those who use them invite credibility problems. Indefinite pronouns such as anybody and someone don’t always satisfy the need for a gender-neutral alternative because they are traditionally regarded as singular anteced ents that call for a third-person singular pronoun. Many people substi- Word Usage 5.259 tute the plural they and their for the singular he or she. They and their have become common in informal usage, but neither is considered fully acceptable in formal writing, though they are steadily gaining ground. For now, unless you are given guidelines to the contrary, be wary of us ing these forms in a singular sense. This advice, like the techniques for achieving gender neutrality discussed in 5.255, applies mainly to generic references, where the identity of the person is unknown or unimportant. For references to a specific person, the choice of pronoun may depend on the individual. Some people identify not with a gender-specific pronoun but instead with the pronoun they and its forms or some other gender neutral singular pronoun; any such preference should generally be re spected. See 5.48.
5.257 Problematic gender-specific suffixes. The trend in American English is toward eliminating sex-specific suffixes. Words with feminine suf fixes such as -ess and -ette are easily replaced with the suffix-free forms, which are increasingly accepted as applying to both men and women. For example, author and testator are preferable to authoress and testatrix. Compounds with -man are more problematic. The word person rarely functions well in such a compound; chairperson and anchorperson sound more pompous and wooden than the simpler (and correct) chair or an chor. Unless a word is established (such as salesperson, which dates from 1901), don’t automatically substitute -person for -man. English has many alternatives that are not necessarily newly coined, including police officer (first recorded in 1797), firefighter (1903), and mail carrier (1788).
5.258 Necessary gender-specific language. It isn’t always necessary or desir able to use gender-neutral terms and phrasings. If you’re writing about something that clearly concerns only one sex (e.g., womens studies; men's golf championship) or an inherently single-sex institution (e.g., a soror ity; a Masonic lodge), trying to use gender-neutral language may lead to absurd prose {be solicitous of a pregnant daughter’s comfort; he or she will need your support}.
5.259 Sex-specific labels as adjectives. When gender is relevant, it’s accept able to use the noun woman as a modifier {woman judge}. In recent de cades, woman has been rapidly replacing lady in such constructions. The adjective female is also often used unobjectionably. In isolated contexts it may strike some readers as being dismissive or derogatory (perhaps because it’s a biological term used for animals as well as humans), but when parallel references to both sexes are required, the adjectives male and female are typically the most serviceable choices {the police force has 834 male and 635 female officers}.
5.260 GRAMMAR AND USAGE 5.260 Avoiding other biased language. Comments that betray a writer’s con scious or unconscious biases or ignorance may cause readers to lose re spect for the writer and interpret the writer’s words in ways that were never intended. In general, emphasize the person, not a characteristic. A characteristic is a label. It should preferably be used as an adjective, not as a noun. Instead of referring to someone as, for instance, a Catholic or a deaf-mute, put the person first by writing a Catholic man or he is Cath olic, and a deaf-and-mute child or the child is deaf and mute. Avoid irrel evant references to personal characteristics such as sex, race, ethnicity, disability, age, religion, sexual orientation, transgender status, or social standing. Such pointless references may affect a reader’s perception of you or the person you are writing about or both. They may also invoke a reader’s biases and cloud your meaning. When it is important to mention a characteristic because it will help the reader develop a picture of the person you are writing about, use care. For instance, in the sentence Shir ley Chisholm was probably the finest African American woman member of the House of Representatives that New York has ever had, the phrase African American woman may imply to some readers that Chisholm was a great representative “for a woman” but may be surpassed by many or all men, that she stands out only among African American members of Congress, or that it is unusual for a woman or an African American to hold high office. But in Shirley Chisholm was the first African American woman to be elected to Congress and one of New York’s all-time best representatives, the purpose of the phrase African American woman is not likely to be misun derstood.
6 • Punctuation OVERVIEW 6.1 PUNCTUATION IN RELATION TO SURROUNDING TEXT 6.2 Punctuation in Relation to Closing Quotation Marks 6.9 PERIODS 6.12 COMMAS 6.16 Series and the Serial Comma 6.19 Commas with Independent Clauses 6.22 Commas with Dependent Clauses 6.24 Commas with Relative Clauses, Appositives, and Descriptive Phrases 6.27 Commas with Participial and Adverbial Phrases 6.30 Commas with Introductory Words and Phrases 6.33 Commas with Two or More Adjectives Preceding a Noun 6.36 Commas with Dates and Addresses 6.38 Commas with Quotations and Questions 6.40 Commas in Personal and Corporate Names 6.43 Commas with Antithetical Elements 6.45 Other Uses of the Comma 6.48 SEMICOLONS 6.56 COLONS 6.61 QUESTION MARKS 6.68 EXCLAMATION POINTS 6.71 HYPHENS AND DASHES 6.75 Hyphens 6.76 En Dashes 6.78 Em Dashes 6.85 2-Em and 3-Em Dashes 6.93 PARENTHESES 6.95 BRACKETS AND BRACES 6.99 SLASHES 6.105 QUOTATION MARKS 6.114 APOSTROPHES 6.116 SPACES 6.119 MULTIPLE PUNCTUATION MARKS 6.122 LISTS AND OUTLINE STYLE 6.127 6.1 PUNCTUATION Overview 6.1 The role of punctuation and the scope of this chapter. Punctuation should be governed by its function, which in ordinary text is to promote ease of reading by clarifying relationships within and between sentences. This function, although it allows for a degree of subjectivity, should in turn be governed by the consistent application of some basic principles lest the subjective element obscure meaning. The principles set forth in this chapter are based on a logical application of traditional practice in the United States. Some of the more significant exceptions have been noted where they apply. For the special requirements of languages other than English, mathematics, source citations (including bibliographies), indexes, and more, see the appropriate chapters in this manual or consult the index.
Punctuation in Relation to Surrounding Text 6.2 Punctuation and italics. All punctuation marks should appear in the same font—roman or italic—as the main or surrounding text, except for punctuation that belongs to a title in a different font (usually italics). So, for example, the word and, which in this sentence is in italics^ is followed by a comma in roman type; the comma, strictly speaking, does not be long to and, which is italicized because it is a word used as a word (see 7.63) . Depending on typeface, it may be difficult to tell whether a comma is in italics or not (to say nothing of periods); for other marks it will be more evident. Readers of this manual online may be able to view the source code for italic text, and all those who prepare manuscripts or pub lications in electronic formats will need to pay attention to this level of detail (see 2.80; see also 2.81-83). In the first four examples that follow, the punctuation marks next to italic text belong with the surrounding sentence and are therefore presented in roman. In the last two examples, the three punctuation marks that belong with the italic titles—the excla mation mark following “Help,” the colon following “Sublime,” and the comma following “Code”—are in italics (the comma following “Beauty” is in roman).
For light amusement he turns to the Principia Mathematical How can they be sure that the temperature was in fact rising? The letters a, b, and c are often invoked as being fundamental. There are two primary audiences for The Chicago Manual of Style: perfectionists and humorists.
Punctuation in Relation to Surrounding Text 6.4 but The Beatles’ Help! was released long before the heyday of the music video.
After reading Geek Sublime: The Beauty of Code, the Code of Beauty, she was in spired to write a program that generates poetry from prose.
For parentheses and brackets, see 6.5; for quotation marks, see 6.6. For a different approach, see 6.4.
6.3 Punctuation and boldface or color. The choice of boldface (or, by exten sion, type in a different color), unlike that of italics (see 6.2), is sometimes an aesthetic rather than a purely logical decision. Punctuation marks fol lowing boldface or color should be dealt with case by case, depending on how the boldface is used. In the first example, the period following “line spacing” belongs with the boldface glossary term and is therefore set in bold; the period following “leading” is part of the surrounding sentence and is therefore not set in bold. In the middle two examples, the punctua tion next to the boldface terms belongs with them, like the first period in the first example. In the final example, the question mark belongs to the surrounding sentence and not to the boldface word (see 7.79).
line spacing. See leading.
Figure 6. Title page from an apocryphal Second Poetics. For sale: a 2005 Subaru Legacy and two gently used sleeping bags. Will the installation remain stalled until I choose I accept?
6.4 Punctuation and font—aesthetic considerations. According to a more traditional system, periods, commas, colons, and semicolons should ap pear in the same font as the word, letter, character, or symbol immedi ately preceding them if different from that of the main or surrounding text. In the third and fourth examples in 6.2, the commas following a and b and the colon following the title of this manual would be italic, as would the comma following the book title in the last example (i.e., after Beauty). A question mark or exclamation point, however, would appear in the same font as the immediately preceding word only if it belonged to that word, as in the title Help! in 6.2. This system, once preferred by Chicago and still preferred by some as more aesthetically pleasing, should be reserved—if it must be used—for publications des tined for print only. In electronic publications, where typeface may be determined by content as well as appearance (e.g., a book title might be tagged as such, separate from any surrounding punctuation), the more logical system described in 6.2 should be preferred. See also 2.82, note 1.
6.5 PUNCTUATION 6.5 Parentheses and brackets in relation to surrounding text. Parentheses and brackets should appear in the same font—roman or italic—as the surrounding text, not in that of the material they enclose. This system, though it may occasionally cause typefitting problems when a slanting italic letter touches a nonslanting roman parenthesis or bracket, has two main virtues: it is easy to use, and it has long been practiced. For printed works, a thin space or a hair space may need to be added between over lapping characters (see 6.120). For electronic works, where type display will vary depending on hardware and software, no such adjustments should normally be made.
The Asian long-horned beetle (Anoplophora glabripennis) attacks maples. The letter stated that my check had been “recieved [sic] with thanks.” When a phrase in parentheses or brackets appears on a line by itself, however, the parentheses or brackets are usually in the same font as the phrase.
[continued on page 72] 6.6 Quotation marks in relation to surrounding text. Like parentheses and brackets (see 6.5), quotation marks should appear in the same font roman or italic—as the surrounding text, which may or may not match that of the material they enclose. In the first two examples, the quotation marks are in roman; in the third example, they have been italicized as part of the italic title.
The approach to the runway was, they reported, “extremely dangerous” (italics in original).
“Hamlet and the Pre-Jazz Hipster Persona” is the fourth article in a series on liter ature and fashionable existentialism.
I just finished reading Sennacherib's “Palace without Rival” at Nineveh, by John Malcolm Russell.
As with parentheses and brackets, when a sentence or phrase in quota tion marks appears on a line by itself, the quotation marks are usually in the same font as the sentence or phrase. See also 13.62.
6.7 Punctuation and space—one space or two? In typeset matter, one space, not two, should be used between two sentences—whether the first ends in a period, a question mark, an exclamation point, or a closing quotation mark or parenthesis. By the same token, one space, not two, should follow a colon. When a particular design layout calls for more Punctuation in Relation to Surrounding Text 6.9 space between two elements—for example, between a figure number and a caption—the design should specify the exact amount of space (e.g., em space). See also 6.119-21.
6.8 Punctuation with URLs and email addresses. Sentences that include an email address or a uniform resource identifier such as a URL should be punctuated normally. Though angle brackets or other “wrappers” are standard in some applications, these are generally unnecessary in nor mal prose (see 6.104). Readers of print sources should assume that any punctuation at the end of an email address or URL belongs to the sen tence. By the same logic, any hypertext markup for electronic formats should exclude the surrounding punctuation. For dividing an email ad dress or a URL at the end of a line, see 7.46.
Chicago’s online forum, which can be found at style.org, continues to attract much constructive commentary.
Write to me at grammar88@parsed-out.edu.
Punctuation in Relation to Closing Quotation Marks 6.9 Periods and commas in relation to closing quotation marks. Periods and commas precede closing quotation marks, whether double or single. (An apostrophe at the end of a word should never be confused with a closing single quotation mark; see 6.118.) This is a traditional style, in use in the United States well before the first edition of this manual (1906). For an exception, see 7.79. See also table 6.1.
He described what he heard as a “short, sharp shock.” “Thus conscience does make cowards of us all,” she replied.
t a b l e 6.1. Punctuation relative to closing quotation marks and parentheses or brackets Closing mark Double or single quotation marks Parentheses or brackets1 Period Inside Inside or outside; see 6.13 Comma Inside Outside Semicolon Outside Outside Colon Outside Outside Question mark or exclamation point Inside or outside; see 6.10 Inside or outside; see 6.70, 6.74 Em dash Inside or outside; see 6.87 Outside See also 6.9, 6.70, 6.74.
fSee also 6.18, 6.98, 6.101, 6.103.
6.10 PUNCTUATION In an alternative system, sometimes called British style (as described in the New Oxford Style Manual; see bibliog. 1.1), single quotation marks are used, and only those punctuation points that appeared in the original material are included within the quotation marks; all others follow the closing quotation marks. (Exceptions to the rule are widespread: for ex ample, periods are routinely placed inside any quotation that begins with a capital letter and forms a grammatically complete sentence.) Double quotation marks are reserved for quotations within quotations. This sys tem or a variation (like the one prescribed by Scientific Style and Format, bibliog. 1.1) may be appropriate in works of textual criticism or in com puter coding and other technical or scientific settings. See also 13.7-8, 13.30-31.
6.10 Other punctuation in relation to closing quotation marks. Colons and semicolons—unlike periods and commas—follow closing quotation marks; question marks and exclamation points follow closing quotation marks unless they belong within the quoted matter. (This rule applies the logic that is often absent from the traditional US style described in 6.9.) See also table 6.1.
Take, for example, the first line of “Filling Station”: “Oh, but it is dirty!” I can’t believe you don’t know “Filling Station”!
I was invited to recite the lyrics to “Sympathy for the Devil”; instead I read from the op-ed page of the New York Times.
Which of Shakespeare’s characters said, “All the world’s a stage”? “Timber!” “What’s the rush?” she wondered.
6.11 Single quotation marks next to double quotation marks. When single quotation marks are nested within double quotation marks, and two of the marks appear next to each other, a space between the two marks, though not strictly required, aids legibility. For print publications, type setters may place a thin space or a hair space between the two marks (as in the print edition of this manual). In electronic environments (includ ing manuscripts submitted for publication), a nonbreaking space can be used (as in the online edition of this manual); such a space will prevent the second mark from becoming stranded at the beginning of a new line. See also 6.120,13.30. In the example that follows, note that the period precedes the single quotation mark (see also 6.9).
“Admit it,” she said. “You haven’t read ‘The Simple Art of Murder.’” Periods 6.14 Periods 6.12 Use of the period. A period marks the end of a declarative or an imper ative sentence. In some contexts, a period is referred to as a dot (as in a URL) or a point (as in decimals). In British usage, a period is called a full stop. Between sentences, it is followed by a single space (see 2.9, 6.7, 6.119). A period may also follow a word or phrase standing alone, as in the third example. For the many other uses of the period, consult the index.
The two faced each other in silence.
Wait here.
My answer? Never.
6.13 Periods in relation to parentheses and brackets. When an entire in dependent sentence is enclosed in parentheses or square brackets, the period belongs inside the closing parenthesis or bracket. When matter in parentheses or brackets, even a grammatically complete sentence, is included within another sentence, the period belongs outside (but see also 6.98). Avoid enclosing more than one complete sentence within an other sentence. In the third example, two periods are required—one for the abbreviation etc. and one for the sentence as a whole, outside the pa rentheses (see also 6.14, 6.123). For periods relative to quotation marks, see 6.9.
Fiorelli insisted on rewriting the paragraph. (His newfound ability to type was both a blessing and a curse.) Felipe had left an angry message for Isadora on the mantel (she noticed it while glancing in the mirror).
His chilly demeanor gave him an affinity for the noble gases (helium, neon, etc.). There were many groundbreaking moments in All in the Family. (The one featur ing “the kiss,” with Sammy Davis Jr., springs to mind.) “All the evidence pointed to the second location [the Laszlos’ studio].” 6.14 When to omit a period. Unless it ends in an abbreviation or other expres sion that normally requires a period, no period should follow a display line (i.e., chapter title, subhead, or similar heading), a running head, a column head in a table, a phrase used as a caption (but see 3.21), a date line in correspondence, a signature, or an address. (Likewise, a comma is sometimes omitted for aesthetic reasons at the end of a line set in display type; see 8.165.) A run-in subhead at the beginning of a paragraph, how ever, is followed by a period (see 1.56). When an expression that ends in a 6.15 PUNCTUATION period (e.g., an abbreviation) falls at the end of a sentence, no additional period follows (see 6.123; but see 6.13). For use or omission of the period in lists and outline style, see 6.127-32. For punctuation with URLs and email addresses, see 6.8.
6.15 Periods in ellipses. An ellipsis—a series of three periods, or dots (some times referred to as suspension points)—may be used to indicate an omis sion in quoted material; for a full discussion of this use, see 13.50-58. An ellipsis may also be used to indicate faltering speech or an incomplete sentence or thought (see 13.41,13.55). For the use of ellipses in languages other than English, see 11.19,11.32,11.49,11.65,11.102. For the use of the em dash to indicate a sudden break or interruption, see 6.87.
Commas 6.16 Use of the comma. The comma, aside from its technical uses in scien tific, bibliographical, and other contexts, indicates the smallest break in sentence structure. It usually denotes a slight pause. In formal prose, however, logical considerations come first. Effective use of the comma involves good judgment, with the goal being ease of reading.
6.17 Commas in pairs. Whenever a comma is placed before an element to set it off from the surrounding text (such as “1920” or “Minnesota” in the first two examples below), a second comma is required if the phrase or sentence continues beyond the element being set off. This principle ap plies to many of the uses for commas described in this section. An excep tion is made for commas within the title of a work (third example); such commas are considered to be independent of the surrounding sentence.
August 18,1920, was a good day for American women.
Sledding in Duluth, Minnesota, is facilitated by that city’s hills and frigid winters. but Look Homeward, Angel was not the working title of Wolfe’s manuscript.
6.18 Commas relative to parentheses and brackets. When the context calls for a comma at the end of material in parentheses or brackets, the comma should follow the closing parenthesis or bracket. A comma never precedes a closing parenthesis. (For its rare appearance before an open ing parenthesis, see the examples in 6.129.) Rarely, a comma may appear inside and immediately before a closing bracket as part of an editorial interpolation (as in the last example; see also 13.59).
Commas 6.19 After several drummers had tried out for the part (the last having destroyed the kit), the band decided that a drum machine was their steadiest option.
Her delivery, especially when she would turn to address the audience (almost as if to spot a long-lost friend), was universally praised.
“Conrad told his assistant [Martin], who was clearly exhausted, to rest.” “The contents of the vault included fennel seeds, tweezers, [straight-edged ra zors,] and empty Coca-Cola cans.” Series and the Serial Comma 6.19 Serial commas. Items in a series are normally separated by commas (but see 6.60). When a conjunction joins the last two elements in a series of three or more, a comma—known as the serial or series comma or the Ox ford comma—should appear before the conjunction. Chicago strongly recommends this widely practiced usage, blessed by Fowler and other authorities (see bibliog. 1.2), since it prevents ambiguity. If the last ele ment consists of a pair joined by and, the pair should still be preceded by a serial comma and the first and (as in the last two examples below).
She posted pictures of her parents, the president, and the vice president. Before heading out the door, he took note of the typical outlines of sweet gum, ginkgo, and elm leaves.
I want no ifs, ands, or buts.
Paul put the kettle on, Don fetched the teapot, and I made tea.
Their wartime rations included cabbage, turnips, and bread and butter.
Ahmed was configuring updates, Jean was installing new hardware, and Alan was running errands and furnishing food.
If the sentence continues beyond the series, add a comma only if one is required by the syntax of the surrounding sentence.
Apples, plums, and grapes can all be used to make wine.
but Apples, plums, and grapes, available at most large grocery stores, can all be used to make wine.
In the rare case where the serial comma does not prevent ambiguity, it may be necessary to reword. In the following example, the repetition of and makes it clear that Lady Gaga is not the writer’s mother (and see the examples at the end of this paragraph). In the second example, “Lady Gaga” might be read as an appositive (see 6.28).
6.20 PUNCTUATION I thanked my mother and Lady Gaga and Madonna.
not I thanked my mother, Lady Gaga, and Madonna.
Note that the phrase as well as cannot substitute for and in a series of items.
The team fielded one Mazda, two Corvettes, and three Bugattis, as well as a bat tered Plymouth Belvedere.
not The team fielded one Mazda, two Corvettes, three Bugattis, as well as a battered Plymouth Belvedere.
In a series whose elements are all joined by conjunctions, no commas are needed unless the elements are long and delimiters would be helpful.
Would you prefer Mendelssohn or Schumann or Liszt?
You can turn left at the second fountain and right when you reach the temple, or left at the third fountain and left again at the statue of Venus, or in whatever direction Google sends you.
6.20 Commas with "etc." and "et al." The abbreviation etc. (et cetera, liter ally “and others of the same kind”) and such equivalents as and so forth and and the like are preceded by a comma; they are followed by a comma only if required by the surrounding text. This small departure from the recommendation in previous editions treats such terms as equivalent to the final element in a series (see 6.19). (According to a more traditional usage, such terms were often set off by two commas.) In formal prose, Chicago prefers to limit the abbreviation etc. to parentheses, notes, and tabular matter. See also 5.250 under etc.
The map was far from complete (lacking many of the streets, alleys, etc. seen in earlieriterations).
The philosopher’s population studies, classic textbooks, stray notes, and so forth were found in the attic.
but For a discussion or periods, commas, and the like, see chapter 6.
The abbreviation etal. (etalia [neut.J, et alii [masc.J, or et aliae [fem.], lit erally “and others”), whether used in regular text or (more often) in bib liographical references, should be treated like etc. If et al. follows a single item, however (e.g., “Jones et al.”), it requires no preceding comma. (Nor Commas 6.23 is a preceding comma required in the rare case that etc. follows a single item.) Note that neither etc. nor et al. is italicized in normal prose (see the first example above).
6.21 Omitting serial commas before ampersands. When an ampersand is used instead of the word and (as in company names), the serial comma is omitted.
Winken, Blinken & Nod is a purveyor of nightwear.
See also 14.88,14.135.
Commas with Independent Clauses 6.22 Commas with independent clauses joined by coordinating conjunc tions. When independent clauses are joined by and, but, or, so, yet, or any other coordinating conjunction, a comma usually precedes the con junction. If the clauses are very short and closely connected, the comma may be omitted (as in the last two examples) unless the clauses are part of a series. These recommendations apply equally to imperative sen tences, in which the subject (you) is omitted but understood (as in the fifth and last examples). (For the use of a semicolon between indepen dent clauses, see 6.56.) We activated the alarm, but the intruder was already inside. All watches display the time, and some of them do so accurately.
Do we want to foster creativity, or are we interested only in our intellectual prop erty?
The bus never came, so we took a taxi.
Wait for me at the bottom of the hill on Buffalo Street, or walk up to Eddy Street and meet me next to the Yield sign.
Donald cooked, Sally poured the wine, and Maddie and Cammie offered hors d’oeuvres.
but Electra played the guitar and Tambora sang. Raise your right hand and repeat after me.
6.23 Commas with compound predicates. A comma is not normally used to separate a two-part compound predicate joined by a coordinating con junction (cf. 6.22). (A compound predicate occurs when a subject that is shared by two or more clauses is not repeated after the first clause.) A 6.24 PUNCTUATION comma may occasionally be needed, however, to prevent a misreading (as in the fourth example).
He printed out a week’s worth of crossword puzzles and arranged them on his clipboard.
Kelleher tried to contact the mayor but was informed that she had stopped ac cepting unsolicited calls.
He stood up and opened his mouth but failed to remember his question. but She recognized the man who entered the room, and gasped.
When then is used as a shorthand for and then, a comma usually precedes the adverb. (See also 6.57.) She filled in the last square in Sunday’s puzzle, then yawned.
but She filled in the last square in Sunday’s puzzle and then yawned.
Compound predicates of three or more parts treated as a series are punc tuated accordingly (see 6.19).
She scrubbed the floors, washed the dishes, and finished her essay on twenty- first-century labor-saving technologies.
Commas with Dependent Clauses 6.24 Commas with introductory dependent clauses. When a dependent clause precedes the main, independent clause, it should be followed by a comma. A dependent clause is generally introduced by a subordinating conjunction such as if, because, or when (see 5.200, 5.201).
If you accept our conditions, we shall agree to the proposal.
Until we have seen the light, we cannot guarantee a safe exit from the tunnel. Whether you agree with her or not, she has a point.
Compare 6.25.
6.25 Commas with dependent clauses following the main clause. A de pendent clause that follows a main, independent clause should not be preceded by a comma if it is restrictive—that is, essential to fully under standing the meaning of the main clause (see also 6.27). For instance, in Commas 6.26 the first example below, it is not necessarily true that “we will agree to the proposal”; the dependent if clause adds essential information.
We will agree to the proposal if you accept our conditions.
Paul sighed when he heard the news.
He wasn’t running because he was afraid; he was running because he was late.
If the dependent clause is merely supplementary or parenthetical (i.e., nonrestrictive, or not essential to the meaning of the main clause), it should be preceded by a comma. Such distinctions are occasionally ten uous. In the fourth example below, the meaning—and whether the sub ject is running or not—depends almost entirely on the presence of the comma (compare with the third example above). If in doubt, rephrase.
I’d like the tom yum, if you don’t mind.
At last she arrived, when the food was cold.
She has a point, whether you agree with her or not.
He wasn’t running, because he was afraid of the dark.
or Because he was afraid of the dark, he wasn’t running.
Compare 6.24.
6.26 Commas with intervening dependent clauses (two consecutive con junctions). When a dependent clause intervenes between two other clauses joined by a coordinating conjunction, causing the coordinating and subordinating conjunctions to appear next to each other (e.g., and if, but if), the conjunctions need not be separated by a comma. See also 6.22, 6.24.
Burton examined the documents for over an hour, and if Smedley had not inter vened, the forgery would have been revealed.
She claimed to have seen the whole film, but when we pressed her for details, she failed to recall the name of Rhett and Scarlett’s only child.
By a similar logic, when a dependent clause intervenes between an inde pendent clause and a dependent clause introduced by a subordinating conjunction, no comma is needed between the two subordinating con junctions (e.g., that if).
They decided that if it rained, they would reschedule the game.
6.27 PUNCTUATION Strictly speaking, it would not be wrong to add a comma between the conjunctions in any of the examples above. Such usage, which would ex tend the logic of commas in pairs (see 6.17), may be preferred in certain cases for emphasis or clarity. See also 6.32.
Commas with Relative Clauses, Appositives, and Descriptive Phrases 6.27 Commas with relative clauses—"that" versus "which." A clause is said to be restrictive (or defining) if it provides information that is essential to understanding the intended meaning of the rest of the sentence. Re strictive relative clauses are usually introduced by that (or by who/whom/ whose) and are never set off by commas from the rest of the sentence. The pronouns that and who or whom may occasionally be omitted (but need not be) if the sentence is just as clear without them, as in the second and fourth examples (before “I” [that] and “we” [whom], respectively).
The manuscript that the editors submitted to the publisher was well formatted. The book I have just finished is due back tomorrow; the others can wait.
I prefer to share the road with drivers who focus on the road rather than on what they happen to be reading.
The drivers we hire to make deliveries must have good driving records.
The author whose work I admire the most is generally the one whose books I have most recently read.
A clause is said to be nonrestrictive (or nondefining or parenthetical) if it could be omitted without obscuring the identity of the noun to which it refers or otherwise changing the intended meaning of the rest of the sentence. Nonrestrictive relative clauses are usually introduced by which (or who/whom/whose) and are set off from the rest of the sentence by commas.
The final manuscript, which was well formatted, was submitted to the publisher on time.
Ulysses, which I finished early this morning, is due back on June 16.
I prefer to share the road with illiterate drivers, who are unlikely to read books while driving.
Boris Pasternak, whose most famous creation was a doctor, wrote what is prob ably the best novel about the Russian Revolution.
Although which can be substituted for that in a restrictive clause (a com mon practice in British English), many writers preserve the distinction Commas 6.29 between restrictive that (with no commas) and nonrestrictive which (with commas). See also 5.250 under that; which.
6.28 Commas with appositives. A word, abbreviation, phrase, or clause that is placed in apposition to a noun (i.e., providing an explanatory equiva lent) is normally set off by commas if it is nonrestrictive—that is, if it can be omitted without obscuring the identity of the noun to which it refers.
K. Lester’s only collection of poems, An Apocryphal Miscellany, first appeared as a series of mimeographs. (The collection has been identified as his only one; the title provides additional rather than essential information.) This year’s poet laureate, K. Lester, spoke first. (There is only one laureate this year.) Ursula’s husband, Jan, is also a writer. (Ursula has only one husband.) Ursula’s son, Clifford, had been a student of Norman Maclean’s. (Ursula has only one son.) If, however, the word or phrase is restrictive—that is, it provides (or may provide) essential information about the noun (or nouns) to which it re fers—no commas should appear.
O’Neill’s play The Hairy Ape was being revived. (O’Neill wrote a number of plays; the title identifies the one being revived.) The renowned poet and historian K. Lester scheduled a six-city tour for April. (K. Lester is not the world’s only renowned poet and historian.) Caligula’s sister Drusilla has been the subject of much speculation. (Caligula had three sisters.) The playwright’s son Julio was there. (Whether the playwright has sons in addi tion to Julio is not known.) Though the possessive may be used with a restrictive appositive (e.g., Caligula’s sister Drusilla’s son), avoid such a construction with a nonre strictive appositive. Instead, reword as needed.
The motorcycle belonging to Ursula’s mother, Hulga, was a Harley.
not Ursula’s mother’s, Hulga’s, motorcycle was a Harley.
See also 5.23, 6.27.
Commas with descriptive phrases. A descriptive phrase that is restric tive—that is, essential to the meaning (and often the identity) of the noun it belongs to—should not be set off by commas. A nonrestrictive phrase, however, should be enclosed in commas (or, if at the end of a sentence, 6.29 6.30 PUNCTUATION preceded by a comma). In the first example, the descriptive phrase (“with the guitar over her shoulder”) is essential information that identifies which woman is the mother. In the second example, the identity of the per son who turned to the drummer (“My mother”) is clear; the fact that she has a guitar over her shoulder is not essential information. See also 6.27.
The woman with the guitar over her shoulder is my mother.
My mother, with her guitar over her shoulder, turned to the drummer and gave the signal to begin.
6.30 Commas with participial phrases. An introductory participial phrase is normally set off from the rest of the sentence by a comma.
Exhilarated by her morning workout, she headed for the ocean.
Having forgotten his lines, the actor was forced to ad-lib.
When such a phrase occurs in the middle of a sentence, it should be set off by commas unless it is used restrictively, providing essential informa tion about the main clause (see also 6.29).
The actor, having forgotten his lines, was forced to ad-lib.
but Actors forgetting their lines may be forced to ad-lib. (The phrase “forgetting their lines” specifies which actors may be forced to ad-lib.) Likewise, a comma sets off such a phrase at the end of a sentence unless the phrase is used restrictively.
She headed for the ocean, exhilarated by her morning workout.
The actor was forced to ad-lib, having forgotten his lines.
but She always headed for the ocean exhilarated by her morning workout. (It is not true that she always headed for the ocean; it is true that she always headed for the ocean in a state of exhilaration from her morning workout.) A comma should not be used if the participial phrase modifies the subject of a sentence by means of a linking verb (see 5.101), even if the sentence is inverted.
Running along behind the wagon was the archduke himself!
Commas 6.32 6.31 Commas with adverbial phrases. Although an introductory adverbial phrase can usually be followed by a comma, it need not be unless mis reading is likely. Shorter adverbial phrases are less likely to merit a comma than longer ones.
On the other hand, his vices could be considered virtues.
With three consecutive swings, Jackson made history.
In 1931 Henrietta turned fifty.
but Before eating, the members of the committee met in the assembly room.
To Anthony, Blake remained an enigma.
When such a phrase occurs in the middle of a sentence, it is normally set off by commas (cf. 6.29).
Jackson, with three consecutive swings, made history.
His vices, on the other hand, could be considered virtues.
At the end of a sentence, a comma is necessary only when the phrase is used in a nonrestrictive sense, providing information that is not essential to the meaning of the rest of the sentence.
Jackson made history with three consecutive swings.
Henrietta turned fifty in 1931.
but Henrietta turned fifty a decade later, in 1931.
A comma should not be used to set off an adverbial phrase that intro duces an inverted sentence.
Before the footlights stood one of the most notorious rakes of the twenty-first century.
6.32 Commas with a participial or adverbial phrase plus a conjunction.
When a participial or adverbial phrase immediately follows a coordinat ing conjunction, the use of commas depends on whether the conjunction joins two independent sentences. If the conjunction is simply a part of the predicate or joins a compound predicate, the first comma follows the conjunction (see also 6.23).
We were extremely tired and, in light of our binge the night before, anxious to go home.
6.33 PUNCTUATION The Packers trailed at halftime but, buoyed by Rodgers’s arm, stormed back to win.
If the conjunction joins two independent clauses, however, the comma precedes the conjunction (see also 6.22).
We were elated, but realizing that the day was almost over, we decided to go to bed.
Strictly speaking, it would not be wrong to add a second comma after but in the last example. Such usage, which would extend the logic of com mas in pairs (see 6.17), may be preferred in certain cases for emphasis or clarity. See also 6.26.
6.33 Commas with introductory phrases. Whether to use a comma to set off an introductory phrase can depend on the type of phrase, its relation ship to the rest of the sentence, and its length. For participial phrases, see 6.30. For adverbial phrases, see 6.31. Some cases involving specific words are discussed below (6.34 and 6.35). For dependent clauses, see 6.24.
6.34 Commas with an introductory "yes," "no," or the like. A comma should follow an introductory yes, no, OK, well, and the like, except in certain instances more likely to be encountered in informal prose or dialogue.
Yes, it is true that 78 percent of the subjects ate 50 percent more than they re ported.
No, neither scenario improved the subjects’ accuracy.
OK, I’ll try the quinoa.
Well then, we shall have to take a vote. but No you will not!
6.35 Commas with an introductory "oh" or "ah." A comma usually follows an exclamatory oh or ah unless it is followed by an exclamation mark (or a dash) or forms part of a phrase (e.g., “oh boy,” “ah yes”). No comma fol lows a vocative oh or (mainly poetic and largely archaic) O. See also 7.31.
Oh, you’re right!
Ah, here we are at last! Oh no! Ah yes! Oh yeah?
My oh my!
Oh mighty king!
“O wild West Wind...” Commas 6.38 Commas with Two or More Adjectives Preceding a Noun 6.36 Commas with coordinate adjectives. As a general rule, when a noun is preceded by two or more adjectives that could, without affecting the meaning, be joined by and, the adjectives are separated by commas. Such adjectives, which are called coordinate adjectives, can also usually be reversed in order and still make sense. If, on the other hand, the ad jectives are not coordinate—that is, if one or more of the adjectives are essential to (i.e., form a unit with) the noun being modified—no commas are used. See also 5.91.
Shelly had proved a faithful, sincere friend. (Shelly’s friendship has proved faith ful and sincere.) It is going to be a long, hot, exhausting summer. (The summer is going to be long and hot and exhausting.) but She has many faithful friends.
He has rejected traditional religious affiliations.
She opted for an inexpensive quartz watch.
6.37 Commas with repeated adjectives. When an adjective is repeated be fore a noun, a comma normally appears between the pair.
Many, many people have enjoyed the book.
Commas with Dates and Addresses 6.38 Commas with dates. In the month-day-year style of dates, commas must be used to set off the year—a traditional usage that not only applies the logic of commas in pairs (see 6.17) but also serves to separate the numerals for day and year. By a similar logic, when the day of the week is given, it is separated from the month and day by a comma. Commas are usually unnecessary, however, between the name for the day and the ordinal in references where the month is not expressed (see also 9.31). Commas are also unnecessary where month and year only are given, or where a named day (such as a holiday) is given with a year. For dates used adjectivally, see 5.83. See also 9.29-36.
The performance took place on February 2,2006, at the State Theatre in Ithaca. The hearing was scheduled for Friday, August 11,2017.
Monday, May 5, was a holiday; Tuesday the sixth was not.
6.39 PUNCTUATION Her license expires sometime in April 2021.
On Thanksgiving Day 1998 they celebrated their seventy-fifth anniversary.
In the day-month-year system—useful in material that requires many full dates (and standard in British English)—no commas are needed to set off the year. For the year-month-day (ISO) date style, see 9.36.
The accused gradually came to accept the verdict. (See his journal entries of 6 October 2015 and 4 January 2017.) 6.39 Commas with addresses. Commas are used to set off the individual el ements in addresses or place-names that are run in to the text (see also 6.17). In a mailing address, commas should be used sparingly, mainly to set off the separate lines of the address, but also to separate city and state or province (but not the postal code), apartment numbers, and the like. If in doubt about the accuracy of an address, consult the applicable postal service. (Preferred postal usage will be tailored for use on address labels and may consist of all capital letters and spare punctuation, a style that need not be emulated in regular text and related contexts.) For place- names used adjectivally, see 5.69.
A printout was sent to the author at 743 Olga Drive NE, Ashtabula, OH 44044, on May 2.
Queries can be sent to the author at 123 Main St., Apt. 10, Montreal, QC H3Z 2Y7. Waukegan, Illinois, is not far from the Wisconsin border.
The plane landed in Kampala, Uganda, that evening.
Some institutional names include place-names set off by commas. When such a name appears in the middle of a clause, a second comma is re quired to set off the place-name. See also 6.81.
California State University, Northridge, has an enrollment of... but The University of Wisconsin-Madison has an enrollment of...
Commas with Quotations and Questions 6.40 Commas with quotations. An independent clause quoted in the form of dialogue or from text and introduced with said, replied, asked, wrote, and the like (including variations of such terms) is usually introduced with a comma (but see 6.65,13.16). This traditional usage considers the gram mar and syntax of the quoted material to be separate from the text that introduces it.
Commas 6.41 It was Thoreau who wrote, “One generation abandons the enterprises of another like stranded vessels.” She replied, “I hope you aren’t referring to us.” Commas are required regardless of the position of the explanatory text relative to the quotation (but see 6.125).
“I hope,” she replied, “you aren’t referring to us.” “I hope you aren’t referring to us,” she replied.
If, however, such a quotation is introduced by that, whether, if, or a simi lar conjunction (see 5.200), no comma is normally needed.
Was it Stevenson who said that “the crudest lies are often told in silence”?
He wondered whether “to think is to live.” For the location of a comma in relation to closing quotation marks, see 6.9. For quoted titles and expressions, see 6.41; for questions, see 6.42. For words such as yes and no, see 13.40. For a more detailed discus sion and illustration of the use or omission of commas before and after quoted material, including dialogue, see 13.13-17,13.50-58, and the ex amples throughout chapter 13.
6.41 Commas with quoted or italicized titles and expressions. Titles or ex pressions set off from the surrounding text with quotation marks or ital ics are usually treated like noun forms; commas are used or omitted as they would be with any other noun.
His favorite story in Joyce’s Dubliners is “Counterparts.” She recites the poem “One Art” every night before bed.
Of her many favorites, “One Art” is the one she knows best.
A common mistake is to use a comma before a title or expression when ever it follows a noun that describes it (e.g., story, novel, or poem). In fact, the rule for appositives applies: the title or expression is set off by com mas only if it is nonrestrictive—that is, if it can be omitted without ob scuring the identity of the noun (i.e., story, novel, etc.) to which it refers (see 6.28). In the first example below, the quoted or italicized titles iden tify which poem by Bishop (she wrote many) and which novel by Weiner (she has published more than one); in the third example, the quoted words tell us which motto appears over the door. In the second and fourth examples, which story (the last one in the collection) and which proverb (Tom’s favorite) have already been identified.
6.42 PUNCTUATION 6.43 Elizabeth Bishop’s poem “One Art” was featured in Jennifer Weiner’s novel In Her Shoes and read by Cameron Diaz in the movie adaptation of the book.
In the collection’s last story, “Negocios,” Junot Diaz gives us a portrait of Papi. The motto “All for one and one for all” appears over the door.
Tom’s favorite proverb, “A rolling stone gathers no moss,” proved wrong.
For quotations marks versus italics for the titles of works, see 8.163. See also 7.62.
6.42 Commas with questions. A direct question is sometimes included within a sentence but not enclosed in quotation marks. Such a question is usually introduced by a comma (unless it comes at the beginning of a sentence) and begins with a capital letter. This slight departure from ear lier editions of the manual recognizes that such a question is analogous to (and can be treated like) a direct quotation (see 6.40; see also 6.65).
She wondered, What am I doing?
Legislators had to be asking themselves, Can the fund be used for the current emergency, or must it remain dedicated to its original purpose?
If the question ends before the end of the sentence, no comma is re quired after the question mark (see also 6.125).
What am I doing? she wondered.
If the result seems awkward, rephrase as an indirect question. An indi rect question does not require a question mark, nor does it need to be set off with a comma. Indirect questions are never capitalized (except at the beginning of a sentence). See also 6.69.
She wondered what she was doing.
The question of how to tell her was on everyone’s mind. Ursula wondered why her watch had stopped ticking. Where to find a reliable clock is the question of the hour.
Commas in Personal and Corporate Names Commas with "Jr.," "Sr.," and the like. Commas are not required with Jr. and Sr., and they are never used to set off II, III, and the like when these are used as part of a name. In an inverted name, however (as in an index; see 16.41), a comma is required before such an element, which comes last.
Commas 6.46 John Doe Sr. continues to cast a shadow over his son.
Jason Deer III has turned over stewardship of the family business to his cousin. but Doe, John, Sr.
Deer, Jason, III If a comma is used to set off Jr. or Sr., a second comma is normally re quired in the middle of a sentence (see 6.17); rephrase as needed to avoid the possessive.
the speech made by John Doe, Sr. (not John Doe, Sr.’s, speech) 6.44 Commas with "lnc.r" "Ltd.," and the like. Commas are not required with Inc., Ltd., and such as part of a company’s name. A particular com pany may use such commas in its corporate documentation; articles and books about such companies, however, should generally opt for a consis tent style rather than make exceptions for particular cases.
QuartzMove Inc. was just one such company named in the suit.
If a comma is used to set off the abbreviation, a second comma is nor mally required in the middle of a sentence (see 6.17); rephrase as needed to avoid the possessive. See also 6.43.
Commas with Antithetical Elements 6.45 Commas with "not" phrases. When a phrase beginning with not is inter jected in order to clarify a particular noun, commas should be used to set off the phrase. See also 6.46.
We hoped the mayor herself, not her assistant, would attend the meeting. They want you, not him.
6.46 Commas with "not... but," "not only... but also," and the like. With an interjected phrase of the type not... but or not only... but also, commas are usually unnecessary.
Works of art are created not by inspiration but by persistence.
Being almost perfectly ambidextrous, she wore not one watch but two.
They marched to Washington not only armed with petitions and determined to get their senators’ attention but also hoping to demonstrate their solidarity with one another.
6.47 PUNCTUATION If, however, such a phrase seems to require special emphasis or clarifica tion (usually a matter of editorial judgment), commas may be used to set off the not phrase. Alternatively, a dash may be used in place of the first comma, in which case a second comma usually becomes unnecessary.
She was in the habit of placing her orders months ahead of the competition—not only as a matter of personal pride but also to bolster her credibility as an early adopter.
6.47 Commas with "the more," "the less," and so on. A comma is customar ily used between clauses of the more. ..the more type. Shorter phrases of that type, however, rarely merit commas.
The more I discover about the workings of mechanical movements, the less I seem to care about the holy grail of perfectly accurate timekeeping.
but The more the merrier.
Other Uses of the Comma 6.48 Commas with parenthetical elements. If only a slight break is intended, commas may be used to set off a parenthetical element inserted into a sentence as an explanation or comment. Such elements are occasionally awkward, especially if they are inserted between an adjective and the noun it modifies; in such cases, rewording may help.
All the test participants, in spite of our initial fears, recovered.
The Hooligan Report was, to say the least, a bombshell.
Most children fail to consider the history behind new technologies, if they think of it at all.
She was the fastest, not to mention the strongest, runner on her team. or, better, She was the fastest runner on her team, not to mention the strongest.
If a stronger break is needed or if there are commas within the paren thetical element, em dashes (6.85) or parentheses (6.95) should be used instead of commas.
6.49 Commas with "however," "therefore," "indeed," and the like. Commas— sometimes paired with semicolons (see 6.57)—are traditionally used to set off conjunctive adverbs such as however, therefore, and indeed. When Commas 6.51 the adverb is essential to the meaning of the clause, or if the emphasis is on the adverb itself, commas are usually unnecessary (as in the last two examples).
A truly efficient gasoline-powered engine remains, however, a pipe dream. Indeed, not one test subject accurately predicted the amount of soup in the bowl. but If you cheat and are therefore disqualified, you may also risk losing your schol arship.
That was indeed the outcome of the study.
6.50 Commas with "such as" and "including." The principles delineated in 6.29 apply also to phrases introduced by such as or including. Nonrestric tive phrases introduced by these terms are set off by commas (because they are not essential to the meaning or identity of the noun they mod ify). When such phrases are restrictive (i.e., essential to the meaning or identity of the noun), commas are not used.
The entire band, including the matutinal lead singer, overslept the noon rehearsal. Some words, such as matutinal and onomatopoetic, are best avoided in everyday speech.
but Words such as matutinal and onomatopoetic are best avoided in everyday speech.
6.51 Commas with "that is," "namely," "for example," and the like. Expres sions of the type that is are traditionally followed by a comma. They are best preceded by an em dash or a semicolon rather than a comma, or the entire phrase they introduce may be enclosed in parentheses or em dashes.
There are simple alternatives to the stigmatized plastic shopping bag—namely, reusable cloth bags and foldable carts.
The committee (that is, its more influential members) wanted to drop the matter. Keesler managed to change the subject; that is, he introduced a tangential issue. Bones from various small animals—for example, a squirrel, a cat, a pigeon, and a muskrat—were found in the doctor’s cabinet.
When or is used in a sense analogous to that is (to mean “in other words”), the phrase it introduces is usually set off by commas.
The compass stand, or binnacle, must be situated within the helmsman’s field of vision.
6.52 PUNCTUATION Note that in formal writing, Chicago prefers to confine the abbreviations i.e. (“that is”) and e.g. (“for example”) to parentheses or notes, where they are followed by a comma.
The most noticeable difference between male and female ginkgo trees (i.e., the presence of berries in the latter) is also the species’ most controversial feature.
6.52 Commas with "too" and "either." The adverbs too and either used in the sense of “also” generally need not be preceded by a comma.
I had my cake and ate it too.
Anders likes Beethoven; his sister does too.
The airport lacked charging stations; there were no comfortable chairs either.
When too comes in the middle of the sentence or clause, however, a comma aids comprehension.
She, too, decided against the early showing.
See also 6.31.
6.53 Commas with direct address. A comma is used to set off names or words used in direct address.
Ms. Jones, please come in.
James, your order is ready.
Hello, Ms. Philips.
Hi, Pratchi. Please sit down.
Take that, you devil.
Kiss me, you fool!
Are you listening, class?
It’s time to go, Marta.
I am not here, my friends, to discuss personalities.
In correspondence, a comma typically follows the greeting, though a co lon may be used instead (especially in formal correspondence; see 6.66).
Dear Lucien,...
If the greeting itself consists of a direct address, two marks of punctua tion are needed (i.e., the comma in the direct address and the colon or comma following the greeting). (The first mark is often left out in casual correspondence.) Semicolons 6.57 Greetings, Board Members:...
Hi, Karel,...
6.54 Commas to indicate elision. A comma is often used to indicate the omis sion of a word or words readily understood from the context.
In Illinois there are seventeen such schools; in Ohio, twenty; in Indiana, thirteen. Thousands rushed to serve him in victory; in defeat, none.
The comma may be omitted if the elliptical construction is clear with out it.
One student excels at composition, another at mathematics, and the third at sports.
Jasper missed her and she him.
6.55 Commas between homonyms. For ease of reading and subject to edito rial discretion, two words that are spelled alike but have different func tions may be separated by a comma if such clarification seems desirable.
Let us march in, in twos.
Whatever is, is good.
but “It depends on what means means.” Semicolons 6.56 Use of the semicolon. In regular prose, a semicolon is most commonly used between two independent clauses not joined by a conjunction to signal a closer connection between them than a period would. (For the similar use of a colon, see 6.61.) She spent much of her free time immersed in the ocean; no mere water-resistant watch would do.
Though a gifted writer, Miqueas has never bothered to master the semicolon; he insists that half a colon is no colon at all.
For the use of the semicolon in index entries, see 16.96,16.17. For its use in parenthetical text citations, see 15.30. For its use with a second sub title of a work, see 14.90.
6.57 Semicolons with "however," "therefore," "indeed," and the like. Cer tain adverbs, when they are used to join two independent clauses, should 6.58 PUNCTUATION be preceded by a semicolon rather than a comma. These conjunctive adverbs include however, thus, hence, indeed, accordingly, besides, and therefore (see also 6.58). A comma usually follows the adverb but may be omitted if the sentence seems just as effective without it (see also 6.31).
The accuracy of Jesse’s watch was never in question; besides, he was an expert at intuiting the time of day from the position of the sun and stars.
Kallista was determined not to miss anything on her voyage; accordingly, she made an appointment with her ophthalmologist.
The trumpet player developed a painful cold sore; therefore plans for a third show were scrapped.
The adverb then is often seen between independent clauses as shorthand for and then, preceded by a comma. This usage is perfectly acceptable, and it is more or less obligatory in the imperative (as in the first example below); some writers, however, may prefer to use a semicolon, which is strictly correct.
Touch and hold the icon, then drag it to the trash.
First we went out for shiitake burgers, then we enjoyed vegan sundaes.
or First we went out for shiitake burgers; then we enjoyed vegan sundaes. but First we went out for shiitake burgers, and then we enjoyed vegan sundaes.
See also 6.22, 6.23.
6.58 Semicolons with "that is," "for example," "namely," and the like. A semicolon may be used before an expression such as that is, for example, or namely when it introduces an independent clause. For an example, see 6.51. See also 6.57.
6.59 Semicolons before a conjunction. Normally, an independent clause introduced by a coordinating conjunction is preceded by a comma (see 6.22). In formal prose, a semicolon may be used instead—either to effect a stronger separation between clauses or when the second independent clause has internal punctuation. Another option is to use a period instead of a semicolon; see 5.203.
Frobisher had always assured his grandson that the house would be his; yet there was no provision for this bequest in his will.
Garrett had insisted on remixing the track; but the engineer’s demands for over Colons 6.61 time pay, together with the band’s reluctance, persuaded him to accept the original mix.
or Garrett had insisted on remixing the track. But the engineer’s demands...
6.60 Semicolons in a complex series. When items in a series themselves con tain internal punctuation, separating the items with semicolons can aid clarity. If ambiguity seems unlikely, commas may be used instead (see 6.19) . See also 6.129. Note that when a sentence continues beyond a se ries (as in the third example), no additional semicolon is required.
The membership of the international commission was as follows: France, 4; Ger many, 5; Great Britain, 1; Italy, 3; United States, 7.
The defendant, in an attempt to mitigate his sentence, pleaded that he had re cently, on doctor’s orders, gone off his medications; that his car—which, in cidentally, he had won in the late 1970s on Let’s Make a Deal—had sponta neously caught fire; and that he had not eaten for several days.
Marilynn, Sunita, and Jared, research assistants; Carlos, programming consul tant; and Carol, audiovisual editor, provided support and prepared these materials for publication.
but She decided to buy three watches—an atomic watch for travel within the United States, a solar-powered, water-resistant quartz for international travel, and an expensive self-winding model for special occasions.
Colons 6.61 Use of the colon. A colon introduces an element or a series of elements illustrating or amplifying what has preceded the colon. Between inde pendent clauses it functions much like a semicolon (see 6.56), and in some cases either mark may work as well as the other; use a colon spar ingly, however, and only to emphasize that the second clause illustrates or amplifies the first. (The colon usually conveys or reinforces the sense of “as follows”; see also 6.64.) The colon may sometimes be used instead of a period to introduce a series of related sentences (as in the third ex ample below).
The watch came with a choice of three bands: stainless steel, plastic, or leather. They even relied on a chronological analogy: just as the Year II had overshadowed 1789, so the October Revolution had eclipsed that of February.
Yolanda faced a conundrum: She could finish the soup, pretending not to care that what she had thought until a moment ago was a vegetable broth was in 6.62 PUNCTUATION fact made from chicken. She could feign satiety and thank the host for a good meal. Or she could use this opportunity to assert her preference for a vegan diet.
For use of the em dash instead of a colon, see 6.85. For colons in ratios, see 9.58. For the use of colons with subtitles, see 14.89. For the use of colons in indexes, see 16.95. For other uses of the colon—in source cita tions, URLs, mathematical expressions, and other settings—consult the index or search the online edition of this manual.
6.62 Space after colon. In typeset matter, no more than one space should fol low a colon. Further, in some settings—as in a source citation between a volume and page number with no intervening date or issue number (see 14.116,14.177), a biblical citation (see 14.239), or a ratio (see 9.58)—no space should follow a colon. See also 6.7.
6.63 Lowercase or capital letter after a colon. When a colon is used within a sentence, as in the first two examples in 6.61, the first word following the colon is lowercased unless it is a proper noun. When a colon introduces two or more sentences (as in the third example in 6.61) or when it intro duces speech in dialogue or a quotation or question (see 6.65), the first word following it is capitalized.
6.64 Colons with "as follows" and other introductory phrases. A colon is normally used after as follows, the following, and similar expressions. (For lists, see 6.127-32.) The steps are as follows: first, make grooves for the seeds; second, sprinkle the seeds; third, push the earth back over the grooves; fourth, water generously.
Kenzie’s results yield the following hypotheses: First,... Second,... Third,...
On the other hand, a colon is not normally used after namely, for example, and similar expressions; these are usually followed by a comma instead (see 6.51).
6.65 Colons to introduce quotations or questions. A colon is often used to introduce speech in dialogue.
Michael: The incident has already been reported.
Timothy: Then, sir, all is lost!
A colon may also be used to introduce a quotation or a direct but un quoted question, especially where the introduction constitutes a gram matically complete sentence.
Question Marks 6.68 The author begins by challenging nature itself: “The trees were tall, but I was taller.” The question occurred to her at once: What if I can’t do this?
For quotations or questions introduced with said, replied, asked, wrote, and the like, where a comma is normally used (see 6.40, 6.42), a colon may be used occasionally for emphasis or to set up a block quotation. See also 13.13-17.
6.66 Colons in formal communication. At the beginning of a speech or a for mal communication, a colon usually follows the identification of those addressed. For use of a comma in direct address, see 6.53.
Ladies and Gentlemen: Dear Credit and Collections Manager: To Whom It May Concern: 6.67 Some common misuses of colons. Many writers assume—wrongly— that a colon is always needed before a series or a list. In fact, if a colon intervenes in what would otherwise constitute a grammatical sentence—even if the introduction appears on a separate line, as in a list (see 6.127-32)—there is a good chance it is being used inappropriately. A colon, for example, should not be used before a series that serves as the object of a verb. When in doubt, apply this test: to merit a colon, the words that introduce a series or list must themselves constitute a gram matically complete sentence.
The menagerie included cats, pigeons, newts, and deer ticks.
not The menagerie included: cats, pigeons, newts, and deer ticks.
An exception may be made when a word or phrase introduces a series or list and the verb is elided or otherwise understood. In such cases a colon is usually required.
Pros: accuracy and water resistance. Cons: cheap-looking exterior,... (The pros included accuracy and water resistance. Among its cons were a cheap-looking exterior,...) Question Marks 6.68 Use of the question mark. The question mark, as its name suggests, is used to indicate a direct question. It may also be used to indicate 6.69 PUNCTUATION editorial doubt or (occasionally) to express surprise, disbelief, or uncer tainty at the end of a declarative or imperative sentence. See also 6.72, 6.122, 6.124,14.132.
Who will represent the poor?
Thomas Kraftig (1610 ?-66) was the subject of the final essay.
This is your reply?
6.69 Direct and indirect questions. A question mark is used to mark the end of a direct but unquoted question within a sentence. This usage is no dif ferent from that of a directly quoted question (see 6.125). See also 6.42.
Is it worth the risk? he wondered.
An indirect question never takes a question mark.
He wondered whether it was worth the risk.
How the two could be reconciled was the question on everyone’s mind.
When a question within a sentence consists of a single word, such as who, when, how, or why, a question mark may be omitted, and the word is sometimes italicized.
She asked herself why.
The question was no longer how but when.
A polite request disguised as a question does not always require a ques tion mark. Such formulations can usually be reduced to the imperative.
Will the audience please rise.
Would you kindly respond by March 1.
or Please respond by March 1.
but Would you mind telling me your age?
6.70 Question marks in relation to surrounding text and punctuation. A question mark should be placed inside quotation marks, parentheses, or brackets only when it is part of (i.e., applies to) the quoted or parentheti cal matter. See also 6.10, 6.125.
The ambassador asked, “Has the Marine Corps been alerted?” “Is it worth the risk?” he asked.
Exclamation Points 6.74 Why was Farragut trembling when he said, “I’m here to open an inquiry”? The man in the gray flannel suit (had we met before?) winked at me. Why did she tell him only on the morning of his departure (March 18)? “What do you suppose he had in mind,” inquired Newman, “when he said, ‘You are all greater fools than I thought’?” Exclamation Points 6.71 Use of the exclamation point. An exclamation point (which should be used sparingly to be effective) marks an outcry or an emphatic or ironic comment. See also 6.122, 6.124.
Heads up! According to one model, Miami will remain above sea level until at least 2100. We should all be so lucky!
6.72 Exclamation rather than question. A sentence in the form of a direct question may be marked as rhetorical by the use of an exclamation point in place of a question mark (see also 6.126).
How could you possibly believe that! When will I ever learn!
6.73 Exclamation point as editorial protest or amusement. Writers and ed itors should be aware that an exclamation point added in brackets to quoted matter to indicate editorial protest or amusement risks being interpreted as contemptuous or arrogant. Unless such a sentiment is in tended, this device should be avoided. Nor is it a substitute for the Latin expression sic (thus), which should be reserved to indicate an error in the source that might otherwise be taken as an error of transcription (see 13.61) .
6.74 Exclamation points in relation to surrounding punctuation. An excla mation point should be placed inside quotation marks, parentheses, or brackets only when it is part of the quoted or parenthetical matter. See also 6.10, 6.125.
The performer walked off the stage amid cries of “Brava!” She actually believes the seller’s claim that the MP3 sounds “as good as the un compressed original”!
Alex Ramirez (I could have had a stroke!) repeated the whole story.
6.75 PUNCTUATION Hyphens and Dashes 6.75 Hyphens and dashes compared. Hyphens and the various dashes all have their specific appearance (shown below) and uses (discussed in the following paragraphs). The hyphen, the en dash, and the em dash are the most commonly used. Though the differences can sometimes be subtle—especially in the case of an en dash versus a hyphen—correct use of the different types is a sign of editorial precision and care. See also 2.13, 2.14, 2.96.
hyphen - en dash - em dash — 2-em dash---- 3-em dash-------Hyphens 6.76 Hyphens in compound words. The use of the hyphen in compound words and names and in word division is discussed in 5.92 and in chapter 7, especially 7.36-47 and 7.81-89. See also 6.80.
6.77 Hyphens as separators. A hyphen is used to separate numbers that are not inclusive, such as telephone numbers (see 9.57), social security num bers, and ISBNs. (For hyphens with dates, see 9.36.) It is also used to sep arate letters when a word is spelled out letter by letter, as in dialogue or in reference to American Sign Language (see 11.125-35).
1-800-621-2376 978-0-226-15906-5 (ISBN) “My name is Phyllis; that’s p-h-y-l-l-i-s.” A proficient signer can fingerspell c -o -l -o -r -a -d -o in less than two seconds.
For hyphens in URLs and email addresses, see 7.46.
En Dashes 6.78 En dash as "to." The principal use of the en dash is to connect numbers and, less often, words. With continuing numbers—such as dates, times, and page numbers—it signifies up to and including (or through). For the sake of parallel construction, the word to or through (or until), never the en dash, should be used if the word from precedes the first element in such a pair; similarly, and should be used if between precedes the first element.
Hyphens and Dashes 6.80 The years 1993-2000 were heady ones for the computer literate.
For source citations and indexing, see chapters 14-16. In Genesis 6:13-21 we find God’s instructions to Noah. Join us on Thursday, 11:30 a.m.-4:00 p.m., to celebrate the New Year. I have blocked out December 2016-March 2017 to complete my manuscript. Her articles appeared in Postwar Journal (3 November 1945-4 February 1946). but She was in college from 2012 to 2016 {not from 2012-16).
He usually naps between 11:30 a.m. and 1:30 p.m. {not between 11:30 a.m.-l:30 p.m.) In other contexts, such as with scores and directions, the en dash signi fies, more simply, to.
The London-Paris train leaves at two o’clock.
On November 20,1966, Green Bay defeated Chicago, 13-6. The legislature voted 101-13 to adopt the resolution.
For more on dates and times, see 9.29-36,9.37-40. For more on number ranges, see 9.60-64. See also 6.107.
6.79 En dash with an unfinished number range. An en dash may be used to indicate a number range that is ongoing—for example, to indicate the dates of a serial publication or to give the birth date of a living person. No space intervenes between the en dash and any mark of punctuation that follows.
The History of Cartography (1987-) is a multivolume work published by Chicago. Jack Stag (1950-) or Jack Stag (b. 1950) 6.80 En dashes with compound adjectives. The en dash can be used in place of a hyphen in a compound adjective when one of its elements consists of an open compound or when both elements consist of hyphenated com pounds (see 7.82). Whereas a hyphen joins exactly two words, the en dash is intended to signal a link across more than two. Because this editorial nicety will almost certainly go unnoticed by the majority of readers, it should be used sparingly, when a more elegant solution is unavailable. As the first two examples illustrate, the distinction is most helpful with proper compounds, whose limits are made clear within the larger context by capitalization. The relationship in the third example depends to some small degree on an en dash that many readers will perceive as a hyphen connecting music and influenced. The relationships in the fourth example are less awkwardly conveyed with a comma.
6.81 PUNCTUATION the post-World War II years Chuck Berry-style lyrics country music-influenced lyrics (or lyrics influenced by country music) a quasi-public-quasi-judicial body (or, better, a quasi-public, quasi-judicial body) A single word or prefix should be joined to a hyphenated compound by another hyphen rather than an en dash; if the result is awkward, reword.
non-English-speaking peoples a two-thirds-full cup (or, better, a cup that is two-thirds full) An abbreviated compound is treated as a single word, so a hyphen, not an en dash, is used in such phrases as “US-Canadian relations” (Chicago’s sense of the en dash does not extend to between).
6.81 En dashes with campus locations. Some universities that have more than one campus use an en dash to link the campus location to the name of the university. Usage varies widely; when in doubt, follow the stated preference of the institution. See also 6.39.
the University of Wisconsin-Madison the University of Wisconsin-Milwaukee but the University of California, San Diego the State University of New York at Buffalo the University of Massachusetts Amherst; UMass Amherst 6.82 En dashes and line breaks. In printed publications, line breaks should generally be made after an en dash but not before, in the manner of hy phens. If possible, avoid carrying over a single character to the next line, as in a number range or score. In reflowable electronic formats, it is usu ally best to let the software determine such breaks. See also 7.36-47.
6.83 En dash as em dash. In British usage, an en dash (with space before and after) is usually preferred to the em dash as punctuation in running text, a practice that is followed by some non-British publications as well (see 6.85).
6.84 En dash as minus sign. The en dash is sometimes used as a minus sign, but minus signs and en dashes are distinct characters (defined by the Unicode standard as U+2212 and U+2013, respectively; see 11.2,12.9). Both the characters themselves and the spacing around them may differ; moreover, substituting any character for another may hinder searches in Hyphens and Dashes 6.87 electronic publications. Thus it is best to use the correct character, espe cially in mathematical copy.
Em Dashes 6.85 Em dashes instead of commas, parentheses, or colons. The em dash, often simply called the dash, is the most commonly used and most ver satile of the dashes. (In British usage, spaced en dashes are used in place of em dashes; see 6.83.) Em dashes are used to set off an amplifying or explanatory element and in that sense can function as an alternative to parentheses (second and third examples), commas (fourth and fifth ex amples), or a colon (first example)—especially when an abrupt break in thought is called for.
It was a revival of the most potent image in modern democracy—the revolution ary idea.
The influence of three impressionists—Monet, Sisley, and Degas—is obvious in her work.
The chancellor—he had been awake half the night—came down in an angry mood. She outlined the strategy—a strategy that would, she hoped, secure the peace. My friends—that is, my former friends—ganged up on me.
To avoid confusion, the em dash should never be used within or imme diately following another element set off by an em dash (or pair of em dashes). Use parentheses or commas instead.
The Whipplesworth conference—which had already been interrupted by three demonstrations (the last bordering on violence)—was adjourned promptly.
or The Whipplesworth conference—which had already been interrupted by three demonstrations, the last bordering on violence—was adjourned promptly.
6.86 Em dash between noun and pronoun. An em dash is occasionally used to set off an introductory noun, or a series of nouns, from a pronoun that refers back to the noun or nouns and introduces the main clause.
Consensus—that was the will-o’-the-wisp he doggedly pursued.
Broken promises, petty rivalries, and false rumors—such were the obstacles she encountered.
6.87 Em dashes for sudden breaks or interruptions. An em dash or a pair of em dashes may indicate a sudden break in thought or sentence structure 6.88 PUNCTUATION or an interruption in dialogue. (Where a faltering rather than sudden break is intended, an ellipsis may be used; see 6.15.) “Will he—can he—obtain the necessary signatures?” asked Mill.
“Well, I don’t know,” I began tentatively. “I thought I might—” “Might what?” she demanded.
If the break belongs to the surrounding sentence rather than to the quoted material, the em dashes must appear outside the quotation marks.
“Someday he’s going to hit one of those long shots, and”—his voice turned huffy—“I won’t be there to see it.” 6.88 Em dashes with "that is," "namely," "for example," and similar ex pressions. An em dash may be used before expressions such as that is or namely. For examples, see 6.51; see also 6.58.
6.89 Em dashes with other punctuation. In modern usage, a question mark or an exclamation point—but never a comma, a colon, or a semicolon may precede an em dash. A period may precede an em dash if it is part of an abbreviation (see also 14.51).
Without further warning—but what could we have done to dissuade her?—she left the plant, determined to stop the union in its tracks.
Only if—heaven forbid!—you lose your passport should you call home. No one—at least not before 11:42 p.m.—could have predicted the outcome.
If the context calls for an em dash where a comma would ordinarily separate a dependent clause from an independent clause, the comma is omitted. Likewise, if an em dash is used at the end of quoted material to indicate an interruption, the comma can be safely omitted before the words that identify the speaker (see also 6.125).
Because the data had not been fully analyzed—let alone collated—the publication of the report was delayed.
“I assure you, we shall never—” Sylvia began, but Mark cut her short.
6.90 Em dashes and line breaks. In printed publications, line breaks should generally be made after an em dash but not before, in the manner of hyphens. In the case of a closing quotation mark (or any other mark of punctuation) immediately following the dash, however, the quotation mark and dash must not be broken at the end of a line (see also 6.87, Hyphens and Dashes 6.94 6.89). In reflowable electronic formats, it is usually best to let the soft ware determine such breaks. See also 7.36-47.
6.91 Em dashes in lieu of quotation marks. Em dashes are occasionally used instead of quotation marks to set off dialogue (a la writers in some Euro pean languages). Each speech starts a new paragraph. No space follows the dash.
—Will he obtain the necessary signatures?
—Of course he will!
6.92 Em dashes in lists, indexes, and tables. In informal settings, em dashes are sometimes used in the manner of bullet points in a vertical list (see 6.130). Such usage is best avoided in formal prose, though em dashes may sometimes be used in a similar manner to organize subentries in an index (see 16.27). In tables, an em dash may be used for an otherwise blank or empty data cell (see 3.67).
2-Em and 3-Em Dashes 6.93 2-em dash. A 2-em dash represents a missing word or part of a word, either omitted to disguise a name (or occasionally an expletive) or else missing from or illegible in quoted or reprinted material. When a whole word is missing, space appears on both sides of the dash. When only part of a word is missing, no space appears between the dash and the existing part (or parts) of the word; when the dash represents the end of a word, a space follows it (unless a period or other punctuation immediately fol lows). See also 7.66,13.59.
“The region gives its---- to the language spoken there.” Admiral N---- and Lady R-----were among the guests.
David H---- h [Hirsch?] voted aye.
Although a 2-em dash sometimes represents material to be supplied, it should not be confused with a blank line to be filled in, which should normally appear as an underscore (e.g.,).
6.94 3-em dash. In a bibliography, a 3-em dash followed by a period rep resents the same author(s) or editor(s) named in the preceding entry. Such usage, because it can obscure important information, is best ap plied by the publisher or manuscript editor rather than by the author. See also 14.67-71,15.17-20.
6.95 PUNCTUATION Chaudhuri, Amit. Odysseus Abroad. New York: Alfred A. Knopf, 2015.
------ . A Strange and Sublime Address. London: Minerva, 1992.
Parentheses 6.95 Use of parentheses. Parentheses—stronger than a comma and similar to the dash—are used to set off material from the surrounding text. Like dashes but unlike commas, parentheses can set off text that has no gram matical relationship to the rest of the sentence.
He suspected that the noble gases (helium, neon, etc.) could produce a similar effect.
Intelligence tests (e.g., the Stanford-Binet) are no longer widely used.
Our final sample (collected under difficult conditions) contained an impurity. Wexford’s analysis (see chapter 3) is more to the point.
Dichtung und Wahrheit (also known as Wahrheit und Dichtung) has been trans lated as Poetry and Truth (or, as at least one edition has it, Truth and Fiction).
The disagreement between Johns and Evans (its origins have been discussed else where) ultimately destroyed the organization.
For the use of parentheses as delimiters for letters or numbers in a list or outline, see 6.129, 6.132. For parenthetical references to a list of works cited, see 15.21-31. For parenthetical references following quoted ma terial, see 13.64-72. For parentheses in notes and bibliographies, see chapter 14. For parentheses in mathematics, see chapter 12, especially 12.26-35. For roman versus italic type, see 6.5.
6.96 Parentheses for glosses or translations. Parentheses are used to enclose glosses of unfamiliar terms or translations from other languages—or, if the term is given in English, to enclose the original word. In quoted mat ter, brackets should be used (see 6.99). See also 7.53,11.9.
A drop folio (a page number printed at the foot of a page) is useful on the opening page of a chapter.
The term you should use for 1,000,000,000 is mil millones (billion), not billon (trillion).
German has two terms for eating—one for the way humans eat (essen) and an other for the way animals eat (fressen).
6.97 Parentheses within parentheses. Although the use of parentheses within parentheses (usually for bibliographic purposes) is permitted in Brackets and Braces 6.99 some publications—especially in law—Chicago prefers brackets within parentheses (see 6.101). (British style is to use parentheses within paren theses.) For parentheses in mathematics, see 12.26.
6.98 Parentheses with other punctuation. An opening parenthesis should be preceded by a comma or a semicolon only in an enumeration (see 6.129); a closing parenthesis should never be preceded by a comma, a semicolon, or a colon. A question mark, an exclamation point, and clos ing quotation marks precede a closing parenthesis if they belong to the parenthetical matter; they follow it if they belong to the surrounding sen tence. A period precedes the closing parenthesis if the entire sentence is in parentheses; otherwise it follows. (Avoid enclosing more than one sentence within another sentence; see 6.13.) Parentheses may appear back to back (with a space in between) if they enclose entirely unrelated material; sometimes, however, such material can be enclosed in a single set of parentheses, usually separated by a semicolon. See also table 6.1. For parentheses in documentation, see chapters 14 and 15.
Having entered (on tiptoe), we sat down on the nearest seats we could find. Come on in (quietly, please!) and take a seat.
If parenthesis is Greek for the act of inserting, is it redundant to insert something in parentheses (i.e., in English)?
On display were the watchmakers’ five latest creations (all of which Sheilahan coveted).
Five new watches were on display. (Shellahan fancied the battery-powered quartz model.) Strabo is probably referring to instruction (SidaoKaXia) (Jones et al. 2017).
Brackets and Braces 6.99 Use of square brackets. Square brackets (often simply called brackets) are used mainly to enclose material—usually added by someone other than the original writer—that does not form a part of the surrounding text. Specifically, in quoted matter, reprints, anthologies, and other non original material, brackets enclose editorial interpolations, explanations, translations of terms from other languages, or corrections. Sometimes the bracketed material replaces rather than amplifies the original word or words. For brackets in mathematical copy, see 12.26. See also 13.59-63.
“They [the free-silver Democrats] asserted that the ratio could be maintained.” “Many CF [cystic fibrosis] patients have been helped by the new therapy.” 6.100 PUNCTUATION Satire, Jebb tells us, “is the only [form] that has a continuous development.” [This was written before the discovery of the Driscoll manuscript.—Ed.] If quoted matter already includes brackets of its own, the editor should so state in the source citation (e.g., “brackets in the original”); see 13.62 for an analogous situation with italics.
6.100 Square brackets in translated text. In a translated work, square brackets are sometimes used to enclose a word or phrase in the original language. (Translators should use this device sparingly.) If quoted matter already includes brackets of its own, the editor should so state in a note or else where (see also 6.99).
The differences between society [Gesellschaft] and community [Gemeinschaft] will now be analyzed.
6.101 Square brackets for parentheses within parentheses. Chicago pre fers square brackets as parentheses within parentheses, usually for bib liographic purposes. For mathematical groupings, see 12.26.
(For further discussion see Richardson’s excellent analysis and Danne- berger’s survey .) 6.102 Square brackets in phonetics. Square brackets may be used to enclose a phonetic transcription.
The verb entretenir [atratnir], like keep, is used in many idioms.
6.103 Square brackets with other punctuation. For brackets with other punc tuation, most of the same principles apply as for parentheses (see 6.98). For their use in enclosing editorial interpolations, however, the appear ance of other punctuation and its position relative to the brackets may depend on the source. In the first example, the comma after “Dear Ja cob” is part of the missing greeting that the editor is interpolating. In most cases, however, material added in brackets should be treated as if it were in parentheses. See also 14.145.
The original letter, the transcription of which was incomplete, probably read as follows: “ [Dear Jacob,] It’s been seventy years since I last set eyes on you [...] ” The report was unambiguous: “The scholars fled Ithaca [New York] and drove south.” not The report was unambiguous: “The scholars fled Ithaca[, New York,] and drove south.” Slashes 6.10 6 6.104 Angle brackets and braces. The term angle brackets is used here to de note the mathematical symbols for less than (<) and greater than (>) paired to work as delimiters (<.. .>). (True mathematical angle brackets, < and >, not readily available in most typefaces, are reserved for mathe matical notation; see, for example, 12.55.) Angle brackets are most often used to enclose tags in XML and related markup languages (see 2.83). By extension, some manuscript editors opt for angle brackets—unlikely to appear elsewhere in a typical word-processed manuscript—to enclose generic instructions for typesetting (see 2.81). Although angle brackets are sometimes used to set off URLs and email addresses (e.g., in mes sage headers in email applications), Chicago discourages this practice for regular prose. Angle brackets are also occasionally used instead of brackets in textual studies to indicate missing or illegible material (see 6.99). Braces, {}, also called curly brackets, provide yet another option for enclosing data and are used in various ways in certain programming languages. They are also used in mathematical and other specialized writing (see, e.g., 12.28). Braces are not interchangeable with parenthe ses or brackets. See the example phrases throughout chapter 5 for one possible use of braces.
Slashes 6.105 Other names for the slash. The slash (/)—also known as virgule, soli dus, slant, or forward slash, to distinguish it from a backward slash, or backslash ()—has various distinct uses. For a discussion of the niceties associated with the various terms, see Richard Eckersley et al., Glossary of Typesetting Terms (bibliog. 2.7).
6.106 Slashes to signify alternatives. A slash most commonly signifies alter natives. In certain contexts it is a convenient (if somewhat informal) shorthand for or. It is also used for alternative spellings or names. Where one or more of the terms separated by slashes is an open compound, a space before and after the slash can make the text more legible.
he/she Hercules/Heracles his/her Margaret/Meg/Maggie and/or World War I / First World War Occasionally a slash can include the sense of and—while still also con veying a sense of alternatives (but see 6.107). (Note that in most cases a hyphen is the better choice for and— e.g., “mother-daughter friend ship?) 6.107 PUNCTUATION an insertion/deletion mutation (a mutation with insertions or deletions or both) an MD/PhD program (a program that offers one or both of these degrees) a Jekyll/Hyde personality (a personality that includes the two alternating traits) 6.107 Slashes with two-year spans. A slash is sometimes used in dates instead of an en dash (see 6.78), or even in combination with an en dash, to indi cate the last part of one year and the first part of the next. See also 9.64.
The winter of 1966/67 was especially severe.
Enrollment has increased between 1998/99 and 2001/2.
The fiscal years 2005/6-2009/10 were encouraging in one or two respects.
6.108 Slashes with dates. Slashes (or periods or hyphens) are used informally in all-numeral dates (e.g., 3/10/02), but this device should be avoided in formal writing and wherever clarity is essential (in the United States the month usually comes first, but elsewhere it is more common for the day to come first). If an all-numeral format must be used, use the ISO standard date format (year, month, day, in the form YYYY-MM-DD; see 9.36).
6.109 Slashes in abbreviations. A slash may stand as shorthand for per, as in “110 km/sec,” “$450/week,” or, in certain abbreviations, in lieu of peri ods, as in “c/o” (in care of) or “n/a” (not applicable; see also 3.67).
6.110 Slashes as fraction bars. A slash can be used to mean “divided by” when a fraction bar is inappropriate or impractical. When available, single glyph fractions may be used (e.g., V2 rather than 1/2). See also 12.45.
6.111 Slashes to show line breaks in quoted poetry. When two or more lines of poetry are quoted in regular text, slashes with space on each side are used to show line breaks. See also 13.29.
“Thou hast not missed one thought that could be fit, / And all that was improper dost omit.” 6.112 Slashes in URLs and other paths. Slashes are used in URLs and other paths to separate directories and file names. Spaces are never used in such contexts. In typeset paths, a line break may occur before a slash but not between two slashes (see 7.46). Some operating systems use back ward slashes (or backslashes, ) or colons rather than, or in addition to, slashes. See also 14.17.
Quotation Marks 6.115 6.113 Slashes and line breaks. In printed publications, where slashes are used to signify alternative terms, a line break should be made after the slash, never before. If possible, avoid carrying over a single character to the next line, as in an expression signifying a two-year span. Fractions should not be broken at the end of a line. If a slash is used to show where a line break occurs in poetry, a break can be made either after or be fore the slash. For URLs, a line break should be made before the slash (see 7.46). In reflowable electronic formats, it is usually best to let the software decide such breaks.
Quotation Marks 6.114 Quotation marks relative to other punctuation and text. For the loca tion of closing quotation marks in relation to other punctuation, see 6.9- 11. For the use of quotation marks with a comma, see 6.40; with a colon, 6.65; with a question mark, 6.70; with an exclamation point, 6.74. For a full discussion of quotation marks with dialogue and quoted matter, see 13.9-10, 13.13-17, 13.30-38. For the use of quotation marks with single words or phrases to signal some special usage, see 7.57, 7.60, 7.63. For quotation marks in French, see 11.29,11.30; in German, 11.41; in Italian, 11.47; in Spanish, 11.63. For quotation marks with titles of certain types of works, see the examples in chapter 8.
6.115 "Smart" quotation marks. Published works should use directional (or “smart”) quotation marks, sometimes called typographer’s or “curly” quotation marks. These marks, which are available in any modern word processor, generally match the surrounding typeface. For a variety of reasons, including the limitations of typewriter-based keyboards and of certain software programs, these marks are often rendered incorrectly. Care must be taken that the proper mark—left or right, as the case may be—has been used in each instance. All software includes a “default” quotation mark (”); in published prose this unidirectional mark, though far more portable than typographer’s marks, signals a lack of typograph ical sophistication. Proper directional characters should also be used for single quotation marks (°). The hexadecimal code points for Unicode are as follows: left double quotation mark (“), U+201C; right double quotation mark (”), U+201D; left single quotation mark C), U+2018; right single quotation mark or apostrophe (’), U+2019 (see 11.2). See also 6.117.
6.116 PUNCTUATION Apostrophes 6.116 Use of the apostrophe. The apostrophe has three main uses: to indicate the possessive case, to stand in for missing letters or numerals, and—in rare instances—to form the plural of certain expressions. For more on the possessive case, see 7.16-29,5.20. For contractions, see 7.30. For plurals, see 7.5-15—especially 7.15.
6.117 "Smart" apostrophes. Published works should use directional (or “smart”) apostrophes. In most typefaces, this mark will appear as a raised (but not inverted) comma. The apostrophe is the same character as the right single, quotation mark (defined for Unicode as U+2019; see 6.115). Owing to the limitations of conventional keyboards and many software programs, the apostrophe continues to be one of the most abused marks in punctuation. There are two common pitfalls: using the “default” uni directional mark ('), on the one hand, and using the left single quotation mark, on the other. The latter usage in particular should always be con strued as an error. Some software programs automatically turn a typed apostrophe at the beginning of a word into a left single quotation mark; authors and editors need to be vigilant in overriding such automation to produce the correct mark, and typesetters need to take care not to in troduce errors of their own. (If necessary, consult your software’s help documentation or special characters menu.) We spent the ’90s (not ‘90s) in thrall to our gadgets.
Where’d you get ’em (not ‘em)?
I love rock ’n’ roll (not rock ‘n’ roll).
6.118 Apostrophes relative to other punctuation. An apostrophe (’) is consid ered part of the word (or number) in which it appears. An apostrophe should not be confused with a single closing quotation mark; when a word ends in an apostrophe, no period or comma should intervene be tween the word and the apostrophe.
The last car in the lot was the Smiths’.
Spaces 6.119 Use of the space. The spaces that occur in running text are not punctua tion per se, but they play a similar if more fundamental role. The primary use of the space is as a separator—for example, between words and sen Multiple Punctuation Marks 6.122 tences (see 6.7) or between a numeral and a unit of measure (see 10.49). Spaces come in different widths (see 6.120), and nonbreaking spaces can be used to control breaks at the end of a line (see 6.121). For other uses of the space, consult the index.
6.120 Spaces with different widths. The regular type of space that is added to a line of text with the Space bar is the one used in almost all contexts (it is the space that occurs between the words and sentences in this paragraph and almost everywhere else in this manual). This type of space varies in width when a line is justified to the left and right margins. In addition, there are a number of spaces with fixed widths, based on the spaces that typesetters have been using for centuries. En spaces and em spaces are wider than the regular space and match the width of en dashes and em dashes in a particular font (see 6.75). Em spaces in particular are some times used as a design element, as between a figure number and caption (see 3.23). A thin space or a hair space may be used between contiguous single and double quotation marks (see 6.11). These fixed-width spaces are usually appropriate only in professionally typeset material intended for print or PDF. Though each is defined for Unicode (see 11.2), they may not be supported by a given device or application and should therefore be used with caution (if at all) in electronic publication formats (cf. 6.121). For tabs and indents, see 2.11.
6.121 Nonbreaking spaces. A regular nonbreaking space, readily available in word processors and page-layout programs, can be used to prevent cer tain elements that contain spaces from breaking over a line. These in clude the spaced ellipses preferred by Chicago (see 13.50) and numerals that use spaces rather than commas as separators (see 9.55, 9.56). Non breaking spaces can also be used to prevent a numeral from being sepa rated from an abbreviated unit of measure (e.g., 11.5 km) or to prevent a break between initials in a name like E. B. White. The nonbreaking space is defined for Unicode (as the “no-break space,” U+00A0; see also 11.2) and is widely supported in electronic publication formats. This space, like the Space bar space, varies in width when a line is justified to the left and right margins. Especially for printed formats, spaces of different fixed widths are sometimes used for specific contexts and defined not to break over the line (see 6.120).
Multiple Punctuation Marks 6.122 Likely combinations for multiple punctuation marks. The use of more than one mark of punctuation at the same location usually involves 6.123 PUNCTUATION quotation marks, em dashes, parentheses, or brackets in combination with periods, commas, colons, semicolons, question marks, or exclama tion points. For quotation marks see 6.9-11,6.40,6.65,6.70,6.74. For em dashes see 6.89, 6.94. For parentheses and brackets see 6.85, 6.97, 6.98, 6.101, 6.103, 6.129. For ellipses see 13.50-58. See also table 6.1.
6.123 Abbreviation-ending periods with other punctuation. When an abbre viation or other expression that ends with a period occurs at the end a sentence, no additional period follows (see 6.14). Of course, when any other mark of punctuation is needed immediately after the period, both the period and the additional mark appear (see also 6.9, 6.13).
The study was funded by Mulvehill & Co.
Johnson et al., in How to Survive, describe such an ordeal.
6.124 Periods with question marks or exclamation points. A period (aside from an abbreviating period; see 6.123) never accompanies a question mark or an exclamation point. The latter two marks, being stronger, take precedence over the period. This principle continues to apply when the question mark or exclamation point is part of the title of a work, as in the final example (cf. 6.125).
Their first question was a hard one: “Who is willing to trade oil for water?” What did she mean when she said, “The foot now wears a different shoe”? She owned two copies of Will You Please Be Quiet, Please?
6.125 Commas with question marks or exclamation points. When a question mark or exclamation point appears at the end of a quotation where a comma would normally appear, the comma is omitted (as in the first ex ample below; see also 6.42). When, however, the title of a work ends in a question mark or exclamation point, a comma should also appear if the grammar of the sentence would normally call for one. This usage recog nizes not only the syntactic independence of titles but also the potential for clearer sentence structure—especially apparent in the final example, where the comma after Help! separates it from the following title. (The occasional awkward result may require rewording.) Compare 6.124. See also 14.96.
“Are you a doctor?” asked Mahmoud.
but “Are You a Doctor?,” the fifth story in Will You Please Be Quiet, Please?, treats mod ern love.
Lists and Outline Style 6.129 All the band’s soundtrack albums—A Hard Day's Night, Help!, Yellow Submarine, and Magical Mystery Tour—were popular.
6.126 Question mark with exclamation point. In the rare case of a question or exclamation ending with a title or quotation that ends in a question mark or exclamation point, include both marks only if they are different and the sentence punctuation seems essential.
Have you seen Help!?
Who shouted, “Long live the king!”?
I just love Who's Afraid of Virginia Woolf?\ but Who starred opposite Richard Burton in Who's Afraid of Virginia Woolf?
Who wrote “Are You a Doctor?” Where were you when you asked, “Why so blue?” The question mark is sometimes used in combination with the excla mation point to express excitement or disbelief, a practice that is best avoided in formal prose; see also 6.72.
Lists and Outline Style 6.127 Lists and outlines—general principles. Items in a list should consist of parallel elements. Unless introductory numerals or letters serve a pur pose—to indicate the order in which tasks should be done, to suggest chronology or relative importance among the items, to facilitate text references, or, in a run-in list, to clearly separate the items—they may be omitted. Where similar lists are fairly close together, consistent treat ment is essential. Note that the advice in this section applies primarily to lists that occur in the text. For lists of illustrations and tables, see 1.39; for lists of abbreviations, see 1.44; for glossaries, see 1.61; for indexes, see chapter 16.
6.128 Run-in versus vertical lists. Lists may be either run in to the text or set vertically. Short, simple lists are usually better run in, especially if the introductory text and the items in the list together form a sentence (see 6.129). Lists that require typographic prominence, that are relatively long, or that contain multiple levels (see 6.132) should be set vertically.
6.129 Run-in lists. If numerals or letters are used to mark the divisions in a run-in list, enclose them in parentheses. If letters are used, they are 6.130 PUNCTUATION sometimes italicized (within roman parentheses; see 6.5). If the intro ductory material forms a grammatically complete sentence, a colon should precede the first parenthesis (see also 6.61,6.64, 6.67). The items are separated by commas unless any of the items requires internal com mas, in which case all the items will usually need to be separated by semicolons (see 6.60). When each item in a list consists of a complete sentence or several sentences, the list is best set vertically (see 6.130).
The qualifications are as follows: a doctorate in physics, five years’ experience in a national laboratory, and an ability to communicate technical matter to a lay audience.
Compose three sentences to illustrate analogous uses of (1) commas, (2) em dashes, and (3) parentheses.
For the duration of the experiment, the dieters were instructed to avoid (a) meat, (b) bottled drinks, (c) packaged foods, and (d) nicotine.
Data are available on three groups of counsel: (1) the public defender of Cook County, (2) the member attorneys of the Chicago Bar Association’s Defense of Prisoners Committee, and (3) all other attorneys.
You are advised to pack the following items: (a) warm, sturdy outer clothing and enough underwear to last ten days; (b) two pairs of boots, two pairs of sneak ers, and plenty of socks; and (c) three durable paperback novels.
6.130 Vertical lists—capitalization, punctuation, and format. A vertical list is best introduced by a grammatically complete sentence, followed by a colon (but see 6.131). There are two basic types of lists: (1) unordered, in which the items are introduced by a bullet or other such marker or by nothing at all, and (2) ordered, in which items are introduced by numbers or letters. If the list is unordered, and unless the items consist of com plete sentences, each item carries no end punctuation and each can usu ally begin lowercase (except for proper nouns). For lists whose items re quire more prominence, capitalization may instead be preferred; choose one approach and follow it consistently. If items run over to one or more new lines, the runover lines are usually assigned a hanging indent (see 2.11). (An alternative to indenting runover lines is to insert extra space between the items.) Your application must include the following documents: a full resume three letters of recommendation all your diplomas, from high school to graduate school a brief essay indicating why you want the position and why you consider yourself qualified for it two forms of identification Lists and Outline Style 6.130 To avoid long, skinny lists, short items may be arranged in two or more columns.
An administrative facility can be judged by eight measures: image quality security functional organization access design efficiency flexibility environmental systems Each of these measures is discussed below.
If the items are numbered (i.e., the list is ordered; see also 6.127), a pe riod follows the numeral. It is customary to capitalize items in a num bered list even if the items do not consist of complete sentences (but see 6.131). Closing punctuation is used only if items consist of complete sen tences. For the use of roman numerals and letters, see 6.132.
Compose three sentences: 1. To illustrate the use of commas in dates 2. To distinguish the use of semicolons from the use of periods 3. To illustrate the use of parentheses within dashes In a numbered list, runover lines are aligned with the first word following the numeral; a tab usually separates the number from the text of the list (see also 2.21).
To change the date display from “31” to “1” on the day following the last day of a thirty-day month, the following steps are recommended: 1. Pull the stem out past the date-setting position to the time-setting position.
2. Make a mental note of the exact minute (but see step 4).
3. Turn the stem repeatedly in a clockwise direction through twenty-four hours.
4. If you are able to consult the correct time, adjust the minute hand accordingly and press the stem all the way in on the exact second. If you are not able to consult the correct time, settle on a minute or so past the time noted in step 2.
Bulleted lists are usually formatted in the same way as numbered lists.
Use the Control Panel to make changes to your computer: • To uninstall or repair a program or to change how it runs, go to Programs, and then choose Programs and Features.
• To adjust the resolution displayed by your monitor, go to Appearance and Per sonalization, and then choose Display. (Lowering the resolution will increase the size of images on the screen.) 6.131 PUNCTUATION • To add a language other than English or to change handwriting options, go to Clock, Language, and Region, and then choose Language.
If none of the items in a bulleted list consist of complete sentences, how ever, each item can usually begin lowercase (except for proper nouns). For bulleted lists whose items require more prominence, capitalization may instead be preferred (as throughout this manual); choose one ap proach and follow it consistently.
The style sheet allows for two types of lists: • ordered lists, marked with numbers or letters • unordered lists, marked by bullets or other ornaments (or unmarked) Sometimes list format may not be the best choice. For example, when items in a list consist of very long sentences, or of several sentences, and the list itself does not require typographic prominence, the items may be formatted like regular paragraphs of text, each paragraph beginning with a number (and formatted with a first-line paragraph indent) and punctu ated as normal prose (see also 2.12).
6.131 Vertical lists punctuated as a sentence. If the items in a vertical list complete a sentence begun in the introductory text, semicolons or com mas may be used between the items, and a period should follow the fi nal item. (If the items include internal punctuation, semicolons are pre ferred; see also 6.60.) Each item begins with a lowercase letter, even if the list is a numbered list (cf. 6.130). A conjunction (and or or) before the final item is optional. Such lists, often better run in to the text, should be set vertically only if the context demands that they be highlighted.
Reporting for the Development Committee, Jobson reported that 1. a fundraising campaign director was being sought; 2. the salary for this director, about $175,000 a year, would be paid out of cam paign funds; and 3. the fundraising campaign would be launched in the spring of 2017.
In the case of an unnumbered or bulleted list, the punctuation and capi talization would remain the same as in the example above.
6.132 Vertical lists with multiple levels (outlines). Where items in a numbered list are subdivided (i.e., into a multilevel list, also called an outline), both numerals and letters may be used. Any runover lines should be aligned with the first word following the numeral or letter; a tab usually separates the number or letter from the text of the list (see also 2.21).
Lists and Outline Style 6.132 Applicants will be tested for their skills in the following areas: 1. Punctuation a. Using commas appropriately b. Deleting unnecessary quotation marks c. Distinguishing colons from semicolons 2. Spelling a. Using a dictionary appropriately b. Recognizing homonyms c. Hyphenating correctly 3. Syntax a. Matching verb to subject b. Recognizing and eliminating misplaced modifiers c. Distinguishing phrases from clauses while singing the “Conjunction Junc tion” song In the following example, note that the numerals and letters denoting the top three levels are set off by periods and those for the lower four by single or double parentheses, thus distinguishing all seven levels by punctuation as well as indentation. Note also that numerals of more than one digit are aligned vertically on the last digit.
I. Historical introduction II. Dentition in various groups of vertebrates A. Reptilia 1. Histology and development of reptilian teeth 2. Survey of forms B. Mammalia 1. Histology and development of mammalian teeth 2. Survey of forms a) Primates (1) Lemuroidea (2) Anthropoidea (a) Platyrrhini (b) Catarrhini i) Cercopithecidae ii) Pongidae b) Carnivora (1) Creodonta (2) Fissipedia (a) Ailuroidea (b) Arctoidea (3) Pinnipedia c) Etc....
6.132 PUNCTUATION In a list with fewer levels, one might dispense with capital roman numer als and capital letters and instead begin with arabic numerals. What is important is that readers see at a glance the level to which each item be longs. Note that each division and subdivision should normally contain at least two items.
7 • Spelling, Distinctive Treatment of Words, and Compounds OVERVIEW 7.1 PLURALS 7.5 POSSESSIVES 7.16 The General Rule 7.16 Exceptions to the General Rule 7.20 Particularities of the Possessive 7.23 CONTRACTIONS AND INTERJECTIONS 7.30 “A” AND “AN” 7.32 LIGATURES 7.34 WORD DIVISION 7.36 ITALICS, CAPITALS, AND QUOTATION MARKS 7.48 Emphasis 7.50 Words from Other Languages 7.53 Highlighting Key Terms and Expressions 7.56 Words as Words and Letters as Letters 7.63 Music: Some Typographic Conventions 7.70 Computer Terms 7.76 COMPOUNDS AND HYPHENATION 7.81 7.1 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS Overview 7.1 Recommended dictionaries. For general matters of spelling, Chicago recommends the dictionaries published by Merriam-Webster—specif ically, Webster's Third New International Dictionary (or its ongoing on line-only revision) and the latest edition of its chief abridgment, Merriam- Webster's Collegiate Dictionary (regularly updated online and referred to below as Webster's). If more than one spelling is given, or more than one form of the plural (see 7.6), Chicago normally opts for the first form listed (even for equal variants), thus aiding consistency. If, as occasionally hap pens, the Collegiate disagrees with the Third International, the Collegiate (or its online counterpart) should be followed, since it represents newer lexical research. For further definitions or alternative spellings, refer to another standard dictionary such as the American Heritage Dictionary of the English Language. At least for spelling, one source should be used con sistently throughout a single work. (For full bibliographic information on these and other English dictionaries, including Canadian, British, and Australian references, see bibliog. 3.1.) 7.2 Spellings peculiar to particular disciplines. Where a variant spelling carries a special connotation within a discipline, the author’s preference should be respected. For example, “archeology,” though it is listed as an equal variant of “archaeology” in Webster's, is the spelling insisted on by certain specialists. In the absence of such a preference, Chicago prefers the first-listed “archaeology.” (Webster's separates equal variants by or; secondary variants are preceded by also.) 7.3 Non-US spelling. In English-language works by non-US authors that are edited and produced in the United States, editors at Chicago generally change spelling used in other English-speaking countries to American spelling (e.g., colour to color, analyse to analyze). Since consistency is more easily maintained by this practice, few authors object. In quoted material, however, spelling is left unchanged (see 13.7).
7.4 Supplementing the dictionary. Much of this chapter is devoted to mat ters not easily found in most dictionaries: how to form the plural and possessive forms of certain nouns and compounds; how to break words at the end of a printed line, especially those that are not listed in the dic tionary; when to use capitals, italics, or quotation marks for distinctive treatment of words and phrases; and, perhaps most important but placed at the end of the chapter for easy reference (7.89), when to use hyphens with compound words, prefixes, and suffixes.
Plurals 7.8 Plurals 7.5 Standard plural forms. Most nouns form their plural by adding s or—if they end in ch,j, s, sh, x, or z—by adding es. Most English speakers will not need help with such plural forms as thumbs, churches, fixes, or boys, and these are not listed in standard dictionary entries, including those in Webster’s Collegiate. (All inflected forms are listed in Webster’s Third New International, and Webster’s and other dictionaries published online generally accommodate the correct plural forms in their search engines.) Most dictionaries do, however, give plural forms for words ending in y that change to ies (baby etc.); for words ending in o (ratio,potato, etc.); for certain words of Latin or Greek origin such as crocus, datum, or alumna', and for all words with irregular plurals (child, leaf, etc.).
7.6 Alternative plural forms. Where Webster’s gives two forms of the plural— whether as primary and secondary variants, like zeros and zeroes, or as equal variants, like millennia and millenniums—Chicago normally opts for the first. In some cases, however, different forms of the plural are used for different purposes. A book may have two indexes and a math ematical expression two indices, as indicated in the Webster’s entry for index.
7.7 Plurals of compound nouns. Webster’s gives the plural form of most com pounds that are tricky (fathers-in-law, coups d’etat, courts-martial, chefs d’oeuvre, etc.). For those not listed, common sense can usually provide the answer.
bachelors of science masters of arts spheres of influence child laborers 7.8 Plurals for centuries. The plural is normally used to refer to more than one century with ordinals (or other modifiers) joined by and; the plural is also used for ranges expressed with through. For ranges expressed with to or for alternatives expressed with or or nor, use the singular. Also use the singular in compound modifiers (as in the last example; see also 7.88).
the eighteenth and early nineteenth centuries in the fifth through eighth centuries but from the twentieth to the twenty-first century as of the fifth or sixth century eighteenth- and nineteenth-century technologies 7.9 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS 7.9 Plurals of proper nouns. Names of persons and other capitalized nouns normally form the plural by adding s or es. Exceptions, including the last example, are generally listed in Webster's (see also 7.10).
Tom, Dick, and Harry;/?/. Toms, Dicks, and Harrys the Jones family, pl. the Joneses the Martinez family,/?/, the Martinezes the Bruno family, pl. the Brunos Sunday, pl. Sundays Germany, pl. Germanys Pakistani,/?/. Pakistanis but Romany,/?/. Romanies An apostrophe is never used to form the plural of a family name: “The Jeffersons live here” (not “Jefferson’s”). For the apostrophe in the posses sive form of proper nouns, see 7.17.
7.10 Plural form for Native American group names. For the plurals of names of Native American groups, Chicago now defers to the first-listed form in Webster's, in the absence of any overriding preference of the author or publisher. For names not found there, check an up-to-date encyclopedia or other trusted resource or, unless the author or publisher has a prefer ence, opt for consistency with other such names mentioned in the text. See also 7.2.
the Hopi of northeastern Arizona the Iroquoian language spoken by the Cherokee the languages of the Iroquois 7.11 Singular form used for the plural. Names ending in an unpronounced s or x are best left in the singular form.
the seventeen Louis of France the two Dumas, father and son two Charlevoix (or, better, two towns called Charlevoix) The class included three Margaux (but two Felixes) 7.12 Plural form of italicized words. If an italicized term such as the title of a newspaper or book or a word used as a word must be written in the plu ral, the s is normally set in roman. A title already in plural form, however, may be left unchanged. In case of doubt, avoid the plural by rephrasing.
Plurals 7.15 two Chicago Tribunes and three Milwaukee Journal Sentinels several Madame Bovarys (or, better, several copies of Madame Bovary) too many sics but four New York Times The plural endings to italicized words in another language should also be set in italics.
Blume, Blumen cheval, chevaux sehor, sehores 7.13 Plural form for words in quotation marks. The plural of a word or phrase in quotation marks may be formed in the usual way (without an apos trophe). If the result is awkward, reword. Chicago discourages a plural ending following a closing quotation mark.
How many more “To be continueds” (not “To be continued”s) can we expect? or, better, How many more times can we expect to see “To be continued”?
7.14 Plurals of noun coinages. Words and hyphenated phrases that are not nouns but are used as nouns usually form the plural by adding s or es. (If in doubt, consult an unabridged dictionary like Webster's Third New Inter national, which indicates the preferred inflected forms for most nouns, including all of the examples below.) ifs and buts dos and don’ts threes and fours thank-yous maybes yeses and nos 7.15 Plurals for letters, abbreviations, and numerals. Capital letters used as words, numerals used as nouns, and abbreviations usually form the plu ral by adding s. To aid comprehension, lowercase letters form the plural with an apostrophe and an s (compare “two as in llama" with “two a's in llama"). For some exceptions beyond those listed in the last three ex amples, see 10.42; see also 10.52 (for the International System). For the omission of periods in abbreviations like “BS,” “MA,” and “PhD,” see 10.4. See also 7.63-69.
the three Rs x’s and j/s the 1990s IRAs URLs but BSs, MAs, PhDs p. (page), pp. (pages) vols. n. (note), nn. (notes) eds. MS (manuscript), MSS (manuscripts) 7.16 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS Possessives The General Rule 7.16 Possessive form of most nouns. The possessive of most singular nouns is formed by adding an apostrophe and an 5. The possessive of plural nouns (except for a few irregular plurals, like children, that do not end in s) is formed by adding an apostrophe only. For the few exceptions to these principles, see 7.20-22. See also 5.20.
the horse’s mouth a bass’s stripes puppies’paws children’s literature a herd of sheep’s mysterious disappearance 7.17 Possessive of proper nouns, abbreviations, and numbers. The general rule stated at 7.16 extends to the possessives of proper nouns, including names ending in s, x, or z, in both their singular and plural forms, as well as abbreviations and numbers.
Singular forms Kansas’s legislature Chicago’s lakefront Marx’s theories Jesus’s adherents Berlioz’s works Tacitus’s Histories Borges’s library Dickens’s novels Malraux’s masterpiece Josquin des Prez’s motets Plural forms the Lincolns’ marriage the Williamses’ new house the Martinezes’ daughter dinner at the Browns’ (that is, at the Browns’place) Abbreviations and numbers FDR’s legacy Apollo Il’s fiftieth anniversary HP Inc.’s latest offerings Avoid forming the possessive of an abbreviation that is followed by a spelled-out form in parentheses (or vice versa).
the long history of IBM (International Business Machines) not IBM’s (International Business Machines’) long history Possessives 7.21 7.18 Possessive of words and names ending in unpronounced "s." Words and names ending in an unpronounced 5 form the possessive in the usual way—with the addition of an apostrophe and an s (which, when such forms are spoken, is usually pronounced).
Descartes’s three dreams the marquis’s mother Francois’s efforts to learn English Vaucouleurs’s assistance to Joan of Arc Albert Camus’s novels 7.19 Possessive of names like "Euripides." Classical proper names of two or more syllables that end in an eez sound form the possessive in the usual way (though when these forms are spoken, the additional s is generally not pronounced).
Euripides’s tragedies the Ganges’s source Xerxes’s armies Exceptions to the General Rule 7.20 Possessive of nouns plural in form, singular in meaning. When the sin gular form of a noun ending in s is the same as the plural (i.e., the plural is uninflected), the possessives of both are formed by the addition of an apostrophe only. If ambiguity threatens, use of to avoid the possessive.
politics’ true meaning economics’ forerunners this species’ first record (or, better, the first record of this species) The same rule applies when the name of a place or an organization or a publication (or the last element in the name) is a plural form ending in s, such as the United States, even though the entity is singular.
the United States’ role in international law Highland Hills’ late mayor Callaway Gardens’ former curator the National Academy of Sciences’ new policy 7.21 "For ... sake" expressions. For the sake of euphony, a few/br... sake expressions used with a singular noun that ends in an s end in an apos trophe alone, omitting the additional s.
7.22 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS for goodness’ sake for righteousness’ sake Aside from these traditional formulations, however, the possessive in for... sake expressions may be formed in the normal way.
for expedience’s sake for appearance’s sake (or for appearances’ sake [plural possessive] or for the sake of appearance [or appearances]) for Jesus’s sake 7.22 An alternative practice for words ending in "s." Some writers and pub lishers prefer the system, formerly more common, of simply omitting the possessive s on all words ending in s—hence “Dylan Thomas’ poetry,” “Etta James’ singing,” and “that business’ main concern.” Though easy to apply and economical, such usage disregards pronunciation in the ma jority of cases and is therefore not recommended by Chicago.
Particularities of the Possessive 7.23 Joint versus separate possession. Closely linked nouns are considered a single unit in forming the possessive when the thing being “possessed” is the same for both; only the second element takes the possessive form.
my aunt and uncle’s house Gilbert and Sullivan’s Iolanthe Minneapolis and Saint Paul’s transportation system When the things possessed are discrete, both nouns take the possessive form.
my aunt’s and uncle’s medical profiles Dylan’s and Jagger’s hairlines New York’s and Chicago’s transportation systems Gilbert’s or Sullivan’s mustache 7.24 Compound possessives. In compound nouns and noun phrases, the fi nal element usually takes the possessive form, even in the plural.
student assistants’ time cards my daughter-in-law’s address my sons-in-law’s addresses Possessives 7.27 7.25 Possessive to mean "of." Analogous to possessives, and formed like them, are certain expressions that would otherwise include of, (Such us age is one of the genitive forms discussed in 5.20.) In the first example below, the literal meaning is “in three days of time.” If the result seems ambiguous or awkward, reword.
in three days’ time (or in three days) an hour’s delay (or a one-hour delay) six months’ leave of absence (or a six-month leave of absence) 7.26 Double possessive. According to a usage that is sometimes referred to as the double possessive or double genitive (see 5.21), a possessive form may be preceded by o/where one of several is implied. Where the meaning is not literally possessive, however, the possessive form should not be used.
a friend of Dick’s (or a friend of his) but a student of Kierkegaard 7.27 Possessive versus attributive forms for groups. Although terms de noting group ownership or participation sometimes appear without an apostrophe (i.e., as an attributive rather than a possessive noun), Chi cago dispenses with the apostrophe only in proper names (often corpo rate names) that do not officially include one. In a few established cases, a singular noun can be used attributively; if in doubt, choose the plural possessive. (Irregular plurals such as children and women must always be in the possessive.) children’s rights (or child rights) farmers’ market women’s soccer team boys’ clubs veterans’ organizations players’ unions taxpayers’ associations (or taxpayer associations) consumers’ group (or consumer group) but Publishers Weekly Diners Club Department of Veterans Affairs 7.28 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS In some cases, the distinction between attributive and possessive is subtle. Of the following two examples, only the first connotes actual pos session.
the Lakers’ game plan (the team’s game plan) but the Lakers game (the game featuring the team) When in doubt, opt for the possessive.
7.28 Possessive with gerund. A noun followed by a gerund (see 5.112) may take the possessive form in some contexts. This practice, usually limited to proper names and personal nouns or pronouns, should be used with caution. For an excellent discussion, see “Possessive with Gerund,” in Fowler’s Modern English Usage (bibliog. 1.2). The possessive is most com monly used when the gerund rather than the noun that precedes it can be considered to be the subject of a clause, as in the first four examples be low. In the fifth example, Fathers is clearly the subject of the sentence and assuming is a participle (verb form) rather than a gerund (noun form); the possessive would therefore be incorrect.
Fathers’ assuming the care of children has changed the traditional household economy.
We all agreed that Jerod’s running away from the tigers had been the right thing to do.
Our finding a solution depends on the nature of the problem.
Eleanor’s revealing her secret (or Eleanor’s revelation) resulted in a lawsuit. but Fathers assuming the care of children often need to consult mothers for advice.
When the noun or pronoun follows a preposition, the possessive is usu ally optional.
She was worried about her daughter (or daughter’s) going there alone.
I won’t stand for him (or his) being denigrated.
The problem of authors (or authors’) finding the right publisher can be solved.
7.29 Possessive with italicized or quoted terms. As with plurals (see 7.12), when an italicized term appears in roman text, the apostrophe and s should be set in roman. When the last element is plural in form, add only an apostrophe, in roman (see 7.20). Chicago discourages, however, at tempting to form the possessive of a term enclosed in quotation marks (a practice that is seen in some periodical publications where most titles are quoted rather than italicized).
“A”and“An” 7.32 the Atlantic Monthly’s editor the New York Times’ new fashion editor Gone with the Wind’s admirers but admirers of “Ode on a Grecian Urn” Contractions and Interjections 7.30 Contractions. In contractions, an apostrophe normally replaces omitted letters. Some contractions, such as won’t or ain’t, are formed irregularly. Colloquialisms such as gonna or wanna take no apostrophe (there being no obvious place for one). Webster’s lists many common contractions, along with alternative spellings and, where appropriate, plurals. Note that an apostrophe—the equivalent of a right single quotation mark (’ not ‘)—is always used to form a contraction (see 6.117).
singin’ gov’t ’tis (not ‘tis) dos and don’ts rock ’n’ roll Contractions that end in a period (e.g., Dr.) are generally referred to as abbreviations (see 10.2).
7.31 Interjections. As with contractions, Webster’s lists such interjections as ugh, er, um, and sh. For those not found in the dictionary—or where a different emphasis is required—plausible spellings should be sought in literature or invented.
atchoo! shhh!
The interjection oh is not to be confused with the vocative O (largely ob solete), which is always capitalized; oh is capitalized only at the begin ning of a sentence. See also 6.35.
Where, oh where, have you been?
Oh! It’s you!
but “Thine arm, O Lord, in days of old...” “A” and “An” 7.32 "A" and "an" before "h." The indefinite article a, not an, is used in En glish before words beginning with a pronounced h. (British English 7.33 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS differs from American English in not pronouncing the h is many cases; when in doubt, check a standard dictionary.) See also 5.74.
a hotel a historical study but an honor an heir 7.33 "A" and "an" before abbreviations, symbols, and numerals. Before an abbreviation, a symbol, or a numeral, the use of a or an depends on (or, conversely, determines) how the term is pronounced. In the first example below, “MS” would be pronounced em ess\ in the second, it would be pronounced manuscript. In the last two examples, “007” would be pro nounced oh oh seven and double oh seven, respectively.
an MS treatment (a treatment for multiple sclerosis) a MS in the National Library an NBC anchor a CBS anchor a URL an @ sign an 800 number an 007 field (in a library catalog) a 007-style agent Ligatures 7.34 When not to use ligatures. The ligatures z (a+ e) and oe(o + e) should not be used in Latin or transliterated Greek words. Nor should they be used in words adopted into English from Latin, Greek, or French (and thus to be found in English dictionaries). Compare 7.35.
aesthetics Encyclopaedia Britannica (contrary to corporate usage) oedipal a trompe l’oeil mural a tray of hors d’oeuvres Emily Dickinson’s oeuvre 7.35 When ligatures should be used. The ligature z(a + e) is needed for spell ing Old English words in an Old English context. And the ligature oe (o Word Division 7.37 + e) is needed for spelling French words in a French context. (See also 11.21.) Compare 7.34.
TElfric es hael le noeud gordien (Euvres completes Word Division 7.36 Dictionary word division. The advice in this section applies only to pub lished works and mainly to print or PDF (and not to reflowable electronic formats); word breaks should not be applied at the manuscript stage (see 2.13) . For end-of-line word breaks, as for spelling and plural forms, Chi cago turns to Webster's as its primary guide. The dots between syllables in Webster's indicate where breaks may be made; in words of three or more syllables, there is usually a choice of breaks. The paragraphs in this sec tion are intended merely to supplement, not to replace, the dictionary’s system of word division—for example, by suggesting preferred breaks where more than one might be possible. These recommendations are also intended to serve as a guide for determining appropriate hyphen ation settings in page-layout applications. Most such programs automate hyphenation relative to a standard dictionary but allow users to define certain rules (e.g., to specify the minimum number of characters to carry to a new line or the maximum number of consecutive lines that can end in a hyphen) and to make exceptions (e.g., for specific words). For di vision of non-English words (other than those that are included in an English-language dictionary), see chapter 11. For end-of-line breaks rel ative to en and em dashes, see 6.82, 6.90; for slashes, see 6.113.
7.37 Word divisions that should be avoided. Single-syllable words, including verb forms such as aimed and helped, are never divided. Since at least two letters must appear before a break, such words as again, enough, and unite also cannot be divided. And at least three letters must appear af ter a break, so divisions that carry only two letters over to the next line, even where indicated by Webster's, are usually also avoided (but see 6.82, 6.113).
women (rather than wom-en) losses (rather than loss-es) sur-prises (rather than surpris-es) In languages other than English, however, it may be not only permissible but customary to carry two-letter word endings to the next line (see, e.g., 11.33,11.34,11.42,11.43,11.50,11.51,11.66,11.67).
7.38 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS 7.38 Dividing according to pronunciation. In the usage preferred by Chicago and reflected in Webster’s, most words are divided according to how the break will affect pronunciation rather than according to derivation.
knowl-edge (not know-ledge) democ-racy or de-mocracy (not demo-cracy) Special attention should be paid to breaks in certain words with multiple meanings and pronunciations, such asproj-ect (noun) andpro-ject (verb), which automatic hyphenation may not properly account for.
7.39 Dividing after a vowel. Unless a resulting break affects pronunciation, words are best divided after a vowel. When a vowel forms a syllable in the middle of a word, it should remain on the first line if possible. Diph thongs are treated as single vowels (e.g., the eu in aneurysm).
criti-cism (rather than crit-icism) liga-ture (rather than lig-ature) an-tipodes or antipo-des (rather than antip-odes) aneu-rysm (rather than an-eurysm) 7.40 Dividing compounds, prefixes, and suffixes. Hyphenated or closed com pounds and words with prefixes or suffixes are best divided at the natural breaks.
poverty- / stricken (rather than pov- / erty-stricken) thanks-giving (rather than thanksgiv-ing) dis-pleasure (rather than displea-sure) re-inforce (rather than rein-force) 7.41 Dividing words ending in "ing." Most gerunds and present participles may be divided before the ing. When the final consonant before the ingis doubled, however, the break occurs between the consonants. For words ending in ling, check the dictionary.
certify-ing giv-ing dab-bing run-ning fiz-zling bris-tling 7.42 Dividing proper nouns and personal names. Proper nouns of more than one element, especially personal names, should be broken, if possible, between the elements rather than within any of the elements. If a break within a name is needed, consult the dictionary. Many proper nouns ap pear, with suggested divisions, in the listings of biographical and geo graphical names in Webster s Collegiate. For fuller treatment, consult the Word Division 7.46 online revision of Webster’s Third New International (bibliog. 3.1). Those that cannot be found in a dictionary should be broken (or left unbroken) according to the guidelines elsewhere in this section. If pronunciation is not known or easily guessed, the break should usually follow a vowel.
Alek-sis Heitor Villa- / Lobos (or, better, Heitor / Villa-Lobos) Ana-stasia A personal name that includes initials should be broken after the initials. A break before a number or Jr. or Sr. should be avoided. A nonbreaking space can prevent such breaks (see 6.121).
Frederick L. / Anderson M. F. K. / Fisher Elizabeth II (or, if necessary, Eliza- / beth II) 7.43 Dividing numerals. Large numbers expressed as numerals are best left intact. To avoid a break, reword the sentence. If a break must be made, however, it should come only after a comma and never after a single digit. See also 12.23.
1,365,- / 000,000 or 1,365,000,- / 000 7.44 Dividing numerals with abbreviated units of measure. A numeral used with an abbreviated unit of measure is best left intact; either the nu meral should be carried over to the next line or the abbreviation should be moved up. A nonbreaking space can prevent such breaks (see 6.121). (Numerals used with spelled-out units of measure, which tend to form longer expressions, may be broken across a line as needed.) 345 m 24 kg 55 BCE 6:35 p.m.
7.45 Division in run-in lists. A number or letter, such as (3) or (c), used in a run-in list (see 6.129) should not be separated from the beginning of what follows it. If it occurs at the end of a line, it should be carried over to the next line. A nonbreaking space can prevent such breaks (see 6.121).
7.46 Dividing URLs and email addresses. In printed works, it is often nec essary to break an email address or a uniform resource identifier such as a URL at the end of a line. Such a break should be made between el ements if at all possible: after a colon or a double slash; before or after an equals sign or an ampersand; or before a single slash, a period, or any 7.47 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS other punctuation or symbol. To avoid confusion, an address that con tains a hyphen should be broken before the hyphen rather than after (so that the hyphen begins a new line); by a similar logic, a hyphen should never be added to break an email address or URL. If a particularly long element must be broken to avoid a seriously loose or tight line, it can be broken between words or syllables according to the guidelines offered elsewhere in this section. Editors, proofreaders, and compositors should use their discretion in applying these recommendations, aiming for a balance between readability and aesthetics. See also 6.8,14.18.
http:// www.chicagomanualofstyle.org/ or .chicagomanualofstyle.org/ or manualofstyle.org/ Authors should not break URLs in their manuscripts (see 2.13).
7.47 Hyphenation and appearance. For aesthetic reasons, no more than three succeeding lines should be allowed to end in hyphens. (Such hyphens are sometimes referred to as a hyphen stack or ladder.) And though hy phens are necessary far more often in justified text, word breaks may be needed in material with a ragged right-hand margin to avoid exceedingly uneven lines. (In manuscript preparation, however, hyphenation should never be applied; see 2.13.) In reflowable electronic publication formats, end-of-line hyphenation may be applied automatically by a particular application or device; such hyphenation can sometimes be suppressed for aesthetic reasons (as for chapter titles and other headings).
Italics, Capitals, and Quotation Marks 7.48 Setting off proper names and titles of works. With the exception of musical and computer terminology, most of the recommendations in this section are related to the distinctive treatment of letters, words, and phrases as such. For the use of italics, capitals, and quotation marks to indicate or set off proper nouns and titles of works, see chapter 8.
7.49 Italics and markup. Italics (or, more rarely, boldface) have been used for centuries as the default means of setting off text from the surrounding Italics, Capitals, and Quotation Marks 7.52 (usually roman) context. Italics as such are used for emphasis, key terms or terms in another language, words used as words, titles of works, and so on. For electronic publication formats, semantic markup can be added that specifies the nature of the term and not just how it will be presented on the screen. Among other advantages, such markup has the potential to enhance accessibility for readers—for example, by determining how a word will be vocalized by text-to-speech applications. Publishers con cerned about providing accessible content are encouraged to consult the guidelines offered by the Web Accessibility Initiative (WAI) of the World Wide Web Consortium (W3C) and, for EPUB (a standard format for e-books), the Accessibility Guidelines available from the International Digital Publishing Forum. See also 2.81-83.
Emphasis 7.50 Italics for emphasis. Use italics for emphasis only as an occasional ad junct to efficient sentence structure. Overused, italics quickly lose their force. Seldom should as much as a sentence be italicized for emphasis, and never a whole passage. In the first example below, the last three words, though clearly emphatic, do not require italics because of their dramatic position at the end of the sentence.
The damaging evidence was offered not by the arresting officer, not by the injured plaintiff, but by the boy’s own mother.
On the other hand, the emphasis in the following example depends on the italics: It was Leo!
7.51 Boldface or underscore for emphasis. Occasionally, boldface or under score (also called underlining) is used for emphasis. In formal prose, es pecially in print, italics are usually more appropriate (see 7.50). See also 7.56,7.79.
7.52 Capitals for emphasis. Initial capitals, once used to lend importance to certain words, are now used only ironically (but see 8.94).
“OK, so I’m a Bad Mother,” admitted Mary cheerfully.
Capitalizing an entire word or phrase for emphasis is rarely appropriate in formal prose. If capitals are wanted—in dialogue or in representing 7.53 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS newspaper headlines, for example—small caps rather than full cap itals look more graceful. Note that “capitalizing” a word means set ting only the initial letter as a capital. Capitalizing a whole word, LIKE THIS, is known as “setting in full caps” (or “all caps”). Setting a word in small capitals—or “small caps”—results in t h is s t y l e . (For the use of small capitals in representing terms in American Sign Language, see 11.125-35.) See also 10.8.
“Be careful—w a t c h o u t !” she yelled.
We could not believe the headline: po l a r ic e c a p r e t u r n s .
Words from Other Languages 7.53 Unfamiliar words and phrases from other languages. Use italics for iso lated words and phrases from another language unless they appear in Webster's or another standard English-language dictionary (see 7.54). If a word from another language becomes familiar through repeated use throughout a work, it need be italicized only on its first occurrence. If it appears only rarely, however, italics may be retained.
Her love for futbol and telenovelas set her apart from her bookish peers.
The word he used, Kaffeetasse (coffee cup), was just accurate enough to gain the desired result.
This rule does not extend to proper nouns, which can generally appear in roman type (except for titles of books and the like). For further dis cussion and examples, including the treatment of translated terms, see 11.3-5. For capitalization in other languages, see 11.18.
7.54 Roman for familiar words from other languages. Words and phrases from another language that are familiar to most readers and listed in Webster's should appear in roman (not italics) if used in an English con text, and they should be spelled as in Webster's. German nouns, if in Web ster's, are lowercased (but see 11.39). See also 7.55.
pasha Weltanschauung in vitro but a priori the kaiser recherche de novo bourgeois eros and agape He never missed a chance to epater les bourgeois.
Italics, Capitals, and Quotation Marks 7.58 If a familiar term, such as mise en scene, should occur in the same context as a less familiar one, such as mise en bouteille (not listed in Webster's), either both or neither should be italicized, so as to maintain internal con sistency. See also 7.53.
7.55 Roman for Latin words and abbreviations. Commonly used Latin words and abbreviations should not be italicized.
ibid. et al. ca. passim Because of its peculiar use in quoted matter, sic is best italicized.
“mindful of what has been done here by we [sic] as agents of principle” Highlighting Key Terms and Expressions 7.56 Italics or boldface for key terms. Key terms in a particular context are often italicized on their first occurrence. Thereafter they are best set in roman.
The two chief tactics of this group, obstructionism and misinformation, require careful analysis.
Occasionally, boldface may be used for key terms, as in a textbook or to highlight terms that also appear in a glossary; such usage should be noted in the text. See also 7.79.
7.57 "Scare quotes." Quotation marks are often used to alert readers that a term is used in a nonstandard (or slang), ironic, or other special sense. Such scare quotes imply “This is not my term” or “This is not how the term is usually applied.” Like any such device, scare quotes lose their force and irritate readers if overused. See also 7.59,7.60.
My rotary simulation app allows me to “dial” phone numbers again. “Child protection” sometimes fails to protect.
7.58 Mixing single and double quotation marks. In works of philosophy or other specialized contexts, single and double quotation marks are sometimes used to signal different things. For example, single quotation marks might be used for special terms and double quotation marks for their definitions. Chicago discourages such a practice, preferring a mix of italics, quotation marks, and parentheses instead.
7.59 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS 7.59 "So-called." A word or phrase preceded by so-called need not be en closed in quotation marks. The expression itself indicates irony or doubt. If, however, it is necessary to call attention to only one part of a phrase, quotation marks may be helpful.
So-called child protection sometimes fails to protect.
Her so-called mentor induced her to embezzle from the company. but These days, so-called “running” shoes are more likely to be seen on the feet of walkers.
7.60 Common expressions and figures of speech. Quotation marks are rarely needed for common expressions or figures of speech (including slang). They should normally be reserved for phrases borrowed verbatim from another context or terms used ironically (see 7.57).
Myths of paradise lost are common in folklore.
I grew up in a one-horse town.
Only techies will appreciate this joke.
but Though she was a lifetime subscriber to the Journal of Infectious Diseases, she was not one to ask “for whom the bell tolls.” 7.61 Signs and notices. Specific wording of common short signs or notices is capitalized headline-style in running text (see 8.159). A longer notice is better treated as a quotation.
The door was marked Authorized Personnel Only.
She encountered the usual Thank You for Not Smoking signs.
We were disturbed by the notice “Shoes and shirt required of patrons but not of personnel.” 7.62 Mottoes. Mottoes may be treated the same way as signs (see 7.61). If the wording is in another language, it is usually italicized and only the first word capitalized. See also 7.53, 6.41.
The flag bore the motto Don’t Tread on Me.
My old college has the motto Souvent me souviens.
The motto “All for one and one for all” appears over the door.
Italics, Capitals, and Quotation Marks 7.65 Words as Words and Letters as Letters 7.63 Words and phrases used as words. When a word or term is not used functionally but is referred to as the word or term itself, it is either itali cized or enclosed in quotation marks. Proper nouns used as words, as in the third example, are usually set in roman (see also 7.64).
The term critical mass is more often used metaphorically than literally.
What is meant by neurobotics?
You rarely see the term iPhone with a capital i.
Although italics are the traditional choice, quotation marks may be more appropriate in certain contexts. In the first example below, italics set off the Spanish term, and quotation marks are used for the English (see also 7.53) . In the second example, quotation marks help to convey the idea of speech.
The Spanish verbs ser and estar are both rendered by “to be.” Many people say “I” even when “me” would be more correct.
7.64 Letters as letters. Individual letters and combinations of letters of the Latin alphabet are usually italicized.
the letter q a lowercase n a capital W The plural is usually formed in English by adding s or es.
He signed the document with an X.
I need a word with two e’s and three s’s.
Roman type, however, is traditionally used in two common expressions (see also 7.15).
Mind your p’s and q’s! dotting the i’s and crossing the t’s Roman type is always used for phonetic symbols. For details, consult Geoffrey K. Pullum and William A. Ladusaw, Phonetic Symbol Guide (bibliog. 5).
7.65 Scholastic grades. Letters used to denote grades are usually capitalized and set in roman type. No apostrophe is required in the plural (see also 7.15) .
7.66 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS She finished with three As, one B, and two Cs.
7.66 Letters standing for names. A letter used in place of a name is usually capitalized and set in roman type. If it bears no relation to an actual name, it is not followed by a period.
Let us assume that A sues B for breach of contract...
If a single initial is used to abbreviate an actual name, it is usually fol lowed by a period; if used to conceal a name, it may be followed by a 2-em dash and no period (see 6.93). If no punctuation follows the dash, it must be followed by a space.
Professor D. will be making his entrance shortly.
Senator K---- and Representative L-----were in attendance.
If two or more initials are used as an abbreviation for an entire name, no periods are needed. See also 8.4,10.12.
Kennedy and Johnson soon became known as JFK and LBJ.
7.67 Letters as shapes. Letters that are used to represent shapes are capital ized and set in roman type (an S curve, an L-shaped room). (Using a sans serif font in a serif context, as is sometimes done, does not necessarily aid comprehension and, unless the sans serif perfectly complements the serif, tends to look clumsy.) 7.68 Names of letters. When legibility cannot be counted on, editors and proofreaders occasionally need to name letters (“a cue, not a gee”). The name of a letter, as distinct from the letter itself, is usually set in roman type, without quotation marks. The following standard spellings are drawn from Webster's Collegiate. With vowels, which are not named in standard dictionaries, it may be best to give an example (“a as in apple"). (For the names of special characters, see chapter 11, esp. tables 11.1 and 11.2.
) b bee k kay 5 ess c cee I el t tee d dee m em V vee f ef n en w double-u S gee P pee X ex h aitch 4 cue y wye j jay r ar z zee Italics, Capitals, and Quotation Marks 7.72 7.69 Rhyme schemes. Lowercase italic letters, with no space between, are used to indicate rhyme schemes or similar patterns.
The Shakespearean sonnet’s rhyme scheme is abab, cdcd, efefigg.
Music: Some Typographic Conventions 7.70 Suggested references for music publishing. Music publishing is too specialized to be more than touched on here. Authors and editors re quiring detailed guidelines may refer to D. Kern Holoman, Writing about Music (bibliog. 1.1). For an illustration of typeset music, see figure 3.5. For styling the titles of musical works, see 8.193-97. For a more general reference work, consult The New Grove Dictionary of Music and Musicians and the other Grove musical dictionaries, available from Oxford Music Online (bibliog. 5).
7.71 Musical pitches. Letters standing for musical pitches (which in turn are used to identify keys, chords, and so on) are usually set as roman capi tals. The terms sharp, flat, and natural, if spelled out, are set in roman type and preceded by a hyphen. Editors unfamiliar with musicological conventions should proceed with caution. In the context of harmony, for example, some authors may regard a hyphenated “C-major triad” as be ing based on the note rather than the key of C. See also 7.74.
middle C the key of G major the D-major triad or D major triad an F augmented triad (an augmented triad on the note F) G-sharp or G# the key of B-flat minor or Bl? minor Beethoven’s E-flat Major Symphony (the Eroica) an E string A series of pitches are joined by en dashes.
The initial F-G-F-Bl?
7.72 Octaves. In technical works, various systems are used to designate oc tave register. Those systems that group pitches by octaves begin each ascending octave on C. In one widely used system, pitches in the octave below middle C are designated by lowercase letters: c, ctf, d,..., a#, b. Octaves from middle C up are designated with lowercase letters bearing 7.73 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS superscript numbers or primes: c1, c2, and so on, or c', c", and so on. Lower octaves are designated, in descending order, by capital letters and capital letters with subscript numbers: C, Ci, C2. Because of the many systems and their variants in current use, readers should be alerted to the system employed (e.g., by an indication early in the text of the symbol used for middle C). Technical works on the modern piano usually desig nate all pitches with capital letters and subscripts, from Ai at the bottom of the keyboard to C8s at the top. Scientific works on music usually desig nate octaves by capital letters and subscripts beginning with Co (middle C = C4). When pitches are otherwise specified, none of these systems is necessary.
middle C. A 440 the soprano’s high C To indicate simultaneously sounding pitches (as in chords), the pitches are listed from lowest to highest and are sometimes joined by plus signs.
C + E + G 7.73 Chords. In the analysis of harmony, chords are designated by roman nu merals indicating what degree of the scale the chord is based on.
V (a chord based on the fifth, or dominant, degree of the scale) V7 (dominant seventh chord) iii (a chord based on the third, or mediant, degree of the scale) Harmonic progressions are indicated by capital roman numerals sepa rated by en dashes: IV-I-V-I. While roman numerals for all chords suffice for basic descriptions of chordal movement, in more technical writing, minor chords are distinguished by lowercase roman numerals, and other distinctions in chord quality and content are shown by additional sym bols and arabic numerals.
7.74 "Major" and "minor." In some works on musical subjects where many keys are mentioned, capital letters are used for major keys and lowercase for minor. If this practice is followed, the words major and minor are usu ally omitted.
7.75 Dynamics. Terms indicating dynamics are usually given in lowercase, often italicized: piano, mezzo forte, and so on. Where space allows, the spelled-out form is preferred in both text and musical examples. Sym bols for these terms are rendered in lowercase boldface italics with no periods: p, mf, and so on. “Editorial” dynamics—those added to a com Italics, Capitals, and Quotation Marks 7.78 poser’s original by an editor—are sometimes distinguished by another font or by parentheses or brackets.
Computer Terms 7.76 Application-specific versus generic usage. Typographic conventions for expressing the names of particular keys, menus, commands, and the like vary not only across devices, operating systems, and applica tions (also called programs or apps) but also across successive iterations of each. In general, any specific reference to a named component or function should follow the usage that appears with the device or soft ware itself. Likewise, the names for recognized standards should follow accepted usage. References to generic components or functions—or to specific components or functions discussed in a generic sense—often re quire no special treatment. For more comprehensive coverage, see the style guides published by Microsoft and Apple (bibliog. 1.1) or consult a user’s guide or similar documentation for a specific context. For the treatment of proper names for software and devices, see 8.155.
7.77 Capitalization for keys, menu items, and file formats. The basic al phabet keys as well as all named keys are capitalized even if they are lowercased on a particular keyboard. Named menu items and labels for toolbars, tabs, buttons, icons, and the like are usually spelled and capi talized as in a particular application or operating system or on a particu lar device. Abbreviations for file formats are rendered in full capitals un less expressed as extensions (see also 10.49). Items or actions that are not specifically labeled can usually be treated generically.
The function key F2 has no connection with the keys F and 2. The Option key on a Mac is similar to the Alt key on a typical PC. One purpose of the Return key (or, on a PC, Enter) is to insert a hard return. To activate Caps Lock, double-tap Shift (the arrow key).
To show your work in Word, use Track Changes.
Airplane Mode can be toggled on and off using the airplane icon in the Control Center.
Choosing Cut from the Edit menu is an alternative to pressing Ctrl+X.
Save the file as a PNG or a GIF, not as a JPEG. but The extensions .html and .htm are both used for HTML files.
7.78 Keyboard combinations and shortcuts. To indicate that different keys are to be pressed simultaneously (as in a keyboard shortcut), use the plus 7.79 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS sign or the hyphen without a space on either side. The choice of plus or hyphen may depend on the operating system; in the examples below, the plus sign is used for examples from Microsoft Windows and the hyphen for Apple’s OS. Spell out Shift, Hyphen, and Space—and anything else that might otherwise be ambiguous. (The capital S in the second and third ex amples does not indicate that the Shift key should be pressed as part of the combination.) To insert a double dagger, press Option-Shift-7; on a PC, press Alt+0135 (on the numeric keypad).
To save, press Ctrl+S.
To save, press Command-S.
If the screen freezes, press Ctrl+Alt+Delete.
To empty the trash without a prompt, press Option-Shift-Command-Delete.
7.79 Setting off file names and words to be typed or selected. In most con texts, a combination of quotation marks and capitalization will be suffi cient for setting off file names and words to be typed or selected or that otherwise require user interaction or input. References to named items that use headline-style capitalization do not usually require quotation marks (see 8.159); items that use sentence-style capitalization usually do (see 8.158). For file names or words or other strings to be typed, quota tion marks can be used; any punctuation that belongs to the surrounding sentence should appear outside the quotation marks (see also 6.9).
To list your music by album title, tap More, then tap Albums.
To start page numbering on the second page, choose the Layout tab from the Page Setup dialog box and select the checkbox labeled “Different first page.” If your server uses “index.html” as its default file name, the name of your own default file cannot be “index.htm”.
To change the directory to your desktop, type “cd Desktop”.
When a greater prominence than capitalization or quotation marks is desired, a different type treatment may be used to highlight or set off elements. A single treatment may be applied across different types of elements, or different treatments may be used to signal different things. For example, boldface or italics could be used for menu items and the like, and a fixed-width (monospaced) font such as Courier could signal items to be typed and directory paths, file names, variables, and other strings. Any punctuation that follows such text should appear in the font of the surrounding text; see also 6.2, 6.3. Any quotation marks, single or double, belonging to text in a fixed-width font should be of the default, nondirectional kind (cf. 6.115).
Compounds and Hyphenation 7.82 To insert a thorn, choose Symbol from the Insert tab, then enter OOFE in the Character code field.
Click Save As; name your file appendix A, draft.
To set the value, type $var = " 1". (not “ 1") Type c: \KindleGen\kindlegen, followed by a space and the file name(s) for your book content.
For additional guidance, consult the latest style guides from Microsoft and Apple (bibliog. 1.1).
7.80 Terms like "web" and "internet." Terms related to the internet are cap italized only if they are trademarked as such or otherwise constitute the proper name of an organization or the like. Generic terms that are capi talized as part of a proper name may be lowercased when used alone or in combination (see also 8.68). For treatment of the names of keys, menu items, and file formats, see 7.77. For terms such as email, see 7.89.
hypertext transfer protocol (HTTP); a transfer protocol; hypertext internet protocol (IP); the internet; net neutrality; intranet Wi-Fi (a trademark); wireless network; Ethernet (a trademark); cellular (or mo bile) networks; NFC (near-field communication) the World Wide Web Consortium (W3C); the World Wide Web; the web; website; web page the Open Source Initiative (the corporation); open-source platforms Compounds and Hyphenation 7.81 To hyphenate or not to hyphenate. Far and away the most common spelling questions for writers and editors concern compound terms— whether to spell as two words, hyphenate, or close up as a single word. Prefixes (and occasionally suffixes) can be troublesome also. The first place to look for answers is the dictionary. This section, including the hyphenation guide in 7.89, offers guidelines for spelling compounds not necessarily found in the dictionary (though some of the examples are drawn from Webster’s) and for treatment of compounds according to their grammatical function (as nouns, adjectives, or adverbs) and their position in a sentence. See also 5.92.
7.82 Compounds defined. An open compound is spelled as two or more words (high school, lowest common denominator). A hyphenated com pound is spelled with one or more hyphens (mass-produced, kilowatt- hour, non-English-speaking). A closed (or solid) compound is spelled as 7.83 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS a single word (birthrate, smartphone). A permanent compound is one that has been accepted into the general vocabulary and can be found in the dictionary (like all but one of the examples in this paragraph thus far). A temporary compound is a new combination created for some spe cific, often onetime purpose (dictionary-wielding, impeachment hound); such compounds, though some eventually become permanent, are not normally found in the dictionary. Not strictly compounds but often discussed with them are words formed with prefixes (antigrammarian, postmodern); these are dealt with in section 4 of 7.89. (For examples of combining forms—a type of prefix in which a word like electric is mod ified to form a combination like electromagnetic—see section 2 of 7.89, under combining forms.) 7.83 The trend toward closed compounds. With frequent use, open or hy phenated compounds tend to become closed (on line to on-line to online). In some cases, one term will become closed (birthrate) while a closely related term remains open (death rate). When in doubt, opt for an open compound.
7.84 Hyphens and readability. A hyphen can make for easier reading by show ing structure and, often, pronunciation. Words that might otherwise be misread, such as re-creation or co-op, should be hyphenated. Hyphens can also eliminate ambiguity. For example, the hyphen in much-needed clothing shows that the clothing is greatly needed rather than abundant and needed. Where no ambiguity could result, as in public welfare admin istration or graduate student housing, hyphenation is unnecessary.
7.85 Compound modifiers before or after a noun. When compound mod ifiers (also called phrasal adjectives) such as high-profile or book-length precede a noun, hyphenation usually lends clarity. With the exception of proper nouns (such as United States) and compounds formed by an adverb ending in ly plus an adjective (see 7.86), it is never incorrect to hyphenate adjectival compounds before a noun. When such compounds follow the noun they modify, hyphenation is usually unnecessary, even for adjectival compounds that are hyphenated in Webster's (such as well- read or ill-humored).
7.86 Adverbs ending in "ly." Compounds formed by an adverb ending in ly plus an adjective or participle (such as largely irrelevant or smartly dressed) are not hyphenated either before or after a noun, since ambiguity is vir tually impossible. (The ly ending with adverbs signals to the reader that the next word will be another modifier, not a noun.) Compounds and Hyphenation 7.89 7.87 Multiple hyphens. Multiple hyphens are usually appropriate for such phrases as an over-the-counter drug or a winner-take-all contest. If, how ever, the compound modifier consists of an adjective that itself modifies a compound, additional hyphens may not be necessary. The expressions late nineteenth-century literature and early twentieth-century growth are clear without a second hyphen. (Similar expressions formed with mid— which Chicago classifies as a prefix—do not follow this pattern; see 7.89, section 4, under mid.) See also 7.89, section 3, under century.
7.88 Suspended hyphens. When the second part of a hyphenated expression is omitted, the suspended hyphen is retained, followed by a space (or, in a series, by a comma).
fifteen- and twenty-year mortgages Chicago- or Milwaukee-bound passengers five-, ten-, and twenty-dollar bills but a five-by-eight-foot rug (a single entity) a three-to-five-year gap (a single range) Omission of the second part of a solid compound follows the same pat tern.
both over- and underfed cats but overfed and overworked mules (not overfed and -worked mules) 7.89 Hyphenation guide. In general, Chicago prefers a spare hyphenation style: if no suitable example or analogy can be found either in this section (7.81-89) or in the dictionary, hyphens should be added only if doing so will prevent a misreading or otherwise significantly aid comprehension. Each of the four sections of the following table is arranged alphabeti cally (by first column). The first section deals with compounds according to category; the second section, with compounds according to parts of speech. The third section lists examples for specific terms. The fourth section lists common prefixes, most of which join to another word to form one unhyphenated word; note especially the hyphenated excep tions. (Compounds formed with suffixes—e.g., nationhood, penniless—are almost always closed.) 7.89 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS Category/specific term Examples Summary of rule 1 COMPOUNDS ACCORDING TO CATEGORY age terms a three-year-old a five-year-old child a fifty-five-year-old woman a test for nine-to-ten-year-olds a group of ten- and eleven-year-olds but seven years old eighteen years of age Hyphenated in both noun and adjective forms (except as in the last two examples); note the space after the first hyphen in the fifth but not the fourth example (see 7.88). The examples apply equally to ages expressed as numerals.
chemical terms sodium chloride sodium chloride solution Open in both noun and adjective forms.
colors emerald-green tie reddish-brown flagstone blue-green algae snow-white dress black-and-white print but his tie is emerald green the stone is reddish brown the water is blue green the clouds are snow white the truth isn’t black and white Hyphenated before but not after a noun.
compass points and directions northeast southwest east-northeast a north-south street the street runs north-south Closed in noun, adjec tive, and adverb forms unless three directions are combined, in which case a hyphen is used after the first. When from... to is implied, an en dash is used (see 6.78).
ethnic terms. See proper nouns and adjectives relating to geography or nation ality in section 2.
foreign phrases. See non-English phrases fractions, compounds formed with a half hour a half-hour session a quarter mile a quarter-mile run an eighth note Noun form open; adjective form hyphenated. See also number entries in this sec tion and half in section 3.
Compounds and Hyphenation 7.89 Category/specific term Examples Summary of rule 1 COMPOUNDS ACCORDING TO CATEGORY (continued) fractions, simple one-half two-thirds three-quarters one twenty-fifth one and three-quarters a two-thirds majority three-quarters done a one twenty-fifth share Hyphenated in noun, adjective, and adverb forms, except when second element is already hyphenated. See also number + noun and 9.14.
money a five-cent raise sixty-four-million-dollar question a deal worth thirty million dollars multimillion-dollar deal but $30 million loan a $50-$60 million loss For amounts with spelled- out units, hyphenate before a noun but leave open after; where units are expressed as symbols, leave open in all positions, except between number ranges. See also number + abbreviation and 9.20-25, 9.60.
non-English phrases an a priori argument a Sturm und Drang drama in vitro fertilization a tete-a-tete approach Open unless hyphens appear in the original language.
number + the 33 m distance Always open. See also num abbreviation a 2 fcg weight a 3 ft. high wall a 7 lb., 8 oz. baby ber + noun.
number + noun a hundred-meter race a 250-page book a fifty-year project a three-inch-high statuette it’s three inches high a one-and-a-half-inch hem one and a half inches a seven-pound, eight-ounce baby a six-foot-two [or six-foot, two-inch] adult six feet two [or six feet, two inches tall] five-to-ten-minute intervals (a single range) but five- or ten-minute intervals (two values) Hyphenated before a noun, otherwise open. Note the space after the first number in the last example (see 7.88). See also number + abbreviation and 9.13.
7.89 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS Category/specific term ■ Examples Summary of rule i ‘ ‘ 1. COMPOUNDS ACCORDING TO CATEGORY (continued) number+percent 50 percent a 10 percent raise a 30-40 percent increase Noun and adjective forms al ways open, except between number ranges. See also 9.18, 9.60.
number, ordinal, + noun third-floor apartment lO3rd-floor view on the third floor fifth-place contestant twenty-first-row seats in the twenty-first row Hyphenated before a noun, otherwise open (but see numbers, spelled out). See also century in section 3.
number, ordinal, + superlative a second-best decision third-largest town fourth-to-last contestant he arrived fourth to last Hyphenated before a noun, otherwise open.
numbers, spelled out twenty-eight three hundred nineteen forty-five five hundred fifty-two contestants twenty-eighth three hundredth five hundred fifty-second contestant Twenty-one through ninety- nine hyphenated; others open. Applies equally to cardinals and ordinals. See also fractions, simple.
relationships. See foster, grand, in-law, and step in section 3.
time at three thirty the three-thirty train a four o'clock train the 5:00 p.m. news Usually open; forms such as “three thirty,” “four twenty,” etc. are hyphenated before the noun.
2. COMPOUNDS ACCORDING TO PARTS OF SPEECH adjective + noun small-state senators a high-quality alkylate a middle-class neighborhood the neighborhood is middle class Hyphenated before but not after a noun.
adjective + participle tight-lipped person high-jumping grasshoppers open-ended question the question was open ended Hyphenated before but not after a noun.
Compounds and Hyphenation 7.89 Category/specific term ; Examples Summary of rule 2. COMPOUNDS ACCORDING TO PARTS OF SPEECH (continued) adverb ending in ly + participle or adjective a highly paid ragpicker a fully open society he was mildly amusing Open whether before or after a noun.
adverb not ending in ly + participle or adjective a much-needed addition it was much needed a very well-read child little-understood rules a too-easy answer the best-known author the highest-ranking officer the worst-paid job a lesser-paid colleague the most efficient method a less prolific artist a more thorough exam a rather boring play the most skilled workers (most in number) but the most-skilled workers (most in skill) a very much needed addition Hyphenated before but not after a noun; certain com pounds, including those with more, most, less, least, and very, can usually be left open unless ambiguity threatens. When the adverb rather than the compound as a whole is modified by another adverb, the entire expression is open.
combining forms electrocardiogram socioeconomic politico-scientific studies the practico-inert Usually closed if permanent, hyphenated if temporary. See 7.82.
gerund + noun running shoes cooking class running-shoe store Noun form open; adjective form hyphenated. See also noun + gerund.
noun + adjective computer-literate accountants HIV-positive men the stadium is fan friendly she is HIV positive Hyphenated before a noun; usually open after a noun.
noun + gerund mountain climbing a mountain-climbing enthusiast time-clock-punching employees a Nobel Prize-winning chemist (see 6.80) decision-making head-hunting bookkeeping caregiving copyediting Noun form usually open; adjective form hyphenated before a noun. Some perma nent compounds hyphen ated or closed (see 7.82).
7.89 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS Category/specific term Examples Summary of rule 2. COMPOUNDS ACCORDING TO PARTS OF SPEECH (continued) noun + noun, single function (first noun modifies second noun) student nurse restaurant owner directory path tenure track tenure-track position home-rule governance shipbuilder gunrunner copyeditor Noun form open; adjective form hyphenated before a noun. Some permanent compounds closed (see 7.82).
noun + noun, two functions (both nouns equal) writer-director philosopher-king city-state city-state governance Both noun and adjective forms always hyphenated.
noun + numeral or enumerator type A a type A executive type 2 diabetes size 12 slacks dipage 1 headline Both noun and adjective forms always open.
noun + participle a Wagner-burdened repertoire flower-filled garden the garden was flower filled but the room was air-conditioned Hyphenated before a noun, otherwise open unless verb form hyphenated in Webster's (see also phrases, verbal).
participle + noun chopped-liver pate cutting-edge methods their approach was cutting edge Adjective form hyphenated before but not after a noun.
participle + up, out, and similar adverbs dressed-up children burned-out buildings ironed-on decal we were dressed up that decal is ironed on we ironed on the decal Adjective form hyphenated before but not after a noun. Verb form always open.
i phrases, adjectival an over-the-counter drug a matter-of-fact reply an up-to-date solution sold over the counter her tone was matter of fact his equipment was up to date Hyphenated before a noun; usually open after a noun.
phrases, noun stick-in-the-mud jack-of-all-trades a flash in the pan Hyphenated or open as listed in Webster's. If not in the dictionary, open.
Compounds and Hyphenation 7.89 Category/specific term Examples i 1 Summary of rule 2. COMPOUNDS ACCORDING TO PARTS OF SPEECH (continued) ‘ f \ < phrases, verbal babysit handcraft air-condition fast-talk strong-arm sucker punch Closed, hyphenated, or open as listed in Webster’s. If not in the dictionary, leave open.
proper nouns and adjectives relating to geography or nation ality African Americans African American president a Chinese American French Canadians South Asian Americans the Scotch Irish the North Central region Middle Eastern countries but Sino-Tibetan languages the Franco-Prussian War the US-Canada border Anglo-American cooperation Anglo-Americans Open in both noun and adjective forms, unless the first term is a prefix or unless between is implied. See also 8.39.
3. COMPOUNDS FORM!ID WITH SPECIFIC TERMS ache toothache stomachache Always closed.
all all out all along all over an all-out effort an all-American player the book is all-encompassing but we were all in [tired] Adverbial phrases open; adjectival phrases usually hyphenated both before and after a noun.
book reference book coupon book checkbook cookbook Closed or open as listed in Webster’s. If not in the dictio nary, open.
borne waterborne food-borne mosquito-borne Closed if listed as such in Webster’s. If not in Webster’s, hyphenated.
7.89 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS Category/specific term ' Examples Summary of rule 3 COMPOUNDS FORMED WITH SPECIFIC TERMS (tOhtinUed) . / century the twenty-first century fourteenth-century monastery twenty-first-century history a mid-eighteenth-century poet late nineteenth-century photographs her style was nineteenth century Noun forms always open; adjectival compounds hyphenated before but not after a noun. See also old (below), mid (in section 4), and 7.87, 9.32.
cross a cross section a cross-reference cross-referenced cross-grained cross-country crossbow crossover Many compounds formed with cross are in Webster’s (as those listed here). If not in Webster’s, leave noun forms open; hyphenate adjective, adverb, and verb forms.
e email e-book e-commerce eBay Hyphenated except for email (a departure from previous editions) and certain proper nouns. See also 8.154.
elect president-elect vice president elect mayor-elect county assessor elect Hyphenated unless the name of the office consists of an open compound. See also vice.
ever ever-ready help ever-recurring problem everlasting he was ever eager Usually hyphenated before but not after a noun; some permanent compounds closed.
ex ex-partner ex-marine ex-corporate executive Hyphenated, but use en dash if ex- precedes an open compound (see 6.80).
fold fourfold hundredfold but twenty-five-fold 150-fold Hyphenated with hyphen ated forms of spelled-out numbers or with numerals; otherwise closed.
foster foster mother foster parents a foster-family background Noun forms open; adjective forms hyphenated.
free toll-free number accident-free driver the number is toll-free the driver is accident-free Compounds formed with free as second element are hyphenated both before and after a noun.
Compounds and Hyphenation 7.89 Category/specific term Examples Summary of rule 3. COMPOUNDS FORMED WITH SPECIFIC TERMS (continued) full full-length mirror the mirror is full length three bags full a suitcase full Hyphenated before a noun, otherwise open. Use ful only in such permanent com pounds as cupful, handful.
general attorney general postmaster general lieutenants general Always open; in plural forms, general remains singular.
grand, great-grand grandfather granddaughter great-grandmother great-great-grandson Grand compounds closed; great compounds hyphen ated.
half half-asleep half-finished a half sister a half hour a half-hour session halfway halfhearted we half expected to fly Adjective forms hyphen ated before and after the noun; noun and verb forms open. Some permanent compounds closed, whether nouns, adjectives, or ad verbs. Check Webster’s. See also fractions in section 1.
house schoolhouse courthouse safe house rest house Closed or open as listed in Webster’s. If not in the dictio nary, open.
in-law sister-in-law parents-in-law All compounds hyphenated; only the first element takes a plural form.
like catlike childlike Christlike bell-like a penitentiary-like institution Closed if listed as such in Webster’s. If not in Webster’s, hyphenated; compounds re tain the hyphen both before and after a noun.
mid. See section 4.
near in the near term a near accident a near-term proposal a near-dead language Noun forms open; adjective forms hyphenated.
odd a hundred-odd manuscripts 350-odd books Always hyphenated.
7.89 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS Category/specific term Examples Summary of rule ’3 COMPOUNDS, FORMED WITH SPECIFIC TERMS (continued) - '•' ' > ’ , old a three-year-old a 105-year-old woman a decade-old union a centuries-old debate a child who is three years old the debate is centuries old Noun forms hyphenated. Adjective forms hyphen ated before a noun, open after. See also age terms in section 1.
on online onstage ongoing on-screen on-site Sometimes closed, some times hyphenated. Check Webster’s and hyphenate if term is not listed. See also 7.83.
percent. See number +percent in section 1.
pseudo. See section 4.
quasi a quasi corporation a quasi-public corporation quasi-judicial quasiperiodic quasicrystal Noun form usually open; adjective form usually hyphenated. A handful of permanent compounds are listed in Webster’s.
self self-restraint self-realization self-sustaining self-conscious the behavior is self-destructive selfless unselfconscious Both noun and adjective forms hyphenated, except where self is followed by a suffix or preceded by un.
step stepbrother stepparent step-granddaughter step-great-granddaughter Always closed except with grand and great.
style dined family-style 1920s-style dancing danced 1920s-style Chicago-style hyphenation according to Chicago style headline-style capitalization capitalized headline-style use headline style Adjective and adverb forms hyphenated; noun form usually open.
Compounds and Hyphenation 7.89 Category/specific term i Examples Summary of rule 3. COMPOUNDS FORMED WITH SPECIFIC TERMS (continued) vice vice-consul vice-chancellor vice president vice-presidential duties a speech that was vice presidential vice admiral viceroy Adjective forms hyphenated before the noun; noun forms sometimes hyphenated, sometimes open, occasion ally closed. Check Webster’s and hyphenate if term is not listed.
web a website a web page web-related matters Noun form open or closed, as shown; if term is not in any dictionary, opt for open. Adjective form hyphenated. See also 7.80.
wide worldwide citywide Chicago-wide the canvass was university-wide Closed if listed as such in Webster’s. If not in Webster’s, hyphenated; compounds re tain the hyphen both before and after a noun.
4. WORDS FORMED WITH PREFIXES Compounds formed with prefixes are normally closed, whether they are nouns, verbs, adjec tives, or adverbs. A hyphen should appear, however, (1) before a capitalized word or a nu meral, such as sub-Saharan, pre-1950; (2) before a compound term, such as non-self-sustaining, pre-Vietnam War (before an open compound, an en dash is used; see 6.80); (3) to separate two i’s, two a’s, and other combinations of letters or syllables that might cause misreading, such as anti-intellectual, extra-alkaline, pro-life', (4) to separate the repeated terms in a double prefix, such as sub-subentry; (5) when a prefix or combining form stands alone, such as over- and underused, macro- and microeconomics. The spellings shown below conform largely to Merriam- Webster’s Collegiate Dictionary. Compounds formed with combining forms not listed here, such as auto, tri, and para, follow the same pattern.
ante antebellum, antenatal, antediluvian anti antihypertensive, antihero, but anti-inflammatory, anti-Hitlerian bi binomial, bivalent, bisexual bio bioecology, biophysical, biosociology cis cisgender, cissexual, cisatlantic, but cis-Victorian, ds-2-pentene (cis in italics), cis male (cis as adjective) co coequal, coauthor, coeditor, coordinate, cooperation, coworker, but co-op, co-opt counter counterclockwise, counterrevolution cyber cyberspace, cyberstore 7.89 SPELLING, DISTINCTIVE TREATMENT OF WORDS, AND COMPOUNDS 4 WORDS F 6 R ME D' WIT H P REFIXES (coritliiued)<88®||J de decompress, deconstruct, deontological, but de-emphasize, de-stress extra extramural, extrafine, but extra-administrative hyper hypertension, hyperactive, hypertext infra infrasonic, infrastructure inter interorganizational, interfaith intra intrazonal, intramural, but intra-arterial macro macroeconomics, macromolecular mega megavitamin, megamall, but mega-annoyance meta metalanguage, metaethical, but meta-analysis (not the same as metanalysis) micro microeconomics, micromethodical mid midthirties, a midcareer event, midcentury, but mid-July, the mid-1990s, the mid-twentieth century, mid-twentieth-century history mini minivan, minimarket multi multiauthor, multiconductor, but multi-institutional neo neonate, neoorthodox, Neoplatonism, neo-Nazi (neo lowercase or capital and hyphenated as in dictionary; lowercase and hyphenate if not in dictionary) non nonviolent, nonevent, nonnegotiable, but non-beer-drinking over overmagnified, overshoes, overconscientious post postdoctoral, postmodernism, posttraumatic, but post-Vietnam, post-World War II (see 6.80) pre premodern, preregistration, prewar, preempt, but pre-Columbian, Pre-Raphaelite (pre lowercase or capital as in dictionary; lowercase if term is not in dictionary) pro proindustrial, promarket, but pro-life, pro-Canadian proto protolanguage, protogalaxy, protomartyr pseudo pseudotechnocrat, pseudomodern, but pseudo-Tudor re reedit, reunify, reproposition, but re-cover, re-creation (as distinct from recover, recreation) semi semiopaque, semiconductor, but semi-invalid sub subbasement, subzero, subcutaneous super superannuated, supervirtuoso, superpowerful Compounds and Hyphenation 7.89 supra supranational, suprarenal, supraorbital, but supra-American trans transgender, transsexual, transmembrane, transcontinental, transatlantic, but trans-American, trans-2-pentene (trans in italics), trans fat (trans as adjective) ultra ultrasophisticated, ultraorganized, ultraevangelical un unfunded, unneutered, but un-English, un-unionized under underemployed, underrate, undercount 8 • Names, Terms, and Titles of Works OVERVIEW 8.1 PERSONAL NAMES 8.3 General Principles 8.3 Non-English Names in an English Context 8.7 TITLES AND OFFICES 8.19 EPITHETS, KINSHIP NAMES, AND PERSONIFICATIONS 8.34 ETHNIC, SOCIOECONOMIC, AND OTHER GROUPS 8.38 NAMES OF PLACES 8.44 Parts of the World 8.45 Political Divisions 8.51 Topographical Divisions 8.53 Public Places and Major Structures 8.56 WORDS DERIVED FROM PROPER NAMES 8.60 NAMES OF ORGANIZATIONS 8.62 Governmental Bodies 8.62 Political and Economic Organizations and Movements 8.66 Institutions and Companies 8.68 Associations 8.70 HISTORICAL AND CULTURAL TERMS 8.71 Periods 8.71 Events 8.75 Cultural Movements and Styles 8.79 Acts, Treaties, and Government Programs 8.80 Legal Cases 8.82 Awards 8.83 Oaths and Pledges 8.84 Academic Subjects, Courses of Study, and Lecture Series 8.85 CALENDAR AND TIME DESIGNATIONS 8.88 RELIGIOUS NAMES AND TERMS 8.91 Deities and Revered Persons 8.91 Religious Groups 8.96 Religious Writings 8.103 Religious Events, Concepts, Services, and Objects 8.108 MILITARY TERMS 8.112 Forces and Troops 8.112 Wars, Revolutions, Battles, and Campaigns 8.113 Military Awards 8.115 NAMES OF SHIPS AND OTHER VEHICLES 8.116 SCIENTIFIC TERMINOLOGY 8.119 Scientific Names of Plants and Animals 8.119 Vernacular Names of Plants and Animals 8.128 Genetic Terms 8.131 Geological Terms 8.134 Astronomical Terms 8.137 Medical Terms 8.143 Physical and Chemical Terms 8.147 BRAND NAMES AND TRADEMARKS 8.153 SOFTWARE AND DEVICES 8.155 TITLES OF WORKS 8.156 Capitalization, Punctuation, and Italics 8.157 Books and Periodicals 8.168 Articles in Periodicals and Parts of a Book 8.177 Poems and Plays 8.181 Fairy Tales and Nursery Rhymes 8.185 Pamphlets, Reports, and Forms 8.186 Unpublished Works 8.188 Movies, Television, Radio, and Podcasts 8.189 Video Games 8.190 Websites and Blogs 8.191 Musical Works 8.193 Works of Art and Exhibitions 8.198 Personal Names 8.3 Overview 8.1 Chicago's preference for lowercase. Proper nouns are usually capital ized, as are some of the terms derived from or associated with proper nouns. For the latter, Chicago’s preference is for sparing use of capitals— what is sometimes referred to as a “down style.” Although Brussels (the Belgian city) is capitalized, Chicago prefers brussels sprouts—which are not necessarily from Brussels (see 8.61). Likewise, President Taft is cap italized, but the president is not (see 8.19-33). (In certain nonacademic contexts—e.g., a press release—such terms as president may be capital ized.) The term “down style” may also refer to sentence-style capitaliza tion for titles of works (see 8.158); in most contexts, Chicago prefers headline-style capitalization (sometimes called “up style”; see 8.159).
8.2 Italics and quotation marks for titles and other terms. Chicago pre fers italics to set off the titles of major or freestanding works such as books, journals, movies, and paintings. This practice extends to cover the names of ships and other craft, species names, and legal cases. Quo tation marks are usually reserved for the titles of subsections of larger works—including chapter and article titles and the titles of poems in a collection. Some titles—for example, of a book series or a website, under which any number of works or documents may be collected—are neither italicized nor placed in quotation marks. For more on the titles of works, including matters of capitalization and punctuation, see 8.156-201. For the use of italics and quotation marks to highlight or set off certain let ters, words, or phrases, see 7.48-69.
Personal Names General Principles 8.3 Personal names—additional resources. For names of well-known de ceased persons, Chicago generally prefers the spellings in the biograph ical entries of Merriam-Webster’s Collegiate Dictionary (referred to be low as Webster’s), which are incorporated into the regularly updated online version (see bibliog. 3.1), or, for names not found there, Ency clopaedia Britannica (bibliog. 4.3). For living persons, consult either Webster’s or Britannica or, for names not found there, Who’s Who or Who’s Who in America, among other resources. (See bibliog. 4.1 for these and other useful works.) Where different spellings appear in different sources (e.g., W. E. B. DuBois versus W. E. B. Du Bois), the writer or editor must 8.4 NAMES, TERMS, AND TITLES OF WORKS make a choice and stick with it. Names of known and lesser-known per sons not in the standard references can usually be checked and cross checked at any number of reputable online resources (e.g., for authors’ names, library catalogs or booksellers). The name of a living person should, wherever possible, correspond to that person’s preferred usage.
8.4 Capitalization of personal names. Names and initials of persons, real or fictitious, are capitalized. A space should be used between any initials, except when initials are used alone. See also 7.66,10.12.
Jane Doe P. D. James Malcolm X George S. McGovern M. F. K. Fisher LBJ Unconventional spellings strongly preferred by the bearer of the name or pen name (e.g., bell hooks) should usually be respected in appropri ate contexts (library catalogs generally capitalize all such names). E. E. Cummings can be safely capitalized; it was one of his publishers, not he himself, who lowercased his name. Most editors will draw the line at be ginning a sentence with a lowercased name and choose either to rewrite or to capitalize the first letter for the occasion. When a personal name includes a lowercase particle, the particle is capitalized if it begins a sen tence or a note (but see 14.21; see also 8.5).
8.5 Names with particles. Many names include particles such as de, d', de la, el, von, van, and ten. Practice with regard to capitalizing and spacing the particles varies widely, and confirmation should be sought in a biograph ical dictionary or other authoritative source. When the surname is used alone, the parti.cle is usually (but not always) retained, capitalized or low ercased and spaced as in the full name (though always capitalized when beginning a sentence or a note; see also 8.4). Le, La, and I’ are always capitalized when not preceded by de', the, which sometimes appears with the English form of a Native American name, can be lowercased unless it begins a sentence or a note. See also 8.7, 8.8,8.9,8.10,8.11,8.14,8.34.
Alfonse D’Amato; D’Amato Diana DeGette; DeGette Walter de la Mare; de la Mare Paul de Man; de Man Thomas De Quincey; De Quincey Page duBois; duBois W. E. B. DuBois; DuBois (but see 8.3) Daphne du Maurier; du Maurier Robert M. La Follette Sr.; La Follette John Le Carre; Le Carre Pierre-Charles L’Enfant; L’Enfant Farouk El-Baz; El-Baz Abraham Ten Broeck; Ten Broeck the Prophet (Tenskwatawa) Robert van Gulik; van Gulik Stephen Van Rensselaer; Van Rensselaer Wernher von Braun; von Braun Personal Names 8.7 8.6 Hyphenated and extended names. A hyphenated last name or a last name that consists of two or more elements should usually retain each element. (In the case of someone who is generally known by a shorter form, that form may be used, but only after the fuller form has been es tablished.) For names of prominent or historical figures, Webster’s and other reliable alphabetical listings usually indicate where the last name begins.
Victoria Sackville-West; Sackville-West Ralph Vaughan Williams; Vaughan Williams (not Williams) Ludwig Mies van der Rohe; Mies van der Rohe (not van der Rohe); Mies but John Hope Franklin; Franklin Charlotte Perkins Gilman; Gilman For unhyphenated compound names of lesser-known persons for whom proper usage cannot be determined, use only the last element (including any particle [s]; see 8.5). But see 8.11.
Non-English Names in an English Context 8.7 French names. The particles de and d’ are lowercased (except at the be ginning of a sentence). When the last name is used alone, de (but not d’) is often dropped. Its occasional retention, in de Gaulle, for example, is suggested by tradition rather than logic. (When a name begins with closed-up de, such as Debussy, the d is always capitalized.) Jean d’Alembert; d’Alembert Alfred de Musset; Musset Alexis de Tocqueville; Tocqueville but Charles de Gaulle; de Gaulle When de la precedes a name, la is usually capitalized and is always re tained when the last name is used alone. The contraction du is usually lowercased in a full name but is retained and capitalized when the last name is used alone. (When a name begins with closed-up Du, such as Dupont, the d is always capitalized.) Jean de La Fontaine; La Fontaine Rene-Robert Cavelier de La Salle; La Salle Philippe du Puy de Clinchamps; Du Puy de Clinchamps 8.8 NAMES, TERMS, AND TITLES OF WORKS When the article le accompanies a name, it is capitalized with or without the first name.
Gustave Le Bon; Le Bon Initials standing for a hyphenated given name are usually hyphenated.
Jean-Paul Sartre; J.-P. Sartre; Sartre There is considerable variation in French usage; the guidelines and ex amples above merely represent the most common forms.
8.8 German and Portuguese names. In the original languages, particles in German and Portuguese names are lowercased and are usually dropped when the last name is used alone. But in English contexts, if another form is widely known, it may be used instead.
Alexander von Humboldt; Humboldt Maximilian von Spee; Spee Heinrich Friedrich Karl vom und zum Stein; Stein Ludwig van Beethoven; Beethoven Agostinho da Silva; Silva but Vasco da Gama; da Gama 8.9 Italian names. Particles in Italian names are most often uppercased and retained when the last name is used alone.
Gabriele D’Annunzio; D’Annunzio Lorenzo Da Ponte; Da Ponte Luca Della Robbia; Della Robbia In many older aristocratic names, however, the particle is traditionally lowercased and dropped when the last name is used alone.
Beatrice d’Este; Este Lorenzo de’ Medici; Medici 8.10 Dutch names. In English usage, the particles van, van den, ter, and the like are lowercased when full names are given but usually capitalized when only the last name is used.
Johannes van Keulen; Van Keulen Pieter van den Keere; Van den Keere Personal Names 8.12 Vincent van Gogh; Van Gogh Gerard ter Borch; Ter Borch 8.11 Spanish names. Many Spanish names are composed of both the father’s and the mother’s family names, usually in that order, sometimes joined byj; (and). When the given name is omitted, persons with such names are usually referred to by both family names but sometimes by only one (usually, but not always, the first of the two family names), according to their own preference (or, sometimes, to established usage). It is never incorrect to use both.
Jose Ortega y Gasset; Ortega y Gasset or Ortega Pascual Ortiz Rubio; Ortiz Rubio or Ortiz Federico Garcia Lorca; Garcia Lorca (popularly known as Lorca) Spanish family names that include an article, a preposition, or both are treated in the same way as analogous French names.
Tomas de Torquemada; Torquemada Manuel de Falla; Falla Bartolome de Las Casas; Las Casas Gonzalo Fernandez de Oviedo; Fernandez de Oviedo Traditionally, a married woman replaced her mother’s family name with her husband’s (first) family name, sometimes preceded by de. If, for example, Maria Carmen Mendoza Salinas married Juan Alberto Pena Montalvo, she could change her legal name to Maria Carmen Mendoza (de) Pena or, if the husband was well known by both family names, to Maria Carmen Mendoza (de) Pena Montalvo. Many women in Spanish speaking countries, however, no longer take their husband’s family name. For alphabetizing, see 16.84.
8.12 Russian names. Russian family names, as well as patronymics (the name preceding the family name and derived from the name of the father), sometimes take different endings for male and female members of the family. For example, Lenin’s real name was Vladimir Ilyich Ulyanov (given name, patronymic, family name); his sister Maria was Maria Ilyinichna Ulyanova. In Russian sources (and, by extension, their En glish translations), often only the given name and patronymic are used; in such instances the patronymic should not be confused either for a middle name or for the family name.
8.13 NAMES, TERMS, AND TITLES OF WORKS 8.13 Hungarian names. In Hungarian practice, the family name precedes the given name—for example, Molnar Ferenc, Kodaly Zoltan. In English contexts, however, such names are usually inverted—Ferenc Molnar, Zoltan Kodaly. In some cases, the family name includes an initial—for example, E. Kiss Katalin. When such a name is inverted for English con texts (i.e., to become Katalin E. Kiss), the initial should not be confused for a middle initial. When such a name is inverted, as for an index, it is properly listed under the initial (see 16.78).
8.14 Arabic names. Surnames of Arabic origin (which are strictly surnames rather than family names) are often prefixed by such elements as Abu, Abd, Ibn, al, or el. Since these are integral parts of a name, just as Me or Fitz forms a part of certain names, they should usually not be dropped when the surname is used alone. Capitalization of such elements varies widely, but terms joined with a hyphen may usually be lowercased. See also 11.79,11.80,16.75.
Syed Abu Zafar Nadvi; Abu Zafar Nadvi Abdul Aziz Ibn Saud; Ibn Saud Tawfiq al-Hakim; al-Hakim but Anwar el-Sadat; Sadat (as commonly known) Names of rulers of older times, however, are often shortened to the first part of the name rather than the second.
Harun al-Rashid; Harun (al-Rashid, “Rightly Guided,” was Harun’s laqab, a de scriptive name he took on his accession to the caliphate) 8.15 Chinese names. In Chinese practice, the family name comes before the given name. (This practice should be followed in English contexts with names of Chinese persons but not with those of persons of Chinese ori gin whose names have been anglicized.) For use of the Pinyin and Wade- Giles systems of transliteration, see 11.82-90.
Chiang Kai-shek; Chiang (Wade-Giles) Mao Tse-tung; Mao (Wade-Giles) Li Bai; Li (Pinyin) Du Fu; Du (Pinyin) but Anthony Yu; Yu Tang Tsou; Tsou Titles and Offices 8.19 8.16 Japanese names. In Japanese usage, the family name precedes the given name. Japanese names are frequently westernized, however, by authors writing in English or persons of Japanese origin living in the West.
Yoshida Shigeru; Yoshida Kanda Nobuo; Kanda but Akira Kurosawa; Kurosawa Shinzo Abe; Abe 8.17 Korean names. In Korean usage, the family name precedes the given name, and this is how it is usually presented even in English-language contexts. Persons of Korean origin living in the West, however, often in vert this order.
Kim Dae-jung; Kim Oh Jung-hee; Oh but Chang-rae Lee; Lee 8.18 Other Asian names. In some Asian countries, people are usually known by their given name rather than by a surname or family name. The In donesian writer Pramoedya Ananta Toer, for example, is referred to in short form as Pramoedya (not as Toer). For further examples, see 16.76, 16.80,16.85,16.86. If in doubt, use the full form of a name in all refer ences or consult an expert or consult the usage in a reputable source that discusses the person in question.
Titles and Offices 8.19 Titles and offices—the general rule. Civil, military, religious, and pro fessional titles are capitalized when they immediately precede a personal name and are thus used as part of the name (traditionally replacing the title holder’s first name). In formal prose and other generic text, titles are normally lowercased when following a name or used in place of a name (but see 8.20). For abbreviated forms, see 10.11-26.
Abraham Lincoln, president of the United States (or President Abraham Lincoln of the United States); President Lincoln; the president General Bradley; the general Cardinal Newman; the cardinal Governors Ige and Brown; the governors 8.20 NAMES, TERMS, AND TITLES OF WORKS Although a full name may be used with a capitalized title (e.g., President Abraham Lincoln)—and though it is perfectly correct to do so—some writers choose to avoid using the title before a full name in formal prose, especially with civil, corporate, and academic titles (see 8.22,8.27,8.28). (For titles used in apposition to a name, see 8.21.) Note also that once a title has been given, it need not be repeated each time a person’s name is mentioned.
Elizabeth Warren, senator from Massachusetts (or Senator Elizabeth Warren of Massachusetts); Senator Warren; Warren; the senator 8.20 Exceptions to the general rule for titles and offices. In promotional or ceremonial contexts such as a displayed list of donors in the front matter of a book or a list of corporate officers in an annual report, titles are usu ally capitalized even when following a personal name. Exceptions may also be called for in other contexts for reasons of courtesy or diplomacy.
Maria Martinez, Director of International Sales A title used alone, in place of a personal name, is capitalized only in such contexts as a toast or a formal introduction, or when used in direct ad dress (see also 6.53, 8.36).
Ladies and Gentlemen, the Prime Minister.
I would have done it, Captain, but the ship was sinking.
Thank you, Mr. President.
8.21 Titles used in apposition. When a title is used in apposition before a per sonal name—that is, not alone and as part of the name but as an equiva lent to it, usually preceded by the or by a modifier—it is considered not a title but rather a descriptive phrase and is therefore lowercased.
the empress Elisabeth of Austria (but Empress Elisabeth of Austria) German chancellor Angela Merkel (but Chancellor Merkel) the Argentinian-born pope Francis former president Carter former presidents Reagan and Ford the then secretary of state Hillary Clinton 8.22 Civil titles. Much of the usage below is contradicted by the official lit erature typically generated by political offices, where capitalization of a title in any position is the norm (see 8.20). In formal prose, however, Titles and Offices 8.22 civil titles are capitalized only when used as part of the name (except as noted). See also 10.13.
the president; George Washington, first president of the United States; Presi dent Washington; the presidency; presidential; the Washington administra tion; Washington; Benigno Aquino III, president of the Philippines; President Aquino; Aquino the vice president; John Adams, vice president of the United States; Vice Presi dent Adams; vice presidential duties the secretary of state; John Kerry, secretary of state; Secretary of State Kerry or Secretary Kerry the senator; the senator from New York; New York senator Kirsten E. Gillibrand (see 8.21); Senator Gillibrand; Senators Gillibrand and Schumer; Senator Mikulski, Democrat from Maryland (or D-MD) the representative; the congressman; the congresswoman; Robin Kelly, represen tative from Illinois or congresswoman from Illinois; Congresswoman Kelly or Rep. Robin Kelly (D-IL); Kay Granger, representative from Texas; Congress woman Granger; the congresswoman or the representative; Representatives Kelly and Granger the Speaker; Paul Ryan, Speaker of the House of Representatives; Speaker Ryan (Speaker is best capitalized in all contexts to avoid conflation with generic speakers) the chief justice; John G. Roberts Jr., chief justice of the United States; Chief Jus tice Roberts (see also 8.64) the associate justice; Elena Kagan, associate justice; Justice Kagan; Justices Kagan and Sotomayor the chief judge; Timothy C. Evans, chief judge; Judge Evans the ambassador; Matthew W. Barzun, ambassador to the Court of St. James’s or ambassador to the United Kingdom; Ambassador Barzun the governor; Earl Ray Tomblin, governor of the state of West Virginia; Governor Tomblin the mayor; Rahm Emanuel, mayor of Chicago; Mayor Emanuel the state senator; Teresa Fedor, Ohio state senator; the Honorable Teresa Fedor the state representative (same pattern as state senator) the governor general of Canada; the Right Honourable David Johnston the minister of finance (or finance minister); Arun Jaitley, finance minister of India; Jaitley the prime minister; the Right Honourable Pierre Elliott Trudeau, former prime minister of Canada; Theresa May, the British prime minister the premier (of a Canadian province); the Honourable Brad Wall the member of Parliament (UK and Canada); Jane Doe, member of Parliament, or, more commonly, Jane Doe, MP; Jane Doe, the member for West Hamage 8.23 NAMES, TERMS, AND TITLES OF WORKS the chief whip; Jackson Mphikwa Mthembu, chief whip of the African National Congress; Mthembu the foreign secretary (UK); the foreign minister (other nations); the British foreign secretary; the German foreign minister the chancellor; Angela Merkel, chancellor of Germany; Chancellor Merkel the chancellor of the exchequer (UK); George Osborne; Chancellor Osborne the Lord Privy Seal (UK; always capitalized) For use of the Honorable and similar terms of respect, see 8.33,10.18.
8.23 Titles of sovereigns and other rulers. Most titles of sovereigns and other rulers are lowercased when used alone. See also 8.32.
King Abdullah II; the king of Jordan Queen Elizabeth; Elizabeth II; the queen (in a British Commonwealth context, the Queen) the Holy Roman emperor Nero, emperor of Rome; the Roman emperor Hamad bin Isa al-Khalifa, king of Bahrain; King Hamad the shah of Iran the sharif of Mecca the paramount chief of Basutoland Wilhelm II, emperor of Germany; Kaiser Wilhelm II; the kaiser the fuhrer (Adolf Hitler) Il Duce (used only of Benito Mussolini; both i and d capitalized) 8.24 Military titles. As is the case with civil titles, military titles are routinely capitalized in the. literature of the organization or government with which they are associated. Nonetheless, in formal prose, most such titles are capitalized only when used as part of a person’s name. Occasional exceptions may be made if ambiguity threatens. See also 10.13.
the general; General Ulysses S. Grant, commander in chief of the Union army; General Grant; the commander in chief the general of the army; Omar N. Bradley, general of the army; General Bradley the admiral; Chester W. Nimitz, fleet admiral; Admiral Nimitz, commander in chief of the Pacific Fleet the chairman; Joseph F. Dunford Jr., chairman of the Joint Chiefs of Staff; General Dunford the captain; Captain Frances LeClaire, company commander the sergeant; Sergeant Carleton C. Singer; a noncommissioned officer (NCO) the warrant officer; Warrant Officer John Carmichael the chief petty officer; Chief Petty Officer Tannenbaum Titles and Offices 8.27 the private; Private T. C. Alhambra the British general; General Sir Guy Carleton, British commander in New York City; General Carleton For abbreviations, often used when a title precedes a name and appropri ate in material in which many military titles appear, see 10.15.
8.25 Quasi-military titles. Titles and ranks used in organizations such as the police, the merchant marine, or the Salvation Army are treated the same way as military titles.
the chief of police; Frederick Day, Parkdale chief of police; Chief Day the warden; Jane Simmons, warden of the state penitentiary; Warden Simmons 8.26 Religious titles. Religious titles are treated much like civil and military titles (see 8.22, 8.24).
the rabbi; Rabbi Avraham Yitzhak ha-Kohen Kuk; the rabbinate the cantor or hazan; Deborah Bard, cantor; Cantor Bard the sheikh; Sheikh Ibrahim el-Zakzaky the imam; Imam Shamil the ayatollah; Ayatollah Khomeini the Dalai Lama (traditionally capitalized); but previous dalai lamas the sadhu; the guru; the shaman the pope; Pope Francis; the papacy; papal the cardinal; John Cardinal Dew (in formal contexts) or Cardinal John Dew; Car dinal Dew; the sacred college of cardinals the patriarch; Cyrillus Lucaris, patriarch of Constantinople; the patriarchate the archbishop; the archbishop of Canterbury; Archbishop Williams (or, in this case, Dr. Williams) the bishop; the bishop of Toledo; Bishop Donnelly; bishopric; diocese the minister; the Reverend Shirley Stoops-Frantz the rector; the Reverend James Williams (see also 10.18,8.33) 8.27 Corporate and organizational titles. Titles of persons holding offices such as those listed below are rarely used as part of a name. If a short form is required, either the generic term or simply a personal name suf fices.
the chief executive officer; Pat Beldos, chief executive officer of Caterham Indus tries; the CEO the director; Gabriel Dotto, director of the Michigan State University Press 8.28 NAMES, TERMS, AND TITLES OF WORKS the school superintendent; Janice Bayder, superintendent of Coriander Township High School District the secretary-treasurer; Georgina Fido, secretary-treasurer of the Kenilworth Kennel Society 8.28 Academic titles. Academic titles generally follow the pattern for civil titles (see 8.22).
the professor; Fran^oise Meltzer, professor of comparative literature; Professor Meltzer the chair; Mark Payne, chair of the Department of Classics; Professor Payne (but see 8.30) the provost; Eric D. Isaacs, provost of the University of Chicago; Isaacs the president; Robert J. Zimmer, president of the University of Chicago; Zimmer or President Zimmer the dean; John W. Boyer, dean of the College at the University of Chicago (the College is an official division of the University of Chicago); Dean Boyer named professorships; Wendy Doniger, Mircea Eliade Distinguished Service Pro fessor of the History of Religions in the Divinity School; Professor Doniger; Anthony Grafton, Dodge Professor of History, Princeton University; Professor Grafton the professor emeritus (masc.); the professor emerita (fem.); professors emer iti (masc. or masc. and fem.); professors emeritae (fem.); Professor Emerita Neugarten (note that emeritus and emerita are honorary designations and do not simply mean “retired”) 8.29 Other academic designations. Terms denoting student status are lower cased.
freshman or first-year student sophomore junior senior Names of degrees, fellowships, and the like are lowercased when re ferred to generically. See also 10.21.
a master’s degree; a doctorate; a fellowship; master of business administration (MBA) 8.30 Descriptive titles. When preceding a name, generic titles that describe a person’s role or occupation—such as philosopher or historian—should be lowercased and treated as if in apposition (see 8.21). Compare 8.28.
the historian William McNeill (not Historian McNeill) Titles and Offices 8.32 8.31 Civic and academic honors. Titles denoting civic or academic honors are capitalized when following a personal name. For awards, see 8.83; for abbreviations, see 10.22.
Roberta Bondar, Fellow of the Royal Society of Canada; the fellows 8.32 Titles of nobility. Unlike most of the titles mentioned in the previous paragraphs, titles of nobility do not denote offices (such as that of a pres ident or an admiral). Whether inherited or conferred, they form an inte gral and, with rare exceptions, permanent part of a person’s name and are therefore usually capitalized. The generic element in a title, however (duke, earl, etc.), is lowercased when used alone as a short form of the name. (In British usage, the generic term used alone remains capitalized in the case of royal dukes but not in the case of nonroyal dukes; in North American usage, such niceties may be disregarded.) For further advice, consult The Times Style and Usage Guide (bibliog. 1.1), and for a compre hensive listing, consult the latest edition of Burke's Peerage, Baronetage, and Knightage (bibliog. 4.1). See also 8.23.
the prince; Prince Charles; the Prince of Wales the duke; the duchess; the Duke and Duchess of Windsor the marquess; the Marquess of Bath; Lord Bath the marchioness; the Marchioness of Bath; Lady Bath the earl; the Earl of Shaftesbury; Lord Shaftesbury; Anthony Ashley Cooper, 7th (or seventh) Earl of Shaftesbury; previous earls of Shaftesbury the countess (wife of an earl); the Countess of Shaftesbury; Lady Shaftesbury the viscount; Viscount Eccles; Lord Eccles Baroness Thatcher; Lady Thatcher Dame Judi Dench; Dame Judi (not Dame Dench) the baron; Lord Rutland the baronet; the knight; Sir Paul McCartney; Sir Paul (not Sir McCartney) Lady So-and-So [husband’s last name] (wife of a marquess, earl, baron, or bar onet) Lady Olivia So-and-So (daughter of a duke, marquess, or earl); Lady Olivia the Honourable Jessica So-and-So (daughter of a baron) the due de Guise (lowercased in accordance with French usage); Francois de Lorraine, due de Guise the count; Count Helmuth von Moltke or Graf Helmuth von Moltke; the Count of Toulouse or the comte de Toulouse Note that marquesses, earls, viscounts, barons, and baronesses are ad dressed, and referred to after first mention, as Lord or Lady So-and-So, at least in British usage. The following entry, drawn from Burke's Peerage, 8.33 NAMES, TERMS, AND TITLES OF WORKS Baronetage, and Knightage, illustrates the complexities of British noble nomenclature: The 5th Marquess of Salisbury (Sir Robert Arthur James Gascoyne-Cecil, K.G., P.C.), Earl of Salisbury, Wilts; Viscount Cranborne, Dorset, and Baron Cecil of Essendon, Rutland; co-heir to the Barony of Ogle 8.33 Honorifics. Honorific titles and respectful forms of address are capital ized in any context. For the use of many such terms in formal correspon dence, see “Forms of Address,” a comprehensive listing at the back of the print edition of Merriam-Webster s Collegiate Dictionary. For abbrevi ations, see 10.18. See also 8.26.
the Honorable Angus Stanley King Jr. (US senator, member of Congress, etc.) the Right Honourable Justin Trudeau (Canadian prime minister) the First Gentleman; the First Lady the Queen Mother Pandit Nehru Mahatma Gandhi Her (His, Your) Majesty; His (Her, Your) Royal Highness the Most Reverend William S. Skylstad (Roman Catholic bishop) Your (Her, His) Excellency Mr. President; Madam President Madam Speaker Your Honor but sir, ma’am my lord, my lady Epithets, Kinship Names, and Personifications 8.34 Epithets (or nicknames) and bynames. A descriptive or characterizing word or phrase used as part of, or instead of, a person’s name is capital ized. A the used as part of such a name is not capitalized (except, e.g., at the beginning of a sentence).
the Great Emancipator (Abraham Lincoln) the Sun King (Louis XIV) the Wizard of Menlo Park (Thomas Edison) Stonewall Jackson Old Hickory (Andrew Jackson) the Young Pretender (Charles Edward Stuart) Epithets, Kinship Names, and Personifications 8.36 the Great Commoner (William Jennings Bryan) Catherine the Great Babe Ruth the Swedish Nightingale (Jenny Lind) Ivan the Terrible When used in addition to a name, an epithet is enclosed in quotation marks and placed either within or after the name. Parentheses are un necessary.
George Herman “Babe” Ruth Jenny Lind, “the Swedish Nightingale” Ivan IV, “the Terrible” 8.35 Epithets as names of characters. In references to works of drama or fic tion, epithets or generic titles used in place of names are normally capi talized.
John Barrymore performed brilliantly as Chief Executioner.
Alice encounters the Red Queen and the Mad Hatter.
8.36 Kinship names and the like. Kinship names are lowercased unless they immediately precede a personal name or are used alone, in place of a personal name. Used in apposition, however, such names are lowercased (see 8.21). This usage extends to certain words that express a similar type of relationship. See also 8.20.
my father and mother the Bronte sisters Let’s write to Aunt Maud.
I believe Grandmother’s middle name was Marie.
Please, Dad, let’s go.
my daughter’s coach Ask Coach Wilson.
You can count on me, Coach.
but She adores her aunt Maud.
Kinship terms used in connection with religious offices or callings are treated similarly.
The note referred to a certain Brother Thomas, one of the brothers from the Fran ciscan monastery.
8.37 NAMES, TERMS, AND TITLES OF WORKS 8.37 Personifications. The poetic device of giving abstractions the attributes of persons, and hence capitalizing them, is rare in today’s writing. The use of capitals for such a purpose is best confined to quoted material.
“The Night is Mother of the Day, / The Winter of the Spring, / And ever upon old Decay / The greenest mosses cling.” (John Greenleaf Whittier) but In springtime, nature is at its best. It was a battle between head and heart; reason finally won.
Ethnic, Socioeconomic, and Other Groups 8.38 Ethnic and national groups and associated adjectives. Names of eth nic and national groups are capitalized. Adjectives associated with these names are also capitalized. For hyphenation or its absence, see 8.39. Note that terms such as black and white, when referring to ethnicity, are usu ally lowercased unless a particular author or publisher prefers otherwise.
Aboriginal peoples; Aboriginals (or Aborigines); an Aboriginal; Aboriginal art African Americans; African American culture American Indians; an American Indian (see text below) Arabs; Arabian Asians; Asian influence in the West; an Asian American the British; a British person or, colloquially, a Britisher, a Brit Caucasians; a Caucasian Chicanos; a Chicano; a Chicana European Americans the French; a Frenchman; a Frenchwoman; French Canadians Hispanics; a Hispanic the Hopi; a Hopi; Hopi customs (see also 7.10) Inuit; Inuit sculpture Italian Americans; an Italian American neighborhood Jews; a Jew; Jewish ethnicity (see also 8.96) Latinos; a Latino; a Latina; Latino immigration Metis; Metis history and culture Native Americans; Native American poetry (see text below) New Zealanders; New Zealand immigration Pygmies; a Pygmy; Pygmy peoples Romanies; a Romany; the Romany people but black people; blacks; people of color white people; whites Ethnic, Socioeconomic, and Other Groups 8.42 Many among those who trace their roots to the Aboriginal peoples of the Americas prefer American Indians to Native Americans, and in certain his torical works Indians may be more appropriate. Canadians often speak of First Peoples (and of First Nations) when not referring to specific groups by name.
8.39 Compound nationalities. Whether terms such as African American, Ital ian American, Chinese American, and the like should be spelled open or hyphenated has been the subject of considerable controversy. But since the hyphen does not aid comprehension in such terms as those men tioned above, it may be omitted unless a particular author or publisher prefers the hyphen. See also the table at 7.89, section 2, under proper nouns and adjectives relating to geography or nationality.
8.40 Class. Terms denoting socioeconomic classes or groups are lowercased.
the middle class; a middle-class neighborhood the upper-middle class; an upper-middle-class family the 1 percent blue-collar workers the aristocracy the proletariat homeless people 8.41 Sexual orientation and gender identity. Terms that refer to individuals or groups according to sexual orientation or gender identity or expres sion are lowercased. For pronoun use with individuals who do not iden tify with a gender-specific pronoun, see 5.48.
lesbians; lesbian history gay men transgender women; transgender men 8.42 Generation. Terms denoting generations are usually lowercased. Fol lowing Webster's and many other sources, however, lettered Generations X, Y, and Z are capitalized.
the me generation baby boomer(s); boomers; baby busters the MTV generation the millennials but Generation X, Generation Y, Generation Z 8.43 NAMES, TERMS, AND TITLES OF WORKS 8.43 Physical characteristics. Terms describing groups or individuals ac cording to a physical characteristic or a disability are usually lowercased.
wheelchair users blind persons deaf children Some writers capitalize deaf when referring to people who identify them selves as members of the distinct linguistic and cultural group whose primary language is ASL—the Deaf community—and lowercase it when referring to people who have a hearing loss or to those deaf people who prefer oral methods of communication. See also 11.125.
Names of Places 8.44 Names of places—additional resources. For the spelling of place-names, consult the geographical listings in Merriam-Webster’s Collegiate Dictio nary (bibliog. 3.1) or Encyclopaedia Britannica (bibliog. 4.3) or, for names not listed there, the United States Board on Geographic Names or one of the other resources listed in the bibliography (bibliog. 4.2). Since names of countries and cities often change, frequently updated online resources should be preferred over print for modern place-names. For country names, the US Central Intelligence Agency’s World Factbook is a good place to start (bibliog. 4.2). For historical works, writers and editors should attempt to use the form of names appropriate to the period under discussion.
Parts of the World 8.45 Continents, countries, cities, oceans, and such. Entities that appear on maps are always capitalized, as are adjectives and nouns derived from them. An initial the as part of a name is lowercased in running text, ex cept in the rare case of an initial the in the name of a city.
Asia; Asian Ireland; Irish California; Californian Chicago; Chicagoan Atlantic Ocean; Atlantic South China Sea the North Pole the Netherlands; Dutch but The Hague 8.46 Points of the compass. Compass points and terms, derived from them are lowercased if they simply indicate direction or location. But see 8.47.
Names of Places 8.47 pointing toward the north; a north wind; a northern climate to fly east; an eastward move; in the southwest of France; southwesterly 8.47 Regions of the world and national regions. Terms that denote regions of the world or of a particular country are often capitalized, as are a few of the adjectives and nouns derived from such terms. The follow ing examples illustrate not only the principles sketched in 8.1 but also variations based on context and usage. For terms not included here or for which no suitable analogy can be made, consult Webster's or an ency clopedia: if an otherwise generic term is not listed there (either capital ized or, for dictionary entries, with the indication capitalized next to the applicable subentry), opt for lowercase. Note that exceptions based on specific regional, political, or historical contexts are inevitable (a few that are generally applicable are included below) and that an author’s strong preference should usually be respected. See also 8.46.
the Swiss Alps; the Australian Alps; the Alps; an Alpine village (if in the European or Australian Alps); Alpine skiing; but alpine pastures in the Rockies (see also 8.53) Antarctica; the Antarctic Circle; the Antarctic Continent the Arctic; the Arctic Circle; Arctic waters; a mass of Arctic air (but lowercased when used metaphorically, as in “an arctic stare”; see 8.61) Central America, Central American countries; central Asia; central Illinois; cen tral France; central Europe (but Central Europe when referring to the political division of World War I) the continental United States; the continent of Europe; but on the Continent (used to denote mainland Europe); Continental cuisine; but continental breakfast the East, eastern, an easterner (referring to the eastern part of the United States or other country); the Eastern Seaboard (or Atlantic Seaboard), East Coast (re ferring to the eastern United States); eastern Massachusetts (but East Tennes see); the East, the Far East, Eastern (referring to the Orient and Asian culture); the Middle East (or, formerly more common, the Near East), Middle Eastern (referring to Iran, Iraq, etc.); the Eastern Hemisphere; eastern Europe (but Eastern Europe when referring to the post-World War II division of Europe); east, eastern, eastward, to the east (directions) the equator; equatorial climate; the Equatorial Current; Equatorial Guinea (for merly Spanish Guinea); the forty-second parallel north (of the equator) the Great Plains; the northern plains; the plains (but Plains Indians) the Midwest, midwestern, a midwesterner (as of the United States); the middle of Texas (but Middle Tennessee) the North, northern, a northerner (of a country); the North, Northern, North erner (in American Civil War contexts); northern Ohio (but Northern Cal ifornia); North Africa, North African countries, in northern Africa; North America, North American, the North American continent; the North Atlantic, 8.48 NAMES, TERMS, AND TITLES OF WORKS a northern Atlantic route; the Northern Hemisphere; the Far North; north, northern, northward, to the north (directions) the Northeast, the Northwest, northwestern, northeastern, a northwesterner, a northeasterner (as of the United States); the Pacific Northwest; the Northwest Passage the poles; the North Pole; the North Polar ice cap; the South Pole; polar regions (see also Antarctica; the Arctic) the South, southern, a southerner (of a country); the South, Southern, a South erner (in American Civil War contexts); the Deep South; southern Minnesota (but Southern California); the South of France (region); Southeast Asia; South Africa, South African (referring to the Republic of South Africa); southern Af rica (referring to the southern part of the continent); south, southern, south ward, to the south (directions) the Southeast, the Southwest, southeastern, southwestern, a southeasterner, a southwesterner (as of the United States) the tropics, tropical; the Tropic of Cancer; the Neotropics, Neotropical (of the New World biogeographical region); the subtropics, subtropical the Upper Peninsula (of Michigan); the upper reaches of the Thames the West, western, a westerner (of a country); the West Coast; western Arizona (but West Tennessee); the West, Western (referring to the culture of the Occi dent, or Europe and the Western Hemisphere; but westernize); west, western, westward, to the west (directions) 8.48 Popular place-names or epithets. Popular names of places, or epithets, are usually capitalized. Quotation marks are not needed. Some of the following examples may be used of more than one place. None should be used in contexts where they will not be readily understood. See also 8.34.
Back Bay the Badger State the Badlands the Bay Area the Beltway the Bible Belt the Big Island the Cape City of Light the Delta the East End the Eastern Shore the Eternal City the Fertile Crescent the Gaza Strip the Gulf the Holy City the Jewish Quarter the Lake District the Left Bank the Loop (Chicago) Midtown (Manhattan) the Old World the Panhandle the Promised Land the Rust Belt Silicon Valley Skid Row the South Seas the South Side the Sun Belt the Twin Cities the Upper West Side the Village (Greenwich Village) the West End the Wild West the Windy City Names of Places 8.51 Certain terms considered political rather than geographical need not be capitalized. Some editorial discretion is advisable, however. In reference to Soviet-era global politics, for example, the following terms might be suitably capitalized: the iron curtain or Iron Curtain the third world or Third World 8.49 Urban areas. Generic terms used for parts of urban areas are not capital ized.
the business district the inner city the metropolitan area; the greater Chicago metropolitan area; Chicagoland the tristate area but Greater London (an official administrative region) On the other hand, a work that treats a specific local culture may choose to favor an established local usage (e.g., Greater Boston).
8.50 Real versus metaphorical names. Mecca is capitalized when referring to the Islamic holy city, as is Utopia when referring to Thomas More’s imaginary country. Both are lowercased when used metaphorically. See also 8.61.
Stratford-upon-Avon is a mecca for Shakespeare enthusiasts.
She is trying to create a utopia for her children.
Political Divisions 8.51 Political divisions—capitalization. Words denoting political divisions— from empire, republic, and state down to ward and precinct— are capital ized when they follow a name and are used as an accepted part of the name. When preceding the name, such terms are usually capitalized in names of countries but lowercased in entities below the national level (but see 8.52). Used alone, they are usually lowercased, though reason able exceptions based on specific regional, political, or historical con texts should be respected. See also 9.46.
the Ottoman Empire; the empire the British Commonwealth; Commonwealth nations; the Commonwealth (but a commonwealth) 8.52 NAMES, TERMS, AND TITLES OF WORKS 8.52 the United States; the republic; the Union (Civil War era); the Confederacy (Civil War era) the United Kingdom; Great Britain; Britain (not the kingdom) the Russian Federation (formerly the Union of Soviet Socialist Republics; the So viet Union); Russia; the federation the Republic of South Africa (formerly the Union of South Africa); South Africa; the republic the Fifth Republic (France) the Republic of Indonesia; the republic the Republic of Lithuania; the republic the Federal Democratic Republic of Ethiopia; the republic; the State of the Gam- bella Peoples; the state the Commonwealth of Australia; the commonwealth; the state of New South Wales; the Australian Capital Territory the Commonwealth of Puerto Rico Washington State; the state of Washington the New England states the province of Ontario Jiangxi Province Massachusetts Bay Colony; the colony at Massachusetts Bay the British colonies; the thirteen colonies the Indiana Territory; the territory of Indiana the Northwest Territory; the Old Northwest the Western Reserve Lake County; the county of Lake; the county; county Kildare (Irish usage) New York City; the city of New York the City (the old city of London, now the financial district, always capitalized) Shields Township; the township the Eleventh Congressional District; the congressional district the Fifth Ward; the ward the Sixth Precinct; the precinct A generic term that is capitalized as part of the name of an official body remains capitalized when it is used in the plural to refer to two or more names and applies to both.
Lake and Cook Counties the Republics of Indonesia and South Africa Governmental entities. In contexts where a specific governmental body rather than the place is meant, the words state, city, and the like are usu ally capitalized when used as part of the full name of the body. See also 8.51.
Names of Places 8.54 She works for the Village of Forest Park.
That is a City of Chicago ordinance.
but Residents of the village of Forest Park enjoy easy access to the city of Chicago.
Topographical Divisions 8.53 Mountains, rivers, and the like. Names of mountains, rivers, oceans, is lands, and so forth are capitalized. The generic term (mountain etc.) is also capitalized when used as part of the name. In the plural, it is capitalized when it is part of a single name (Hawaiian Islands) and when it is used with two or more names, whether beginning with the generic term (Mounts Washington and Rainier) or when the generic term comes second and applies to two or more names (e.g., the Illinois and the Chicago Rivers).
Walden Pond Silver Lake Lake Michigan; Lakes Michigan and Erie; the Great Lakes the Illinois River; the Illinois and the Chicago Rivers the Nile River valley; the Nile valley; the Nile delta; the Mississippi River valley; the Mississippi delta (where river forms part of the proper names but valley and delta do not; see also 8.54) the Bering Strait the Mediterranean Sea; the Mediterranean the Pacific Ocean; the Pacific and the Atlantic Oceans the Great Barrier Reef the Hawaiian Islands; Hawaii; but the island of Hawaii (the Big Island) the Windward Islands; the Windwards the Iberian Peninsula Cape Verde the Black Forest Stone Mountain Mount Washington; Mount Rainier; Mounts Washington and Rainier the Rocky Mountains; the Rockies (see also 8.47) Death Valley; the Valley of Kings the Continental Divide the Horn of Africa; the Horn (to avoid confusion with a different kind of horn) the Indian subcontinent (a descriptive rather than proper geographical name) 8.54 Generic terms for geographic entities. When a generic term is used de scriptively (or in apposition; see 8.21) rather than as part of a name, or when used alone, it is lowercased.
8.55 NAMES, TERMS, AND TITLES OF WORKS the Amazon basin along the Pacific coast (but the West Coast; see 8.47) the California desert the river Thames the Hudson River valley 8.55 Non-English terms for geographic entities. When a generic term from a language other than English forms part of a geographic name, the equiv alent English term should not be included. See also 11.4.
the Rio Grande (not the Rio Grande River) Fujiyama (not Mount Fujiyama) Mauna Loa (not Mount Mauna Loa) the Sierra Nevada (not the Sierra Nevada Mountains) Public Places and Major Structures 8.56 Thoroughfares and the like. The names of streets, avenues, squares, parks, and so forth are capitalized. The generic term is lowercased when used alone but capitalized when used as part of a plural name.
Broadway Fifty-Fifth Street; Fifty-Seventh and Fifty-Fifth Streets Hyde Park Boulevard; the boulevard Interstate 80; 1-80; an interstate highway; the interstate the Ishtar Gate; the gate Jackson Park; the park London Bridge; the bridge the Mall (in London) Park Lane Pennsylvania Avenue; Carnegie and Euclid Avenues Piccadilly Circus the Spanish Steps; the steps Tiananmen Square; the square US Route 66; Routes 1 and 2; a state route See also 9.50-52.
8.57 Buildings and monuments. The names of buildings and monuments are generally capitalized. The generic term is usually lowercased when used alone but capitalized when used as part of a plural name (as in the fifth example).
Names of Places 8.59 the Babri Mosque; the mosque the Berlin Wall; the wall Buckingham Fountain; the fountain the Capitol (where the US Congress meets, as distinct from the capital city) the Chrysler Building; the building; the Empire State and Chrysler Buildings the Houses of Parliament the Jefferson Memorial; the memorial the Leaning Tower of Pisa the Pyramids (but the Egyptian pyramids) Shedd Aquarium; the aquarium the Stone of Scone Symphony Center; the center Tribune Tower; the tower the Washington Monument; the monument Westminster Abbey; the abbey the White House Though major works of art are generally italicized (see 8.198), some massive works of sculpture are regarded primarily as monuments and therefore not italicized.
the Statue of Liberty; the statue Mount Rushmore National Memorial; Mount Rushmore the Colossus of Rhodes; the colossus 8.58 Rooms, offices, and such. Official names of rooms, offices, and the like are capitalized.
the Empire Room (but room 421) the Amelia Earhart Suite (but suite 219) the Lincoln Bedroom the Oval Office the West Wing of the White House 8.59 Non-English names for places and structures. Non-English names of thoroughfares and buildings are not italicized and may be preceded by English the if the definite article would appear in the original language. See also 11.4.
the Champs-Elysees the Bibliotheque nationale the Bois de Boulogne Unter den Linden (never preceded by the) the Marktstrasse the Piazza delle Terme 8.60 NAMES, TERMS, AND TITLES OF WORKS Words Derived from Proper Names 8.60 When to capitalize words derived from proper names. Adjectives de rived from personal names are normally capitalized. Those in common use may be found in Webster's, sometimes in the biographical names sec tion (e.g., Aristotelian, Jamesian, Machiavellian, Shakespearean). If not in the dictionary, adjectives can sometimes be coined by adding ian (to a name ending in a consonant) or an (to a name ending in e or 0—or, fail ing these, esque. As with Foucault and Shaw, the final consonant some times undergoes a transformation as an aid to pronunciation. If a name does not seem to lend itself to any such coinage, it is best avoided. See also 8.61, 8.79.
Baudelaire; Baudelairean Bayes; Bayesian Dickens; Dickensian Foucault; Foucauldian Jordan; Jordanesque (a la Michael Jordan) Kafka; Kafkaesque Marx; Marxist Mendel; Mendelian Rabelais; Rabelaisian Sartre; Sartrean Shaw; Shavian 8.61 When to lowercase words derived from proper names. Personal, na tional, or geographical names, and words derived from such names, are often lowercased when used with a nonliteral meaning. For example, the cheese known as “gruyere” takes its name from a district in Switzer land but is not necessarily from there; “swiss cheese” (lowercase s) is a cheese that resembles Swiss emmentaler (which derives its name from the Emme River valley). Although some of the terms in this paragraph and the examples that follow are capitalized in Webster's, Chicago prefers to lowercase them in their nonliteral use. See also 8.79.
anglicize arabic numerals arctics (boots) bohemian bordeaux brie brussels sprouts burgundy champagne cheddar delphic diesel engine dutch oven epicure Names of Organizations 8.62 frankfurter pasteurize french dressing Pharisaic french fries philistine, philistinism french windows platonic (but see 8.79) gruyere quixotic herculean roman numerals homeric roman type india ink scotch (but Scotch whisky, a product of Scotland) italicize Stilton italic type swiss cheese (not made in Switzerland) jeremiad Venetian blinds lombardy poplar vulcanize manila envelope wiener morocco leather Names of Organizations Governmental Bodies 8.62 Legislative and deliberative bodies. The full names of legislative and deliberative bodies, departments, bureaus, and offices are capitalized (but see 8.65). Adjectives derived from them are usually lowercased, as are many of the generic names for such bodies when used alone (as on subsequent mentions). For generic names used alone but not listed here, opt for lowercase. For administrative bodies, see 8.63; for judicial bodies, see 8.64. See also 11.4.
the United Nations General Assembly; the UN General Assembly; the assembly the League of Nations; the league the United Nations Security Council; the Security Council; the council the United States Congress; the US Congress; the 115th Congress; Congress; 115th Cong.; congressional (see also 9.45) the United States Senate; the Senate; senatorial; the upper house of Congress the House of Representatives; the House; the lower house of Congress the Electoral College the Committee on Foreign Affairs; the Foreign Affairs Committee; the committee the Illinois General Assembly; the assembly; the Illinois legislature; the state sen ate the Chicago City Council; the city council the British Parliament (or UK Parliament); Parliament; an early parliament; par liamentary; the House of Commons; the Commons; the House of Lords; the Lords 8.63 NAMES, TERMS, AND TITLES OF WORKS the Crown (the British monarchy); Crown lands the Privy Council (but a Privy Counsellor) the Parliament of Canada; Parliament; the Senate (upper house); the House of Commons (lower house) the Legislative Assembly of British Columbia; the National Assembly of Quebec or Assemblee nationale du Quebec the Oireachtas (Irish parliament); Seanad Eireann (Irish upper house); Dail Eireann (Irish lower house) the Assemblee nationale or the National Assembly (present-day France); the (French) Senate; the parliamentary system; the Parlement de Paris (historical) the States General or Estates General (France and Netherlands, historical) the Cortes Generales; the Cortes (Spain); Cortes Espanolas (Franco era) the Camara de Diputados (the lower house of Mexico’s congress) the Bundestag (German parliament); the Bundesrat (German upper house); the Reichstag (imperial Germany) the House of People’s Representatives; the House of Federation; the Council of Ministers (Ethiopia) the Dewan Perwakilan Rakyat or the House of Representatives; the Majelis Per- musyawaratan Rakyat or People’s Consultative Assembly (Indonesia) the European Parliament; the Parliament 8.63 Administrative bodies. The full names of administrative bodies are cap italized. Adjectives derived from them are usually lowercased, as are many of the generic names for such bodies when used alone. See also 8.62.
the United States Census Bureau; census forms; the census of2000 the Centers for Disease Control and Prevention; the CDC (abbreviation did not change when “and Prevention” was added to name) the Department of the Interior; the Interior the Department of State; the State Department; the department; departmental the Department of the Treasury; the Treasury the Eunice Kennedy Shriver National Institute of Child Health and Human De velopment; the NICHD the Federal Bureau of Investigation; the bureau; the FBI the Federal Reserve System; the Federal Reserve Board; the Federal Reserve the United States Foreign Service; Foreign Service Officer; officer in the Foreign Service the National Institutes of Health; the NIH; the National Institute of Mental Health; the NIMH the Occupational Safety and Health Administration; OSHA the Office of Human Resources; Human Resources Names of Organizations 8.65 the Peace Corps the United States Postal Service; the Postal Service; the post office the Illinois State Board of Education; the board of education the Ithaca City School District; the school district; the district 8.64 Judicial bodies. The full name of a court, often including a place-name, is capitalized. Subsequent references to a court (or district court, supreme court, etc.) are lowercased, except for the phrase “Supreme Court” at the national level.
the United States (or US) Supreme Court; the Supreme Court; but the court the United States Court of Appeals for the Seventh Circuit; the court of appeals the Arizona Supreme Court; the supreme court; the supreme courts of Arizona and New Mexico the District Court for the Southern District of New York; the district court the Court of Common Pleas (Ohio); the court the Circuit Court of Lake County, Family Division (Illinois); family court the Supreme Court of Canada; the Supreme Court; the court the Birmingham Crown Court; Dawlish Magistrates’ Court (England) the Federal Supreme Court (Ethiopia) States, counties, and cities vary in the way they name their courts. For example, court of appeals in New York State and Maryland is equivalent to supreme court in other states; and such terms as district court, circuit court, superior court, and court of common pleas are used for similar court systems in different states. Generic names should therefore be used only after the full name or jurisdiction has been stated.
8.65 Government entities that are lowercased. Certain generic terms asso ciated with governmental bodies are lowercased. Compare 8.51.
administration; the Carter administration brain trust cabinet (but the Kitchen Cabinet in the Jackson administration) city hall (the municipal government and the building) civil service court (a royal court) executive, legislative, or judicial branch federal; the federal government; federal agencies government monarchy parlement (French; but the Parlement of Paris) 8.66 NAMES, TERMS, AND TITLES OF WORKS parliament, parliamentary (but Parliament, usually not preceded by the, in the United Kingdom) state; church and state; state powers Political and Economic Organizations and Movements 8.66 Organizations, parties, alliances, and so forth. Official names of na tional and international organizations, alliances, and political move ments and parties are capitalized (e.g., “the Labor Party in Israel”). Words like party, union, and movement are capitalized when they are part of the name of an organization. Terms identifying formal members of or adherents to such groups are also usually capitalized (e.g., “a Social ist”; “a Republican”). Names of the systems of thought and references to the adherents to such systems, however, are often lowercased (e.g., “an eighteenth-century precursor of socialism”; “a communist at heart”). Nonliteral or metaphorical references are also lowercased (e.g., “fascist parenting techniques”; “nazi tendencies”). For consistency, however—as in a work about communism in which the philosophy and its adherents, the political party, and party members and adherents are discussed- capitalizing the philosophy, together with the organization and its adher ents, in both noun and adjective forms, will prevent editorial headaches.
the African National Congress party (party is not part of the official name); the ANC Arab Socialist Bacth Party; the Bacth Party; the party; Bacthists Bahujan Samaj Party; the BSP Bolshevik(s); the Bolshevik (or Bolshevist) movement; bolshevism or Bolshevism (see text above) Chartist; Chartism the Communist Party (but communist parties); the party; Communist(s); Com munist countries; communism or Communism (see text above) the Democratic Party; the party; Democrat(s) (party members or adherents); de mocracy; democratic nations the Entente Cordiale (signed 1904); the Entente; but an entente cordiale the Ethiopian Somali Democratic League; the league; the party the European Union; the EU; the Common Market the Fascist Party; Fascist(s); fascism or Fascism (see text above) the Federalist Party; Federalist(s) (US history); federalism or Federalism (see text above) the Free-Soil Party; Free-Soiler(s) the General Agreement on Tariffs and Trade; GATT the Green Party; the party; Green(s); the Green movement Names of Organizations 8.67 the Hanseatic League; Hansa; a Hanseatic city the Holy Alliance the Know-Nothing Party; Know-Nothing(s) the Labour Party; Labourite (s) (members of the British party) the League of Arab States; the Arab League; the league the Libertarian Party; Libertarian(s); libertarianism or Libertarianism (see text above) Loyalist(s) (American Revolution; Spanish Civil War) Marxism-Leninism; Marxist-Leninist(s) the National Socialist Party; National Socialism; the Nazi Party; Nazi(s); Nazism the North American Free Trade Agreement; NAFTA the North Atlantic Treaty Organization; NATO the Organisation for Economic Co-operation and Development; the OECD; the organization the Popular Front; the Front; but a popular front the Populist Party; Populist(s); populism or Populism (see text above) the Progressive Party; Progressive movement; Progressive(s); progressivism or Progressivism (see text above) the Quadruple Alliance; the alliance the Rashtriya Janata Dal; the RJD (National People’s Party) the Republican Party; the party; the GOP (Grand Old Party); Republican(s) (party members or adherents); republicanism; a republican form of government the Social Democratic Party; the party; Social Democrat(s) the Socialist Party (but socialist parties); the party; Socialist(s) (party members or adherents); socialism or Socialism (see text above) the United Democratic Movement; the movement the World Health Organization; WHO 8.67 Adherents of unofficial political groups and movements. Names for ad herents of political groups or movements other than recognized parties are usually lowercased.
anarchist(s) centrist(s) independent(s) moderate (s) mugwump (s) opposition (but the Opposition, in British and Canadian contexts, referring to the party out of power) but the Left; members of the left wing; left-winger(s); on the left the Right; members of the right wing; right-winger (s); on the right the Far Left 8.68 NAMES, TERMS, AND TITLES OF WORKS the Far Right the radical Right the Tea Party; Tea Partiers (modeled on names for established parties) Institutions and Companies 8.68 Institutions and companies—capitalization. The full names of institu tions, groups, and companies and the names of their departments, and often the shortened forms of such names (e.g., the Art Institute), are cap italized. A the preceding a name, even when part of the official title, is lowercased in running text. Such generic terms as company and university are usually lowercased when used alone (though they are routinely capi talized in promotional materials, business documents, and the like).
the University of Chicago; the university; the University of Chicago and Har vard University; Northwestern and Princeton Universities; the University of Wisconsin-Madison (see also 6.81) the Department of History; the department; the Law School the University of Chicago Press; the press the Board of Trustees of the University of Chicago; the board of trustees; the board the Art Institute of Chicago; the Art Institute the Beach Boys; the Beatles; the Grateful Dead, the Dead; the Who (but Tha Eastsidaz) Captain Beefheart and His Magic Band; the band the Cleveland Orchestra; the orchestra the General Foods Corporation; General Foods; the corporation the Green Bay Packers; the Packers the Hudson’s Bay Company; the company the Illinois Central Railroad; the Illinois Central; the railroad the Library of Congress; the library the Manuscripts Division of the library the Museum of Modern Art; MOM A; the museum the New School (see also 8.69) the New York Stock Exchange; the stock exchange Skidmore, Owings & Merrill; SOM; the architectural firm the Smithsonian Institution; the Smithsonian Miguel Juarez Middle School; the middle school 8.69 Corporate names with unusual capitalization. Corporate names that appear in all lowercase in logotype and other promotional settings can often be capitalized in the usual way. A copyright or “About Us” state Names of Organizations 8.70 ment on a corporate website can be helpful in determining a usage that might be suitable for regular text. Words that would normally be lower case in headline-style capitalization can usually be lowercased (see 8.159) . Spellings that begin lowercase but include a capital letter are usu ally appropriate for running text, even at the beginning of a sentence, as are names with additional internal capitals (see 8.154). A preference for all uppercase should be respected. If a company appears to prefer all lowercase even in running text, an initial capital can be applied as a mat ter of editorial expediency.
Intel (not intel) Adidas (not adidas) AT&T (not at&t) Ebrary (not ebrary) Parsons the New School for Design (lowercase the, contrary to corporate usage) but GlaxoSmithKline HarperCollins RAND Corporation eBay Associations 8.70 Associations, unions, and the like. The full names of associations, socie ties, unions, meetings, and conferences, and often the shortened forms of such names, are capitalized. A the preceding a name, even when part of the official title, is lowercased in running text. Such generic terms as society and union are usually lowercased when used alone.
the Congress of Industrial Organizations; CIO; the union Girl Scouts of the United States of America; a Girl Scout; a Scout the Independent Order of Odd Fellows; IOOF; an Odd Fellow Industrial Workers of the World; IWW; the Wobblies the International Olympic Committee; the IOC; the committee the League of Women Voters; the league the National Conference for Community and Justice; the conference the National Organization for Women; NOW; the organization the New-York Historical Society (the hyphen is part of the official name of the society); the society the 130th Annual Meeting of the American Historical Association; the annual meeting of the association 8.71 NAMES, TERMS, AND TITLES OF WORKS the Quadrangle Club; the club the Textile Workers Union of America; the union On the other hand, a substantive title given to a single meeting, confer ence, speech, or discussion is enclosed in quotation marks. For lecture series, see 8.87.
“Inside the Mind of a Master Procrastinator,” a TED talk by Tim Urban posted in March 2016.
Historical and Cultural Terms Periods 8.71 Numerical designations for periods. A numerical designation for a pe riod is usually lowercased; however, certain periods may be treated as proper nouns (in which both the numerical designation and the term for the period are capitalized) to avoid any confusion with the generic mean ing of the same term (see also 8.51). For the use of numerals, see 9.33, 9.45.
the twenty-first century the second millennium BCE the nineteen hundreds but the nineties the Eighteenth Dynasty (Egypt) the quattrocento 8.72 Descriptive designations for periods. A descriptive designation of a pe riod is usually lowercased, except for proper names or to avoid ambiguity with a generic term. For traditionally capitalized forms, see 8.73.
ancient Greece the antebellum period antiquity the baroque period the colonial period a golden age the Hellenistic period imperial Rome modern history the Romantic period (see also 8.79) the Shang dynasty (considered an era rather than a political division; see 8.51) the Victorian era Historical and Cultural Terms 8.75 8.73 Traditional period names. Some names of periods are capitalized, either by tradition or to avoid ambiguity. See also 8.75.
the Augustan Age the Common Era the Counter-Reformation the Dark Ages the Enlightenment the Gay Nineties the Gilded Age the Grand Siecle the High Middle Ages (but the early Middle Ages, the late Middle Ages) the High Renaissance the Jazz Age the Mauve Decade the Middle Ages (but the medieval era) the Old Kingdom (ancient Egypt) the Old Regime (but the ancien regime) the Progressive Era the Reformation the Renaissance the Restoration the Roaring Twenties 8.74 Cultural periods. Names of prehistoric cultural periods are capitalized. For geological periods, see 8.134-36.
the Bronze Age the Iron Age the Ice Age the Stone Age Similar terms for modern periods are often lowercased (but see 8.73).
the age of reason the information age the age of steam the nuclear age Events 8.75 Historical events and programs. Names of many major historical events and programs are conventionally capitalized. Others, more recent or known by their generic descriptions, are often lowercased but may be capitalized to prevent ambiguity. If in doubt, opt for lowercase. For wars and battles, see 8.113-14; for religious events, 8.108; for acts and treaties, 8.80.
8.76 NAMES, TERMS, AND TITLES OF WORKS the Arab Spring Black Lives Matter Boston Tea Party the Boxer Rebellion the Cold War (but a cold war, used generically) the Cultural Revolution the Great Chicago Fire; the Chicago fire; the fire of 1871 the Great Depression; the Depression the Great Fire of London; the Great Fire the Great Plague; the Plague (but plague [the disease]) (President Johnson’s) Great Society the Industrial Revolution the Long March the May 18 Democratic Uprising (or Gwangju Uprising) the New Deal Occupy Wall Street; the Occupy movement Prohibition Reconstruction the Reign of Terror; the Terror the South Sea Bubble the War on Poverty but the baby boom the Black September attacks the civil rights movement the crash of 1929 the Dreyfus affair the gold rush the Moroccan crises the Tiananmen Square protests the war on terror 8.76 Speeches. Titles of a select few speeches are traditionally capitalized. Others are usually lowercased (but see 8.188).
Washington’s Farewell Address the Gettysburg Address the annual State of the Union address Franklin Roosevelt’s second inaugural address the Checkers speech Martin Luther King Jr.’s “I Have a Dream” speech Historical and Cultural Terms 8.79 g.77 Meteorological and other natural phenomena. Named hurricanes and other tropical cyclones are capitalized, as are many other named meteo rological phenomena. If in doubt, consult a dictionary or encyclopedia. Natural phenomena identified generically by a place-name or a year are usually lowercased.
Cyclone Becky; the 2007 cyclone Hurricane Katrina; the 2005 hurricane El Nino the Northridge earthquake of 1994 the Arctic polar vortex Use the pronoun it, not he or she, when referring to named storms, hur ricanes, and the like (notwithstanding the practice of using male and fe male proper names to refer to such events).
8.78 Sporting events. The full names of major sporting events are capital ized.
the Kentucky Derby; the derby the NBA Finals; the finals the Olympic Games; the Olympics; the Winter Olympics the World Cup Cultural Movements and Styles 8.79 Movements and styles—capitalization. Nouns and adjectives designat ing cultural styles, movements, and schools—artistic, architectural, mu sical, and so forth—and their adherents are capitalized if derived from proper nouns. (Words such as school and movement remain lowercased.) Others may be lowercased, though a few (e.g., Beat, Cynic, Scholastic, New Criticism) are capitalized to distinguish them from the generic words used in everyday speech. Some of the terms lowercased below may appropriately be capitalized in certain works if done consistently— especially those that include the designation “often capitalized” in Web ster's. (But if, for example, impressionism is capitalized in a work about art, other art movements must also be capitalized—which could result in an undesirable profusion of capitals.) For religious movements, see 8.97. See also 8.60.
8.79 NAMES, TERMS, AND TITLES OF WORKS abstract expressionism Aristotelian art deco art nouveau baroque Beat movement; the Beats (but beatnik) Beaux-Arts (derived from Ecole des Beaux-Arts) British Invasion camp Cartesian Chicago school (of architecture, of economics, of literary criticism) classicism, classical conceptualism cubism Cynicism; Cynic Dadaism; Dada deconstruction Doric Epicurean (see text below) existentialism fauvism formalism Gothic (but gothic fiction) Gregorian chant Hellenism Hudson River school humanism idealism imagism impressionism Keynesianism mannerism miracle play modernism mysticism; mystic naturalism neoclassicism; neoclassical Neoplatonism New Criticism nominalism op art Peripatetic (see text below) philosophe (French) Platonism pop art postimpressionism postmodernism Pre-Raphaelite Reaganomics realism rococo Romanesque Romanticism; Romantic Scholasticism; Scholastic; Schoolmen scientific rationalism Sophist (see text below) Stoicism; Stoic (see text below) structuralism Sturm und Drang (but storm and stress) surrealism symbolism theater of the absurd transcendentalism Some words capitalized when used in reference to a school of thought are lowercased when used metaphorically.
epicurean tastes peripatetic families she’s a sophist, not a logician a stoic attitude Historical and Cultural Terms 8.80 Acts, Treaties, and Government Programs 8.80 Formal names of acts, treaties, and so forth. Formal or accepted titles of pacts, plans, policies, treaties, acts, programs, and similar documents or agreements are capitalized. Incomplete or generic forms are usually low ercased. For citing the published text of a bill or law, see 14.282,14.283.
the Articles of Confederation the Bill of Rights the Brady law the Constitution of the United States; the United States (or US) Constitution; the Constitution (usually capitalized in reference to the US Constitution); Article VI; the article (see also 9.28) the Illinois Constitution; the constitution the Constitution Act, 1982 (Canada) the Corn Laws (Great Britain) the Declaration of Independence the due process clause the Equal Rights Amendment (usually capitalized though not ratified); ERA; but an equal rights amendment the Family and Medical Leave Act of 1993; FMLA; the 1993 act the Fifteenth Amendment (to the US Constitution); the Smith Amendment; the amendment the Food Stamp Act of 1964; food stamps the Hawley-Smoot (or Smoot-Hawley) Tariff Act; the tariff act Head Start impeachment; the first and second articles of impeachment the Kyoto Protocol; the protocol the Marshall Plan the Mayflower Compact; the compact Medicare (lowercase in Canada); Medicaid the Monroe Doctrine; the doctrine the Munich agreement (1938); Munich the New Economic Policy; NEP (Soviet Union) the Open Door policy the Peace of Utrecht the Reform Bills; the Reform Bill of 1832 (Great Britain) the Social Security Act; Social Security (or, generically, social security) Temporary Assistance for Needy Families; TANF Title VII or Title 7 Treaty for the Renunciation of War, known as the Pact of Paris or the Kellogg- Briand Pact; the pact 8.81 NAMES, TERMS, AND TITLES OF WORKS the Treaty of Versailles; the treaty the Treaty on European Union (official name); the Maastricht treaty (informal name) the Wilmot Proviso 8.81 Generic terms for pending legislation. Informal, purely descriptive ref erences to pending legislation are lowercased.
The anti-injunction bill was introduced on Tuesday. (See also the table at 7.89, section 4.) Legal Cases 8.82 Legal cases mentioned in text. The names of legal cases are italicized when mentioned in text. The abbreviation v. (versus) occasionally ap pears in roman, but Chicago recommends italics. In footnotes, legal dictionaries, and contexts where numerous legal cases appear, they are sometimes set in roman. For legal citation style, see 14.269-305.
Bloomfield Village Drain Dist. v. Keefe Miranda v. Arizona In discussion, a case name may be shortened.
the Miranda case (or simply Miram/#) Awards 8.83 Capitalization for names of awards and prizes. Names of awards and prizes are capitalized, but some generic terms used with the names are lowercased. For military awards, see 8.115.
the 2017 Nobel Prize in Physiology or Medicine; a Nobel Prize winner; a Nobel Prize-winning physiologist (see 6.80); a Nobel Peace Prize; the Nobel Prize in Literature the 2017 Pulitzer Prize for Commentary (but a Pulitzer in journalism) an Academy Award; the Academy Award for Best Picture; an Oscar an Emmy Award for Outstanding Supporting Actor in a Comedy Series; she has three Emmys a Webby Award; the Webbys; the Webby Award for Activism (Web); a Webby the Presidential Medal of Freedom Historical and Cultural Terms 8.87 a Guggenheim Fellowship (but a Guggenheim grant) an International Music Scholarship National Merit Scholarship awards; Merit Scholarships; Merit Scholar Oaths and Pledges 8.84 Formal oaths and pledges. Formal oaths and pledges are usually lower cased.
the oath of citizenship the Hippocratic oath the presidential oath of office marriage vows but the Pledge of Allegiance Academic Subjects, Courses of Study, and Lecture Series 8.85 Academic subjects. Academic subjects are not capitalized unless they form part of a department name or an official course name (see 8.86) or are themselves proper nouns (e.g., English, Latin).
She has published widely in the history of religions. They have a wide variety of courses in gender studies. He is majoring in comparative literature.
She is pursuing graduate studies in philosophy of science. but Jones is chair of the Committee on Comparative Literature.
8.86 Courses of study. Official names of courses of study are capitalized.
I am signing up for Archaeology 101.
A popular course at the Graham School of General Studies is Basic Manuscript Editing.
but His ballroom dancing classes have failed to civilize him.
8.87 Lectures. Names of lecture series are capitalized. Titles for individual lectures are capitalized and usually enclosed in quotation marks. See also 8.70.
This year’s Robinson Memorial Lectures were devoted to the nursing profession. The first lecture, “How Nightingale Got Her Way,” was a sellout.
8.88 NAMES, TERMS, AND TITLES OF WORKS Calendar and Time Designations 8.88 Days of the week, months, and seasons. Names of days and months are capitalized. The four seasons are lowercased (except when used to denote an issue of a journal; see 14.171). For centuries and decades, see 8.71.
Tuesday spring the vernal (or spring) equinox November fall the winter solstice 8.89 Holidays. The names of secular and religious holidays or officially desig nated days or seasons are capitalized.
All Fools’ Day Christmas Day Earth Day Election Day Father’s Day the Fourth of July, the Fourth Good Friday Halloween Hanukkah Holy Week Inauguration Day Independence Day Kwanzaa Labor Day Lent Lincoln’s Birthday Martin Luther King Jr. Day Memorial Day Mother’s Day National Poetry Month New Year’s Day New Year’s Eve Passover Presidents’ Day Ramadan Remembrance Day (Canada) Rosh Hashanah Saint Patrick’s Day Thanksgiving Day Veterans Day Yom Kippur Yuletide but D-day a bank holiday 8.90 Time and time zones. When spelled out, designations of time and time zones are lowercased (except for proper nouns). Abbreviations are capi talized. See also 9.37-40,10.41.
eastern standard time; EST central daylight time; CDT mountain standard time; MST Pacific daylight time; PDT Greenwich mean time; GMT daylight saving time; DST Religious Names and Terms 8.93 Religious Names and Terms Deities and Revered Persons 8.91 Deities. Names of deities, whether in monotheistic or polytheistic reli gions, are capitalized.
Allah Jehovah Astarte Mithra Freyja Satan (but the devil) God Serapis Itzamna Yahweh Some writers follow a pious convention of not fully spelling out the name of a deity (e.g., G-d). This convention should be respected when it is practical to do so.
8.92 Alternative names. Alternative or descriptive names for God as supreme being are capitalized. See also 8.93.
Adonai the Almighty the Deity the Holy Ghost or the Holy Spirit or the Paraclete the Lord Providence the Supreme Being the Trinity 8.93 Prophets and the like. Designations of prophets, apostles, saints, and other revered persons are often capitalized.
the Buddha the prophet Isaiah Jesus; Christ; the Good Shepherd; the Son (or son) of man John the Baptist the Messiah Muhammad; the Prophet Saint John; the Beloved Apostle the Virgin Mary; the Blessed Virgin; Mother of God but the apostles 8.94 NAMES, TERMS, AND TITLES OF WORKS the patriarchs the psalmist 8.94 Platonic ideas. Words for transcendent ideas in the Platonic sense, es pecially when used in a religious context, are often capitalized. See also 7.52.
Good; Beauty; Truth; the One 8.95 Pronouns referring to religious figures. Pronouns referring to God or Jesus are not capitalized unless a particular author or publisher prefers otherwise. (Note that they are lowercased in most English translations of the Bible.) They prayed to God that he would deliver them.
Jesus and his disciples Religious Groups 8.96 Major religions. Names of major religions are capitalized, as are their adherents and adjectives derived from them.
Buddhism; Buddhist Christianity; Christian; Christendom (see also 8.98) Confucianism; Confucian Hinduism; Hindu Islam; Islamic; Muslim Judaism; Jew; Jewry; Jewish Shinto; Shintoism; Shintoist Taoism; Taoist; Taoistic but atheism agnosticism 8.97 Denominations, sects, orders, and religious movements. Like the names of major religions, names of denominations, communions, sects, orders, and religious movements are capitalized, as are their adherents and ad jectives derived from them. See also 8.99.
the Amish; Amish communities Anglicanism; the Anglican Communion (see also Episcopal Church) Religious Names and Terms 8.97 Baptists; a Baptist church; the Baptist General Convention; the Southern Baptist Convention Catholicism (see Roman Catholicism) Christian Science; Church of Christ, Scientist; Christian Scientist the Church of England (but an Anglican church) the Church of Ireland Community of Christ Conservative Judaism; a Conservative Jew Dissenter (lowercased when used in a nonsectarian context) Druidism; Druid (sometimes lowercased) Eastern Orthodox churches; the Eastern Church (but an Eastern Orthodox church) the Episcopal Church; an Episcopal church; an Episcopalian the Episcopal Church of Scotland Essenes; an Essene Gnosticism; Gnostic Hasidism; Hasid (singular); Hasidim (plural); Hasidic Jehovah’s Witnesses Jesuit(s); the Society of Jesus; Jesuitic(al) (lowercased when used pejoratively) Methodism; the United Methodist Church (but a United Methodist church); Wesleyan Mormonism; Mormon; the Church of Jesus Christ of Latter-day Saints Nonconformism; Nonconformist (lowercased when used in a nonsectarian con text) Old Catholics; an Old Catholic church the Order of Preachers; the Dominican order; a Dominican Orthodox Judaism; an Orthodox Jew Orthodoxy; the (Greek, Serbian, etc.) Orthodox Church (but a Greek Orthodox church) Protestantism; Protestant (lowercased when used in a nonsectarian context) Puritanism; Puritan (lowercased when used in a nonsectarian context) Quakerism; Quaker; the Religious Society of Friends; a Friend Reform Judaism; a Reform Jew Roman Catholicism; the Roman Catholic Church (but a Roman Catholic church) Satanism; Satanist Seventh-day Adventist; Adventist; Adventism Shiism; Shia; Shiite Sufism; Sufi Sunnism; Sunni; Sunnite Theosophy; Theosophist; the Theosophical Society Vedanta Wicca; Wiccan Zen; Zen Buddhism 8.98 NAMES, TERMS, AND TITLES OF WORKS 8.98 "Church" as institution. When used to refer to the institution of religion or of a particular religion, church is usually lowercased unless a particular author or publisher prefers otherwise.
church and state the church in the twenty-first century the early church the church fathers Church is capitalized when part of the formal name of a denomination (e.g., the United Methodist Church; see other examples in 8.97) or con gregation (e.g., the Church of St. Thomas the Apostle).
8.99 Generic versus religious terms. Many terms that are lowercased when used generically, such as animism, fundamentalism, or spiritualism, may be capitalized when used as the name of a specific religion or a sect.
a popular medium in turn-of-the-century Spiritualist circles but liberal versus fundamentalist Christians 8.100 Religious jurisdictions. The names of official divisions within organized religions are capitalized. The generic terms used alone are lowercased.
the Archdiocese of Chicago; the archdiocese the Eastern Diocese of the Armenian Church the Fifty-Seventh Street Meeting; the (Quaker) meeting the Holy See the Missouri Synod; the synod 8.101 Places of worship. The names of the buildings in which religious con gregations meet are capitalized. The generic terms used alone are lower cased.
Babri Mosque; the mosque Bethany Evangelical Lutheran Church; the church Temple Emanuel; the temple; the synagogue Nichiren Buddhist Temple; the temple 8.102 Councils, synods, and the like. The accepted names of historic councils and the official names of modern counterparts are capitalized.
the Council of Chalcedon (or the Fourth Ecumenical Council) the General Convention (Episcopal) Religious Names and Terms 8.104 the Second Vatican Council; Vatican II the Synod of Whitby Religious Writings 8.103 Scriptures. Names of scriptures and other highly revered works are capitalized but not usually italicized (except when used in the title of a published work).
the Bhagavad Gita (or Bhagavad Gita) the Bible (but biblical) the Book of Common Prayer the Dead Sea Scrolls the Hebrew Bible Koran; Koranic (or Qur’an; Qur’anic) the Mahabharata (or Mahabharata) Mishnah; Mishnaic Sunna Talmud; Talmudic Tao Te Ching Tripitaka the Upanishads the Vedas; Vedic but sutra (s) 8.104 Other names and versions for bibles. Other names and versions of the Hebrew and Christian bibles are usually capitalized but not italicized.
the Authorized Version or the King James Version the Breeches (or Geneva) Bible Codex Sinaiticus Complutensian Polyglot Bible the Douay (or Rheims-Douay) Version the Holy Bible Holy Writ (sometimes used figuratively) the New English Bible the New Jerusalem Bible the New Revised Standard Version Peshitta the Psalter (but a psalter) 8.105 NAMES, TERMS, AND TITLES OF WORKS the Septuagint the Vulgate but scripture (s); scriptural 8.105 Books of the Bible. The names of books of the Bible are capitalized but never italicized. The word book is usually lowercased, and the words gospel and epistle are usually capitalized. But in a work in which all three terms are used with some frequency, they may all be treated alike, either lowercased or capitalized. See also 9.26,10.44-48.
Genesis; the book of Genesis Job; the book of Job 2 Chronicles; Second Chronicles; the second book of Chronicles Psalms (but a psalm) John; the Gospel according to John Acts; the Acts of the Apostles 1 Corinthians; the First Epistle to the Corinthians 8.106 Sections of the Bible. Names of sections of the Bible are usually capital ized but not italicized.
the Hebrew scriptures or the Old Testament the Christian scriptures or the New Testament the Apocrypha; Apocryphal (or, generically, apocryphal) the Epistles; the pastoral Epistles the Gospels; the synoptic Gospels the Pentateuch dr the Torah; Pentateuchal Hagiographa or Ketuvim; hagiographic 8.107 Prayers, creeds, and such. Named prayers, canticles, creeds, and such, as well as scriptural terms of special importance, are usually capitalized. Parables and miracles are usually lowercased.
the Decalogue; the Ten Commandments; the first commandment Kaddish; to say Kaddish the Lord’s Prayer; the Our Father Luther’s Ninety-Five Theses the Nicene Creed; the creed Salat al-Fajr the Sermon on the Mount the Shema ReligiousNames and Terms 8.110 but the doxology the miracle of the loaves and fishes the parable of the prodigal son the star of Bethlehem Religious Events, Concepts, Services, and Objects 8.108 Religious events and concepts. Religious events and concepts of ma jor theological importance are often capitalized. Used generically, such terms are lowercased.
the Creation the Crucifixion the Diaspora the Exodus the Fall the Hegira the Second Coming but Most religions have creation myths.
For the Romans, crucifixion was a common form of execution.
Doctrines and principles are usually lowercased.
atonement dharma original sin resurrection 8.109 Heaven, hell, and so on. Terms for divine dwelling places, ideal states, places of divine punishment, and the like are usually lowercased (though they are often capitalized in a purely religious context). See also 8.50.
heaven hell limbo nirvana outer darkness paradise the pearly gates purgatory but Eden Elysium Hades Olympus 8.110 Services and rites. Names of services and rites are usually lowercased (though they may be capitalized in strictly religious contexts; if in doubt, consult Webster's).
8.111 NAMES, TERMS, AND TITLES OF WORKS baptism bar mitzvah bat mitzvah confirmation morning prayer; matins the seder the sun dance vespers Terms denoting the Eucharistic sacrament, however, are traditionally capitalized, though certain terms may be lowercased in nonreligious contexts or when used generically.
the Eucharist Holy Communion High Mass; Low Mass; attend Mass; but an afternoon mass 8.111 Objects. Objects of religious use or significance are usually lowercased, especially in nonreligious contexts.
altar ark chalice and paten mandala mezuzah rosary sacred pipe sanctuary stations of the cross Military Terms Forces and Troops 8.112 Armies, battalions, and such. Titles of armies, navies, air forces, fleets, regiments, battalions, companies, corps, and so forth are capitalized. Unofficial but well-known names, such as Green Berets, are also capital ized. Words such as army and navy are lowercased when standing alone, when used collectively in the plural, or when not part of an official title. Many of the lowercased terms below are routinely capitalized in official or promotional contexts (see 8.19). See also 9.47.
the Allies (World Wars I and II); the Allied forces American Expeditionary Force; the AEF Army Corps of Engineers; the corps Army of Northern Virginia; the army Army of the Potomac Army Special Forces the Axis powers (World War II) Canadian Army (a branch of the Canadian Forces) Canadian Forces or Canadian Armed Forces the Central powers (World War I) Military Terms 8.113 Combined Chiefs of Staff (World War II) Confederate army (American Civil War) Continental navy (American Revolution) Eighth Air Force; the air force Fifth Army; the army First Battalion, 178th Infantry; the battalion; the 178th French Foreign Legion Green Berets Joint Chiefs of Staff the Luftwaffe; the German air force National Guard Pacific Fleet (US, World War II) Red Army (Russian, World War II); Russian army the Resistance; the French Resistance; a resistance movement Rough Riders Royal Air Force; RAF; British air force Royal Canadian Air Force (a branch of the Canadian Forces) Royal Canadian Mounted Police; the Mounties; a Mountie Royal Canadian Navy (a branch of the Canadian Forces) Royal Navy; the British navy Royal Scots Fusiliers; the fusiliers Seventh Fleet; the fleet Thirty-Third Infantry Division; the Thirty-Third Division; the division Union army (American Civil War) United States (or US) Army; the army United States Army Signal Corps; the Signal Corps or the signal corps United States Coast Guard; the Coast Guard or the coast guard United States Marine Corps; the Marine Corps or the marine corps; the US Ma rines; a marine United States Navy; the navy Wars, Revolutions, Battles, and Campaigns 8.113 Wars and revolutions. Names of most major wars and revolutions are capitalized. The generic terms are usually lowercased when used alone. More recent, unresolved conflicts can usually be lowercased.
American Civil War; the War between the States American Revolution; American War of Independence; the revolution (some times capitalized); the Revolutionary War Crusades; the Sixth Crusade; a crusader French Revolution; the Revolution of 1789; the Revolution (usually capitalized 8.114 NAMES, TERMS, AND TITLES OF WORKS to distinguish the Revolution of 1789 from the revolutions in 1830 and 1848); revolutionary France Great Sioux War; the Sioux war Iran-Iraq War Iraq War Korean War; the war Mexican Revolution; the revolution Napoleonic Wars Norman Conquest; the conquest of England Persian Gulf War or Gulf War the revolution(s) of 1848 Russian Revolution; the revolution Seven Years’ War Shays’s Rebellion Six-Day War Spanish-American War Spanish Civil War Vietnam War War of 1812 Whiskey Rebellion World War I; the First World War; the Great War; the war World War II; the Second World War; World Wars I and II; the First and Second World Wars; the two world wars but the South Sudanese civil war; the civil war in South Sudan the Ukrainian unrest 8.114 Battles and campaigns. Some of the names of major battles and cam paigns that have entered the general lexicon are capitalized. In other, more generic descriptions, only proper names are capitalized. For names not included here, consult an encyclopedia.
Battle of Britain Battle of Bunker Hill; Bunker Hill; the battle Battle of the Bulge (or Battle of the Ardennes) battle of Vimy Ridge the Blitz European theater of operations; ETO Mexican border campaign Operation Devil Siphon third battle of Ypres Vicksburg Campaign western front (World War I) Names of Ships and Other Vehicles 8.116 Military Awards 8.115 Medals and awards. Specific names of medals and awards are capital ized. For civil awards, see 8.83.
Croix de Guerre (sometimes lowercased) Distinguished Flying Cross; DFC Distinguished Service Order; DSO Medal of Honor (US congressional award); the medal Purple Heart Silver Star Victoria Cross; VC Names of Ships and Other Vehicles 8.116 Ships and other named vessels. Names of specific ships and other vessels are both capitalized and italicized. Note that when such abbre viations as USS (United States ship) or HMS (Her [or His] Majesty’s ship) precede a name, the word ship or other vessel type should not be used. The abbreviations themselves are not italicized. For much useful information, consult Eric Wertheim, The Naval Institute Guide to Combat Fleets of the World (bibliog. 5).
Mars global surveyor; Mars polar lander; Phoenix Mars lander; Phoenix the space shuttle Discovery the Spirit of St. Louis HMS Frolic; the British ship Frolic SS United States; the United States USS SC-530; the US ship SC-530 Every US Navy ship is assigned a hull number (according to a system for mally implemented in 1920), consisting of a combination of letters (in dicating the type of ship) and a serial number. Where necessary to avoid confusion between vessels of the same name—in a work on naval history, for example—the numbers should be included at first mention. Smaller ships such as landing craft and submarine chasers are individually num bered but not named.
USS Enterprise (CVN-65) was already on its way to the Red Sea.
8.117 NAMES, TERMS, AND TITLES OF WORKS 8.117 Other vehicle names. Names of makes and classes of aircraft, models of automobiles and other vehicles, names of trains or train runs, and names of space programs are capitalized but not italicized.
Acela Express Boeing 787 Dreamliner Concorde Project Apollo Metroliner Subaru Forester 8.118 Pronouns referring to vessels. When a pronoun is used to refer to a vessel, the neuter it or its (rather than she or her) is preferred. See also 5.43, 8.77.
Scientific Terminology Scientific Nantes of Plants and Animals 8.119 Scientific style—additional resources. The following paragraphs offer only general guidelines. Writers or editors requiring detailed guidance should consult Scientific Style and Format (bibliog. 1.1). The ultimate au thorities are the International Code of Nomenclature for Algae, Fungi, and Plants (ICN), whose guidelines are followed in the botanical examples below, and the International Code of Zoological Nomenclature (ICZN) (see bibliog. 5). Note that some fields, such as virology, have slightly different rules. Writers and editors should try to follow the standards established within those fields.
8.120 Genus and specific epithet. Whether in lists or in running text, the Latin names of species of plants and animals are italicized. Each binomial con tains a genus name (orgeneric name), which is capitalized, and a species name (also called specific name or specific epithet), which is lowercased (even if it is a proper adjective). Do not confuse these names with phyla, orders, and such, which are not italicized; see 8.126.
The Pleistocene saber-toothed cats all belonged to the genus Smilodon.
Many species names, such as Rosa caroliniana and Styrax californica, reflect the locale of the first specimens described.
The pike, Esox lucius, is valued for food as well as sport.
For the grass snake Natrix natrix, longevity in captivity is ten years.
Certain lizard taxa, such as Basiliscus and Crotaphytus, are bipedal specialists.
8.121 Abbreviation of genus name. After the first use the genus name may be abbreviated to a single capital letter. If two or more species of the same Scientific Terminology 8.124 genus are listed together, the abbreviation may be doubled (to indicate the plural) before the first species, though repeating the abbreviation with each species is more common. But if species of different genera be ginning with the same letter are discussed in the same context, abbrevi ations may not be appropriate.
Two methods allow us to estimate the maximum speeds obtained by Callisaurus draconoides in the field. Irschick and Jayne (1998) found that stride durations of both C. draconoides and Uma scoparia do not change dramatically after the fifth stride during accelerations from a standstill.
The “quaking” of the aspen, Populus tremuloides, is due to the construction of the petiole; an analogous phenomenon has been noted in the cottonwood, P. del- toides.
Among popular species of the genus Cyclamen are CC. coum, hederifolium, and persicum... [or, more commonly, C. coum, C. hederifolium, and C.persicum...] Studies of Corylus avellana and Corokia cotoneaster...; in further studies it was noted that Corylus avellana and Corokia cotoneaster...
8.122 Subspecies and varieties. A subspecific zoological name or epithet, when used, follows the binomial species name and is also italicized. If the two names are the same, the first one may be abbreviated.
Noctilio labialis labialis (or Noctilio l. labialis) Trogon collaris puella In horticultural usage, the abbreviations “subsp.” (or “ssp.”), “var.,” and “f.” (none of them italicized) are inserted before the subspecific epithet or variety or form name. See also 8.123.
Buxus microphy Ila var.japonica Hydrangea anomala subsp. petiolaris Rhododendron arboreum f. album 8.123 Unspecified species and varieties. The abbreviations “sp.” and “var.,” when used without a following element, indicate that the species or va riety is unknown or unspecified. The plural “spp.” is used to refer to a group of species. The abbreviations are not italicized.
Rhododendron spp. Rosa rugosa var. Viola sp.
8.124 Author names. The name of the person who proposed a specific epithet is sometimes added, often abbreviated, and never italicized. A capital L. stands for Linnaeus; Mill, stands for Miller.
8.125 NAMES, TERMS, AND TITLES OF WORKS Diaemusyoungi cypselinus Thomas Molossus coibensis J. A. Allen Euchistenes hartii (Thomas) Quercus alba L.
Felis leo Scop. Linaria spuria (L.) Mill.
The parentheses in the second example, from zoology, mean that Thomas originally described the species E. hartii but referred it to a dif ferent genus. In botanical usage, the name of the person who referred it to the new genus is added after the parentheses, as in the last example.
8.125 Plant hybrids. The crossing of two species is indicated by a multiplica tion sign (x; not the letter x) between the two species names, with space on each side. Many older primary plant hybrids are indicated by a multi plication sign immediately before the specific epithet of the hybrid, with space only before it.
Magnolia denudata x M. liliiflora (crossing of species) Magnolia xsoulangeana (hybrid name) 8.126 Higher divisions. Divisions higher than genus—phylum, class, order, and family—are capitalized but not italicized. (The terms order, family, and so on are not capitalized.) Intermediate groupings are treated similarly.
Chordata (phylum) Chondrichthyes (class) Monotremata (order) Ruminantia (suborder) Hominidae (family) Felinae (subfamily) Selachii (term used of various groups of cartilaginous fishes) The new species Gleichenia glauca provides further details about the history of Gleicheniaceae.
8.127 English derivatives. English words derived from the taxonomic system are lowercased and treated as English words.
carnivore (s) (from the order Carnivora) hominid(s) (from the family Hominidae) irid(s) (from the family Iridaceae) feline (s) (from subfamily Felinae) astilbe(s) (from the genus Astilbe) mastodon (s) (from the genus Mastodon) Scientific Terminology 8.131 Vernacular Names of Plants and Animals 8.128 Plants and animals—additional resources. For the correct capitalization and spelling of common names of plants and animals, consult a dictio nary or the authoritative guides to nomenclature, the ICN and the ICZN, mentioned in 8.119. In general, Chicago recommends capitalizing only proper nouns and adjectives, as in the following examples, which con form to Merriam-Webster's Collegiate Dictionary: Dutchman’s-breeches j ack-in-the-pulpit mayapple Cooper’s hawk rhesus monkey Rocky Mountain sheep 8.129 8.130 8.131 Domestic animals and horticultural categories. Either a dictionary or the guides to nomenclature ICZN and ICN should be consulted for the proper spelling of breeds of domestic animals and broad horticultural categories.
German shorthaired pointer Hereford Maine coon or coon cat Rhode Island Red boysenberry rambler rose Thoroughbred horse (but purebred dog) Horticultural cultivars. Many horticultural cultivars (cultivated varie ties) have fanciful names that must be respected since they may be reg istered trademarks.
the Peace rose a Queen of the Market aster In some horticultural publications, such names are enclosed in single quotation marks; any following punctuation is placed after the closing quotation mark. If the English name follows the Latin name, there is no intervening punctuation. For examples of this usage, consult any issue of the magazine Horticulture (bibliog. 5).
The hybrid Agastache ‘Apricot Sunrise’, best grown in zone 6, mingles with sheaves of cape fuchsia (Phygelius ‘Salmon Leap’).
Genetic Terms Genetic nomenclature—additional resources. Only the most basic guidelines can be offered here. Writers or editors working in the field of 8.132 NAMES, TERMS, AND TITLES OF WORKS genetics should consult the AMA Manual of Style or Scientific Style and Format (both in bibliog. 1.1) and online databases including the HGNC (HUGO Gene Nomenclature Committee) database of human gene names and the Mouse Genome Database (both in bibliog. 5).
8.132 Genes. Names of genes, or gene symbols, including any arabic numerals that form a part of such symbols, are usually italicized. (Italicization helps differentiate genes from entities with similar names.) Symbols for genes contain no Greek characters or roman numerals. Human gene symbols are set in full capitals, as are the gene symbols for other primates. Mouse and rat gene symbols are usually spelled with an initial capital. Gene no menclature systems for other organisms (yeast, fruit flies, nematodes, plants, fish) vary. Symbols for proteins, also called gene products and often derived from the symbols of the corresponding genes, are set in roman.
Human genes BRCA1 GPC3 IGH(a) (the symbol @ indicates a family or cluster) SNRPN Mouse genes Cmvl Fgfl2 Roml Wntl NLP3 (gene symbol); NLP3p (encoded protein; notep suffix) GIF (gene symbol); GIF (gastric intrinsic factor) Only a very few gene symbols contain hyphens.
HLA-DRB1, for human leukocyte antigen D-related p chain 1 8.133 Enzymes. Enzyme names consist of a string of italic and roman charac ters. The first three letters, which represent the name of the organism (usually a bacterium) from which the enzyme has been isolated, are ital icized. The roman numeral that follows represents the series number. Sometimes an upper- or lowercase roman letter or an arabic numeral (or both), representing the strain of bacterium, intervenes between the name and series number.
Aval BamHl Clal EcoRl Hindlll Sau3AI Scientific Terminology 8.137 Geological Terms 8.134 Geological terms—additional resources. The following paragraphs offer only the most general guidelines. Writers or editors working in geologi cal studies should consult US Geological Survey, Suggestions to Authors of the Reports of the United States Geological Survey, and Scientific Style and Format (both listed in bibliog. 1.1).
8.135 Formal versus generic geological terms. Formal geological terms are capitalized in both noun and adjective forms; terms used generically are not. The generic terms eon, era, and the like are lowercased or omitted immediately following a formal name. Eons are divided into eras, eras into periods, periods into epochs, and epochs into stages. The term ice age is best lowercased in scientific contexts because of the uncertainty surrounding any formal use of the term (cf. Little Ice Age); but see 8.74.
the Archean (eon) the Mesoproterozoic (era) the Tertiary period of the Cenozoic (era) the Paleocene (epoch) Pleistocene-Holocene transition the second interglacial stage or II interglacial Illinoian glaciation The modifiers early, middle, or late are capitalized when used formally but lowercased when used informally.
Early Archean Middle Cambrian Late Quaternary but early Middle Cambrian in late Pleistocene times 8.136 Stratigraphy. Formal stratigraphic names are capitalized. For prehistoric cultural terms, see 8.74.
Fleur de Lys Supergroup Niobrara Member Ramey Ridge Complex Morrison Formation Astronomical Terms 8.137 Astronomical terms—additional resources. The following paragraphs offer only the most general guidelines. Writers or editors working in 8.138 NAMES, TERMS, AND TITLES OF WORKS astronomy or astrophysics should consult Scientific Style and Format (bib liog. 1.1) and the website of the International Astronomical Union.
8.138 Celestial bodies. The names of galaxies, constellations, stars, planets, and such are capitalized. For earthy sun, and moon, see 8.140, 8.141.
Aldebaran Alpha Centauri or a Centauri the Big Dipper or Ursa Major or the Great Bear Cassiopeia’s Chair the Crab Nebula 85 Pegasi the Magellanic Clouds the Milky Way the North Star or Polaris, polestar Saturn but Halley’s comet the solar system 8.139 Catalog names for celestial objects. Celestial objects listed in well- known catalogs are designated by the catalog name, often abbreviated, and a number.
Bond 619 Lalande5761 Lynds 1251 or L1251 NGC 6165 8.140 "Earth." In nontechnical contexts, the word earth, in the sense of our planet, is usually lowercased when preceded by the or in such idioms as “down to earth” or “move heaven and earth.” When used as the proper name of our planet, especially in context with other planets, it is capital ized and the is usually omitted.
Some still believe the earth is flat.
The gender accorded to the moon, the sun, and the earth varies in different my thologies.
Where on earth have you been?
The astronauts have returned successfully to Earth.
Does Mars, like Earth, have an atmosphere?
8.141 "Sun" and "moon." The words sun and moon are usually lowercased in nontechnical contexts and always lowercased in the plural.
Scientific Terminology 8.145 The moon circles the earth, as the earth circles the sun.
Some planets have several moons.
Some publications in the fields of astronomy and related sciences, how ever, routinely capitalize these words when used as proper nouns. (See also 8.138.) 8.142 Descriptive terms. Merely descriptive terms applied to celestial objects or phenomena are not capitalized.
aurora borealis or northern lights interstellar dust gegenschein the rings of Saturn Medical Terms 8.143 Medical terms—additional resources. The following paragraphs offer only the most general guidelines. Medical writers or editors should con sult the AMA Manual of Style or Scientific Style and Format (both in bib liog. 1.1).
8.144 Diseases, procedures, and such. Names of diseases, syndromes, diag nostic procedures, anatomical parts, and the like are lowercased, except for proper names forming part of the term. Acronyms and initialisms are capitalized.
acquired immunodeficiency syndrome or AIDS Alzheimer disease (see below) computed tomography or CT Down syndrome (see below) finger-nose test islets of Langerhans non-Hodgkin lymphoma (see below) ultrasound; ultrasonography The possessive forms Alzheimer's, Down's, Hodgkin's, and the like, though less common in medical literature, may be preferred in a general con text. For x-rays and radiation, see 8.151.
8.145 Infections. Names of infectious organisms are treated like other specific names (see 8.119-27). Common forms of such names and the names of conditions based on such names are neither italicized nor capitalized, except in the case of a proper noun.
8.146 NAMES, TERMS, AND TITLES OF WORKS Microorganisms of the genus Streptococcus are present in the blood of persons with streptococcal infection.
The larvae of Trichinella spiralis are responsible for the disease trichinosis.
The Ebola virus (which refers to the species Zaire ebolavirus) derives its name from the Ebola River.
8.146 Drugs. Generic names of drugs, which should be used wherever pos sible in preference to brand names, are lowercased. Brand names must be capitalized; they are often enclosed in parentheses after the first use of the generic name. For guidance, consult the AMA Manual of Style and Scientific Style and Format (bibliog. 1.1) and USP Dictionary of USAN and International Drug Names (bibliog. 3.3). For brand names and trade marks, see 8.153.
The patient takes weekly injections of interferon beta-la (Avonex) to control his multiple sclerosis.
Physical and Chemical Terms 8.147 Physical and chemical terms—additional resources. The following para graphs offer only the most general guidelines for nontechnical editors. Writers or editors working in physics should consult AIP Publishing’s online author resources (bibliog. 1.1) or, among other journals, Physical Review Letters (bibliog. 5); those working in chemistry should consult The ACS Style Guide (bibliog. 1.1).
8.148 Laws and theories. Though usage varies widely, Chicago recommends that names of laws, theories, and the like be lowercased, except for proper names attached to them.
Avogadro’s hypothesis (or Avogadro’s law) the big bang theory Boyle’s law (Einstein’s) general theory of relativity Newton’s first law 8.149 Chemical names and symbols. Names of chemical elements and com pounds are lowercased when written out. Symbols, however, are capital ized and set without periods; the number of atoms in a molecule appears as a subscript. For a list of symbols for the elements, including atomic numbers, see 10.63.
Brand Names and Trademarks 8.153 ozone; O3 sulfuric acid; H2SO4 sodium chloride; NaCl tungsten carbide; WC 8.150 Mass number. In formal chemical literature, the mass number appears as a superscript to the left of the symbol. In work intended for a general audience, however, it may follow the symbol, after a hyphen, in full size.
14C (formal style); C-14 or carbon-14 (informal style) 238U (formal style); U-238 or uranium-238 (informal style) 8.151 Radiations. Terms for electromagnetic radiations may be spelled as fol lows: P~ray (noun or adjective) or beta ray (in nonscientific contexts, noun or adjective) y-ray (noun or adjective) or gamma ray (in nonscientific contexts, noun or adjec tive) x-ray (noun, verb, or adjective) cosmic ray (noun); cosmic-ray (adjective) ultraviolet ray (noun); ultraviolet-ray (adjective) Note that the verb to x-ray, though acceptable in a general context, is not normally used in scholarly medical literature, where writers would more likely speak of obtaining an x-ray image, or a radiograph, of something, or of subjecting something to x-ray analysis.
8.152 Metric units. Although the spellings meter, liter, and so on are widely used in the United States, some American business, government, or professional organizations have adopted the European spellings (metre, litre, etc.). Chicago’s publications show a preference for the traditional American spellings. For abbreviations used in the International System of Units, see 10.51-59.
Brand Names and Trademarks 8.153 Trademarks. Brand names that are trademarks—often so indicated in dictionaries—should be capitalized if they must be used. A better choice is to substitute a generic term when available. Although the symbols ® and ™ (for registered and unregistered trademarks, respectively) often accompany trademark names on product packaging and in promotional material, there is no legal requirement to use these symbols, and they should be omitted wherever possible. (If one of these symbols must be used at the end of a product name, it should appear before any period, 8.154 NAMES, TERMS, AND TITLES OF WORKS comma, or other mark of punctuation.) Note also that some companies encourage the use of both the proper and the generic term in reference to their products (“Kleenex facial tissue,” not just “Kleenex”) and discour age turning product names into verbs, but these restrictions, while they may be followed in corporate documentation, are not legally binding. (In fact, Webster's includes entries for lowercase verbs google, photoshop, and xerox.) See also 8.155.
Bufferin; buffered aspirin Coca-Cola; cola Google; search engine; search Jacuzzi; whirlpool bath Kleenex; (facial) tissue Levi’s; jeans Photoshop; image-editing software Ping-Pong; table tennis Post-it Note; sticky note Pyrex; heat-resistant glassware Scrabble Sharpie; permanent marker Vaseline; petroleum jelly Xerox; photocopier; copy More information about registered trademarks can be found on the web sites of the US Patent and Trademark Office and the International Trade mark Association.
8.154 Brand names or trademarks with an initial lowercase letter. Brand names or trademarks spelled with a lowercase initial letter followed by a capital letter need not be capitalized at the beginning of a sentence or heading; the existing capital letter is sufficient to signal that these are proper nouns. Likewise, names that begin with a capital letter and in clude additional capitals in the middle of the word should be left un changed. (In either scenario, such capital letters are sometimes referred to as intercaps or midcaps.) Chicago draws the line, however, at names in all lowercase; in order to signal that such a term is in fact a proper noun, an initial capital should be applied even midsentence (as for Mini in the last example; see also 8.69, 8.155).
iTunes is both an app and a media service.
Does your iPhone have an AccuWeather app?
PowerPoint has become virtually synonymous with presentation software. but The Mac Mini is a good solution for cluttered workspaces.
In text that is set in all capitals, such distinctions are usually overridden (e.g., POWERPOINT); with a mix of capitals and small capitals, they are preserved (e.g., iPh o n e ).
Titles of Works 8.156 Software and Devices 8.155 Names for applications, operating systems, and devices. References to specific applications (also called programs or apps) and the operating systems and devices they run on are set in roman type without quota tion marks (but see 8.190); capitalization can usually reflect the usage displayed by the software or the device itself. If in doubt, consult a help menu or a user’s guide. Occasionally, an apparent preference for low ercase can be overridden (as in the first example below; see also 8.154). Generic references can be treated as ordinary text.
OS X; Macintosh; Mac Mini (contrary to corporate usage); Mac Windows 10; HP desktop computer; notebook computer; PC Microsoft Word; Apple Pages for Mac; LibreOffice Writer; Google Docs; word processor iOS 10; iPhone; the Maps app for iOS 10; the Sleep/Wake button Firefox; a browser; the Firefox app for Android the Messenger app for Android; Apple’s Messages app; a messaging app Kindle; the Kindle app for Apple devices; Android’s Kindle app The iPhone’s Clock app includes a stopwatch, a countdown timer, and an alarm. Does your phone have a clock app?
I prefer the New Yorker's iPhone app to the printed magazine.
Use your word-processing program to track changes and insert comments.
For typographic conventions for the names of particular keys, menus, commands, and the like, see 7.76-80.
Titles of Works 8.156 Treatment of titles in text and notes—overview. The following guide lines apply primarily to titles as they are mentioned or cited in text or notes. They apply to titles of books, journals, newspapers, and websites as well as to shorter works (stories, poems, articles, etc.), divisions of lon ger works (parts, chapters, sections), unpublished works (lectures etc.), plays and films, radio and television programs, musical works, and art works. For details on citing titles in bibliographies and reference lists, see chapters 14 and 15. For the treatment of titles in languages other than English, see 11.6-10.
8.157 NAMES, TERMS, AND TITLES OF WORKS Capitalization, Punctuation, and Italics 8.157 Capitalization of titles of works—general principles. Titles mentioned or cited in text or notes are usually capitalized headline-style (see 8.159). For aesthetic purposes, titles appearing on the cover or title page or at the head of an article or chapter may deviate from Chicago’s rules for the capitalization of titles. For capitalization of non-English titles, see 11.6. For the use of quotation marks versus italics, see 8.163.
8.158 Principles and examples of sentence-style capitalization. In sentence style capitalization, only the first word in a title, the first word in a sub title, and any proper names are capitalized. This style or some variant of it is commonly used in library catalogs and in the reference lists of some journals (see 15.13) and is the style recommended for most titles from other languages (see 11.6). It is also useful for some types of subheads (see 2.18), including those that include terms (such as species names) that require their own internal capitalization (but note that the specific epithet remains lowercase in headline style; see 8.159, rule 7). See also 8.162.
The house of Rothschild: The world’s banker, 1849-1999 Crossing Magnolia denudata with M. liliiflora to create a new hybrid: A success story 8.159 Principles of headline-style capitalization. The conventions of head line style are governed mainly by emphasis and grammar. The following rules, though occasionally arbitrary, are intended primarily to facilitate the consistent styling of titles mentioned or cited in text and notes: 1. Capitalize the first and last words in titles and subtitles (but see rule 7), and capitalize all other major words (nouns, pronouns, verbs, adjectives, adverbs, and some conjunctions—but see rule 4).
2. Lowercase the articles the, a, and an.
3. Lowercase prepositions, regardless of length, except when they are used adverbially or adjectivally (up in Look Up, down in Turn Down, on in The On Button, to in Come To, etc.) or when they compose part of a Latin expression used adjectivally or adverbially (De Facto, In Vitro, etc.).
4. Lowercase the common coordinating conjunctions and, but, for, or, and nor.
5. Lowercase to not only as a preposition (rule 3) but also as part of an infinitive (to Run, to Hide, etc.), and lowercase as in any grammatical function.
6. Lowercase the part of a proper name that would be lowercased in text, such as de or von.
Titles of Works 8.161 7. Lowercase the second part of a species name, such as fulvescens in Acipenser fulvescens, even if it is the last word in a title or subtitle.
For examples, see 8.160. For hyphenated compounds in titles, see 8.161.
8.160 Examples of headline-style capitalization. The following examples il lustrate the numbered rules in 8.159. All of them demonstrate the first rule; the numbers in parentheses refer to rules 2-7.
Mnemonics That Work Are Better Than Rules That Do Not Singing While You Work A Little Learning Is a Dangerous Thing (2) Four Theories concerning the Gospel according to Matthew (2,3) Taking Down Names, Spelling Them Out, and Typing Them Up (3, 4) Tired but Happy (4) The Editor as Anonymous Assistant (5) From Homo erectus to Homo sapiens: A Brief History (3,7) Defenders of da Vinci Fail the Test: The Name Is Leonardo (2, 3, 6) Sitting on the Floor in an Empty Room (2, 3), but Turn On, Tune In, and Enjoy (3,4) Ten Hectares per Capita, but Landownership and Per Capita Income (3) Progress in In Vitro Fertilization (3) 8.161 Hyphenated compounds in headline-style titles. The following rules apply to hyphenated terms appearing in a title capitalized in headline style. For rules of hyphenation, see 7.81-89.
1. Always capitalize the first element.
2. Capitalize any subsequent elements unless they are articles, prepositions, coordinating conjunctions (and, but, for, or, nor), or such modifiers as flat or sharp following musical key symbols.
3. If the first element is merely a prefix or combining form that could not stand by itself as a word (anti, pre, etc.), do not capitalize the second element unless it is a proper noun or proper adjective.
4. Capitalize the second element in a hyphenated spelled-out number (twenty- one or twenty-first, etc.) or hyphenated simple fraction (two-thirds in two-thirds majority).
The examples that follow demonstrate the numbered rules (all the ex amples demonstrate the first rule; the numbers in parentheses refer to rules 2-4).
8.162 NAMES, TERMS, AND TITLES OF WORKS Under-the-Counter Transactions and Out-of-Fashion Initiatives (2) Bed-and-Breakfast Options in Upstate New York (2) Record-Breaking Borrowings from Medium-Sized Libraries (2) Cross-Stitching for Beginners (2) A History of the Chicago Lying-In Hospital (2; “In” functions as an adverb, not a preposition) The E-flat Concerto (2) Self-Sustaining Reactions (2) Anti-intellectual Pursuits (3) Why Solar Is the Future of E-books (3) A Two-Thirds Majority of Non-English-Speaking Representatives (3,4) Ninety-Fifth Avenue Blues (4) Atari’s Twenty-First-Century Adherents (4) Under another, simplified practice that is not recommended by Chicago, only the first element and any subsequent element that is a proper noun or adjective are capitalized.
8.162 Titles containing quotations. When a direct quotation of a sentence or an independent clause is used as a title, headline-style capitalization may be imposed, even for longer quotations. See also 14.94.
“We All Live More like Brutes Than like Humans”: Labor and Capital in the Gold Rush 8.163 Italics versus quotation marks for titles. The choice of italics or quota tion marks for a title of a work cited in text or notes is determined by the type of work. Titles of books and periodicals are italicized (see 8.168); titles of articles, chapters, and other shorter works are set in roman and enclosed in quotation marks (see 8.177).
Many editors use The Chicago Manual of Style.
Refer to the article titled “A Comparison of MLA and APA Style.” For treatment of book series and editions, see 8.176; for poems and plays, see 8.181-84; for fairy tales and nursery rhymes, see 8.185; for pamphlets and forms, see 8.186-87; for unpublished works, see 8.188; for movies, television, radio, and podcasts, see 8.189; for video games, see 8.190; for websites and blogs, see 8.191-92; for musical works, see 8.193-97; for works of art and exhibitions, see 8.198-201. For titles from other lan guages, see 11.6-10.
Titles of Works 8.165 8.164 Subtitle capitalization. A subtitle, whether in sentence-style or headline style capitalization, always begins with a capital letter. Although on a title page or in a chapter heading a subtitle is often distinguished from a title by a different typeface, when referred to it is separated from the title by a colon. When an em dash rather than a colon is used, what follows the em dash is not normally considered to be a subtitle, and the first word is not necessarily capitalized. See also 14.90.
“Manuals of Style: Guidelines, Not Strangleholds” (headline style) Tapetum character states: Analytical keys (sentence style) but Chicago—a Metropolitan Smorgasbord 8.165 Permissible changes to titles. When a title is referred to in text or notes or listed in a bibliography or reference list, its original spelling (including non-Latin letters such as 7t or y) and hyphenation should be preserved, re gardless of the style used in the surrounding text. Capitalization may be changed to headline style (8.159) or sentence style (8.158), as applicable. As a matter of editorial discretion, an ampersand (&) may be changed to and, or, more rarely, a numeral may be spelled out (see 14.88). On title pages, commas are sometimes omitted from the ends of lines for aes thetic reasons. When such a title is referred to, such commas should be added, including any comma omitted before a date that appears on a line by itself at the end of a title or subtitle. (Serial commas should be added only if it is clear that they are used in the work itself; see 6.19.) If title and subtitle on a title page are distinguished by typeface alone, a colon must be added when referring to the full title. A dash in the original should be retained; however, a semicolon between title and subtitle may usually be changed to a colon. (For two subtitles in the original, see 14.90. For older titles, see 14.97.) The following examples illustrate the way titles and subtitles are normally punctuated and capitalized in running text, notes, and bibliographies using headline capitalization. The first three are books, the fourth an article.
Disease, Pain, and Sacrifice: Toward a Psychology of Suffering Melodrama Unveiled: American Theater and Culture, 1800-1850 Browning’s Roman Murder Story: A Reading of “The Ring and the Book” “Milton Friedman’s Capitalism and Freedom—a Best-Seller for Chicago” For titles within titles (as in the third and fourth examples above), see 8.173, 8.177. For double titles connected by or, see 8.167.
8.166 NAMES, TERMS, AND TITLES OF WORKS 8.166 Titles in relation to surrounding text. A title, which is considered to be a singular noun, always takes a singular verb. Moreover, any punctuation that is part of the title should not affect the punctuation of the surround ing text (with the exception of a sentence-ending period, which should be omitted after a title ending in a question mark or exclamation point; see 6.124). See also 6.28, 6.125, 8.174.
The Waves is not a typical novel, (singular verb in spite of plural in title) Her role in Play It Again, Sam confirmed her stature, (no comma after Sam) Three stories she never mentioned were “Are You a Doctor?,” “The Library of Babel,” and “The Diamond as Big as the Ritz.” (comma after first title in spite of the question mark) 8.167 Double titles connected by "or." Old-fashioned double titles (or titles and subtitles) connected by or have traditionally been punctuated in a variety of ways. When referring to such titles, prefer the punctuation on the title page or at the head of the original source. In the absence of such punctuation (e.g., when the title is distinguished from the subtitle by ty pography alone), or when the source is not available to consult, prefer the simpler form shown in the first example (see also 8.165). This small departure from advice in earlier editions of this manual recognizes the importance of balancing editorial expediency with fidelity to original sources. See also 14.91.
The Tempest, or The Enchanted Island but Moby-Dick; or, The Whale Books and Periodicals 8.168 Treatment of book and periodical titles. When mentioned in text, notes, or bibliography, the titles and subtitles of books and periodicals are ital icized and capitalized headline-style (see 8.159), though some publica tions may require sentence style for reference lists (see 8.158,15.13). A book title cited in full in the notes or bibliography may be shortened in text (e.g., a subtitle may be omitted). For short titles in notes, see 14.30.
8.169 An initial "a," "an," or "the" in book titles. An initial a, an, or the in run ning text may be dropped from a book title if it does not fit the surround ing syntax. When in doubt, or if the article seems indispensable, it should be retained.
Titles of Works 8.170 Fielding, in his introduction to The History of Tom Jones, a Foundling, announces himself as a professional author.
Fielding’s History of Tom Jones...
That dreadful Old Curiosity Shop character, Quilp...
but In The Old Curiosity Shop, Dickens...
In L’Amour’s The Quick and the Dead...
8.170 An initial "the" in periodical titles. When newspapers and other peri odicals are mentioned in text, an initial the, even if part of the title on the masthead, is usually lowercased (unless it begins a sentence) and not italicized. Most newspapers and many journals (and, by analogy with journals, some magazines) are referred to with a definite article whether or not it might be considered part of the official title; treating the definite article as part of the surrounding text facilitates consistency in discus sions that mention a variety of periodical titles. In the examples below, The is included on the masthead or cover of all but the Los Angeles Times, the Chicago Tribune, and the Journal of Labor Economics (an article never appears with Forbes). As with book titles, use of the definite article will depend on the syntax of the surrounding sentence (see 8.169).
I read the Los Angeles Times on my phone and the Chicago Tribune on my com puter, but I prefer the paper edition of the New York Times.
She reads the Journal of Labor Economics at work and the American Naturalist at home.
Do you get your information from the Wall Street Journal or Forbes?
The New Yorker's cartoons generally have nothing to do with the surrounding text. but I’m a New Yorker fan for the cartoons alone.
Her Wall Street Journal subscription expired last month.
Some editors may prefer to retain an initial The with the titles of peri odicals that otherwise consist of a single word (but not with a title that is actually a short form). This practice should be used consistently, and only with titles that include the article.
Have you read The Week this week? How about The Believer? Would you believe The Onion?
I saw it in The Times (referring to the British publication). but I read Harper's mainly for its puzzles.
Does the Times (referring to the New York Times) publish a crossword?
8.171 NAMES, TERMS, AND TITLES OF WORKS Because it may govern the inflection of the following word, non-English titles retain the article in the original language—but only if it is an official part of the title (see also 11.27).
We read Le Monde and DieZeit while traveling in Europe. but Did you see the review in the Frankfurter Allgemeine?
In citation form, an initial the is dropped from periodical titles (except for those, if any, for which the The has been incorporated into the title in the text; see above); for languages other than English, articles are retained (see 14.193,14.194).
8.171 "Magazine" and other descriptive terms. A word like magazine Journal, or review should be italicized only when it forms part of the official title of a particular periodical. When such a word functions as an added descrip tive term, it is treated as part of the surrounding text. See also 8.170.
I read it both in Time magazine and in the Wall Street Journal, but His article was reprinted in the New York Times Magazine.
8.172 Periodical titles in awards, buildings, and so forth. When the title of a newspaper or periodical is part of the name of a building, organization, prize, or the like, it is not italicized.
Los Angeles Times Book Prize Chicago Defender Charities Tribune Tower 8.173 Italicized terms and titles within titles. Any term within an italicized title that would itself be italicized in running text—such as a word from another language, a genus name, or the name of a ship—should be set in roman type (reverse italics). See also 8.116, 8.120, 8.165,14.95.
From Tyrannosaurus rex to King Kong: Large Creatures in Fact and Fiction The BigE: The Story of the USS Enterprise A title of a work within another title, however, should remain in italics and be enclosed in quotation marks. See also 14.94.
A Key to Whitehead’s “Process and Reality” 8.174 Title not interchangeable with subject. The title of a work should not be used to stand for the subject of a work.
Titles of Works 8.177 Dostoevsky wrote a book about crime and punishment (not... about Crime and Punishment).
Edward Wasiolek’s book on Dostoevsky’s Crime and Punishment is titled “Crime and Punishment” and the Critics. (See also 8.173.) In their book The Craft of Translation, Biguenet and Schulte ... (not In discussing The Craft of Translation, Biguenet and Schulte...) 8.175 Titles of multivolume works. Titles of multivolume books are treated in the same manner as titles of single-volume works, as are named titles of individual volumes. The word volume may be abbreviated in paren theses and notes; it is capitalized (and never abbreviated) only if part of the title. For treatment of multivolume works in bibliographies and ref erence lists, see 14.116-22. See also 8.176.
The Day of the Scorpion, volume 2 of The Raj Quartet Art in an Age of Counterrevolution, 1815-1848 (vol. 3, A Social History of Modern Art) the fourth volume of the landmark eleventh edition of Encyclopaedia Britannica 8.176 Titles of series and editions. Titles of book series and editions are capi talized but not italicized. The words series and edition are capitalized only if part of the title. See also 14.123-26.
the Loeb Classical Library a Modern Library edition Late Editions: Cultural Studies for the End of the Century the Crime and Justice series a book in the Heritage of Sociology Series Numbered or named editions of a specific publication (as in a cited source) are usually not part of a title and are set in roman and lowercase (if in doubt, consult a library catalog). See also 14.113.
Black's Law Dictionary, 10th ed.
Kuhn, The Structure of Scientific Revolutions, 50th anniversary ed. but Hayek, The Constitution of Liberty: The Definitive Edition Articles in Periodicals and Parts of a Book 8.177 Articles, stories, chapters, and so on. Titles of articles and features in periodicals and newspapers, chapter and part titles, titles of short stories 8.178 NAMES, TERMS, AND TITLES OF WORKS or essays, and individual selections in books are set in roman type and enclosed in quotation marks.
John S. Ellis’s article “Reconciling the Celt” appeared in the Journal of British Studies.
In chapter 3 of The Footnote, “How the Historian Found His Muse,” Anthony Grafton...
“Tom Outland’s Story,” by Willa Cather...
Book titles and other normally italicized terms remain italicized within an article title. A term quoted in the original title is enclosed in single quotation marks (since it is already within double quotation marks). See also 14.94.
The article “Schiller’s ‘Ode to Joy’ in Beethoven’s Ninth Symphony” received un expected attention.
Neuberger’s “Sergei Eisenstein’s Ivan the Terrible as History” calls the filmmaker’s approach to history “serious and nuanced.” Titles of regular departments or columns in periodicals are set in roman with no quotation marks (see also 14.190,14.195).
In this week’s Talk of the Town, Lizzie Widdicombe features...
8.178 Collected works. When two or more works originally published as sep arate books are included in a single volume, often as part of an author’s collected works, they are best italicized rather than placed in quotation marks.
The introduction to the Critique of Pure Reason in Kant’s Collected Works...
8.179 Terms like "foreword," "preface," and so on. Such generic terms as foreword, preface, acknowledgments, introduction, appendix, bibliography, glossary, and index, whether used in cross-references or in reference to another work, are lowercased and set in roman type.
The author states in her preface that...
For further documentation, see the appendix.
Full details are given in the bibliography.
The book contains a glossary, a subject index, and an index of names.
8.180 Numbered chapters, parts, and so on. The words chapter, part, ap pendix, table, figure, and the like are lowercased and spelled out in text Titles of Works 8.184 (though sometimes abbreviated in parenthetical references). Numbers can usually be given in arabic numerals, regardless of how they appear in the original. (Subject to editorial discretion, an exception is sometimes made for references within a work to other parts of the same work num bered with roman numerals; for other exceptions, see 9.26, 9.28.) If let ters are used, they may be upper- or lowercase (following the original) and are sometimes put in parentheses. See also 3.9, 3.50.
This matter is discussed in chapters 4 and 5.
The Latin text appears in appendix B.
The range is presented numerically in table 4.2 and diagrammed in figure 4.1. These connections are illustrated in table A3.
Turn to section 5(a) for further examples.
Poems and Plays 8.181 Titles of poems. Titles of most poems are set in roman type and enclosed in quotation marks. A very long poetic work, especially one constituting a book, is italicized and not enclosed in quotation marks.
Robert Frost’s poem “The Housekeeper” in his collection North of Boston Dante’s Inferno 8.182 Poems referred to by first line. Poems referred to by first line rather than by title are capitalized sentence-style, even if the first word is lowercased in the original, but any words capitalized in the original should remain capitalized. See also 8.158,16.145.
E. E. Cummings, in “My father moved through dooms of love,”... (“my” is lower cased in the original) “Shall I compare thee to a Summer’s day?” 8.183 Titles of plays. Titles of plays, regardless of the length of the play, are italicized.
Shaw’s Arms and the Man, in volume 2 of his Plays: Pleasant and Unpleasant 8.184 Divisions of plays or poems. Words denoting parts of long poems or acts and scenes of plays are usually lowercased, neither italicized nor enclosed in quotation marks. Numbers are arabic, regardless of the orig inal.
canto 2 stanza 5 act 3, scene 2 8.185 NAMES, TERMS, AND TITLES OF WORKS Fairy Tales and Nursery Rhymes 8.185 Titles of folktales, fables, nursery rhymes, and the like. Folktales, fables, fairy tales, nursery rhymes, and the like are usually treated in the manner of shorter poems and set in roman type and enclosed in quotation marks. Italics should be used to refer to fairy tales published as books, plays, and the like.
“Aladdin” is arguably the most well-known tale in A Thousand and One Nights. “Rumpelstiltskin” originally appeared in the Grimm brothers’ Children’s and Household Tales.
Everybody knows at least one verse of “Jack and Jill.” Ella Enchanted is a retelling by Gail Carson Levine of “Cinderella.” The opera Hansel and Gretel (Hansel und Gretel) is based on the fairy tale of the same name.
Pamphlets, Reports, and Forms 8.186 Titles of pamphlets and reports. Titles of pamphlets, reports, and simi lar freestanding publications are, like books, italicized when mentioned or cited in text or notes (see also 8.168,14.220).
Payne’s Common Sense, first published anonymously...
Young Adult Migration: 2007-2009 to 2010-2012, a report published in 2015 as part of the US Census Bureau’s American Community Survey...
8.187 Titles of forms. Government, departmental, and other titled or num bered forms can usually be capitalized according to the usage in the form itself; wording should follow the usage in the document itself but may be shortened if necessary.
Form 1040-ES, Estimated Tax for Individuals (or 1040-ES) DHS TRIP Traveler Inquiry Form (or DHS TRIP) United States Census 2010; the Census (but a census form) Unpublished Works 8.188 Titles of unpublished works. Titles of unpublished works—theses, disser tations, manuscripts in collections, unpublished transcripts of speeches, Titles of Works 8.189 and so on—are set in roman type, capitalized as titles, and enclosed in quotation marks. Titles of manuscript collections take no quotation marks. The title of a not-yet-published book that is under contract may be italicized, but the word forthcoming (or in press or some other equiva lent term), in parentheses, must follow the title. For speeches, see 8.76. See also 15.45.
In a master’s thesis, “Charles Valentin Alkan and His Pianoforte Works,” ...
“A Canal Boat Journey, 1857,” an anonymous manuscript in the Library of Con gress Manuscripts Division, describes...
Letters and other material may be found in the Collis P. Huntington Papers at the George Arents Library of Syracuse University.
Gianfranco’s Fourth Millennium (forthcoming) continues this line of research.
Movies, Television, Radio, and Podcasts 8.189 Titles for movies, television, radio, and podcasts. Titles of movies (or films) and movie series and of television, radio, and podcast programs and series are italicized. A single episode in a television, radio, or podcast series is set in roman and enclosed in quotation marks. Sequels should be numbered as in the source itself; if in doubt, prefer arabic numerals (see also 9.43). The names of networks, channels, streaming services, and the like are set in roman.
Gone with the Wind The Godfather, Part II (see also 9.43) The Hunger Games; The Hunger Games: Mockingjay—Part 1; the Hunger Games film series Sesame Street on PBS The Ten O’clock News, WGBH’s long-running program “The Alibi,” the first episode in the podcast Serial, a This American Life spin-off NPR’s Ask Me Another, a podcast hosted by Ophira Eisenberg Performance Today, hosted by Fred Child and produced by American Public Me dia; broadcast by Minnesota Public Radio and others and also available as a podcast “Thirsty Bird,” the first episode in the second season of the Netflix series Orange Is the New Black Season 5, episode 4, of the Masterpiece series Downton Abbey, originally broadcast on ITV (UK) and PBS (US) and also available from Amazon Video but the ten o’clock news 8.190 NAMES, TERMS, AND TITLES OF WORKS Video Games 8.190 Titles of video games. Though video games are technically software applications, or apps (see 8.155), they may be treated like movies, a us age that recognizes the narrative and audiovisual similarities between the two art forms. Older video games—despite being technically simpler than the majority of today’s apps—are treated in the same way. (Note that this usage does not apply to other types of games, which are set in roman rather than italics and capitalized only if trademarked: e.g., Monopoly but poker.) See also 8.189.
Pong Ms. Pac-Man Tetris Angry Birds; the Angry Birds app Nintendo’s Mario Bros.; Mario Kart Wii; Mario Kart 7 for Nintendo 3DS Call of Duty; Call of Duty 2; Call of Duty: Modern Warfare 3; the Call of Duty series The Sims 4; The Sims, a series that debuted in 2000 Websites and Blogs 8.191 Titles of websites and web pages. Titles of websites mentioned or cited in text or notes are normally set in roman, headline-style, without quota tion marks. An initial The should be lowercased except at the beginning of a sentence. Titled sections, pages, or special features on a website should be placed in quotation marks. Titles of the types of works discussed else where in this chapter (i.e., books, journals, etc.) should usually be treated the same whether they are published in print or online. But in a departure from the recommendations in the previous edition, the title of a website that does not have (and never had) a printed counterpart, even if it is analogous to a traditionally printed work, can be treated like the titles of other websites, subject to editorial discretion. (When in doubt, treat the source as if there is no printed counterpart.) See also 14.205-10.
Project Gutenberg; Jane Austen’s Emma, available as an audiobook from Project Gutenberg the Internet Movie Database; IMDb (note lowercase b); IMDb’s page for Live and Let Die; “Roger Moore (I)”; the page for early Bond portrayer Roger Moore Google; Google Maps; the “Maps Help Center” Facebook, Twitter, Instagram, and other social-networking sites; the Facebook app (see 8.155) Titles of Works 8.192 Wikipedia; Wikipedia’s “Let It Be” entry; Wikipedia’s entry on the Beatles’ album Let It Be but The Chicago Manual of Style Online; the online edition of The Chicago Manual of Style; “Chicago Style Q&A”; “New Questions and Answers” Encyclopaedia Britannica Online; the online version of Encyclopaedia Britannica the Oxford English Dictionary Online; the OED Online; the online version of the Oxford English Dictionary the Onion (but see 8.170) Many websites either do not have a formal title or do not have a title that distinguishes it as a website. These can usually be identified according to the entity responsible for the site along with a description of the site and, in some cases, a short form of the URL. For example, .com/ might be referred to in running text as Apple.com.
The website for Apple Inc.; Apple.com Microsoft’s website; Microsoft.com the website for the University of Chicago Press, Books Division 8.192 Blogs and blog posts. Titles of named blogs (and video blogs), like the titles of journals and other periodicals (see 8.163), can usually be ital icized. An initial “the” can be treated as part of the title (an exception may be made for news blogs whose titles are styled like those of newspa pers; see 8.170). Titles of blog posts should be placed in quotation marks (untitled posts should be referred to by date). See also 14.205-10.
Sinosphere: Dispatches from China; “Q. and A.: Chang-rae Lee on His Tale of Mi grants from an Environmentally Ruined China,” by Edward Wong, in Sino sphere , a blog about China in the New York Times Wasted Food; “Vancouver Hoovers Up Food Waste,” in Wasted Food, a blog by Jonathan Bloom “Surprise of the Day: Hungry Great White Shark Steals Fisherman’s Catch,” in I Can Has Cheezburger?, a video blog known for its “Lolcats” the Huffington Post; “Why the 99 Percent Keeps Losing,” by Robert Kuttner In many cases, the distinction between a blog and a website (not to men tion social-networking service or app) will be blurry (as in the case of the Huffington Post). When in doubt, treat the source like a website (see 8.191). For podcasts, see 8.189.
8.193 NAMES, TERMS, AND TITLES OF WORKS Musical Works 8.193 Musical works—additional resources. The following paragraphs are intended only as general guidance for citing musical works. Writers or editors working with highly musicological material should consult D. Kern Holoman, Writing about Music (bibliog. 1.1). For a more general reference work, consult The New Grove Dictionary of Music and Musicians and the other Grove musical dictionaries, available from Oxford Music Online (bibliog. 5). For typographic conventions used in musicology, see 7.70-75.
8.194 Operas, songs, and the like. Titles of operas, oratorios, tone poems, and other long musical compositions are italicized and given standard title capitalization. Titles of songs and other shorter musical compositions are set in roman and enclosed in quotation marks, capitalized in the same way as poems (see 8.181, 8.182).
“La vendetta, oh, la vendetta” from The Marriage of Figaro the “Anvil Chorus” from Verdi’s II Trovatore Handel’s Messiah Rhapsody in Blue Finlandia “All You Need Is Love” (a song by the Beatles) “So What” (a composition by Miles Davis) “The Star-Spangled Banner” “Oh, What a Beautiful Mornin’” from Oklahoma!
“Wohin?” from Die schone Miillerin 8.195 Instrumental works. Many instrumental works are known by their ge neric names—symphony, quartet, nocturne, and so on—and often a num ber or key or both. Such names are capitalized but not italicized. A de scriptive title, however, is usually italicized if referring to a full work, set in roman and in quotation marks if referring to a section of a work. The abbreviation no. (number; plural nos.) is set in roman and usually lower cased. (For letters indicating keys, see 7.71.) B-flat Nocturne; Chopin’s nocturnes the Menuetto from the First Symphony; the third movement Concerto no. 2 for Piano and Orchestra; the second movement, Allegro appas sionato, from Brahms’s Second Piano Concerto; two piano concertos Bartok’s Concerto for Orchestra (or Concerto for Orchestra) Titles of Works 8.198 Bach’s Mass in B Minor Hungarian Rhapsody no. 12; the Twelfth Hungarian Rhapsody Charles Ives’s Piano Sonata no. 2 (Concord, Mass., 1840-60); the Concord Sonata Symphony no. 6 in F Major (Pastoral); the Sixth Symphony; the Pastoral Sym phony; the Pastoral Air with Variations (“The Harmonious Blacksmith”) from Handel’s Suite no. 5 in E Elliott Carter’s String Quartet no. 5 and his Figment for cello Augusta Read Thomas’s Triple Concerto (Night’s Midsummer Blaze) 8.196 Opus numbers. The abbreviation op. (opus; plural opp. or opera) is set in roman and usually lowercased. An abbreviation designating a catalog of a particular composer’s works is always capitalized (e.g., BWV [Bach- Werke-Verzeichnis]; D. [Deutsch] for Schubert; K. [Kochel] for Mozart; WoO [Werke ohne Opuszahl], assigned by scholars to certain unnum bered works). When op. or a catalog number is used restrictively (see 6.29), no comma precedes it.
Sonata in E-flat, op. 31, no. 3; Sonata op. 31 Fantasy in C Minor, K. 475; Fantasy K. 475 8.197 Recordings. The official title of an album (and sometimes a title under which it has come to be known) is italicized; that of the performer or ensemble is set in roman. Individual items on the album—songs, move ments, and the like—are treated as illustrated in the paragraphs above. See also 14.263.
On The Art of the Trumpet, the New York Trumpet Ensemble plays...
The single “Revolution” should not be confused with “Revolution 1,” an earlier take of the song that appeared on The Beatles (a.k.a. The White Album).
Miles Davis’s Kind of Blue is one of the most influential jazz records ever made. His Majestie’s Clerkes’ Hear My Prayer: Choral Music of the English Romantics in cludes Vaughan Williams’s Mass in G Minor.
Works of Art and Exhibitions 8.198 Paintings, photographs, statues, and such. Titles of paintings, draw ings, photographs, statues, and other works of art are italicized, whether the titles are original, added by someone other than the artist, or trans lated. The names of works of antiquity (whose creators are often un known) are usually set in roman. See also 8.57.
8.199 NAMES, TERMS, AND TITLES OF WORKS Rothko’s Orange Yellow Orange Leonardo da Vinci’s Mona Lisa and The Last Supper North Dome, one of Ansel Adams’s photographs of Kings River Canyon Hogarth’s series of drawings The Rake’s Progress Michelangelo’s David the Winged Victory the Venus de Milo 8.199 Maps. Maps can often be referred to in text with generic descriptive titles. If a map is known by a formal title rather than a generic descrip tion, use italics. See also 11.9.
a fifteenth-century reconstruction of Ptolemy’s world map (ca. AD 150) the Yu ji tu (Map of the tracks of Yu), from 1136 the Tabula Hungariae, Lazar’s map of Hungary Arno Peters’s projection of the world map; the Peters projection map Google Maps, satellite view of metropolitan Los Angeles Maps as illustrations require captions and credit lines as discussed in chapter 3. To cite a map included in a book, see 14.158. To cite a stand alone map, see 14.237.
8.200 Cartoons. Titles of regularly appearing cartoons or comic strips are ital icized.
The Far Side Doonesbury Rudy Park Dilbert 8.201 Exhibitions and such. Titles of world’s fairs and other large-scale exhi bitions and fairs are capitalized but not italicized. Smaller exhibitions (e.g., at museums) and the titles of exhibition catalogs (often one and the same) are italicized.
the Great Exhibition of the Works of All Nations; the Great Exhibition of 1851; London’s Crystal Palace Exhibition; the exhibition the World’s Columbian Exposition the Century-of-Progress Expositions (included more than one fair) the New York World’s Fair but A remarkable exhibition, Motor Cycles, was mounted at the Guggenheim Mu seum.
We saw the exhibition Ansel Adams at 100 when visiting the Museum of Mod ern Art.
We decided to buy the catalog Ansel Adams at 100, by John Szarkowski.
9 • Numbers OVERVIEW 9.1 NUMERALS VERSUS WORDS 9.2 General Principles 9.2 Large Numbers 9.8 Physical Quantities 9.13 Percentages and Decimal Fractions 9.18 Money 9.20 Numbered Divisions in Publications and Other Documents 9.26 Dates 9.29 Time of Day 9.37 Numbers with Proper Names and Titles 9.41 Addresses and Thoroughfares 9.50 PLURALS AND PUNCTUATION OF NUMBERS 9.53 INCLUSIVE NUMBERS 9.60 ROMAN NUMERALS 9.65 9.1 NUMBERS Overview 9.1 Overview and additional resources. This chapter summarizes some of the conventions Chicago observes in handling numbers, especially in making the choice between spelling them out and using numerals. Such a choice should be governed by various factors, including whether the number is large or small, whether it is an approximation or an exact quantity, what kind of entity it stands for, and what context it appears in. Sometimes the goal of consistency must give way to readability (e.g., at the beginning of a sentence; see 9.5). The guidelines in this chapter apply mainly to general works and to scholarly works in the humanities and social sciences, where numeric quantities are relatively infrequent. But even in scientific and other technical contexts, numerals can never to tally replace spelled-out numbers. For more detailed treatment of num bers in technical contexts, consult Scientific Style and Format (bibliog. 1.1). See also 9.13-17.
Numerals versus Words General Principles 9.2 Chicago's general rule—zero through one hundred. In nontechnical contexts, Chicago advises spelling out whole numbers from zero through one hundred and certain round multiples of those numbers (see 9.4).
Thirty-two children from eleven families were packed into eight vintage Beetles. Many people think that seventy is too young to retire.
The property is held on a ninety-nine-year lease. According to a recent appraisal, my house is 103 years old. The three new parking lots will provide space for 540 more cars. The population of our village now stands at 5,893.
Most of the rest of this chapter deals with exceptions and special cases. For hyphens used with spelled-out numbers, see 7.89, section 1. For some additional considerations, consult the index, under numbers. For numer als in direct discourse, see 13.44. For an alternative rule, see 9.3.
9.3 An alternative rule—zero through nine. Many publications, including those in scientific or journalistic contexts, follow the simple rule of spell ing out only single-digit numbers and using numerals for all others (but Numerals versus Words 9.5 see 9.7). Most of the exceptions to the general rule (9.2) also apply to this alternative rule. Round multiples of hundreds, thousands, and hundred thousands, however, are typically expressed as numerals when the alter native rule is in force (cf. 9.4).
9.4 Hundreds, thousands, and hundred thousands. The whole numbers one through one hundred followed by hundred, thousand, or hundred thousand are usually spelled out (except in the sciences or with mone tary amounts)—whether used exactly or as approximations. See also 9.8, 9.24.
Most provincial theaters were designed to accommodate large audiences—from about seven hundred spectators in a small city like Lorient to as many as two thousand in Lyon and Marseille.
A millennium is a period of one thousand years.
The population of our city is more than two hundred thousand.
Some forty-seven thousand persons attended the fair. but The official attendance at this year’s fair was 47,122.
In a context with many large numbers—especially if round numbers oc cur alongside numerals that are not round—it may be best to opt for nu merals for all such numbers. See also 9.7.
9.5 Number beginning a sentence. When a number begins a sentence, it is always spelled out. To avoid awkwardness, a sentence can often be re cast. In the first example, some writers prefer the form one hundred and ten; Chicago’s preference is to omit the and.
One hundred ten candidates were accepted.
or In all, 110 candidates were accepted.
If a year must begin a sentence, spell it out; it is usually preferable, how ever, to reword. Avoid and in such expressions as two thousand one, two thousand ten, two thousand fifty, and the like (see also 9.29).
Nineteen thirty-seven was marked, among other things, by the publication of the eleventh edition of Bartlett’s Familiar Quotations.
or, better, The year 1937...
9.6 NUMBERS If a number beginning a sentence is followed by another number of the same category, spell out only the first or reword.
One hundred eighty of the 214 candidates had law degrees; the remaining 34 were doctoral candidates in fish immunology.
or, better, Of the 214 candidates, 180 had law degrees; the remaining 34 were doctoral can didates in fish immunology.
9.6 Ordinals. The general rule applies to ordinal as well as cardinal numbers. Note that Chicago prefers, for example, 122nd and 123rd (with an n and an r) over 122d and 123d. The latter, however, are common especially in legal style (see 14.269-305). The letters in ordinal numbers should not appear as superscripts (e.g., 122nd, not 122nd).
Gwen stole second base in the top half of the first inning. The restaurant on the forty-fifth floor has a splendid view of the city. She found herself in 125th position out of 360.
The 122nd and 123rd days of the strike were marked by a rash of defections. The ten thousandth child to be born at Mercy Hospital was named Mercy.
In the expression “nth degree,” Chicago style is to italicize the n (see also 7.64) .
9.7 Consistency and flexibility. Where many numbers occur within a para graph or a series of paragraphs, maintain consistency in the immediate context. If according to a given rule you must use numerals for one of the numbers in a given category, use them for all in that category. (An exception should be made at the beginning of a sentence; see 9.5.) In the same sentence or paragraph, however, items in one category may be given as numerals and items in another spelled out. According to the general rule, in the first example, the numerals 50, 3, and 4 would nor mally be spelled out (see 9.2); in the second and third examples, 30,000 and 2,000, respectively, would normally be spelled out (see 9.4; see also 9.8). According to the alternative rule, in the fourth and fifth examples, 9 and 1, respectively, would normally be spelled out (see 9.3).
General Rule A mixture of buildings—one of 103 stories, five of more than 50, and a dozen of only 3 or 4—has been suggested for the area.
In the second half of the nineteenth century, Chicago’s population exploded, from just under 30,000 in 1850 to nearly 1.7 million by 1900.
Between 1,950 and 2,000 people attended the concert.
Numerals versus Words 9.8 Alternative Rule Though most of the test subjects were between 13 and 18, two were 11 and one was 9.
The movie lasted 1 hour and 36 minutes, a typical length for a romantic comedy.
An exception to either rule may also be made to avoid a thickly clustered group of spelled-out numbers, regardless of category. And in some cases, an exception may be applied not only to a paragraph or passage of text but to a work as a whole. If, for example, a book includes many mentions of ages, all ages might be given as numerals. For numerals in direct dis course, see 13.44.
Large Numbers 9.8 Millions, billions, and so forth. Whole numbers used in combination with million, billion, and so forth usually follow the general rule (see 9.2). See also 9.4. For monetary amounts, see 9.20-25; for the use of super scripts in scientific contexts, see 9.9.
The city had grown from three million in 1960 to fourteen million in 1990.
The survey was administered to more than half of the city’s 220 million inhabi tants.
The population of the United States recently surpassed three hundred million.
To express fractional quantities in the millions or more, a mixture of nu merals and spelled-out numbers is used. In the second example below, the number fourteen is expressed as a numeral for the sake of consis tency (see 9.7).
By the end of the fourteenth century, the population of Britain had probably reached 2.3 million.
According to some scientists, the universe is between 13.5 and 14 billion years old.
Note that billion in some countries (including, until recently, Great Brit ain) means a million million (a trillion in American usage), not, as in American usage, a thousand million; in this alternate system, the pre fix bi- indicates twelve zeros (rather than the American nine), or twice the number of zeros in one million. Likewise, trillion indicates eighteen zeros (rather than the American twelve), quadrillion twenty-four (rather than the American fifteen), and so on. Editors working with material by writers who may not be familiar with English usage for these terms may need to query how they are used. See 5.250 under billion; trillion.
9.9 NUMBERS 9.9 Powers of ten. Large round numbers may be expressed in powers of ten, especially in scientific writing. This system is known as scientific notation. For further examples, consult Scientific Style and Format (bib liog. 1.1).
102 = 100 103 = 1,000 106 = 1,000,000 109 = 1,000,000,000 1012 = 1,000,000,000,000 5.34 X io8 = 534,000,000 Inversely, very small numbers may be expressed in negative powers of ten.
IO’2 = 0.01 10_3 = 0.001 10“6 = 0.000001 10“9 = 0.000000001 10_12 = 0.000000000001 5.34 X IO-8 = 0.00 0 0 0 00534 9.10 "Mega-," "giga-," "tera-," and so forth. According to the International System of Units (Systeme international d’unites, abbreviated internation ally as SI), very large quantities may be indicated in some contexts by the use of the prefixes mega- (million), giga- (billion), tera- (trillion), and so on, as part of the unit of measure. Inversely, very small numbers may be expressed by milli- (thousandth), micro- (millionth), nano- (billionth), and so on. These expressions are often formed with symbols (e.g., M, for mega-, as in MB, megabytes). In astrophysical contexts, the abbreviations Myr and Gyr, standing for megayear (one million years) and gigayear (one billion years), are sometimes used. See also 9.9. For a complete list of SI prefixes, see 10.56. See also 9.11.
3 terahertz = 3 x 1012 hertz 7 Gyr = 7 x 109 years 9.11 Binary systems. Bases other than ten are common especially in comput ing, where numbers are usually expressed with bases that are powers of two (e.g., binary, octal, or hexadecimal). When such numbers are used, the base if other than ten should be indicated. Abbreviations b (binary), o (octal), and h (hexadecimal) may precede the number with no interven ing space. Alternatively, the base can be expressed as a subscript. In the following example, the four-digit base-ten number is expressed without a comma, following SI usage (but see 9.56): bllllOOlOOOl = 1937 or 111100100012 = 19371O Note that terms such as megabyte, when used as binary multiples, are approximations—a megabyte was originally equal to 1,048,576 bytes.
Numerals versus Words 9.14 Current SI usage dictates that such prefixes refer to positive powers of ten (where a megabyte is equal to 1,000,000 bytes). If binary multiples must be referred to, the first two letters of the prefix plus bi should be used (kibibyte, mebibyte,gibibyte, etc.).
9.12 Use of "dex." The term dex is sometimes used in scientific notation as shorthand for decimal exponent.
Errors of 3 dex (i.e., 103) can lead to dangerous misconceptions.
Physical Quantities 9.13 Physical quantities in general contexts. In nontechnical material, phys ical quantities such as distances, lengths, areas, and so on are usually treated according to the general rule (see 9.2). See also 9.15.
Within fifteen minutes the temperature dropped twenty degrees.
The train approached at seventy-five miles an hour.
Some students live more than fifteen kilometers from the school.
Three-by-five-inch index cards are now seldom used in index preparation.
She is five feet nine (or, more colloquially, five foot nine or five nine).
It is occasionally acceptable to depart from the general rule for certain types of quantities that are commonly (or more conveniently) expressed as numerals; such a departure, subject to editorial discretion, must be consistently applied for like quantities across a work. See also 9.7. For the absence of the hyphen in the second example below, see 7.89, section 2, under noun + numeral or enumerator.
a 40-watt bulb a size 14 dress a 32-inch inseam a fuel efficiency of 80 miles per gallon (or 3 liters per 100 kilometers) 9.14 Simple fractions. Simple fractions are spelled out. For the sake of read ability and to lend an appearance of consistency, they are hyphenated in noun, adjective, and adverb forms. In the rare event that individual parts of a quantity are emphasized, however, as in the last example, the ex pression is unhyphenated. See also 7.89, section 1, under fractions, simple. For decimal fractions, see 9.19.
9.15 NUMBERS She has read three-fourths of the book.
Four-fifths of the students are boycotting the class.
I do not want all of your material; two-thirds is quite enough.
A two-thirds majority is required.
but We divided the cake into four quarters; I took three quarters, and my brother one.
9.15 Whole numbers plus fractions. Quantities consisting of whole numbers and simple fractions may be spelled out if short but are often better ex pressed in numerals (especially if a symbol for the fraction is available, as in the examples here). For decimal fractions, see 9.19. For fractions in mathematical text, see 12.45. See also 9.17,10.66.
We walked for three and one-quarter (or three and a quarter) miles.
I need 67s yards of the silk fabric. Lester is exactly 3 feet 5inches tall. Letters are usually printed on ST?" x 11" paper.
9.16 Numbers with abbreviations and symbols. If an abbreviation or a sym bol is used for the unit of measure, the quantity is always expressed by a numeral. Such usage is standard in mathematical, statistical, technical, and scientific text, where physical quantities and units of time are ex pressed in numerals, whether whole numbers or fractions, and almost always followed by an abbreviated form of the unit (see also 10.49-68). Note that hyphens are never used between the numeral and the abbrevi ation or symbol, even when they are in adjectival form (see 7.89, section 1, under number + abbreviation). In the last example (which can express inches and feet or minutes and seconds, respectively), note the use of symbols for prime and double prime, which are not equivalent to the apostrophe and quotation mark. A space is normally used between the numeral and the unit of measure, except in a few cases—for example, with degree, percent, and prime symbols (but see 10.58). See also 7.44.
50 km (kilometers); a 50 km race 240 V (volts) 21 ha (hectares) 10°C, 10.5°C 4.5 L (liters) 3'6" 85 g (grams) A unit of measurement used without a numeral should always be spelled out, even in scientific contexts.
We took the measurements in kilojoules (not kJ).
Numerals versus Words 9.19 9.17 Units for repeated quantities. For expressions including two or more quantities, the abbreviation or symbol is repeated if it is closed up to the number but not if it is separated. See also 10.49. For the use of spaces with SI units and abbreviations or symbols, see 10.58.
35%-5O% 3°C-7°C 63/Zx 9" 2x5cm Percentages and Decimal Fractions 9.18 Percentages. Except at the beginning of a sentence, percentages are usually expressed in numerals. In nontechnical contexts, the word per cent is generally used; in scientific and statistical copy, the symbol % is more common.
Fewer than 3 percent of the employees used public transportation.
With 90-95 percent of the work complete, we can relax.
A 75 percent likelihood of winning is worth the effort.
Her five-year certificate of deposit carries an interest rate of 5.9 percent.
Only 20% of the ants were observed to react to the stimulus.
The treatment resulted in a 2O%-25% increase in reports of night blindness. (See also 9.17.) but Thirty-nine percent identified the “big bang” as the origin of the universe; 48 per cent said they believed in human evolution. (See also 9.7.) Note that percent, an adverb, is not interchangeable with the noun per centage (1 percent is a very small percentage). Note also that no space appears between the numeral and the symbol %.
9.19 Decimal fractions and use of the zero. Large or complex fractions are expressed as numeric decimal fractions (cf. 9.14). When a quantity equals less than 1.00, a zero normally appears before the decimal point as an aid to readability, particularly in scientific contexts and especially if quan tities greater than 1.00 appear in the same context. Note that a unit of measure with a quantity of less than one is generally treated as if it were plural (see 10.65,10.53). See also 9.55, 9.58.
a mean of 0.73 the ratio 0.85 In Cyprus, there were 0.96 females for every male in the general population; in the sixty-five-and-over age group, the number was 1.30.
9.20 NUMBERS In contexts where decimal quantities must be 1.00 or less, as in prob abilities, batting averages, and the like, or between -1.00 and 1.00, as in correlation coefficients, a zero is typically omitted before the decimal point. For zeros with decimal points in tables, see 3.72.
p < .05 R = .10 Ty Cobb’s career batting average was .367.
By a similar token, the zero is routinely omitted from firearm calibers expressed as fractions of an inch.
They found and confiscated a .38 police special and a .22-caliber single-shot rifle.
Money 9.20 Words versus monetary symbols and numerals. Isolated references to amounts of money are spelled out for whole numbers of one hundred or less, in accordance with the general principle presented in 9.2. See also 9.3.
seventy-five cents = 75<£ fifteen dollars = $15 seventy-five pounds = £75 Whole amounts expressed numerically should include zeros and a dec imal point only when they appear in the same context with fractional amounts (see also 9.19). Note the singular verb in the second example.
Children can ride for seventy-five cents.
The eighty-three dollars was quickly spent.
The instructor charged €125 per lesson.
Prices ranged from $0.95 up to $10.00.
For larger amounts, see 9.24.
9.21 Non-US currencies using the dollar symbol. In contexts where the sym bol $ may refer to non-US currencies, these currencies should be clearly identified.
three hundred Canadian dollars = C$300 or Can$300 $749 in New Zealand dollars = NZ$749 If you subtract A$15.69 from US$25,...
ninety-eight Mexican pesos = Mex$98 Numerals versus Words 9.24 In more formal usage, the International Organization for Standardiza tion’s three-letter currency codes (e.g., USD for United States dollars, CAD for Canadian dollars, NZD for New Zealand dollars, AUD for Aus tralian dollars, and MXN for Mexican pesos) may be more appropriate. See also 9.23. For a complete list, consult ISO 4217, available from the ISO website. See also Scientific Style and Format or the Style Manual from the US Government Publishing Office (bibliog. 1.1). Where the context makes clear what currency is meant, the dollar sign alone is enough.
9.22 British currency. The basic unit of British currency is the pound, or pound sterling, for which the symbol is £. One-hundredth of a pound is a penny (pluralpence), abbreviated asp (no period).
fifteen pounds = .£15 fifty pence = 50p .£4.75, .£5.00, and .£5.25 Until the decimalization of British currency in 1971, the pound was di vided into shillings (s.) and pence (d.).
Ten pounds, fifteen shillings, and sixpence = .£10 15s. 6d. twopence halfpenny = 2x/2d.
9.23 Other currencies. Most other currencies are handled the same way as US currency, with a decimal point between the main unit and subunits (e.g., EUR 10.75). When letters rather than symbols are used, a space sepa rates the letter(s) from the numeral.
forty euros (or, in European Union documents, 40 euro) = €40 (or EUR 40) 95 (euro) cents (or, in European Union documents, 95 cent) 725 yen = ¥725 (orIPY725) 100 yuan renminbi (or 100 yuan) = ¥100 or RMB100 (or CNY100) 65.50 Swiss francs = SF 65.50 (or CHF 65.50) 12.5 bitcoins = BTC 12.5 (or XBT 12.5) Before adoption of the euro, monetary symbols included F (French franc), DM (deutsche mark), and Lit (Italian lira), among others. The In ternational Organization for Standardization defines three-letter codes (including EUR) for most countries; these may be more appropriate in formal or technical contexts. See also 9.21.
9.24 Large monetary amounts. Sums of money of more than one hundred dollars are normally expressed by numerals or, for numbers of a million or more, by a mixture of numerals and spelled-out numbers, even for whole numbers (cf. 9.4, 9.8).
9.25 NUMBERS An offer of $1,000 once seemed high; we eventually agreed to pay more than fifteen times that amount.
Most of the homes that went into foreclosure were valued at more than $95,000. She signed a ten-year, $250 million contract.
The military requested an additional $7.3 billion.
The marquess sold his ancestral home for .£25 million.
In certain financial contexts, thousands are sometimes represented by K.
Three-bedroom condominiums are priced at $35OK.
9.25 Currency with dates. In contexts where the value of a currency in any particular year is relevant to the discussion, the date may be inserted in parentheses, without intervening space, after the currency symbol. When letters alone are used, spaces intervene before and after the pa rentheses (see also 9.21, 9.23).
US$(1992)2.47 £(2002)15,050 but USD (1992) 2.47 Numbered Divisions in Publications and Other Documents 9.26 Page numbers, chapter numbers, and so forth. Numbers referring to pages, chapters, parts, volumes, and other divisions of a book, as well as numbers referring to illustrations or tables, are set as numerals. Pages of the front matter are usually in lowercase roman numerals; those for the rest of the book are in arabic numerals (see 1.5-9). For the use of en dashes with number ranges, see 6.78. For documentation style, see chap ters 14 and 15. See also 8.180.
The preface will be found on pages vii-xiv and the introduction on pages 1-35.
See part 3, especially chapters 9 and 10, for further discussion; see also volume 2, table 15 and figures 7-9.
Upon completion of step 3, on page 37, the reader is asked to consult appendix B, table 7.
Biblical references are given in numerals only; chapter and verse are separated by a colon with no space following it. For abbreviations, see 10.44-48.
Numerals versus Words 9.29 Acts 27:1 2 Corinthians 11:29-30 Exodus 20:3-17 Gen. 47:12 Psalm 121; Psalms 146-50 9.27 Volume, issue, and page numbers for periodicals. References to vol umes, issues, and pages of a journal are usually made, in that order, with arabic numerals; the words volume and page are usually omitted. See also 14.164-204.
Their article appeared in Modern Philology 112, no. 3 (2015): 554-68.
9.28 Numbered divisions in legal instruments. Arabic or roman numerals are commonly used to distinguish divisions within legal instruments and other documents. When in doubt about a reference to a legal docu ment, prefer arabic numerals or, if possible, consult the document itself for guidance. A mixture of arabic and roman numerals sometimes distin guishes smaller from larger divisions. For legal style in source citations, see 14.269-305.
They have filed for Chapter 11 protection from creditors.
Proposition 20 will be voted on next week.
A search of Title IX (of the Education Amendments of 1972) turns up no mention of athletics.
Do you have a 401(k)?
In paragraph 14(vi) of the bylaws,...
According to the Constitution of the United States, article 2, section 4 (or Article II, Section 4),...
but the Fifth Amendment (or Amendment V) Dates 9.29 The year used alone. Years are expressed in numerals unless they stand at the beginning of a sentence (see 9.5), in which case rewording may be a better option. For eras, see 9.34.
We all know what happened in 1776.
Records for solar eclipses go back at least as far as 3000 BCE.
Twenty twenty (or Two thousand twenty) should be a good year for clairvoyants. or, better, The year 2020 should be a good year for clairvoyants.
9.30 NUMBERS 9.30 The year abbreviated. In informal contexts, the first two digits of a par ticular year are often replaced by an apostrophe (not an opening single quotation mark). See also 6.117.
the spirit of ’ 76 (not ‘ 76) the class of ’06 9.31 Month and day. When specific dates are expressed, cardinal numbers are used, although these may be pronounced as ordinals. For the month- day-year date form versus the day-month-year form, see 9.35; see also 6.38.
August 12,2014, was a sad day for film buffs.
The Watchmaker’s Digest (11 November 2011) praised the new model’s precision.
When a day is mentioned without the month or year, the number is usu ally spelled out in ordinal form.
On November 5, McManus declared victory. By the twenty-fifth, most of his sup porters had deserted him.
9.32 Centuries. Particular centuries referred to as such are spelled out and lowercased. For the use of the singular versus the plural, see 7.8. See also 7.15, 9.34.
the twenty-first century the eighth and ninth centuries from the ninth to the eleventh century but the 1800s (the nineteenth century) Note that expressions such as “turn of the twenty-first century” are po tentially ambiguous; prefer “turn of the century,” and only where the context makes the period absolutely clear.
9.33 Decades. Decades are either expressed in numerals or spelled out (as long as the century is clear) and lowercased. Chicago calls for no apos trophe to appear between the year and the s (see 7.15).
the 1940s and 1950s (or, less formally, the 1940s and ’50s) or the forties and fifties Numerals versus Words 9.35 Note that the first decade of any century cannot be treated in the same way as other decades. “The 2000s,” for example, could easily be taken to refer to the whole of the twenty-first century. To refer to the second decade (i.e., without writing “second decade”), prefer numerals (e.g., 1910s); the expression “the teens” should be avoided, at least in formal contexts.
the first decade of the twenty-first century (or the years 2000-2009) the second decade of the twenty-first century or the 2010s (or the years 2010-19) Note that some consider the first decade of, for example, the twenty-first century to consist of the years 2001-10; the second, 2011-20; and so on. Chicago defers to the preference of its authors in this matter. See also 8.71,9.64.
9.34 Eras. Era designations, at least in the Western world, are usually ex pressed in one of two ways: either CE (“of the Common Era”) and BCE (“before the Common Era”), or AD (anno Domini, “in the year of the Lord”) and BC (“before Christ”). Other forms include AH (anno Hegirae, “in the year of [Muhammad’s] Hegira,” or anno Hebraico, “in the Hebrew year”); AUC (ab urbe condita, “from the founding of the city [Rome]”); and—for archaeological purposes—BP (“before the present”). Note that the Latin abbreviations AD and AH precede the year number, whereas the others follow it. Choice of the era designation depends on tradition, academic discipline, or personal preference. These abbreviations often appear in small capitals, sometimes with periods following each letter. For consistency with the guidelines in chapter 10, Chicago recommends full capitals and no periods; see also 10.38.
Herod Antipas (21 BCE-39 CE) was tetrarch of Galilee from 4 BCE until his death. Britain was invaded successfully in 55 BC and AD 1066.
The First Dynasty appears to have lasted from 4400 BP to 4250 BP in radio carbon years.
Mubarak published his survey at Cairo in 1886 (AH 1306). The campsite seems to have been in use by about 13,500 BP. Rome, from its founding in the eighth century BCE,...
Note that the second half of a pair of inclusive dates used with BCE or BC, where the higher number comes first, should be given in full to avoid confusion (e.g., “350-345 BCE”). See also 9.64.
9.35 All-numeral dates and other brief forms. For practical reasons, all numeral styles of writing dates (e.g., 5/10/99 or 5.10.99) should not be 9.36 NUMBERS used in formal writing (except with certain dates that may be known that way: e.g., 9/11, for September 11,2001). Whereas in American usage the first numeral refers to the month and the second to the day, in much of the rest of the world it is the other way around. When quoting letters or other material dated, say, 5/10/03, a writer must first ascertain and then make it clear to readers whether May 10 or October 5 is meant (not to mention 1903 or 2003). In text, therefore, the full date should always be spelled out (see 9.31). In documentation and in tables, if numerous dates occur, months may be abbreviated, and the day-month-year form, re quiring no punctuation, may be neater (e.g., 5 Oct 2003). See also 10.39. For ISO style, see 9.36.
9.36 ISO style for dates. The International Organization for Standardization (ISO) recommends an all-numeral style consisting of year-month-day (i.e., from largest component to smallest), hyphenated. The year is given in full, and the month or day, if one digit only, is preceded by a zero. Thus July 14, 2018, would appear as 2018-07-14. Among other advantages, this style allows dates to be sorted correctly in an electronic spreadsheet and other applications. See also 9.40.
Time of Day 9.37 Numerals versus words for time of day. Times of day in even, half, and quarter hours are usually spelled out in text. With o’clock, the number is always spelled out. In the third example, the a before quarter is optional.
Her day begins at five o’clock in the morning.
The meeting continued until half past three.
He left the office at a quarter of four (or a quarter to four).
We will resume at ten thirty.
Cinderella almost forgot that she should leave the ball before midnight. (See also 9.38.) Numerals are used when exact times are emphasized. Chicago recom mends lowercase a.m. (ante meridiem) and p.m. (post meridiem), though these sometimes appear in small capitals, with or without periods. (Note that the abbreviations a.m. and p.m. should not be used with morning, afternoon, evening, night, or o’clock.) The first train leaves at 5:22 a.m. and the last at 11:00 p.m.
She caught the 6:20 p.m. flight.
Numerals versus Words 9.40 Please attend a meeting in Grand Rapids, Michigan, on December 5 at 10:30 a.m. (EST).
For more on time zones, see 10.41.
9.38 Noon and midnight. Except in the twenty-four-hour system (see 9.39), numbers should never be used to express noon or midnight (except, in formally, in an expression like twelve o'clock at night). Although noon can be expressed as 12:00 m. (m. = meridies), very few use that form. And the term 12:00 p.m. is ambiguous, if not illogical. In the second example below, note the double date for clarity.
The meeting began at 9:45 a.m. and was adjourned by noon. Rodriguez was born at midnight, August 21-22.
9.39 The twenty-four-hour system. In the twenty-four-hour system of ex pressing time (used in military and scientific contexts and considered regular usage in many countries outside the United States, English- speaking Canada, and several other regions that still use the twelve-hour system), four digits always appear, often with no punctuation between hours and minutes. In settings where hours is not used, or where the time may be confused with a year, a colon may be used (as in the twelve-hour system). See also 9.40.
12:00 = noon 00:00 or 24:00 = midnight (24:00 generally refers to the end of a given day) 00:01 = 12:01 a.m.
14:38 = 2:38 p.m.
At 1500 hours (or 150 Oh) we started off on our mission.
General quarters sounded at 0415.
9.40 ISO style for time of day. The International Organization for Standard ization (ISO) recommends the twenty-four-hour system (see 9.39), with or without colons, with the addition of seconds following minutes; frac tions of a second follow a period. To avoid ambiguity, colons should be used between hours and minutes and between minutes and seconds in running text and similar contexts. This format may be preceded by an ISO-style date (see 9.36). (Note that when the time of day is spelled out, a comma between minutes and seconds denotes and; cf. 6.38.) 09:27:08.6 = 27 minutes, 8.6 seconds after 9:00 a.m.
2018-07-1416:09:41.3 = July 14,2018, at 9 minutes, 41.3 seconds after 4:00 p.m.
9.41 NUMBERS Numbers with Proper Names and Titles 9.41 Numerals for monarchs, popes, and so forth. Sovereigns, emperors, popes, and Orthodox patriarchs with the same name are differentiated by numerals, traditionally roman.
Elizabeth II Benedict XVI In continental European practice, the numeral is sometimes followed by a period (e.g., Wilhelm II.) or a superscript (e.g., Francois Ier) indicating that the number is an ordinal. In an English context, the roman numeral alone should appear. See also 11.27.
9.42 Numerals with personal names. Some personal names are followed by a roman numeral or an arabic ordinal numeral. No punctuation precedes the numeral unless the name is inverted (as in an index entry). For Jr., see 6.43.
Adlai E. Stevenson III Michael F. Johnson 2nd but Stevenson, Adlai E., Ill 9.43 Numbers for sequels. Numerals are often used to designate the sequel to a novel or a movie or to differentiate two chapter titles dealing with the same subject matter. When quoting such titles, follow the usage—roman or arabic (or spelled out)—reflected in the source itself. (For the use of sic, see 13.61.) The Godfather, The Godfather, Part II; The Godfather, Part III Jaws; Jaws 2; Jaws 3-D Dumb and Dumber To [sic] chapter 9, “Alligator Studies in the Everglades—I” chapter 10, “Alligator Studies in the Everglades—II” 9.44 Vehicle and vessel numbers. Boats and the like differentiated by a num ber usually take a roman numeral, spacecraft an arabic numeral. See also 8.116-18.
Bluebird III Mariner 9 Numerals versus Words 9.50 9.45 Numbers for successive governments. Ordinal numbers designating successive dynasties, governments, and other governing bodies are spelled out if one hundred or less. See also 8.51, 8.62, 8.71.
Eighteenth Dynasty Second International Fifth Republic Ninety-Seventh United States Congress Second Continental Congress 115th Congress 9.46 Numbered political and judicial divisions. Ordinal numbers designating political or judicial divisions are spelled out if one hundred or less. See also 8.51.
Fifth Ward Twelfth Congressional District Tenth Circuit 101st Precinct 9.47 Numbered military units. Ordinal numbers designating military units are spelled out if one hundred or less. See also 8.112.
Fifth Army First Corps Support Command Fourth Infantry Division 101st Airborne Division 9.48 Numbered places of worship. Ordinal numbers that are part of the names of places of worship are spelled out.
Fourth Presbyterian Church Twenty-First Church of Christ, Scientist 9.49 Unions and lodges. Numbers designating local branches of labor unions and fraternal lodges are usually expressed in arabic numerals after the name. Commas can usually be omitted.
Chicago Typographical Union No. 16 United Auto Workers Local 890 American Legion Post 21 Addresses and Thoroughfares 9.50 Numbered highways. State, federal, and interstate highways are desig nated by arabic numerals. Names for state routes vary from state to state. See also 8.56.
US Route 41 (or US 41) Illinois Route 50 (or Illinois 50; IL 50); Route 50 Interstate 90 (or 1-90) M6 motorway (England) 9.51 NUMBERS 9.51 Numbered streets. Names of numbered streets, avenues, and so forth are usually spelled out if one hundred or less. For the use of N, E, SW, and the like, see 10.34. See also 8.56.
First Avenue Ninety-Fifth Street 122nd Street 9.52 Building and apartment numbers. Building numbers, in arabic numer als, precede the street name. For preferred forms of mailing addresses, consult the applicable postal service; for readability and to conform to the style of the surrounding text, however, usage may differ slightly from what might be appropriate for a mailing label or the like. See also 6.39, 10.34.
They lived in Oak Park, at 1155 South Euclid Avenue, for almost ten years.
She now lives in unit 114A, 150 Ninth Avenue, with an unrivaled view of the city. Our office is at 1427 East Sixtieth Street, Chicago, Illinois.
Please mail a copy of the German-language edition to 1427 E. 60th St., Chicago, IL 60637.
When a building is referred to in running text by its address, the number is often spelled out.
One Thousand Lake Shore Drive One IBM Plaza Plurals and Punctuation of Numbers 9.53 Plural numbers. Spelled-out numbers form their plurals as other nouns do (see 7.5).
The contestants were in their twenties and thirties.
The family was at sixes and sevens.
Numerals form their plurals by adding s. No apostrophe is needed (see also 7.15).
Among the scores were two 240s and three 238s.
Jazz forms that were developed in the 1920s became popular in the 1930s.
9.54 Comma between digits. In a style followed in most general contexts in the United States and most other English-speaking parts of the world, for numerals of one thousand or more, commas are used between groups of Plurals and Punctuation of Numbers 9.56 three digits, counting from the right. Commas are not used for figures to the right of the decimal marker.
1,512 32,987 4,000,500 but 0.32987 Nor are commas used in page numbers, line numbers (e.g., in poetry and plays), addresses, and years (though years of five digits or more do in clude the comma). See also 9.34.
Punctuation conventions can be found on page 1535 of the tenth edition.
Our business office is at 11030 South Langley Avenue.
Human artifacts dating from between 35,000 BP and 5000 BP have been found there.
In scientific writing, commas are often omitted from four-digit numbers. See also 9.56.
9.55 The decimal marker. According to the predominant usage in the United States and elsewhere, where commas are used between groups of three digits for numerals of one thousand or more (see 9.54), a period is used as the decimal marker (and called a decimal point).
33,333.33 In many other countries, including France (and French-speaking Can ada), Germany, Italy, and Russia, the decimal marker is represented by a comma. Where this is the case, a thin, fixed space, not a comma, sep arates groups of three digits. (In electronic publications, a regular non breaking space may be used; see 6.121.) 33 333,33 This practice reflects SI usage, which allows either the comma or the dec imal point as a decimal marker (see 9.56). English-speaking Canadians increasingly follow SI usage, using spaces rather than commas to sepa rate groups of digits (while retaining the decimal point). In US publica tions, US style should be followed, except in direct quotations and except where SI style is required. See also 10.58.
9.56 Space between digits (SI number style). In the International System of Units (SI units), thin, fixed spaces rather than commas are normally used to mark off groups of three digits, both to the left and to the right of the 9.57 NUMBERS decimal marker. (In electronic publications, a regular nonbreaking space may be used; see 6.121.) No space is used for groups of only four digits either to the left or to the right of the decimal marker (except in table columns that also include numbers having five or more digits, where it is needed for alignment). To mark the decimal, either a decimal point or a comma may be used, according to what is customary in a given context or region (the examples that follow show the decimal point). See also 9.55, 9.54.
3 426 869 0.000 007 2501.4865 (or 2 501.486 5) For more on SI units, see Ambler Thompson and Barry N. Taylor, Guide for the Use of the International System of Units (SI), and The International System of Units (SI), a brochure published in English and French by the Bureau International des Poids et Mesures (see bibliog. 2.4).
9.57 Telephone numbers. In the United States and Canada, telephone num bers consist of the prefix 1 (also called the trunk prefix), an area code (also called the trunk code), and a seven-digit number consisting of a three-digit exchange prefix followed by a four-digit line number. When written, the seven-digit number is conventionally separated by a hy phen; to signal that it may be optional for local calls, the area code is often placed in parentheses. The prefix 1, the same for all numbers (and not always necessary to place a call), can usually be omitted. (Its appear ance with toll-free numbers beginning with 800 and the like is custom ary but not mandatory.) An extension follows the number, separated by a comma.
(000) 000-0000 or (1-000) 000-0000 (000) 000-0000, ext. 0000 An alternative style, which recognizes the increasing need to use the area code even for local calls, drops the parentheses. Either style is accept able, as long as it is used consistently. (On the other hand, the use of pe riods or other punctuation as separators in place of hyphens is generally not recommended.) 000-000-0000 or 1-000-000-0000 For international numbers, use spaces rather than hyphens as separators. A plus symbol, which stands in for the international prefix (e.g., Oil for international calls from the United States or Canada), is placed immedi Plurals and Punctuation of Numbers 9.59 ately before the country code (e.g., 52 for Mexico, 66 for Thailand, or 44 for the United Kingdom) with no intervening space. Because their mean ing may not be clear, parentheses should not be used for international numbers (e.g., to enclose a national access code that is not needed for international calls).
+52 55 0 0 0 0 0 0 00 (for a number in Mexico City, Mexico) +66 20000000 (for a number in Bangkok, Thailand) +44 20 0000 0000 (for a number in London, UK) not +44 (0) 20 0000 0000 If the international prefix must be expressed, it precedes the country code and is separated from it by a space (e.g., for a call initiated from the United States or Canada, 011 44 20 ...). A US or Canadian number writ ten for an international audience follows a similar pattern (the country code for both is 1).
+1607 000 0000 (for a number in New York State) Spaces or hyphens as separators are the norm within most countries; however, the use of parentheses and number groupings varies widely, not only across countries but also within countries, and for landline ver sus mobile numbers. For more guidance, consult the latest standard from the International Telecommunication Union (bibliog. 5) or, for more spe cific advice, a local or international directory.
9.58 Ratios. Ratios composed of whole numbers may generally be expressed using to and spelled out in ordinary text according to one of the rules stated at 9.2 and 9.3. In contexts where numerals are preferred, a colon may be used as a shorthand for to, with no space on either side.
a three-to-one ratio a 13-to-2 ratio (see 9.3; see also 9.7) or a 13:2 ratio In some contexts, ratios may be expressed as decimal fractions, in the manner of percentages (see 9.19).
9.59 Numbered lists and outline style. For the use of numerals (arabic and roman) and letters to distinguish items in lists, see 6.127-32.
9.60 NUMBERS Inclusive Numbers 9.60 The en dash for inclusive numbers. An en dash used between two num bers implies up to and including, or through.
Please refer to pages 75-110.
Here are the figures for 2000-2009.
Campers were divided into age groups 5-7, 8-10,11-13, and 14-16.
The en dash should not be used iffrom or between is used before the first of a pair of numbers; instead, from should be followed by to or through (or until), and between should be followed by and.
from 75 to 110 (not from 75-110) from 1898 to 1903 from January 1,1898, through December 31,1903 between about 150 and 200 Inclusive spelled-out numbers should be joined by to, not by an en dash.
participants aged forty-five to forty-nine years sixty-to-seventy-year-olds For more on the use of the en dash, see 6.78-84. See also 7.89, section 1, under age terms.
9.61 Abbreviating, or condensing, inclusive numbers. Inclusive numbers are abbreviated according to the principles illustrated below (the examples show page numbers, which do not require commas). This system, used by Chicago in essentially this form since the first edition of this manual, is efficient and unambiguous. See also 9.62, 9.60,14.148.
First number Less than 100 Second number Use all digits 100 or multiples of 100 Use all digits Examples 3-10 71-72 96-117 100-104 1100-1113 101 through 109, 201 through 209, etc.
Use changed part only 101-8 808-33 1103-4 Inclusive Numbers 9.64 110 through 199, 210 through 299, etc.
Use two digits unless more are needed to include all changed parts 321-28 498-532 1087-89 1496-500 11564-615 12991-3001 To avoid ambiguity, inclusive roman numerals are always given in full.
xxv-xxviii cvi-cix 9.62 Alternative systems for inclusive numbers. A foolproof system is to give the full form of numbers everywhere (e.g., 234-235, 25039-25041). An other practice, more economical, is to include in the second number only the changed part of the first (e.g., 234-5, 25000-1). Chicago, however, prefers the system presented in 9.61.
9.63 Inclusive numbers with commas. When inclusive numbers with com mas are abbreviated, and only numbers in the hundreds place and below change, the rules described in 9.61 should apply. If a change extends to the thousands place or beyond, it is best to repeat all digits.
6,000-6,018 12,473-79 1,247,689-710 1,247,689-1,248,125 9.64 Inclusive years. Inclusive years may be abbreviated following the pat tern illustrated in 9.61. When the century changes, however, or when the sequence is BCE, BC, or BP (diminishing numbers), all digits must be presented. See also 9.34.
1897-1901 the war of 1914-18 fiscal year 2017-18 (or 2017/18; see 6.107); FY 2017-18 the winter of2000-2001 in 1504-5 327-321 BCE (seven years, inclusively) 327-21 BCE (307 years, inclusively) 115 BC-AD 10 15,000-14,000 BP In book titles it is customary but not obligatory to repeat all digits; when a title is mentioned or cited, the form of the original should be respected (see also 8.165).
9.65 NUMBERS t a b l e 9.1. Roman and arabic numerals Arabic Roman Arabic Roman Arabic Roman 1 I 17 XVII 200 CC 2 II 18 XVIII 300 CCC 3 III 19 XIX 400 CD 4 IV 20 XX 500 D 5 V 21 XXI 600 DC 6 VI 22 XXII 700 DCC 7 VII 23 XXIII 800 DCCC 8 VIII 24 XXIV 900 CM 9 IX 30 XXX 1,000 M 10 X 40 XL 2,000 MM 11 XI 50 L 3,000 MMM 12 XII 60 LX 4,000 MV 13 XIII 70 LXX 5,000 V 14 XIV 80 LXXX 10,000 X 15 XV 90 XC 100,000 C 16 XVI 100 C 1,000,000 M Roman Numerals 9.65 Roman numerals—general principles. Table 9.1 shows the formation of roman numerals with their arabic equivalents. The general principle is that a smaller letter before a larger one subtracts from its value, and a smaller letter after a larger one adds to it; a bar over a letter multiplies its value by one thousand. Roman numerals may also be written in lower case letters (i, ii, iii, iv, etc.). In older sources, a final i was often made like a j (vij, viij), and sometimes a v appeared as a u (uj); citations to roman numeral page numbers in older works should follow the original usage.
9.66 The advent of subtrahends (back counters). The use of subtrahends (back counters) was introduced during the Renaissance. Note that IIII, not IV, still appears on some clock faces. The Romans would have ex pressed the year 1999, for example, as MDCCCCLXXXXVIIII. A more modern form, approved by the US government and accepted (if re luctantly) by classical scholars, is MCMXCIX (not MIM, considered a barbarism).
9.67 Chicago's preference for arabic rather than roman numerals. Chi cago uses arabic numerals in many situations where roman numerals were formerly common, as in references to volume numbers of books and journals or chapters of books (see 9.27). Most of the exceptions are Roman Numerals 9.67 treated elsewhere, as follows: for the use of roman numerals in the front matter of books, see 1.4, 1.7, 9.26; in legal instruments, 9.28; with the names of monarchs, prelates, and such, 9.41; with personal names, 9.42; in titles of sequels, 9.43; with names of certain vessels, 9.44; and in out line style, 6.132.
10 ♦ Abbreviations OVERVIEW 10.1 NAMES AND TITLES 10.11 Personal Names, Titles, and Degrees 10.11 Companies and Other Organizations 10.23 GEOGRAPHICAL TERMS 10.27 Names of Countries 10.31 Addresses 10.33 Compass Points, Latitude, and Longitude 10.35 DESIGNATIONS OF TIME 10.37 SCHOLARLY ABBREVIATIONS 10.42 BIBLICAL ABBREVIATIONS 10.44 TECHNOLOGY AND SCIENCE 10.49 The International System of Units 10.51 Astronomy 10.60 Chemical Elements 10.63 US Measure 10.64 BUSINESS AND COMMERCE 10.69 10.1 ABBREVIATIONS Overview 10.1 Abbreviations—additional resources. This chapter provides guidance for using abbreviations and symbols in general and scholarly writing. It also offers some guidance in technical work, especially for the general ist editor confronted with unfamiliar terms. For abbreviations not listed here, Chicago recommends Merriam-Webster's Collegiate Dictionary (bibliog. 3.1) and the multivolume Acronyms, Initialisms & Abbreviations Dictionary (bibliog. 4.7). Authors and editors of technical material will need to refer to more specialized manuals, starting with Scientific Style and Format (bibliog. 1.1).
10.2 Acronyms, initialisms, contractions. The word acronym refers to terms based on the initial letters of their various elements and read as single words (AIDS, laser, NASA, scuba); initialism refers to terms read as a se ries of letters (IRS, NBA, XML); and contraction refers to abbreviations that include the first and last letters of the full word (Mr., amt.). (For the type of contractions normally formed with apostrophes, see 7.30.) These definitions are not perfect. For example, sometimes a letter in an initialism is formed not, as the term might imply, from an initial letter, but rather from its initial sound (as the X in XML, for extensible markup language) or from the application of a number (W3C, for World Wide Web Consortium). Furthermore, an acronym and an initialism are occa sionally combined (JPEG), and the line between initialism and acronym is not always clear (FAQ, which can be pronounced either as a word or as a series of letters). In this chapter the umbrella term abbreviation will be used for all three, as well as for shortened (i.e., abbreviated) forms (ibid., vol., prof., etc.), except where greater specificity is required. (Occasion ally, a symbol abbreviates a term, as in © for copyright. On the other hand, abbreviations for units are often referred to as symbols in SI usage; see 10.51-59.) 10.3 When to use abbreviations. Outside the area of science and technol ogy, abbreviations and symbols are most appropriate in tabular matter, notes, bibliographies, and parenthetical references. A number of ex pressions are almost always abbreviated, even in regular prose, and may be used without first spelling them out. Many of these will be listed as main entries with pronunciation (labeled as nouns rather than as abbre viations) in the latest edition of Webster's (e.g., ATM, DIY, DNA, GPS, HMO, HTML, IQ, JPEG, laser, Ms., NASA). Others, though in more or less common use (CGI, FDA, HVAC, MLA), should generally be spelled out at first occurrence—at least in formal text—as a courtesy to those Overview 10.4 readers who might not easily recognize them. The use of less familiar abbreviations should be limited to terms that occur frequently enough to warrant abbreviation—roughly five times or more within an article or chapter—and the terms must be spelled out on their first occurrence. The abbreviation usually follows immediately, in parentheses, but it may be introduced in other ways (see examples). Such an abbreviation should not be offered only once, never to be used again, except as an alternative form that may be better known to some readers.
Among recent recommendations of the Federal Aviation Administration (FAA) are...
According to the weak law of large numbers (WLLN)...
The National Aeronautics and Space Administration was founded in 1958. Since its inception, NASA has...
The benefits of ERISA (Employee Retirement Income Security Act) are familiar to many.
The debate over genetically modified organisms, or GMOs, is by no means lim ited to the United States.
Writers and editors should monitor the number of different abbrevia tions used in a document; readers trying to keep track of a large number of abbreviations, especially unfamiliar ones, will benefit from a list of abbreviations (see 1.44, 2.23). In a work with few abbreviations and no list, when an abbreviation reappears after a long interval in which it is not used, it may be helpful to repeat the spelled-out name as a reminder. For rules concerning the plural form of various abbreviations, see 7.15. For abbreviations preceded by a, an, or the, see 7.33,10.9. For abbreviations in charts or tables, see 3.45, 3.62, 3.81.
10.4 Periods with abbreviations. In using periods with abbreviations, Chi cago recommends the following general guidelines in nontechnical set tings. For the use of space between elements, see 10.5.
1. Use periods with abbreviations that end in a lowercase letter: p. (page), vol., e.g., i.e., etc., a.k.a., a.m., p.m., Ms., Dr., et al. (et is not an abbreviation; al. is). An exception may be made for the few academic degrees that end in a lower case letter (e.g., DLitt, DMin); see 10.21 and rule 3.
2. Use periods for initials standing for given names: E. B. White; do not use peri ods for an entire name replaced by initials: JFK.
3. Use no periods with abbreviations that include two or more capital letters, even if the abbreviation also includes lowercase letters: VP, CEO, MA, MD, PhD, UK, US, NY, IL (but see rule 4).
4. In publications using traditional state abbreviations, use periods to abbreviate 10.5 ABBREVIATIONS United States and its states and territories: U.S., N.Y., Ill. Note, however, that Chicago recommends using the two-letter postal codes (and therefore US) wherever abbreviations are used; see 10.27. For Canadian provinces and terri tories, see 10.28. See also 14.274.
Note that the British and the French (among others) omit periods from contractions (Dr, assn, Mme). Note also that a slash is occasionally used instead of periods (as in c/o or n/a) but more often denotes per (see 6.109). Units of measure in nontechnical settings are usually spelled out. In scientific usage, periods are generally omitted for abbreviated units of measure and other technical terms (see 10.49-68).
10.5 Abbreviations and spaces. No space is left between the letters of initial-isms and acronyms, whether lowercase or in capitals. Space is usually left between abbreviated words, unless an abbreviated word is used in combination with a single-letter abbreviation. For personal names, see 10.12.
RN C-SPAN YMCA Gov. Gen.
Mng. Ed.
Dist. Atty.
but S.Dak. (but see 10.27) S.Sgt.
10.6 Capitals versus lowercase for acronyms and initialisms. Initialisms tend to appear in all capital letters, even when they are not derived from proper nouns (HIV, VP, LCD). With frequent use, however, acronyms— especially those of five or more letters—will sometimes become low ercase (scuba); those that are derived from proper nouns will retain an initial capital (Unicef). (In British usage, it is common to retain an initial capital even for acronyms derived from generic terms.) Chicago gener ally prefers the all-capital form (e.g., UNICEF), unless the term is listed otherwise in Webster’s. In the sciences, however, it is common to encoun ter forms with a mix of lowercase and capital letters or in all lowercase (e.g., mRNA, IgG, bp); where such forms are considered standard, they should generally be respected.
NAFTA (not Nafta) Note that the words in a spelled-out version of an acronym or initialism are capitalized only if they are considered to be proper nouns (as in the official name of an organization or a trademark); otherwise, they should generally be lowercased, even when they appear alongside the abbrevi ated form.
Overview 10.9 North American Free Trade Agreement (NAFTA) KFC (Kentucky Fried Chicken) transmission-control protocol/internet protocol (TCP/IP) OCD (obsessive-compulsive disorder) 10.7 Italic versus roman type for abbreviations. Chicago italicizes abbre viations only if they stand for a term that would be italicized if spelled out—the title of a book or periodical, for example. Common Latin abbre viations are set in roman (see also 7.55).
OED (Oxford English Dictionary) JAMA (Journal of the American Medical Association) ibid. etc. e.g. i.e.
10.8 Small versus full-size capitals for acronyms and initialisms. Some de signs call for small capitals rather than full-size capitals for acronyms and initialisms (e.g., NASA rather than NASA). Though such usage may be considered desirable for a work that includes many acronyms and initialisms, Chicago does not generally recommend it. If small capitals must be used, the decision of what to mark should be made by the ed itor, who should apply small capitals on the final manuscript (e.g., us ing a word processor’s small-capitals feature). In general, small capitals should be limited to acronyms or initialisms mentioned in running text. Avoid applying small capitals to such items as two-letter postal codes in notes or bibliographies or to roman numerals (e.g., following a personal name). It should be noted that small capitals are not treated in the same way across all software applications and markup systems; small capitals should therefore be checked after conversion for publication to make sure they have not reverted to lowercase or full capitals (see also 2.81- 83). See also 10.38,10.41. For the use of small capitals for emphasis, see 7.52.
10.9 "A," "an," or "the" preceding an abbreviation. When an abbreviation follows an indefinite article, the choice of a or an is determined by the way the abbreviation would be read aloud. Acronyms are read as words and are rarely preceded by a, an, or the (“member nations of NATO”), ex cept when used adjectivally (“a NATO initiative”; “the NATO meeting”). See 10.2; see also 7.33.
anHMO a UFO a NATO member 10.10 ABBREVIATIONS a LOOM parade an AA meeting a AA battery (pronounced double A) an NAACP convention an NBA coach an HIV test an MS symptom (a symptom of multiple sclerosis) but a MS by... (would be read as a manuscript by...) Initialisms, which are read as a series of letters, are often preceded by a definite article (“member nations of the EU”). Whether to include the ar ticle may depend on established usage. For example, one would refer to the NBA and the NAACP, on the one hand, but to W3C, PBS, and NATO, on the other—though all these organizations include the definite article in spelled-out form. If no established usage can be determined, use the definite article if it would be used with the spelled-out form. Some terms, such as DIY (do it yourself), do not ordinarily require a definite article in spelled-out form and therefore do not require one as an initialism.
10.10 Abbreviations containing ampersands. No space is left on either side of an ampersand used within an initialism. See also 10.24.
R&D Texas A&M Names and Titles Personal Names, Titles, and Degrees 10.11 Abbreviations for personal names. Normally, abbreviations should not be used for given names. A signature, however, should be transcribed as the person wrote it.
Benj. Franklin Geo. D. Fuller Ch. Virolleaud 10.12 Initials in personal names. Initials standing for given names are followed by a period and a space. A period is normally used even if the middle ini tial does not stand for a name (as in Harry S. Truman).
Roger W. Shugg P. D. James M. F. K. Fisher Names and Titles 10.15 If an entire name is abbreviated, spaces and periods can usually be omitted.
FDR (Franklin Delano Roosevelt) MJ (Michael Jordan) JLo (Jennifer Lopez) but J.Lo (the title of Lopez’s 2001 album) 10.13 Abbreviating titles before names. Many civil or military titles preceding a full name may be abbreviated. When preceding a surname alone, how ever, they should be spelled out. See also 8.19.
Rep. Dan Lipinski; Representative Lipinski Sen. Kirsten E. Gillibrand; Senator Gillibrand Vice Adm. Carol M. Pottenger; Vice Admiral Pottenger 10.14 Abbreviations for civil titles. The following abbreviations, among oth ers, may precede a full name where space is tight: Aid.
Atty. Gen.
Insp. Gen.
Prof.
Assoc. Prof.
Fr. (father) Judge Adv. Gen.
Sr. (sister) Asst. Prof.
Gov.
Pres.
Supt.
10.15 Abbreviations for military titles. The US military omits periods in the official abbreviated forms of its ranks. The abbreviations for a given title may vary across branches. The army, for example, uses SSG for staff sergeant] the air force and marines prefer SSgt. (In the examples below, such variants are not presented.) In general contexts, however, including military history, traditional abbreviations—which tend not to vary across the armed forces—are preferred. The following very selective list merely illustrates the difference between military usage and traditional forms. Where no traditional abbreviation is appropriate before a name, use the full form.
GEN Gen.
A1C Airman First Class LT Lt.
ADM Adm.
BG Brig. Gen.
1LT 1st Lt.
CDR Cdr.
2LT 2nd Lt.
COL Col.
LG Lt. Gen.
CPT Capt.
LTC Lt. Col.
CWO Chief Warrant Officer MAJ Maj.
10.16 ABBREVIATIONS MG Maj. Gen.
SGT Sgt.
MSG M.Sgt. (master sergeant) SSG S.Sgt. (staff sergeant) PO Petty Officer WO Warrant Officer For the latest official forms of rank insignia, consult the website of the US Department of Defense. In addition, there are many reference books containing more detailed lists of abbreviations and terms, some of which are published regularly. See, for example, Timothy Zurick, Army Dictio nary and Desk Reference (bibliog. 5). For Canadian military ranks and abbreviations, start with the website of the National Defence and the Canadian Armed Forces. For the United Kingdom, consult the websites of the various forces. See also 8.112.
10.16 Abbreviations for social titles. Social titles are always abbreviated, whether preceding the full name or the surname only. The spelled-out forms Mister or Doctor might be used without a name—as in direct ad dress (see also 8.20).
Ms. Mr. but Mrs. Mx. Thank you, Doctor.
Messrs. Dr. Jekyll Social titles are routinely omitted in most prose, though a few periodicals in particular persist in using them. When an academic degree or profes sional designation follows a name, such titles are always omitted.
Jennifer James, MD (not Dr. Jennifer James, MD) Similarly, the now somewhat archaic abbreviation Esq. (Esquire) is used only after a full name and never when Mr., Dr., or the like precedes the name.
10.17 Abbreviations for French social titles. Note the presence or absence of periods after the following abbreviations for French social titles (which can be used with either a full name or a surname only). Mme and Mlle are considered contractions (see 10.2) and therefore do not take a period. This usage should be observed when such forms appear untranslated in English-language settings.
M. MM. Mme Mlle When Monsieur, Messieurs, Madame, or Mademoiselle is used without a name, in direct address, it is spelled out (and, in French usage, generally lowercased).
Names and Titles 10.20 10.18 Abbreviations for "Reverend" and "Honorable." The abbreviations Rev. and Hon. are traditionally used before a full name when the does not pre cede the title. With the, such titles should be spelled out.
Rev. Sam Portaro; the Reverend Sam Portaro Hon. Henry M. Brown; the Honorable Henry M. Brown With a last name only, such titles are normally omitted. The construc tion “Reverend So-and-So,” however, is common, especially in informal prose or speech.
Rev. Jane Schaefer; Schaefer (or Reverend Schaefer) the Honorable Patricia Birkholz; Birkholz 10.19 Abbreviations for "Junior," "Senior," and the like. The abbreviations Jr. and Sr., as well as roman or arabic numerals such as III or 3rd after a per son’s name, are part of the name and so are retained in connection with any titles or honorifics. Note that these abbreviations are used only with the full name, never with the surname only. See also 6.43,9.42.
Jordan Balfence Jr. spoke first. After Mr. Balfence relinquished the podium,... Zayd Zephyr III, MBA, spoke last. In closing, Mr. Zephyr reiterated...
In some contexts—for example, a biography that includes frequent men tions of a father and son who share the same name—it may be appropri ate to use Jr. or Sr. or the like with a first name alone.
Henry Jr., in his later years (and despite the publication of The Golden Bowl and other masterpieces), was never again to enjoy the kind of wealth that Henry Sr. had once taught him to take for granted.
10.20 Abbreviations for the names of saints. The word Saint is often abbrevi ated (St., pl. SS.) before the name of a Christian saint; it should normally be spelled out in formal prose but need not be if space is at a premium. The choice for one or the other should be implemented consistently.
Saint (or St.) Teresa Saints (or SS.) Francis of Paola and Francis of Sales When Saint or St. forms part of a personal name, the bearer’s usage is followed. See also 10.30.
Augustus Saint-Gaudens Muriel St. Clare Byrne 10.21 ABBREVIATIONS 10.21 Abbreviations for academic degrees. Chicago recommends omitting periods in abbreviations of academic degrees (BA, DDS, etc.) unless they are required for reasons of tradition or consistency with, for example, a journal’s established style. In the following list of some of the more com mon degrees, periods are shown only where uncertainty might arise as to their placement. Spelled-out terms, often capitalized in institutional settings (and on business cards and other promotional items), should be lowercased in normal prose. See also 8.29.
AB artium baccalaureus (bachelor of arts) AM artium magister (master of arts) BA bachelor of arts BD bachelor of divinity BFA bachelor of fine arts BM bachelor of music BS bachelor of science DB divinitatis baccalaureus (bachelor of divinity) DD divinitatis doctor (doctor of divinity) DDS doctor of dental surgery DLitt or DLit doctor litterarum (doctor of letters; doctor of literature) DMD dentariae medicinae doctor (doctor of dental medicine) DMin doctor of ministry DO doctor of osteopathy or osteopathic physician DVM doctor of veterinary medicine EdM educationis magister (master of education) JO juris doctor (doctor of law) LHD litterarum humaniorum doctor (doctor of humanities) LittD litterarum doctor (doctor of letters) LLB (LL.B.) legum baccalaureus (bachelor of laws) LLD (LL.D.) legum doctor (doctor of laws) LLM (LL.M.) legum magister (master of laws) MA master of arts MBA master of business administration MD medicinae doctor (doctor of medicine) MDiv master of divinity MFA master of fine arts MS master of science MSN master of science in nursing MSW master of social welfare or master of social work PhB philosophiae baccalaureus (bachelor of philosophy) PhD philosophiae doctor (doctor of philosophy) PhG graduate in pharmacy SB scientiae baccalaureus (bachelor of science) Names and Titles 10.23 SM STB scientiae magister (master of science) sacrae theologiae baccalaureus (bachelor of sacred theology) 10.22 10.22 10.23 10.23 These designations are set off by commas when they follow a personal name.
Ariel Z. Lee, JD, attended the University of Chicago Law School.
Abbreviations for professional, religious, and other designations. Ab breviations for many other designations, professional and otherwise, fol low the pattern of academic degrees (see 10.21), for which Chicago rec ommends dispensing with periods. Spelled-out terms, often capitalized in institutional settings, are lowercase unless they designate the proper name of an organization.
CNM certified nurse midwife FAIA fellow of the American Institute of Architects FRS fellow of the Royal Society JP justice of the peace LPN licensed practical nurse MP member of Parliament OFM Order of Friars Minor OP Ordo Praedicatorum (Order of Preachers) RN registered nurse SJ Society of Jesus These designations, like academic degrees, are set off by commas when they follow a personal name.
Joan Hotimlanska, LPN, will be working on the second floor.
Companies and Other Organizations Commonly used generic abbreviations for firms and companies. All of the abbreviations in the following list may be found in Webster’s and other standard dictionaries. Use periods, or not, according to the recom mendations in 10.4. See also 10.69.
Assoc. Inc.
Bros. LLC (limited liability company) Co. LLP (limited liability partnership) Corp. LP (limited partnership) 10.24 ABBREVIATIONS RR (railroad) Rwy. or Ry. (railway) Ltd.
Mfg.
PLC (public limited company) In certain languages other than English, periods are omitted from abbre viations if they are contractions (see 10.2).
Cia (Sp. compahia) Cie (Fr. compagnie) 10.24 Abbreviations and ampersands in company names. Abbreviations and ampersands are appropriate in notes, bibliographies, tabular matter, and the like. See also 14.135.
Ginn & Co. JPMorgan Chase & Co. Moss Bros. RAND Corp.
In running text, company names are best given in their full forms. It should be noted, however, that some full forms include ampersands and abbreviations. If in doubt, especially with reference to contemporary firms, look up the company name at a corporate website or other author itative source. Such elements as Inc.y & Co.y and LLC may be omitted unless relevant to the context.
Johnson & Johnson was founded in 1886.
JPMorgan Chase operates in more than sixty countries.
AT&T Corporation was once known as the American Telephone and Telegraph Company.
Abbreviations for companies and other organizations that use initialisms as described in 10.4, rule 3, generally appear without periods; the occa sional exception may be made in the case of a clear and established pref erence (as in the band name R.E.M.).
10.25 Abbreviations for media companies. Abbreviations for media compa nies often take the form of call letters used for broadcasting. These are always capitalized and do not take periods.
ABC CBS HBO KFTV MTV NBC TBS WFMT WTTW 10.26 Abbreviations for associations and the like. Both in running text (pref erably after being spelled out on first occurrence) and in tabular mat ter, notes, and so forth, the names of many agencies and organizations, governmental and otherwise, are commonly abbreviated. Whether aero- Geographical Terms 10.27 nyms or initialisms (see 10.2), such abbreviations appear in full capitals and without periods. For a, an, or the with abbreviations, see 10.9.
AAUP EU (European Union) AFL-CIO HMO (pl. HMOs) EPA WHO WTO (formerly GATT) Geographical Terms 10.27 Abbreviations for US states and territories. In running text, the names of states, territories, and possessions of the United States should always be spelled out when standing alone and preferably (except for DC) when following the name of a city: for example, “Lake Bluff, Illinois, was in corporated in 1895.” In bibliographies, tabular matter, lists, and mailing addresses, they are usually abbreviated. In all such contexts, Chicago prefers the two-letter postal codes to the conventional abbreviations. Note that if traditional abbreviations must be used, some terms may not be subject to abbreviation. See also 10.4.
AK Alaska or Alas.
LA La.
AL Ala.
MA Mass.
AR Ark.
MD Md.
AS American Samoa ME Maine AZ Ariz.
MH Marshall Islands CA Calif.
MI Mich.
CO Colo.
MN Minn.
CT Conn.
MO Mo.
DC D.C.
MP Northern Mariana Islands DE Del.
MS Miss.
FL Fla.
MT Mont.
FM Federated States of NC N.C.
Micronesia ND N.Dak.
GA Ga.
NE Neb. or Nebr.
GU Guam NH N.H.
HI Hawaii NJ N.J.
IA Iowa NM N.Mex.
ID Idaho NV Nev.
IL Ill.
NY N.Y.
IN Ind.
OH Ohio KS Kans.
OK Okla.
KY Ky.
OR Ore. or Oreg.
10.28 ABBREVIATIONS PA Pa.
UT Utah PR P.R. or Puerto Rico VA Va.
PW Palau VI V.I. or Virgin Islands RI R.I.
VT Vt.
SC S.C.
WA Wash.
SD S.Dak.
WI Wis. or Wise.
TN Tenn.
WV W.Va.
TX Tex.
WY Wyo.
10.28 Abbreviations for Canadian provinces and territories. Canadian prov inces and territories are normally spelled out in text (e.g., “Kingston, On tario, is worth a visit”) but may be abbreviated in bibliographies and the like—using the two-letter postal abbreviations, which have the advantage of applying to both the English and French forms.
AB Alberta BC British Columbia or Colombie-Britannique MB Manitoba NB New Brunswick or Nouveau-Brunswick NL Newfoundland and Labrador or Terre-Neuve-et-Labrador NS Nova Scotia or Nouvelle-Ecosse NT Northwest Territories or Territoires du Nord-Ouest NU Nunavut ON Ontario PE Prince Edward Island or Ile-du-Prince-Edouard QC Quebec or Quebec SK Saskatchewan YT Yukon 10.29 Comma with city plus state abbreviation. When following the name of a city, the names of states, provinces, and territories are enclosed in com mas, whether they are spelled out (as in running text) or abbreviated (as in tabular matter or lists). In an exception to the rule, no comma appears between the postal code and a zip code. See also 6.39,6.17.
Bedford, PA, and Jamestown, NY but Send the package to J. Sprocket, 3359 Fob Dr., Quartz, IL 60000.
10.30 Abbreviations for place-names with "Fort," "Mount," and "Saint." Ge neric terms as elements of geographic names are usually spelled out in formal prose (and in mailing addresses) but can be abbreviated where space is at a premium or to reflect predominant usage. San and Santa Geographical Terms 10.32 (e.g., San Diego, Santa Barbara) are never abbreviated. For French place- names with Saint, see 11.26.
Fort (Ft.) Myers Port (Pt.) Arthur Saint (St.) Paul Mount (Mt.) Airy Saint (St.) Louis Names of Countries 10.31 Abbreviating country names. Names of countries are usually spelled out in text but may be abbreviated in tabular matter, lists, and the like. Use discretion in forming the abbreviations; for tables, make sure they are defined in a note to the table if there is any possibility of confusion (see 3.76-80). The examples below reflect entries in standard dictionaries (all are listed in Webster's, with the exception of Swed., which is listed in American Heritage).
Fr. Ger. Isr. It. Neth. Russ. Sp. Swed.
Certain initialisms, on the other hand, may be appropriate in regular text, especially after the full form has been established (see 10.2,10.3). For more on US, see 10.32.
UAE (United Arab Emirates) US UK GDR (the former German Democratic Republic, or East Germany) or DDR (Deut sche Demokratische Republik) FRG (the former Federal Republic of Germany) or BRD (Bundesrepublik Deutsch land) USSR (the former Union of Soviet Socialist Republics) In certain technical applications, it may be advisable to use either the two-letter or three-letter standard abbreviations based on the English names of countries as defined by the International Organization for Standardization (ISO 3166-1, alpha-2 and alpha-3, respectively). For these lists, consult the ISO website.
10.32 "US" versus "United States." Where necessary, initialisms for country names can be used in running text according to the guidelines set forth in 10.3 (see also 10.31). Note that, as a matter of editorial tradition, this manual has long advised spelling out United States as a noun, reserving US for the adjective form only (where it is preferred) and for tabular 10.33 ABBREVIATIONS matter and the like. In a departure, Chicago now permits the use of US as a noun, subject to editorial discretion and provided the meaning is clear from context.
US dollars US involvement in China China’s involvement in the United States or China’s involvement in the US See also 10.4.
Addresses 10.33 Mailing addresses—postal versus standard abbreviations. Standard abbreviations preferred by the US Postal Service (first column) are in all caps and do not use periods; these forms are most appropriate for mail ing addresses. In tabular matter and the like, Chicago prefers the form of abbreviations presented in the second column. For those not listed here, consult a dictionary. For standard postal abbreviations, consult the USPS or other regional postal service. In running text, spell out rather than ab breviate. See also 10.34.
AVE Ave.
PO BOX PO Box RD Rd.
CT Ct.
RM Rm.
DR Dr.
RTE Rte.
BLDG Bldg.
EXPY Expy.
SQ Sq.
HWY Hwy.
ST St.
LN Ln.
STE Ste. (or Suite) PKWY Pkwy.
TER Ter. (or Terr.) PL Pl.
10.34 Abbreviations for compass points in mailing addresses. Single-letter compass points accompanying a street name are normally followed by a period; two-letter compass points are not. (The US Postal Service does not use periods for either; see 10.33; see also 10.4.) Note that when used in an address, the abbreviations NE,NW, SE, and SW remain abbreviated even in running text (there is no comma before them when they follow a street name). The N in the third example is a street name and not a com pass point.
Geographical Terms 10.36 1060 E. Prospect Ave. (or, in running text, 1060 East Prospect Avenue) 456 NW Lane St. (or, in running text, 456 NW Lane Street) I stayed in a building on N Street SW, close to the city center.
A compass point that is the name (or part of the name) of a street or a place-name must never be abbreviated (e.g., South Ave., Northwest Hwy., South Shore Dr., West Bend, East Orange). For the use of numerals in addresses, see 9.51, 9.52.
Compass Points, Latitude, and Longitude 10.35 Abbreviations for compass points. Points of the compass may be abbre viated as follows, without periods (but see 10.34). In formal, nontechni cal text, however, these terms are usually spelled out.
N, E, S, W, NE, SE, SW, NW, NNE, ENE, ESE, etc.
N by NE, NE by N, NE by E, etc.
10.36 Abbreviations for "latitude" and "longitude." In nontechnical contexts, the words latitude and longitude are never abbreviated in running text or when standing alone.
longitude 90° west the polar latitudes Global positioning coordinates are expressed in a variety of ways (though latitude is always given first). Some systems use a minus sign (or hyphen) to indicate south or west. Others use decimal minutes. The following three coordinates are equivalent. The comma is often omitted.
36 25.217, -44 23.017 N 36°25T3", W 44°23'01" N 36 25.217, W 44 23.017 In technical work, the abbreviations lat and long, usually without periods, may be used when part of a coordinate. They can sometimes be dropped, since the compass point identifies the coordinate.
lat 42°15'09" N, long 89°17'45" W lat 45°16T7" S, long 116°40T8" E The chart showed shoal water at 19°29'59" N, 107°45'36" W.
Note that primes (') and double primes ("), not quotation marks, are used. For greater detail, consult Scientific Style and Format (bibliog. 1.1).
10.37 ABBREVIATIONS Designations of Time 10.37 Other discussions related to time. For units of time (seconds, minutes, etc.), see 10.68. For numerical designations of dates and times of day, see 9.30, 9.33, 9.35, 9.37-40.
10.38 Abbreviations for chronological eras. The following abbreviations are used in running text and elsewhere to designate chronological eras. Al though these have traditionally appeared in small capitals (with or with out periods), Chicago recommends full capitals without periods, in keep ing with the general guidelines in this chapter (see 10.4; see also 10.8). The first four precede the year number; the others follow it. See also 9.34.
AD anno Domini (in the year of [our] Lord) AH anno Hegirae (in the year of the Hegira); anno Hebraico (in the Hebrew year) AM anno mundi (in the year of the world) (not to be confused with ante me ridiem; see 10.41) AS anno salutis (in the year of salvation) AUC ab urbe condita (from the founding of the city [Rome, in 753 BCE]) BC before Christ BCE before the Common Era BP before the present CE Common Era MYA million years ago YBP years before the present 10.39 Abbreviations for months. Where space restrictions require that the names of months be abbreviated, one of the following systems is often used. The second and third, which take no periods, are used respectively in computer systems and indexes of periodical literature. In formal prose, Chicago prefers the first.
Jan.
Jan Ja July Jul JI Feb.
Feb F Aug.
Aug Ag Mar.
Mar Mr Sept.
Sep S Apr.
Apr Ap Oct.
Oct 0 May May My Nov.
Nov N June Jun Je Dec.
Dec D 10.40 Abbreviations for days of the week. Where space restrictions require that days of the week be abbreviated, one of the following systems is Scholarly Abbreviations 10.42 often used. The second (common in computer code) and third use no periods. In formal prose, Chicago recommends the first.
Sun.
Sun Su Thurs.
Thu Th Mon.
Mon M Fri.
Fri F Tues.
Tue Tu Sat.
Sat Sa Wed.
Wed W 10.41 Abbreviations for time of day. The following abbreviations are used in text and elsewhere to indicate time of day. Though these sometimes appear in small capitals (with or without periods), Chicago prefers the lowercase form, with periods, as being the most immediately intelligible. For further explanation and examples, see 9.37,9.39. See also 10.4.
a.m. ante meridiem (before noon) m. meridies (noon [rarely used]) p.m. post meridiem (after noon) The abbreviations a.m. an&p.m. should not be used with morning, after noon, evening, night, or o'clock. (See also 7.89, section 1, under time.) 10:30 a.m. or ten thirty in the morning 11:00 p.m. or eleven o’clock at night Time zones, where needed, are usually given in parentheses—for ex ample, 4:45 p.m. (CST).
MST mountain standard time GMT Greenwich mean time EST eastern standard time MDT mountain daylight time EDT eastern daylight time PST Pacific standard time CST central standard time PDT Pacific daylight time CDT central daylight time It should be noted that Greenwich mean time has long been superseded by the nearly identical coordinated universal time (UTC) as the basis of international time. References to GMT, however, remain widespread not only in the United Kingdom but also in the United States and Canada and elsewhere.
Scholarly Abbreviations 10.42 Scholarly abbreviations. Scholarly abbreviations and symbols such as those listed in this section are typically found in bibliographic references, 10.42 ABBREVIATIONS glossaries, and other scholarly apparatus. Some of them are no longer widely used and are listed here mainly as an aid to interpreting older texts. In formal prose, Chicago prefers to confine such abbreviations to parentheses or notes. Some can stand for several terms; only the terms likely to be encountered in scholarly works (mainly in the humanities) and serious nonfiction are included here. The choice between different abbreviations for one term (e.g., L. and Lat. for Latin) depends on the writer’s preference, context, readership, and other factors; if in doubt, choose the longer form. Note that Latin abbreviations are normally set in roman. Note also that ab, ad, et, and other Latin terms that are complete words take no periods. See also 7.55, 10.4. For terms used more com monly in science and technology, see 10.49.
abbr. ab init. abl. abr.
abbreviated, -ion ab initio, from the beginning ablative abridged, abridgment acc. act. add. ad inf. ad init. ad int.
adj. ad lib. ad loc. adv.
aet. or aetat. AFr.
AN accusative active addendum ad infinitum ad initium, at the beginning ad interim, in the intervening time adjective ad libitum, at will (often used without a period) ad locum, at the place adverb aetatis, aged Anglo-French Anglo-Norman anon.
anonymous (see 14.79) app. arch, art. AS b. Bd. bib. bibl. bibliog. biog. biol.
appendix archaic article Anglo-Saxon born; brother Band (Ger.), volume Bible, biblical bibliotheca, library bibliography, -er, -ical biography, -er, -ical biology, -ist, -ical Scholarly Abbreviations 10.42 bk.
c.
c. or cop. ca. or c. Cantab, cet. par.
cf.
chap, or ch. col. colloq. comp, compar, con. conj.
cons, constr.
cont. contr. corr.
cp.
d.
Dan. dat. def. dept, deriv. d. h.
d. i.
dial, diet, dim.
diss.
dist. div.
do. dram. pers. Dr. u. Vrl. DV ea.
book century; chapter (in law citations) copyright (see 10.43) circa, about, approximately (ca. preferred for greater clarity) Cantabrigiensis, of Cambridge ceteris paribus, other things being equal confer, compare (“see, by way of comparison”; should not be used when see alone is meant) chapter color (best spelled out); column colloquial, -ly, -ism compiler (pl. comps.), compiled by comparative contra, against conjunction; conjugation consonant construction continued contraction corrected compare (rarely used; cf. is far more common) died; daughter Danish dative definite; definition department derivative das heifit (or das heisst), namely (used only in German text; note the space between initials) das ist, that is (used only in German text; note the space be tween initials) dialect dictionary diminutive dissertation district division; divorced ditto dramatis personae Druck und Verlag, printer and publisher Deo volente, God willing; Douay Version (see 10.48) each 10.42 ABBREVIATIONS ed.
EE e. g. ellipt.
ency. or encyc. eng.
Eng.
engr.
enl.
eq.
esp.
et al.
etc.
et seq.
ex.
f. or fem.
f.
fasc.
ff.
fig fl.
fol.
Fr. fr.
frag, fut.
f.v. Gael, gen. geog.
geol.
geom.
Ger. or G. ger.
Gk. hist.
editor (pl. eds.), edition, edited by (never add by after ed.: ei ther “ed. Jane Doe” or “edited by Jane Doe”; use eds. only after, never before, the names of two or more editors; see examples throughout chapter 14) Early English exempli gratia, for example (not to be confused with i.e.) elliptical, -ly encyclopedia engineer, -ing English engraved, -ing enlarged equation (pl. eqq. or eqs.; see also 10.43) especially et alii (or et alia), and others (normally used of persons; no pe riod after et) et cetera, and so forth (normally used of things) et sequentes, and the following example (pl. exx. or exs.) feminine; female fir (Ger.), for fascicle and following (see 14.149) figure floruit, flourished (used with a date to indicate the productive years of a historical figure whose birth and death dates are unknown) folio French from fragment future folio verso, on the back of the page Gaelic genitive; genus geography, -er, -ical geology, -ist, -ical geometry, -ical German gerund Greek history, -ian, -ical Scholarly Abbreviations 10.42 HQ ibid.
id. i.e.
IE ill. imp. or imper. incl. indef. indie.
inf. infin. in pr. inst. instr, inter j. intrans. introd, or intro, irreg.
It.
L.
1.
lang.
Lat. or L. lit.
loc. loc. cit. loq.
m.
m. or masc.
marg, math. MHG mimeo.
mise. MM m. m. Mod.E.
MS (pl. MSS) mus.
n.
headquarters ibidem, in the same place (see 14.34) idem, the same (see 14.35) id est, that is (not to be confused with e.g.) Indo-European illustrated, -ion, -or imperative including indefinite indicative infra, below (best spelled out) infinitive in principio, in the beginning instant (this month); institute, -ion instrumental interjection intransitive introduction irregular Italian Latin; left (in stage directions) left; line (pl. 11., but best spelled out to avoid confusion with numerals 1 and 11) language Latin literally locative loco citato, in the place cited (best avoided; see 14.36) loquitur, he or she speaks male; married; measure (pl. mm.) masculine margin, -al mathematics, -ical Middle High German mimeograph, -ed miscellaneous Maelzel’s metronome mutatis mutandis, necessary changes being made Modern English manuscriptum (pl. manuscripta), manuscript museum; music, -al natus, born; note, footnote (pl. nn.); noun 10.42 ABBREVIATIONS nat.
NB, n.b.
national; natural nota bene, take careful note (capitals are illogical but often used for emphasis) n.d.
no date; not determined neg. neut.
no. (pl. nos.) nom.
non obs.
negative neuter number nominative non obstante, notwithstanding non seq.
n.p.
n.s.
NS ob. obs.
non sequitur, it does not follow no place; no publisher; no page new series New Style (dates) obiit, died obsolete occas.
OE OFr.
OHG ON op. cit.
o. s.
OS Oxon.
P-para, or par. pass.
pa. t. path.
perf. perh.
pers.
pers. comm.
pl. posth.
p. p. ppi.
PPS prep, pres.
pron. pro tern.
occasional, -ly Old English Old French Old High German Old Norse opere citato, in the work cited (best avoided; see 14.36) old series Old Style (dates) Oxoniensis, of Oxford page (pl. pp.); past (also pa.) paragraph (see 10.43) passive past tense pathology, -ist, -ical perfect perhaps person, -al personal communication plate (best avoided; see 3.9); plural posthumous, -ly past participle participle post postscriptum, a later postscript preposition present pronoun pro tempore, for the time being (often used without a period) Scholarly Abbreviations 10.42 prox.
PS Pt pub.
QED quar. or quart.
q. v.
R.
r.
refl.
repr. rev.
RIP s.
s.a.
sc.
Sc. or Scot.
s.d.
sd.
sec.
ser.
s.h.
sing, or sg.
s.l.
s.n.
sociol.
Sp.
s.p.
st.
subj. subst. or s.
sup. superl.
supp. or suppl.
s. v. (pl. s.w.) syn.
t.
techn.
theol.
proximo, next month postscriptum, postscript part publication, publisher, published by quod erat demonstrandum, which was to be demonstrated quarter, -ly quod vide, which see (used only in a cross-reference after the term referred to; cf. s.v.) rex, king; regina, queen; right (in stage directions) right; recto; reigned reflexive reprint, -ed review; revised, revised by, revision (never add by after rev.: either “rev. Jane Doe” or “revised by Jane Doe”) requiescat in pace, may he or she rest in peace son; substantive, -ival sine anno, without year; sub anno, under the year scene; scilicet, namely; sculpsit, carved by Scottish sine die, without setting a day for reconvening; stage direction sound section (see 10.43); secundum, according to series speech heading singular sine loco, without place (of publication) sine nomine, without name (of publisher) sociology, -ist, -ical Spanish speech prefix stanza subject, -ive; subjunctive substantive, -al supra, above superlative supplement sub verbo, sub voce, under the word (used in a cross-reference before the term referred to; cf. q.v.) synonym, -ous tome (Fr.), tomo (Sp.), volume technical, -ly theology, -ian, -ical 10.43 ABBREVIATIONS t.p. trans.
title page translated by, translator(s) (never add by after trans.: either “trans. Jane Doe” or “translated by Jane Doe”); transitive treas. TS ult. univ. usw.
treasurer typescript ultimatus, ultimate, last; ultimo, last month university und so weiter, and so forth (equivalent to etc.; used only in Ger man text) ut sup.
V.
v. or vb.
v.i.
ut supra, as above verse (pl. w.); verso; versus; vide, see verb verbum intransitivum, intransitive verb; vide infra, see below viz. voc.
vol.
vs. or v.
v.t.
yr.
videlicet, namely vocative volume versus (in legal contexts use v.) verbum transitivum, transitive verb year; your 10.43 A few scholarly symbols. The symbols below often appear in bibliogra phies and other scholarly apparatus rather than their equivalent abbrevi ations (see 10.42).
© copyright equals, the same as (for examples, see 10.46) 11 (pl. 1111) § (pl- §§) paragraph section Biblical Abbreviations 10.44 Biblical abbreviations—an overview. In running text, books of the Bible are generally spelled out. See also 9.26.
The opening chapters of Ephesians constitute a sermon on love. Jeremiah, chapters 42-44, records the flight of the Jews to Egypt. According to Genesis 1:27, God created man in his own image.
In parenthetical citations or in notes, or where many such references ap pear in the text, abbreviations are appropriate.
My concordance lists five instances of the word nourish: Gen. 47:12, Ruth 4:15, Isa. 44:14, Acts 7:21, and 1 Tim. 4:6.
Biblical Abbreviations 10.45 For authoritative guidance in many biblical areas not covered here, con sult The SBL Handbook of Style (bibliog. 1.1). For citing scriptural refer ences in notes and bibliographies, see 14.238-41.
10.45 Abbreviations for the Old Testament. These are the traditional abbre viations and commonly used shorter forms for books of the Old Testa ment. (Note that the shorter forms have no periods.) The listing is alpha betical, both for easier reference and because the order varies slightly in different versions of the Bible. Alternative names for the same books are indicated by an equals sign (see 10.43). For the New Testament, see 10.47.
Amos or Am 1 Chron. or 1 Chr 2Chron. or 2 Chr Dan. or Dn Deut. or Dt Eccles, or Eccl Esther or Est Exod. or Ex Ezek. or Ez Ezra or Ezr Gen. or Gn Hab. or Hb Hag. or Hg Hosea or Hos Isa. or Is Jer. or Jer Job or Jb Joel or Jl Jon. or Jon Josh, or Jo Judg. or Jgs 1 Kings or 1 Kgs 2 Kings or 2 Kgs Lam. or Lam Lev. or Lv Mai. or Mai Mic. or Mi Nah. or Na Neh. or Neh Num. or Nm Obad. or Ob Amos 1 Chronicles 2 Chronicles Daniel Deuteronomy Ecclesiastes Esther Exodus Ezekiel Ezra Genesis Habakkuk Haggai Hosea Isaiah Jeremiah Job Joel Jonah Joshua Judges 1 Kings 2 Kings Lamentations Leviticus Malachi Micah Nahum Nehemiah Numbers Obadiah 10.46 ABBREVIATIONS Prov. or Prv Ps. (pl. Pss.) or Ps (pl. Pss) Ruth or Ru 1 Sam. or 1 Sm 2 Sam. or 2 Sm Song of Sol. or Sg Zech, or Zee Zeph. or Zep Proverbs Psalms Ruth 1 Samuel 2 Samuel Song of Solomon (= Song of Songs) Zechariah Zephaniah 10.46 Abbreviations for the Apocrypha. The books of the Apocrypha are ac cepted in Roman Catholic versions of the Bible, though not in Jewish and Protestant versions. Some are not complete in themselves but are continuations of books listed in 10.45. These are the traditional abbrevi ations and commonly used shorter forms. (Note that the shorter forms have no periods.) Alternative names for the same books are indicated by an equals sign (see 10.43). Where no abbreviation is given, the full form should be used.
Bar. or Bar Ecclus.
lEsd.
2Esd.
Jth. or Jdt 1 Macc. or 1 Me 2 Macc. or 2 Me Pr. of Man.
Sir. or Sir Sus.
Tob. or Tb Ws Wisd. of Sol.
Baruch Ecclesiasticus (= Sirach) 1 Esdras 2 Esdras Judith 1 Maccabees 2 Maccabees Prayer of Manasses (= Manasseh) Sirach (= Ecclesiasticus) Susanna Tobit Wisdom (= Wisdom of Solomon) Wisdom of Solomon (= Wisdom) 10.47 Abbreviations for the New Testament. These are the traditional abbre viations and commonly used shorter forms for books of the New Testa ment. (Note that the shorter forms have no periods.) The listing is alpha betical, both for easier reference and because the order varies slightly in different versions of the Bible. Alternative names for the same books are indicated by an equals sign (see 10.43). For the Old Testament, see 10.45.
Acts Apoc.
Col. or Col Acts of the Apostles Apocalypse (= Revelation) Colossians Biblical Abbreviations 10.48 1 Cor. or 1 Cor 1 Corinthians 2 Cor. or 2 Cor 2 Corinthians Eph. or Eph Ephesians Gal. or Gal Galatians Heb. or Heb Hebrews James or Jas James John or Jn John (Gospel) 1 John or 1 Jn 1 John (Epistle) 2 John or 2 Jn 2 John (Epistle) 3 John or 3 Jn 3 John (Epistle) Jude Jude Luke or Lk Luke Mark or Mk Mark Matt, or Mt Matthew 1 Pet. or 1 Pt 1 Peter 2 Pet. or 2 Pt 2 Peter Phil, or Phil Philippians Philem. or Phlm Philemon Rev. or Rv Revelation (= Apocalypse) Rom. or Rom Romans 1 Thess. or 1 Thes 1 Thessalonians 2 Thess. or 2 Thes 2 Thessalonians 1 Tim. or 1 Tm 1 Timothy 2 Tim. or 2 Tm 2 Timothy Titus or Ti Titus 10.48 Abbreviations for versions and sections of the Bible. Versions and sec tions of the Bible are usually abbreviated in the form of initialisms, espe cially when they consist of more than one word.
Apoc.
Apocrypha ARV American Revised Version ASV American Standard Version AT American Translation AV Authorized (King James) Version CEV Contemporary English Version DV Douay Version ERV English Revised Version EV English version (s) HB Hebrew Bible JB Jerusalem Bible LXX Septuagint MT Masoretic Text 10.49 ABBREVIATIONS NAB New American Bible NEB NJB NRSV NT New English Bible New Jerusalem Bible New Revised Standard Version New Testament OT RSV RV Syr. Vulg. WEB Old Testament Revised Standard Version Revised Version Syriac Vulgate World English Bible Technology and Science 10.49 Miscellaneous technical abbreviations. The following list, which cannot aim to be comprehensive, includes some abbreviations used in various branches of the physical and biological sciences and in technical writing. Some, such as PC and DVD, are also in wide general use. Abbreviations used in highly specialized areas have generally been omitted, as have most adjectival forms. Many of the abbreviations for units are identical to or compatible with those used in the International System of Units, or SI (see 10.51-59). Periods are omitted in any context (compare 10.4). The capitalization given below, based largely on current usage, sometimes departs from that used in Merriam-Webster s Collegiate Dictionary (see bibliog. 3.1). The first letter of abbreviations derived from proper names (e.g., A [ampere], V, Wb, and the C in °C) are usually capitalized (though the spelled-out term is lowercased—unless, like Celsius, it forms a unit name with another term, as in “degree[s] Celsius”), as are the prefix let ters for mega- (M),giga- (G), tera- (T), and so on (see 10.56). Plurals do not add an s (10 A, 5 ha). With few exceptions (mainly abbreviations with degree symbols), a space usually appears between a numeral and an ab breviation (22 m but 36°C); see also 10.58. For units with repeated quan tities, see 9.17. For statistical abbreviations, see 10.50. For traditional US units of measure, see 10.64-68. See also 9.16.
A A ac AF Ah AM ASCII ampere; adenine (in genetic code) angstrom alternating current audio frequency ampere-hour amplitude modulation American Standard Code for Information Interchange Technology and Science 10.49 atm av or avdp bar BD Be or°Be bhp BMI bp bps Bps Bq Btu C °C cal Cal cc cd CD cgs Ci cm cM cm3 cp CP cps or c/s CPU cu d Da dB de DNS DOI DOS dpi DVD dyn emf erg eV F atmosphere, -ic avoirdupois bar (no abbreviation) Blu-ray Disc degree Baume brake horsepower body mass index boiling point; base pair bits per second bytes per second becquerel British thermal unit coulomb; cytosine (in genetic code) degree Celsius calorie kilocalorie (in nonscientific contexts; see also kcal) cubic centimeter (in clinical contexts; see also cm3) candela compact disc centimeter-gram-second system (SI) curie centimeter centimorgan cubic centimeter (in scientific contexts; see also cc) candlepower chemically pure cycles per second central processing unit cubic day; deuteron dalton decibel direct current domain name system Digital Object Identifier (DOI is a registered trademark) disk operating system dots per inch digital versatile (or video) disc dyne electromotive force erg (no abbreviation) electron volt farad 10.49 ABBREVIATIONS °F FM fp fps FTP g G Gb GB Gbps GeV GIF GIS GPS Gy H h ha hp HTML HTTP Hz IP IR IU J JPEG K kat kb kb or kbit KB orK Kbps kc kcal KE kg kHz kJ km kmh orkmph kn kW degree Fahrenheit frequency modulation freezing point frames per second; feet per second file transfer protocol gram; gas guanine (in genetic code) gigabit gigabyte gigabits per second 109 electron volts graphic interchange format geographic information system global positioning system gigayear; gray (joule per kilogram) henry (pl. henries) hour; helion hectare horsepower hypertext markup language hypertext transfer protocol hertz internet protocol infrared international unit joule from Joint Photographic Experts Group (file format) kelvin (no degree symbol); kilobyte (in commercial contexts) katal kilobar (DNA); kilobase (RNA) kilobit kilobyte kilobits per second kilocycle kilocalorie (in scientific contexts; see also Cal) kinetic energy kilogram kilohertz kilojoule kilometer kilometers per hour knot (nautical mph) kilowatt Technology and Science 10.49 kWh kilowatt-hour L liter (capitalized to avoid confusion with numeral 1) lm lumen lx lux m meter M molar; metal Mb megabase; megabit MB megabyte Mbps megabits per second Me megacycle mCi millicurie MeV million electron volts mg milligram MIDI musical instrument digital interface mks meter-kilogram-second system (SI) mL milliliter mol mole mp melting point MPEG from Moving Pictures Experts Group (file format) mpg miles per gallon mph miles per hour MP3 from MPEG-1 Audio Layer 3 (file format) MP4 from MPEG-4 Part 14 (file format) ms millisecond N newton; number (often italic; see also 10.50) neg negative nm nanometer; nautical mile Q ohm OCR optical character recognition OS operating system Pa pascal pc parsec PC personal computer PDF portable document format PE potential energy PF picofarad PH negative log of hydrogen ion concentration (measure of acidity) PNG from portable network graphics (file format) pos positive ppb parts per billion ppm parts per million ppt parts per trillion; precipitate R electrical resistance 10.50 ABBREVIATIONS °R rad RAM RF ROM rpm or r/min s S SGML soln spgr sq sr std STP Sv t T Tb TB Tbps TCP/IP temp U UCS URI URL USB UV V W Wb wt w/v w/w XML y Z degree Reaumur radian random-access memory radio frequency read-only memory revolutions per minute second siemens standard generalized markup language solution specific gravity square steradian standard standard temperature and pressure sievert metric ton (103 kg); triton (nucleus of tritium) tesla; thymine (in genetic code) terabit terabyte terabits per second transmission-control protocol/internet protocol temperature uracil (in genetic code) universal character set uniform resource identifier uniform resource locator universal serial bus ultraviolet volt watt weber weight weight per volume weight per weight extensible markup language year atomic number (often italic) 10.50 Statistical abbreviations. The following abbreviations are used in sta tistical material, especially in tables. They are often italicized. See also 12.57,12.58, and table 12.3.
Technology and Science 10.51 ANCOVA ANOVA CI CL CLT df, DF, or dof GLIM HR IQR LS MLE MS N analysis of covariance analysis of variance confidence interval confidence limit central limit theorem degrees of freedom generalized linear model hazard ratio interquartile range least squares maximum likelihood estimate mean square number (of population or sample) n number (of sample or subsample) ns OLS OR not (statistically) significant ordinary least squares odds ratio P r R R2 RMS sd or SD se or SE sem or SEM SS SSE SST WLLN x orX X2 probability bivariate correlation coefficient multivariate correlation coefficient coefficient of determination root mean square standard deviation standard error standard error of the mean sum of squares error sum of squares total sum of squares weak law of large numbers mean value chi-square distribution The International System of Units 10.51 SI units—overview. The International System of Units (Systeme interna tional d’unites, abbreviated internationally as SI) is an expanded version of the metric system. It is in general use among the world’s scientists and in many other areas. The following paragraphs discuss only the basics. For the latest official guidelines, consult The International System of Units, a brochure published in French and English by the Bureau International des Poids et Mesures and available online. For further guidance, see 10.52 ABBREVIATIONS Ambler Thompson and Barry N. Taylor, Guide for the Use of the Interna tional System of Units (bibliog. 2.4); and Scientific Style and Format (bib liog. 1.1).
10.52 SI units—form. No periods are used after any of the SI symbols for units, and the same symbols are used for both the singular and the plural. Most symbols are lowercased; exceptions are those that stand for units derived from proper names (e.g., A, for ampere) and those that must be distin guished from similar lowercased forms. All units are lowercased in their spelled-out form except for terms like Celsius (which follows the word “degree” in its unit name; see also 10.49). See also 10.53.
10.53 Plurals for SI units. Though abbreviations for SI units are the same for plu ral and singular forms, the noun forms for such units would generally be written out or pronounced in the plural (e.g., 3 m=three meters; but a three- meter span). The only exception is for a quantity of exactly 1; for quanti ties such as 0.5 m or 1.6 m, the unit would generally be read as if it were plural (zero point five meters; one point six meters). See also 9.19,10.65.
10.54 SI base units. There are seven fundamental, or base, SI units. Note that although weight and mass are usually measured in the same units, they are not interchangeable. Weight is a force due to gravity that depends on an object’s mass. Note also that no degree sign is used with the symbol K. See also 10.55.
luminous intensity Quantity Unit Symbol length meter m mass kilogram kg time second s electric current ampere A thermodynamic temperature kelvin K amount of substance mole mol candela cd Not to be confused with the symbols for base units are the correspond ing symbols for base quantities. These symbols, which represent variable quantities, appear in italic type (e.g., /, length; m, mass; t, time).
10.55 Kilogram versus gram as SI base unit. Although for historical reasons the kilogram rather than the gram was chosen as the base unit, prefixes are applied to the term gram—megagram (Mg), milligram (mg), nano gram (ng), and so forth. See also 10.56.
Technology and Science 10.57 10.56 SI prefixes. Prefixes, representing a power of ten, are added to the name of a base unit, a derived unit, or an accepted non-SI unit (see 10.57, 10.59) to allow notation of very large or very small numerical values. The units so formed are called multiples and submultiples of SI units. For ex ample, a kilometer, or km, is equal to a thousand meters (or 103 m), and a millisecond, or ms, is equal to one-thousandth of a second (or 10-3 s). The following prefixes, with their symbols, are used in the international system. Note that in three cases the final vowel of an SI prefix is omitted: kfl, kilohm (not kiloohm); MQ, megohm (not megaohm); ha, hectare (not hectoare).
Factor Prefix Symbol Factor Prefix Symbol 1024 yotta Y IO’1 deci d 1021 zetta Z IO"2 centi c 1018 exa E IO’3 milli m 1015 peta P IO"6 micro P 1012 tera T IO’9 nano n 109 giga G 10“12 pico P IO6 mega M 10~15 femto f 103 kilo k IO’18 atto a 102 hecto h IO’21 zepto z IO1 deka da 10-24 yocto y These prefixes should not be used to indicate powers of two (as in the field of electrical technology, or computing). If binary multiples must be used, the first two letters of the SI prefixes must be followed by bi, to form kibi- (Ki), mebi- (Mi), gibi- (Gi), tebi- (Tifpebi- (Pi), and exbi- (Ei). See also 9.11.
10.57 Units derived from SI base units. Derived units are expressed algebra ically in terms of base units or other derived units.
meter per second m/s meter per second squared m/s2 kilogram per cubic meter kg/m3 Derived unit In terms of SI base units square meter cubic meter m2 m3 Certain derived units have special names and symbols. Several of the most common—hertz (Hz), volt (V), watt (W), and so forth—are listed in 10.58 ABBREVIATIONS 10.49. These are used in algebraic expressions to denote further derived units. A few are listed below.
Derived unit joule per kelvin newton meter newton per meter Symbol J/K N m or N • m N/m In terms of SI base units m2 kg s-2 K"1 m2 kg s-2 kg s"2 A derived unit can often be expressed in different ways. For example, the weber may be expressed either as Wb or, in another context, in terms of the volt second (V s or V • s).
10.58 SI units and abbreviations—spacing. Only numbers between 0.1 and 1,000 should be used to express the quantity of any SI unit. Thus 12,000 meters is expressed as 12 km (not 12 000 m), and 0.003 cubic cen timeters as 3 mm3 (not 0.003 cm3). (For the use of spaces rather than commas between groups of digits in SI units, see 9.56.) In SI usage as in general usage, a space usually appears between the numeral and any abbreviation or symbol. Contrary to general usage, however, SI usage also stipulates a space before a percentage sign (%) or before a degree symbol used for temperature (compare the advice in the introduction to the table at 10.49). In expressions of degrees, minutes, and seconds, SI usage shows (but does not stipulate) a space between quantities. Many publications do not observe these exceptions, and Chicago does not re quire them in its publications.
SI style 22 °C 22°14' 33" 0.5% Chicago style 22°C 22°14'33" 0.5% (see also 9.18) 10.59 Non-SI units accepted for use. Certain widely used units such as liter (L, capitalized to avoid confusion with the numeral 1), metric ton (t), and hour (h) are not officially part of the international system but are accepted for use within the system.
Astronomy 10.60 Astronomical abbreviations—additional resources. Astronomers and astrophysicists employ the International System of Units (see 10.51-59) supplemented with special terminology and abbreviations. The para graphs in this section offer a minimum of examples for the generalist.
Technology and Science 10.63 Additional guidelines may be found at the website of the International Astronomical Union.
10.61 Celestial coordinates. Right ascension, abbreviated RA or a, is given in hours, minutes, and seconds (abbreviations set as superscripts) of sidereal time. Declination, abbreviated 6, is given in degrees, minutes, and seconds (using the degree symbol, prime, and double prime) of arc north (marked + or left unmarked) or south (marked -) of the celestial equator. Note the abbreviations (set as superscripts) and symbols used.
14h6m7s -49°8'22" Decimal fractions of the basic units are indicated as shown.
14h6m7s.2 +34°.26 10.62 Some other astronomical abbreviations. A few of the more commonly used astronomical abbreviations are listed here. A more extensive list is available in Scientific Style and Format (bibliog. 1).
AU or ua lt-yr Pc kpc Mpc UT astronomical unit (mean earth-sun distance) light-year (9.46 x 1012 km) parsec (parallax second: 3.084 x 1013 km) 103pc 106pc universal time (see also 10.41) Chemical Elements 10.63 Naming conventions for chemical elements. The International Union of Pure and Applied Chemistry (IUPAC) is the recognized body that for mally approves element names. Each element bears a number (reflecting the number of protons in its nucleus) as well as a name—as in “element 106,” also known as seaborgium. This number is an important identifier in cases where formal names are in dispute; between 1995 and 1997, for example, the American Chemical Society and IUPAC adopted different names for some of the same elements. The differences were reconciled, and the list that follows reflects names and symbols approved by IUPAC. Names for undiscovered or unconfirmed elements are provisionally as signed using Latin for the digits of their atomic number (e.g., ununoctium, one-one-eight, for element 118, which was confirmed in 2015 and named oganesson the following year). The elements in the following list are 10.63 ABBREVIATIONS arranged in alphabetical order by common name. If the symbol is based on a term other than the common name—for example, Sb (stibium) for antimony—the term is added in parentheses. Although the names of ele ments are always lowercased, the symbols all have an initial capital. No pe riods are used. In specialized works, the abbreviations commonly appear in text as well as in tables, notes, and so forth. See also 8.149, 8.150.
89 Ac actinium 114 Fl flerovium 13 Al aluminum (US), 9 F fluorine aluminium (IUPAC) 87 Fr francium 95 Am americium 64 Gd gadolinium 51 Sb antimony (stibium) 31 Ga gallium 18 Ar argon 32 Ge germanium 33 As arsenic 79 Au gold (aurum) 85 At astatine 72 Hf hafnium 56 Ba barium 108 Hs hassium 97 Bk berkelium 2 He helium 4 Be beryllium 67 Ho holmium 83 Bi bismuth 1 H hydrogen 107 Bh bohrium 49 In indium 5 B boron 53 I iodine 35 Br bromine 77 Ir iridium 48 Cd cadmium 26 Fe iron (ferrum) 20 Ca calcium 36 Kr krypton 98 Cf californium 57 La lanthanum 6 C carbon 103 Lr lawrencium 58 Ce cerium 82 Pb lead (plumbum) 55 Cs cesium (US), 3 Li lithium caesium (IUPAC) 116 Lv livermorium 17 Cl chlorine 71 Lu lutetium 24 Cr chromium 12 Mg magnesium 27 Co cobalt 25 Mn manganese 112 Cn copernicium 109 Mt meitnerium 29 Cu copper 101 Md mendelevium 96 Cm curium 80 Hg mercury (hydrargyrum) 110 Ds darmstadtium 42 Mo molybdenum 105 Db dubnium 115 Me moscovium 66 Dy dysprosium 60 Nd neodymium 99 Es einsteinium 10 Ne neon 68 Er erbium 93 Np neptunium 63 Eu europium 28 Ni nickel 100 Fm fermium 113 Nh nihonium Technology and Science 10.65 41 Nb niobium 106 Sg seaborgium 7 N nitrogen 34 Se selenium 102 No nobelium 14 Si silicon 118 Og oganesson 47 Ag silver (argentum) 76 Os osmium 11 Na sodium (natrium) 8 O oxygen 38 Sr strontium 46 Pd palladium 16 S sulfur 15 P phosphorus 73 Ta tantalum 78 Pt platinum 43 Tc technetium 94 Pu plutonium 52 Te tellurium 84 Po polonium 117 Ts tennessine 19 K potassium (kalium) 65 Tb terbium 59 Pr praseodymium 81 T1 thallium 61 Pm promethium 90 Th thorium 91 Pa protactinium 69 Tm thulium 88 Ra radium 50 Sn tin (stannum) 86 Rn radon 22 Ti titanium 75 Re rhenium 74 W tungsten (wolfram) 45 Rh rhodium 92 U uranium 111 Rg roentgenium 23 V vanadium 37 Rb rubidium 54 Xe xenon 44 Ru ruthenium 70 Yb ytterbium 104 Rf rutherfordium 39 Y yttrium 62 Sm samarium 30 Zn zinc 21 Sc scandium 40 Zr zirconium US Measure 10.64 Periods with abbreviations of US measure. In the rare instances in which abbreviations for US units of measure are used in scientific copy, they are usually set without periods; in nonscientific contexts, periods are customary. See also 10.4.
10.65 Plural forms for abbreviations of US measure. Abbreviations of US units of measure, like their scientific counterparts, are identical in the singular and the plural (but see 10.68).
10 yd. 5 lb. 8 sq. mi.
Note that the unit of measure in such expressions as 0.5 yd. and 1.5 yd. is generally pronounced as if it were plural (i.e., point five yards; one point 10.66 ABBREVIATIONS five yards); the singular is reserved for measures of exactly one. See also 10.53.
10.66 US abbreviations for length, area, and volume. In the following ex amples, note that the proper symbols for foot and inch are prime (') and double prime ("), not the single (’) and double (”) quotation marks: Length Area Volume in. or " inch sq. in.
square inch cu. in.
cubic inch ft. or ’ foot sq. ft.
square foot cu. ft.
cubic foot yd.
yard sq. yd.
square yard cu. yd.
cubic yard rd.
rod sq. rd.
square rod mi.
mile sq. mi.
square mile As in expressions of latitude and longitude (see 10.35), there is no space in such expressions as the following (for a height or length of 6ft., 1 in.): 6'1" Exponents are sometimes used with abbreviations to designate area or volume, but only when no ambiguity can occur.
425 ft.2 (= 425 sq. ft. not 425 ft. by 425 ft.) 638 ft.3 (= 638 cu. ft.) 10.67 US abbreviations for weight and capacity. The US system comprises three systems of weight and mass: avoirdupois (the common system), troy (used mainly by jewelers), and apothecaries’ measure. Although confusion is unlikely, an abbreviation can, if necessary, be referred to the appropriate system thus: lb. av., lb. t., lb. ap. Also, the systems of capacity measure used in the United States and the British Commonwealth differ (an American pint being more than three ounces smaller than a British pint, for example), but the same abbreviations are used.
Weight or mass Dry measure gr-grain pt.
pint s.
scruple qt.
quart dr.
dram pk.
peck dwt.
pennyweight bu.
bushel oz.
ounce lb. or # pound cwt.
hundredweight tn.
ton Business and Commerce 10.69 Liquid measure minim fluid dram fluid ounce min. or Ity fl. Dr. or f. 3 fl. oz. or f. 5 gi-gill Pt-pint qt.
quart gal.
gallon bbl.
barrel As with length and so forth, abbreviations do not change in the plural.
12 gal. 3 pt.
10.68 US and general abbreviations for time. The following abbreviations, though not limited to the US system of measure, are used mainly in non technical contexts: sec. second h. or hr. hour mo. month min. minute d. or day day yr. year In nontechnical writing, the plurals of these abbreviations, unlike those of length, area, weight, and the like, are often formed by adding an s.
5 secs. 12 hrs. or 12 h. 15 yrs.
Business and Commerce 10.69 Commercial abbreviations—some examples. As for many other abbre viations in nonscientific contexts, periods for abbreviations of commer cial terms are normally used in lowercased forms (see 10.4). See also 10.42-43,10.64-68. For company names, see 10.23.
acct.
account, -ant agt.
agent a.k.a.
also known as amt.
amount AP amounts payable APR annual percentage rate AR amounts receivable ASAP as soon as possible 10.69 ABBREVIATIONS att.
attached, -ment attn.
attention a.v. or AV ad valorem bal.
balance bbl.
barrel (s) bcc blind carbon copy or blind copy, -ies bdl. or bdle.
bundle bl.
bale(s) BS bill of sale bu.
bushel(s) c. or ct.
cent cc carbon copy or copy, -ies c.l. or CL carload c/o in care of COD cash on delivery COGS cost of goods sold COLA cost-of-living adjustment CPI consumer price index CPM cost per thousand (mille) cr.
credit, -or ctn.
carton cttee. or comm.
committee d/b/a doing business as dis.
discount dist.
district distr.
distributor, -ion DJIA Dow Jones Industrial Average doz.
dozen dr.
debtor dstn.
destination ea.
each EEO equal employment opportunity EOE equal opportunity employer EOM end of month exec.
executive f.a.s. or FAS free alongside ship f.o.b. or FOB free on board FY fiscal year GAAP generally accepted accounting principles GL general ledger GM general manager; genetically modified gro.
gross inst.
instant (this month) Business and Commerce 10.69 inv.
invoice IPO initial public offering JIT just in time LBO leveraged buyout LCL less-than-carload lot LIFO last in, first out M and A or M&A mergers and acquisitions mdse.
merchandise mfg-manufacturing mfr.
manufacturer mgmt.
management mgr.
manager MO mail order; money order msg.
message mtg.
meeting mtge.
mortgage NA or n/a not applicable; not available NGO nongovernmental organization nt. wt.
net weight OJT on-the-job training OS operating system; out of stock OTC over the counter P and H or P&H postage and handling pd.
paid Pkg-package POE port of embarkation; port of entry POP point of purchase POS point of sale; point of service PP parcel post ppd.
postpaid; prepaid pr.
pair QA quality assurance Q&A question and answer QC quality control qtr.
quarter qty-quantity ® registered trademark (see 8.153) recd. or rec’d received S and H or S&H shipping and handling SM unregistered service mark std.
standard TBA to be announced TBD to be determined 10.69 ABBREVIATIONS TM unregistered trademark (see 8.153) treas. ult. VAT whsle.
treasurer, -y ultimo (last month) value-added tax wholesale 11 • Languages Other than English OVERVIEW 11.1 GENERAL PRINCIPLES 11.3 Words and Phrases from Other Languages 11.3 Titles of Works from Other Languages 11.6 Quotations from Other Languages 11.11 LANGUAGES USING THE LATIN ALPHABET 11.18 African Languages 11.23 French 11.25 German 11.38 Italian 11.46 Latin 11.54 Spanish 11.60 Other Languages Using the Latin Alphabet 11.70 LANGUAGES USUALLY TRANSLITERATED (OR ROMANIZED) 11.71 Arabic 11.76 Chinese and Japanese 11.82 Hebrew 11.91 Russian 11.98 South Asian Languages 11.109 CLASSICAL GREEK 11.110 Breathings and Accents 11.112 Punctuation and Numbers 11.116 Word Division 11.118 OLD ENGLISH AND MIDDLE ENGLISH 11.122 AMERICAN SIGN LANGUAGE (ASL) 11.125 11.1 LANGUAGES OTHER THAN ENGLISH Overview 11.1 Scope and organization. This chapter provides guidelines for present ing text from languages other than English in English-language con texts. These guidelines are general: authors or editors working with languages in which they are not expert should seek additional guidance from someone who is. More than two dozen languages are covered, with those languages that commonly appear and those that present complex problems being considered most fully. The chapter begins with the treat ment of words and phrases, titles of works, and quotations, the principles of which apply to most of the languages discussed (see 11.3-17). It then addresses languages using the Latin alphabet, transliterated (or roman- ized) languages, classical Greek, Old English and Middle English, and American Sign Language. Individual languages or groups of languages are presented in alphabetical order within their particular sections. (For the treatment of personal names, see 8.7-18.) 11.2 Unicode. Many of the letters and symbols required by the world’s lan guages are included in a widely used standard for character encoding called Unicode. The Unicode standard (published by the Unicode Con sortium; bibliog. 2.7) is widely supported by modern operating systems and browsers and many other applications (including word processors) and is required by such standards as XML and EPUB. Unicode assigns a unique identifying hexadecimal number (or code point) and description to tens of thousands of characters. Even fonts with Unicode character mapping, however, typically support only a subset of the Unicode char acter set. For this reason, it is desirable to determine at the outset which characters will be needed for a publication. Table 11.1 lists special char acters, with Unicode numbers and abbreviated descriptions, needed for each of the languages treated in this chapter that use the Latin alphabet. Table 11.2 lists special characters that may be needed for certain translit erated languages. For Russian (Cyrillic) and Greek characters, see tables 11.3,11.4, and 11.5. Unicode numbers mentioned in text should be pre fixed by U+ (e.g., U+OOEO for d).
General Principles Words and Phrases from Other Languages 11.3 Non-English words and phrases in an English context. Italics are used for isolated words and phrases from another language, especially if General Principles 11.4 they are not listed in a standard English-language dictionary like Mer riam-Webster's Collegiate (see 7.1) or are likely to be unfamiliar to read ers (see also 7.54). (For proper nouns, see 11.4.) If such a word or phrase becomes familiar through repeated use throughout a work, it need be italicized only on its first occurrence. If it appears only rarely, however, italics may be retained.
The greve du zele is not a true strike but a nitpicking obeying of work rules.
She preferred to think of it optimistically as a suefio reparador—rather than, as in English, a sleep that was merely restful.
Unless the term appears in a standard English-language dictionary and is being used as such, observe the capitalization conventions of the orig inal language. In the following examples, the German word for com puter (which is the same as the English word) is capitalized because it is a noun, and the French adjective franpaise is lowercase even though it would be capitalized in English (as “French”). See also 11.18.
The German word for computer is Computer. The French word is ordinateur. In Spanish, the word is either computadora or ordenador, depending on region or context.
We were prepared to learn the nuances of la langue franfaise.
The plurals of non-English words should be formed as in the original lan guage (see also 7.12).
We were sent off with some beautiful Blumen (not Blumes [italic ess] and not Blumes [roman ess]).
An entire sentence or a passage of two or more sentences in another lan guage is usually set in roman and, unless it is set as a block quotation or extract (see 13.9-29), enclosed in quotation marks (see 11.11).
11.4 Non-English proper nouns in an English context. With the exception of titles of books and the like, proper nouns from other languages are gen erally not italicized, even on first mention (cf. 11.3). This usage extends to named places and structures, institutions and companies, brand names, and other categories as discussed in chapter 8. (For titles of works, see 11.6-10.) Capitalization should follow predominant usage in the original language. In some cases, this may entail observing a preference for cap italization that runs counter to the conventions for generic text. If the editor is unfamiliar with the language, an expert, or the author, should be consulted; when in doubt, opt for sentence-style capitalization (see 11.5 LANGUAGES OTHER THAN ENGLISH 8.158). See also 11.18. An initial the may be used if the definite article would appear in the original language.
She won the Premio Nadal for her second novel, Viento del norte.
Mexico City’s Angel de la Independencia is known familiarly as “El Angel.” The Real Academia Espanola was founded in 1713.
A history of the Comedie-Fran^aise has just appeared.
The Academie fran^aise dates to the reign of Louis XIII.
I prefer the Bibliotheque nationale by day and the Bois de Boulogne by night.
He is a member of the Societe d’entraide des membres de l’ordre national de la Legion d’honneur.
Leghorn—in Italian, Livorno—is a port in Tuscany.
When he asked her to meet him along Unter den Linden, she was amused by the consecutive prepositions—one in English and one in German, just like them.
Translations of proper nouns from other languages should be capitalized headline-style (see 8.159).
He is a member of the Mutual Aid Society for Members of the National Order of the Legion of Honor.
Original (or transliterated) names of proper nouns presented as glosses should not be italicized (but see 11.5).
The number of cases adjudicated by the Supreme People’s Court of the People’s Republic of China (Zhonghua renmin gongheguo zuigao renmin fayuan) has increased sharply.
11.5 Translations of terms from other languages. A translation following a word, phrase, or title from another language is enclosed in parentheses or quotation marks. See also 6.96,11.3,11.4,11.9,14.99.
The word she wanted was pecker (to sin), not pecker (to fish).
The Prakrit word majjao, “the tomcat,” may be a dialect version of either of two Sanskrit words: madjaro, “my lover,” or marjaro, “the cat” (from the verb mrij, “to wash,” because the cat constantly washes itself).
A group of German expressionists known as Die Briicke (The Bridge) were influ ential in the decade leading up to the First World War.
Leonardo Fioravanti’s Compendio de i secreti rationali (Compendium of rational secrets) became a best seller.
If a non-English word other than a proper noun is presented as a paren thetical gloss, it should be presented in italics as in running text (but see 11.4).
General Principles 11.9 He said that to fish (pecker) was to sin (pecker).
For quotations from other languages, see 11.11-17.
Titles of Works from Other Languages 11.6 Capitalization of titles from other languages. For titles of works from other languages, whether these appear in text, notes, or bibliographies, Chicago recommends a simple rule: capitalize only the words that would be capitalized in normal prose—the first word of the title and subtitle and all proper nouns or any term that would be capitalized under the conven tions of the original language. That is, use sentence style (see 8.158). This rule applies equally to titles using the Latin alphabet and to transliterated titles. For examples, see 14.98. For special considerations related to Ger man capitalization, see 11.39. For variations in French, see 11.27.
11.7 Punctuation of titles from other languages. When a non-English title is included in an English-language context, the following changes are per missible: a period (or, more rarely, a semicolon) between title and subti tle may be changed to a colon (and the first word of the subtitle may be capitalized); guillemets (« ») or other non-English styles for quotation marks may be changed to regular quotation marks (“ ” or ‘ ’); and any space between a word and a mark of punctuation that follows may be eliminated. Commas should not be inserted (even in a series or before dates) or deleted, nor should any other mark of punctuation be added or deleted. See also 8.165.
11.8 Italic versus roman type for titles from other languages. Titles of works in languages that use the Latin alphabet (including transliterated titles) are set in italic or roman type according to the principles set forth in 8.156-201—for example, books and periodicals in italic; poems and other short works in roman.
Stendhal’s Le rouge et le noir was required reading in my senior year. We picked up a copy of the Neue Zurcher Zeitung to read on the train. She published her article in the Annales de demographie historique. Strains of the German carol “Es ist ein’ Ros’ entsprungen” reached our ears. Miguel Hernandez’s poem “Casida del sediento” has been translated as “Lament of the Thirsting Man.” 11.9 Non-English titles with English translation. When the title of a work in another language is mentioned in text, an English gloss may follow 11.10 LANGUAGES OTHER THAN ENGLISH in parentheses (see 6.96). If the translation has not been published, the English should be capitalized sentence-style (as in the first example be low; see 8.158) and should appear neither in italics nor within quotation marks. A published translation, however, is capitalized headline-style (as in the second and third examples; see 8.159) and appears in italics or quotation marks depending on the type of work (see 8.156-201). Some editorial discretion may be required, especially if the translation is in corporated into running text (as in the third example). For translations of non-English titles in notes and bibliographies, see 14.99. See also 11.10.
Leonardo Fioravanti’s Compendio de i secreti rationali (Compendium of rational secrets) became a best seller.
Proust’s A la recherche du temps perdu (Remembrance of Things Past) was the subject of her dissertation.
but La ciudad y losperros, which literally means “the city and the dogs,” was published in English under the title The Time of the Hero.
11.10 Original-language title of work versus translation. Readership and con text will determine whether to use the original or the translated title of a non-English work mentioned in running text. In a general work, titles that are widely known in their English translation could be cited in En glish first, with the original following in parentheses; in some cases, the original can be omitted entirely. Some authors prefer to cite all non English titles in an English form, whether or not they have appeared in English translation. As long as the documentation clarifies what has been published in English and what has not, translated titles standing in for the original may be capitalized headline-style and treated like other English-language titles (see 8.159, 8.163). See also 11.9.
“The West” in the title of the Chinese classic Journey to the West (Xiyou ji) refers mainly to the Indian subcontinent.
Moliere’s comedy The Miser may have drawn on an obscure late-medieval French treatise, The Evils of Greed, recently discovered in an abandoned chateau.
Quotations from Other Languages 11.11 Typographic style of quotations from other languages. Quotations from a language other than English that are incorporated into an English text are normally treated like quotations in English, set in roman type and run in or set off as block quotations according to their length. (For a complete discussion of quotations, see chapter 13.) They are punctu- General Principles 11.13 ated as in the original except that quotation marks can usually replace guillemets (or their equivalents), and punctuation relative to quotation marks and spacing relative to punctuation are adjusted to conform to the surrounding text (see 11.19). For isolated words and phrases, see 11.3. For excerpts from the original language following an English translation, see 11.12.
The narrator’s “treinta o cuarenta molinos de viento” become Quixote’s “treinta, o pocos mas, desaforados gigantes,” a numerical correspondence that lets the reader trust, at the very least, the hero’s basic grasp of reality.
If em dashes rather than quotation marks are used for dialogue in the original (see 11.31,11.47,11.64,11.101), they should be retained in a block quotation but may be replaced by quotation marks if only a phrase or sentence is quoted in running text.
11.12 Translations relative to quotations. A translation may follow the origi nal in parentheses—or, as in 11.13, the original may follow a translation. Quotation marks need not be repeated for the parenthetical translation (or parenthetical original, as the case may be); any internal quotation marks, however, should be included (as in the second example). See also 6.96,11.5. If a long sentence or more than one sentence appears in paren theses or brackets, as in the second example, closing punctuation of the original and the translation should remain distinct.
A line from Goethe, “ Wer nie sein Brot mit Tranen aE” (Who never ate his bread with tears), comes to mind.
A vrai dire, Abelard n’avoue pas un tel rationalisme: “je ne veux pas etre si philo sophe, ecrit-il, que je resiste a Paul, ni si aristotelicien que je me separe du Christ.” (As a matter of fact, Abelard admits no such rationalism. “I do not wish to be so much of a philosopher,” he writes, “that I resist Paul, nor so much of an Aristote lian that I separate myself from Christ.”) Whether to provide translations of quoted passages depends on the lin guistic abilities of the intended audience. For example, in a work to be read by classicists, Latin or Greek sources may be quoted freely in the original. Or in a literary study of, say, Goethe, quotations from Goethe’s work may be given in the original German only. For a wider readership, translations should be furnished.
11.13 Source of quotation plus translation. When both a source and a transla tion are required in text, the source may be placed in parentheses, with 11.14 LANGUAGES OTHER THAN ENGLISH the original (or translation, as the case may be) following, separated by a semicolon. The following example quotes a thirteenth-century author writing in Middle Dutch. See also 13.68-69.
Hadewijch insists that the most perfect faith is “unfaith,” which endlessly stokes desire and endlessly demands love from God. “Unfaith never allows desire to rest in any faith but always distrusts her, [feeling] that she is not loved enough” (letter 8:39; Ende ontrowe en laet gegherten niewers ghedueren in gheenre trowen, sine mestrout hare altoes, datse niet ghenoech ghemint en es).
If adding a translation or the original in text creates too much clutter, it may be placed in a note, in which case it is enclosed in quotation marks but not in parentheses or brackets. If the parenthetical passage in the sec ond example in 11.12 were to appear in text without the French, as either a run-in or a block quotation, a note could read as follows: 1. “A vrai dire, Abelard n’avoue pas un tel rationalisme: ‘je ne veux pas etre si philosophe, ecrit-il, que je resiste a Paul, ni si aristotelicien que je me separe du Christ.’” See also 13.30.
11.14 Crediting the translation of a quoted passage. When quoting a passage from a language that requires a translation, authors should use a pub lished English translation if one is available and give credit to the source of that translation, including the title of the translation, the translator’s name, relevant bibliographic details, and page number (see 14.99). Au thors providing their own translations should so state, in parentheses following the translation, in a note, or in the prefatory material—for example, “my translation” or “Unless otherwise noted, all translations are my own.” If an individual other than the author provided the trans lations, that person should be credited in a similar manner, but by name. See also 11.9.
11.15 Adjusting translated quotations. An author using a published transla tion may occasionally need to adjust a word or two; “translation mod ified” or some such wording must then be added in parentheses or in a note (see also 13.62). In addition, it is recommended that such modifica tions be indicated by square brackets (see 13.59,13.60). These devices should be used sparingly. If a published translation is unsuitable for the author’s purpose, it should be abandoned and all quoted passages newly translated.
Languages Using the Latin Alphabet 11.19 11.16 Editing translated quotations. Quotations from published translations can be modified only with respect to the permissible changes described in 13.7. In new translations furnished by the author, however, capitaliza tion, punctuation, spelling, and idiom may be adjusted for consistency with the surrounding text.
11.17 The sin of retranslation. Never should a passage from a work originally published in English (or any other language, for that matter) be retrans lated from a version that has been translated into another language. For example, an author quoting from a German study of Blackstone’s Com mentaries that quotes from Blackstone in German must track down the original Blackstone passages in English and reproduce them. If unable to locate the original, the author must resort to paraphrase.
Languages Using the Latin Alphabet 11.18 Capitalization—English versus other languages. Capitalization is ap plied to more classes of words in English than in any other Western lan guage (but see 8.1). Most of the other languages discussed in this chapter follow a simpler set of rules. Except where stated to the contrary, the lan guage in question is assumed to lowercase all adjectives (except those used as proper nouns), all pronouns, months, and days of the week. In addition, capitals are used more sparingly than in English for names of offices, institutions, and so on. Translated terms, however, are subject to Chicago’s recommendations for capitalization of names and terms (see chapter 8). For personal names, see 8.7-18.
11.19 Punctuation—original language versus English context. The remarks in this chapter related to punctuation point out the more obvious depar tures from what is familiar to readers of English. For the purposes of illus tration, quotation marks in the style of the original language have been preserved in the examples; however, spacing relative to these and other punctuation marks has been adjusted to conform to the typographic style of this manual. In quotations from other languages (and in translations), regular English-style quotation marks can usually replace the guillemets or whatever is used in the original (with the placement of periods and commas adjusted as needed; see 6.9-11). Dashes used to mark dialogue, however, should be preserved in block quotations presented in the orig inal language. See 11.11. Another exception is the punctuation at the be ginning of Spanish questions and exclamations (see 11.62), which should 11.20 LANGUAGES OTHER THAN ENGLISH be preserved for quotations in Spanish (but omitted when the passage is translated).
11.20 Word division for languages other than English. Though conventions for dividing words at the ends of lines vary widely, the following general rules apply to non-English languages as well as to English: (1) Single syllable words should never be broken. (2) No words should be broken after one letter, nor should a single letter be carried over to another line (see also 7.37). (3) Hyphenated words and solid compounds should be broken at the hyphen or between elements, if at all possible. See also 7.40; for proper nouns, see 7.42. Specific rules for some of the languages covered in this chapter appear in the relevant sections below.
11.21 Special characters in the Latin alphabet. Words, phrases, or titles from another language that occur in an English-language work must include any special characters that appear in the original language. Those lan guages that use the Latin alphabet may include letters with accents and other diacritical marks, ligatures, and, in some cases, alphabetical forms that do not normally occur in English. Table 11.1 lists the special char acters that might be required for each language treated in this section. Most authors will have access to Unicode-compliant software (see 11.2) and will therefore be able to reproduce each of these characters without the addition of any specialized fonts. Authors should nonetheless supply a list of special characters used within a manuscript (see 2.16) to ensure the correct conversion to a particular font required for publication or, for electronic projects, to ensure compatibility across systems that may not support Unicode. If type is to be reproduced from an author’s hard copy, marginal clarifications may be needed for handwritten accents or special characters. In either case, use table 11.1 to correctly identify the character by name and Unicode number (e.g., for D or d, indicate “D with stroke [U+0110]” or “d with stroke [U+0111]”). For diacritical marks used in transliteration, see 11.74.
11.22 International Phonetic Alphabet (IPA). Phonetic symbols using IPA no tation are based on the Latin alphabet and are defined for Unicode (see 11.2). For the latest version of the IPA alphabet, consult the website of the International Phonetic Association. For additional information on the subject of phonetics, including treatment of other systems of notation, consult Geoffrey K. Pullum and William A. Ladusaw, Phonetic Symbol Guide (bibliog. 5).
t a b l e 11.1. Special characters (and Unicode numbers) for languages using the Latin alphabet Character (and Unicode number) Description Languages that use it « (201E), (OOAB), (2018) » (201C) (OOBB) double low-9 quotation mark, left double quotation mark double angle quotation marks (guillemets) ‘okina (represented by left single quotation mark) German French, German (reversed), Italian, Spanish Hawaiian A (OOCO), a (OOEO) A/a with grave French, Italian, Portuguese A (00C1), a (00E1) A/a with acute Czech, Hungarian, Icelandic, Portuguese, Spanish A (00C2), a (00E2) A/a with circumflex French, Moldovan, Portuguese, Roma nian, Turkish A (00C3), a (00E3) A/a with tilde Portuguese A (00C4), a (00E4) A/a with diaeresis Finnish, German, Swedish, Turkmen A (00C5), a (00E5) A/a with ring above Finnish, Danish, Norwegian, Swedish A (0100), a (0101) A/a with macron Hawaiian, Latin A (0102), a (0103) A/a with breve Latin, Moldovan, Romanian A (0104), 4 (0105) A/a with ogonek Polish JE (00C6), ae (00E6) ligature 2E/ae Danish, Icelandic, Norwegian, Old En glish and Middle English B (0181), 6 (0253) B/b with hook Hausa Q (00C7), 9 (00E7) C/c with cedilla Albanian, Azeri, French, Portuguese, Turkish, Turkmen c (0106), c (0107) C/c with acute Bosnian, Croatian, Montenegrin, Polish, Serbian c (010C), c (010D) C/c with caron (hacek) Bosnian, Croatian, Czech, Montenegrin, Serbian D (OODO), d (OOFO) eth Old English and Middle English, Ice landic b (010E), d (010F) D/d with caron (hacek) Czech D (0110), d (0111) D/d with stroke Bosnian, Croatian, Montenegrin, Serbian D (018A), cf (0257) D/d with hook Hausa E (00C8), e (00E8) E/e with grave French, Italian, Portuguese E (00C9), e (00E9) E/e with acute Czech, French, Hungarian, Icelandic, Italian, Portuguese, Spanish E (OOCA), e (OOEA) E/e with circumflex French, Portuguese E (OOCB), e (OOEB) E/e with diaeresis Albanian, French E (0112), e (0113) E/e with macron Hawaiian, Latin E (0114), e (0115) E/e with breve Latin ?
(0118), ?
(0119) E/e with ogonek Polish E (011A), e (011B) E/e with caron (hacek) Czech 3 (021C), 3 (021D) yogh Old English and Middle English a (018F), 9 (0259) schwa Azeri G (011E), g (01 IF) G/g with breve Azeri, Turkish i (OOCC), i (OOEC) I/i with grave Italian, Portuguese (continued) t a b l e li.i. (continued) Character (and Unicode number) Description Languages that use it i (OOCD), 1' (OOED) I/i with acute !
(OOCE), i (OOEE) I/i with circumflex i (OOCF), i (OOEF) I/i with diaeresis I (012A), i (012B) I/i with macron I (012C), i (012D) I/i with breve i (0130) I with dot above i (0131) dotless i K (0198), k (0199) K/k with hook E (0141), 1 (0142) L/l with stroke N (00D1), n (00F1) N/n with tilde N (0143), ri (0144) N/n with acute N (0147), h (0148) N/n with caron (hacek) O (00D2), 0 (00F2) O/o with grave 6 (00D3), 6 (00F3) O/o with acute o (00D4), 6 (00F4) O/o with circumflex o (00D5), 6 (00F5) O/o with tilde d (00D6), 0 (00F6) O/o with diaeresis 0 (00D8), 0 (00F8) O/o with stroke d (0140, 0 (014D) O/o with macron 0 (014E), 6 (014F) O/o with breve 6 (0150), 0 (0151) O/o with double acute CE (0152), oe (0153) ligature CE/oe R (0158), r (0159) R/r with caron (hacek) S (015A), s (015B) S/s with acute § (015E), ?
(015F) S/s with cedilla ?
(0218), S (0219) S/s with comma below s (0160), s (0161) S/s with caron (hacek) E (OODF) sharp S (eszett) T (021A), t (02 IB) T/t with comma below T (0164), t' (0165) T/t with caron (hacek) I> (OODE), i> (OOFE) thorn U (00D9), u (00F9) U/u with grave U (OODA), u (OOFA) U/u with acute u (OODB), u (OOFB) U/u with circumflex u (00DC), ii (OOFC) U/u with diaeresis u (016E), u (016F) U/u with ring above u (016A), u (016B) U/u with macron u (0160, u (016D) U/u with breve u (0170), u (0171) U/u with double acute Y (00DD), y (OOFD) Y/y with acute Czech, Hungarian, Icelandic, Portuguese, Spanish French, Moldovan, Romanian, Turkish French, Portuguese Hawaiian, Latin Latin Azeri, Turkish Azeri, Turkish Hausa Polish Spanish Polish Czech, Turkmen Italian, Portuguese Czech, Hungarian, Icelandic, Polish, Portuguese, Spanish French, Portuguese Portuguese Azeri, Finnish, German, Hungarian, Ice landic, Swedish, Turkish, Turkmen Danish, Norwegian Hawaiian, Latin Latin Hungarian French Czech Polish, Montenegrin Azeri, Turkish, Turkmen Moldovan, Romanian Bosnian, Croatian, Czech, Montenegrin, Serbian German Moldovan, Romanian Czech Old English and Middle English, Ice landic French, Italian, Portuguese Czech, Hungarian, Icelandic, Portuguese, Spanish French, Turkish Azeri, French, German, Hungarian, Por tuguese, Spanish, Turkish, Turkmen Czech Hawaiian, Latin Latin Hungarian Czech, Icelandic, Turkmen Languages Using the Latin Alphabet 11.24 t a b l e 11.1. (continued) Character (and Unicode number) Description Languages that use it Y (01B3), y (01B4) Y/y with hook Hausa Z (0179), z (017A) Z/z with acute Polish, Montenegrin Z (017B), z (017C) Z/z with dot above Polish Z (017D), z (017E) Z/z with caron (hacek) Bosnian, Croatian, Czech, Montenegrin, Serbian, Turkmen African Languages 11.23 African capitalization and punctuation. Most African languages—with the exception, most notably, of Arabic (see 11.76-81)—use the Latin al phabet and follow English capitalization and punctuation. The most widespread of these is Swahili, spoken by many different ethnic groups in eastern and central Africa. Hausa, Fulfulde, Yoruba, Igbo, Wolof, and Bambara are also spoken by millions, largely in western Africa; the same is true for Kikongo (or Kongo) and Lingala in the Congo-Zaire region and of Amharic and Somali in the Horn of Africa region. Amharic and other Ethiopian Semitic languages such as Tigrinya use the Geez alphabet, not covered here. Xhosa and other “click” languages spoken in southern Af rica do not follow English capitalization. The names of African languages themselves vary widely from ethnic group to ethnic group and from re gion to region. It is now standard practice to capitalize the names of African languages in the traditional way—for example, Kiswahili rather than KiSwahili or KISwahili. Xhosa speakers refer to and spell their lan guage “isiXhosa” but “Isixhosa” (sometimes “Isizhosa”) is also found in English-language publications.
11.24 African special characters. Swahili uses no additional letters or diacrit ics. Among the more than two thousand other African languages, how ever, many rely on diacritics and phonetic symbols to stand for sounds that cannot be represented by letters or combinations of letters. Hausa, which is spoken by millions of people across western Africa, requires the following special characters (see also table 11.1): B6, Def, Kk, Yy In Nigeria, both the upper- and the lowercasey with a “hook” are rep resented instead with an apostrophe (’Y^). Additional diacritics, too numerous to be listed here, may be needed in other African languages.
11.25 LANGUAGES OTHER THAN ENGLISH Languages such as French, Portuguese, and Arabic that are used in Af rica are addressed in separate sections in this chapter.
French 11.25 French—additional resources. As is the case with many languages, there is considerable variation in French publications with respect to capital ization and punctuation. For excellent advice, with frequent reference to the Academie fran^aise and numerous examples from literature, consult the latest edition of Le bon usage, known to many by the name of its origi nal editor, Maurice Grevisse (bibliog. 5). Further guidance maybe had at the website of the Academie fran^aise.
11.26 French capitalization. Generic words denoting roadways, squares, and the like are lowercased, whether used alone or with a specific name as part of an address. Only the proper name is capitalized.
le boulevard Saint-Germain la place de l’Opera 13, rue des Beaux-Arts In most geographical names, the generic word is lowercased and the modifying word capitalized.
la mer Rouge le pic du Midi Names of buildings are usually capitalized.
1’Hotel des Invalides le Palais du Louvre In names of organizations and institutions, only the first substantive and any preceding modifier are capitalized, but not the preceding article (ex cept at the beginning of a sentence).
l’Academie fran^aise la Legion d’honneur le Grand Theatre de Quebec In hyphenated names, both elements are capitalized.
la Comedie-Fran^aise la Haute-Loire Names of religious groups are usually lowercased.
Languages Using the Latin Alphabet 11.28 un chretien des juifs In names of saints, the word saint is lowercased. But when a saint’s name is used as part of a place-name or the name of a church or other institu tion, saint is capitalized and hyphenated to the following element.
le supplice de saint Pierre but l’eglise de Saint-Pierre Adjectives formed from proper nouns are usually lowercased.
une imagination baudelairienne See also 11.18.
11.27 Titles of French works. French publications vary in the way they cap italize titles of works. In general, Chicago recommends sentence-style capitalization (see 8.158), the rule followed by Grevisse, Le bon usage (see 11.6,11.25). Note that a superscript ordinal letter should remain in the superior position, as in the last example (cf. 14.88). An exception may be made for the French newspaper Le Monde, which always appears thus.
L’ Apollon de Bellac: Piece en un acte Lepere Goriot L’assommoir Paris au XXe siecle L’exil et le royaume but Les Rougon-Macquart Le Monde According to an alternative practice advocated by the Academie fran^aise and others (and exemplified by the title Le Monde), for titles beginning with a definite article (Le, La, L’, Les), the article and the first substantive (noun or noun form) and any intervening modifier are capitalized (e.g., La Grande Illusion). Titles that begin with a modifier are treated in the same way, with the modifier and first substantive capitalized (e.g., Mau- vais Sang); any other titles, including those beginning with an indefinite article (Un, Une) are capitalized sentence-style (e.g., “Un coeur simple”). This style, if adopted for French titles, should be used consistently. For punctuation in titles, see 11.7.
11.28 Spacing with French punctuation. In French typeset material, fixed thin spaces generally occur before colons, semicolons, question marks, and exclamation marks; between guillemets (« ») and the text they enclose (see 11.29); and after an em dash used to introduce dialogue (see 11.31). In electronic documents, fixed (i.e., nonbreaking) spaces can be used to 11.29 LANGUAGES OTHER THAN ENGLISH avoid stranding a mark at the beginning of a line or, in the case of an opening guillemet, at the end (see 6.121). In an English context, the typo graphic conventions of the publication as a whole can be observed, and such spacing need not be duplicated. (If for any reason French spacing is required, however, it must be followed consistently and according to French practice for all marks.) See also 11.19.
11.29 French use of guillemets. For quotation marks, the French use guille mets (« »), often with a fixed thin space (or, especially in electronic docu ments, a regular nonbreaking space; see 6.121) to separate the guillemets from the quoted matter. If such guillemets are retained in an English context, as for a quotation in French (but see 11.19), they can usually be spaced like regular quotation marks (see also 11.28). Such tags as ecrit-il or dit-elle are often inserted within the quoted matter without additional guillemets. Only punctuation belonging to the quoted matter is placed within the closing guillemets; other punctuation follows them.
«Mission accomplie?» a-t-il demande.
En ce sens, «avec» signifie «au moyen de».
A vrai dire, Abelard n’avoue pas un tel rationalisme: «je ne veux pas etre si phi losophe, ecrit-il, que je resiste a Paul, ni si aristotelicien que je me separe du Christ».
As in English (see 13.32), when a quotation (other than a block quotation) continues for more than one paragraph, opening guillemets appear at the beginning of each additional paragraph; closing guillemets appear only at the end of the last paragraph. See also 11.30.
11.30 Quotation marks in French. For quotations within quotations, double (or sometimes single) quotation marks are used. Formerly, additional guille mets were used, with opening guillemets repeated on each runover line. (Note that when guillemets are used, if the two quotations end simulta neously, only one set of closing guillemets appears.) See also 11.29.
«Comment peux-tu dire, “Montre-nous le pere”?» Regular quotation marks are sometimes seen in French contexts in lieu of guillemets—especially in email correspondence and other electronic settings. This usage is considered informal.
11.31 French dialogue. In dialogue, guillemets are often replaced by em dashes. In French publications, the dash is usually followed by a thin space; in English publications, the space is not necessary (see 11.28).
Languages Using the Latin Alphabet 11.34 Such dashes are used before each successive speech but are not repeated at the end of a speech. To set off a quotation within a speech, guillemets may be used. See also 11.29.
—Vous viendrez aussitot que possible? a-t-il demande.
—Tout de suite.
—Bien. Bonne chance!
—Tu connais sans doute la parole «De l’abondance du coeur la bouche parle». —Non, je ne la connais pas.
11.32 French ellipses. The French often use an ellipsis to indicate an interrup tion or break in thought. An ellipsis is also sometimes used in lieu of and so forth. In French practice, an ellipsis consists of three unspaced dots closed up to the word they follow (like this... rather than this ...); in En glish contexts, they may be spaced in the manner recommended else where in this manual (see 13.50-58) and shown in the examples below. See also 11.19.
«Ce n’est pas que je n’aime plus l’Algerie ... mon Dieu! un ciel! des arbres!... et le reste!... Toutefois, sept ans de discipline...» To indicate omissions, the French use unspaced ellipses enclosed in brackets, with thin spaces between the brackets and the dots. In English contexts, spaced periods may be used (but with no space between the brackets and the periods they enclose; see 13.58).
«Oh, dit-elle avec un mepris ecrasant, des changements intellectuels! [...]» Les deux amis se reunissaient souvent chez Luc [...].
11.33 French word division—vowels. In French, a word is divided after a vowel wherever possible. One-letter syllables at the ends or beginnings of lines should be avoided (see 11.20).
ache-ter (not a-cheter) in-di-vi-si-bi-li-te tri-age Two or more vowels forming a single sound, or diphthong, are never bro ken.
eeri-vain fouet-ter Gau-guin eloi-gner vieux 11.34 French word division—consonants. A division is normally made be tween two adjacent consonants, whether the same or different.
11.35 LANGUAGES OTHER THAN ENGLISH der-rie-re feuil-le-ter ba-lan-cer Mal-raux ob-jet par-ler but qua-tre ta-bleau Groups of three adjacent consonants are normally divided after the first.
es-prit res-plen-dir 11.35 French words containing apostrophes. Division should never be made immediately after an apostrophe.
jus-qu’au au-jour-d’hui 11.36 French words best left undivided. Since there are as many syllables in French as there are vowels or diphthongs (even if some are unsounded except in poetry), the French break words that appear to English speak ers to be of only one syllable (e.g.yfid-te,guer-re, sor-tent). French practice also permits division after one letter (e.g., e-tait). In English-language publications, however, such breaks should be avoided, since they may confuse readers not fluent in French. Words of four or fewer letters should in any case be left undivided. See also 7.37.
11.37 French accents and ligatures. French employs the following special characters (see also table 11.1): Aa, Aa, Q9, Ee, Ee, Ee, Ee, Ii, Ii, 06, CEoe, Uu, Uu, Uu Although French publishers have often omitted accents on capital letters (especially A) and may set the ligature CE as two separate letters (OE), all the special characters needed for French—including capitalized forms— are widely available, and they should be retained wherever needed in English-language contexts. This practice, advocated by the Academie fran^aise, is especially helpful to readers who may not be familiar with French typographic usage.
German 11.38 The new German orthography. The new rules for German orthography (including spelling and capitalization) adopted in 1998 and made man datory for schools and public documents in 2005 (subject to certain re visions) have been controversial. Some publications have continued to follow traditional rules, or a combination of house style and traditional Languages Using the Latin Alphabet 11.40 rules, whereas others have adopted the new rules. Some book publish ers honor the preference of their authors and, by a similar token, do not update spelling when reprinting older works. Material quoted from Ger man should therefore reflect the spelling in the source. For principles and details of the new orthography, consult the latest edition of Duden: Die deutsche Rechtschreibung (bibliog. 5). The recommendations and ex amples in this section reflect the new orthography.
11.39 German capitalization. In German, all nouns and words used as nouns are capitalized, whether in ordinary sentences or in titles of works (see 11.6).
ein Haus Deutsch (the German language) die Weltanschauung eine Deutsche (a German woman) das Sein etwas Schones Adjectives derived from proper names are generally lowercased. Excep tions include invariable adjectives ending in er (often referring to a city or region) and adjectives that themselves are part of a proper name. For further exceptions, consult Duden (see 11.38).
die deutsche Literatur nordamerikanische Sprachen die platonischen Dialoge but eine beruhmte Berliner StraEe der Nahe Osten der Deutsch-Franzdsische Krieg The pronouns Sie, Ihr, and Ihnen, as polite second-person forms, are capitalized. As third-person pronouns they are lowercased. The familiar second-person forms du, dich, dein, ihr, euch, and so on—once routinely capitalized—are now lowercased.
11.40 German apostrophes. An apostrophe is used to denote the colloquial omission of e.
wie geht’s was gibt’s hab’ ich Although an apostrophe rarely appears before a genitive s, an apostrophe is used to denote the omission of the s after proper names ending in an 5 sound (ce, s, ss,fi, tz, x, or z) or in a silent s, x, or z.
11.41 LANGUAGES OTHER THAN ENGLISH Alice’ Geburtstag Jaspers’ Philosophic Cixous’ Theaterstiicke Leibniz’ Meinung 11.41 German quotation marks. In German, quotations usually take reversed guillemets (» «); split-level inverted quotation marks or, in Switzer land, regular guillemets (see 11.29). Other punctuation is placed outside the closing quotation marks unless it belongs to the quoted matter.
Eros bedeutet fiir sie primar »zusammen-sein mit« und nicht »anschauen«. Denn: „An die Pferde“, hieE es: „Aufgesessen!“ 11.42 German word division—vowels. In German, division is made after a vowel wherever possible. See also 11.20.
Fa-brik hii-ten Bu-Ee Two vowels forming a single sound, or diphthong, are never broken.
Lau-ne blei-ben Further, a break should never be made after a single vowel at the begin ning or end of a word (aber, Ofen, Treue}.
11.43 German word division—consonants. Two or more adjacent consonants, whether the same or different, are divided before the last one unless they belong to different parts of a compound (see also 11.20).
klir-ren Was-ser Verwand-te Meis-ter but Morgen-stern The consonant combinations ch, ck,ph, sch, and th are not divided unless they belong to separate syllables. (Until the 1998 spelling change, st was subject to this rule. The combination ck, on the other hand, used to be changed at the end of a line to kk and divided between the k’s.) Mad-chen Zu-cker Philo-so-phie rau-schen but Klapp-hut Haus-chen Languages Using the Latin Alphabet 11.46 11.44 German word division—compounds. Compound words should be di vided between their component elements whenever possible (see also 11.20).
Meeres-ufer mit-einander Rasier-apparat Tiir-angel 11.45 German special characters. For setting German in roman type (the old Gothic or Fraktur type having long been out of use), the eszett, or sharp s (y?), and three umlauted vowels are needed (see also table 11.1).
Aa, Oo, £, Uii Although umlauted vowels are occasionally represented by omitting the accent and adding an e (ae, Oe, etc.), the availability of umlauted char acters in text-editing software makes such a practice unnecessary. The eszett (y?), also widely available, must not be confused with, or replaced by, the Greek beta ([3). In the new spelling it is replaced by ss in certain words. Consult a German dictionary published after 1998. In German speaking areas of Switzerland, the eszett is rarely used.
Italian 11.46 Italian capitalization. In Italian, a title preceding a proper name is nor mally lowercased.
il commendatore Ugo Emiliano la signora Rossi In commercial correspondence, the formal second-person pronouns are capitalized in both their nominative forms, Lei (singular) and Voi (plural), and their objective forms, La (accusative singular), Le (dative singular), and Vi (accusative and dative plural). The older singular and plural forms Ella (Le, La) and Loro (Loro, Loro) are handled the same way. These pro nouns are capitalized even in combined forms.
Posso pregarLa di farmi una cortesia?
Vorrei darLe una spiegazione.
See also 11.6,11.18. For a fuller treatment of this and other matters of style, consult Roberto Lesina, Il nuovo manuale di stile (bibliog. 5).
11.47 LANGUAGES OTHER THAN ENGLISH 11.47 Italian quotations and dialogue. Italian uses guillemets (« ») to denote quoted matter, but usually without the space between guillemets and quoted text that appears in many French publications. Regular quotation marks (double or single) are also frequently used in Italian—sometimes as scare quotes (see 7.57) in the same text in which guillemets are used for quotations. Note that periods and commas are correctly placed after the closing guillemet or quotation mark.
«Cosa pensi del fatto che io possa diventare “un qualcosa di imperial regio”? Questo non e proprio possibile».
In dialogue, em dashes are sometimes used, as in French. The dash is used before each successive speech. Unlike in French, however, another dash is used at the end of the speech if other matter follows in the same paragraph. The spaces that typically surround the dashes in Italian texts need not be used in English contexts (see 11.19).
—Avremo la neve,—annunzid la vecchia.
—E domani?—chiese Alfredo, voltandosi di scatto dalla finestra.
11.48 Italian apostrophes. An apostrophe is used to indicate the omission of one or more letters. A space should appear after an apostrophe that fol lows a vowel; after an apostrophe that follows a consonant, however, no space should appear.
po’ duro de’ malevoli Fonda all’aura 11.49 Italian ellipses. Italian, like French (see 11.32), uses ellipses to indicate interruptions or breaks in thought. To indicate omitted material, the dots are enclosed in brackets. Though Italian typography usually calls for un spaced dots, in English publications Chicago recommends spaced peri ods wherever ellipses occur (see 13.50-58). See also 11.19.
Voglio... quattro milioni. Dawero? [...] Non ci avevo pensato.
11.50 Italian word division—vowels. In Italian, division is made after a vowel wherever possible. One-letter syllables at the ends or beginnings of lines should be avoided (see 11.20).
acro-po-li (not a-cropoli) mi-se-ra-bi-le ta-vo-li-no Consecutive vowels are rarely divided, and two vowels forming a single sound, or diphthong, are never divided.
Languages Using the Latin Alphabet 11.54 miei pia-ga Gio-van-ni Giu-sep-pe pau-sa gio-iel-lo 11.51 Italian word division—consonants. Certain consonant groups must never be broken: ch, gh, gli, gn, qu, sc, and r or I preceded by any conso nant other than itself.
ac-qua-rio la-ghi pa-dre ri-flet-te-re fi-glio na-sce rau-che so-gna-re Three groups of consonants, however, may be divided: double conso nants; the group cqu; and any group beginning with /, m, n, or r.
bab-bo ac-qua cam-po den-tro af-fre-schi cal-do com-pra par-te 11.52 Italian word division—words containing apostrophes. Division should never be made immediately after an apostrophe (but see 11.48).
dal-l’accusa del-l’or-ga-no quel-l’uomo un’ar-te l’i-dea 11.53 Italian special characters. In Italian, the following special characters are required (see also table 11.1): Aa, Ee, Ee, Ii, do, Uu Although the grave accent on capitalized vowels is sometimes dropped, in stressed final syllables it must be retained to avoid confusion.
CANTO (he sang) CANTO (I sing) PAPA (daddy) PAPA (pope) Especially in older works, an apostrophe is sometimes seen with a capital letter in place of the accent on a stressed final (or single) vowel. In direct quotations, such usage should be retained.
E’ (it is) E (and) PAPA’ (daddy) Latin 11.54 Latin capitalization—titles of works. Titles of ancient and medieval Latin works should usually be capitalized in sentence style—that is, only the first word in the title and subtitle, proper nouns, and proper adjec tives are capitalized (see 8.158).
11.55 LANGUAGES OTHER THAN ENGLISH De bello Gallico De viris illustribus Cur Deus homo?
Renaissance and modern works or works in English with Latin titles, on the other hand, can usually be capitalized headline-style (see 8.159). (If there is any doubt about the era to which the title belongs, opt for sen tence style.) Novum Organum Religio Medici See also 11.6.
11.55 Latin word division—syllables. A Latin word has as many syllables as it has vowels or. diphthongs (ae, au, ei, eu, oe, ui, and, in archaic Latin, ai, oi, ou) and should be divided between syllables (see also 11.20).
na-tu-ra cae-li-co-la in-no-cu-us 11.56 Latin word division—single consonants. When a single consonant oc curs between two vowels, the word is divided before the consonant un less it is an x. Note that i and u sometimes act as consonants (and, when they do, are sometimes written as j and v).
Cae-sar me-ri-di-es in-iu-ri-or (or in-ju-ri-or) but lex-is 11.57 Latin word division—multiple consonants. When two or more conso nants come together, the word is divided before the last consonant, ex cept for the combinations noted below.
om-nis cunc-tus The combinations ch, gu, ph, qu, and th are treated as single consonants and thus never separated.
co-phi-nus lin-gua ae-qua-lis The following consonant groups are never broken: bl, br, chi, chr, cl, cr, dl, dr, gl, gr, phi, phr, pl, pr, thl, thr, tl, and tr.
pan-chres-tus li-bris ex-em-pla pa-tris 11.58 Latin word division—compounds. Compound words are divided be tween parts; within each part the rules detailed elsewhere in this section Languages Using the Latin Alphabet 11.61 apply. The commonest type of compound word begins with a preposition or a prefix (e.g., ab-, ad-, in-, re[d]-).
ab-rum-po ad-est red-eo trans-igo 11.59 Latin special characters. Latin requires no special characters for setting ordinary copy. Elementary texts, however, usually mark the long vowels with a macron and, occasionally, the short vowels with a breve, as fol lows. (See also table 11.1.) Aa, Aa, Ee, Ee, Ii, II, 06, 06, Uu, Uu Spanish 11.60 Spanish—additional resources. There is considerable variation in Spanish-language publications throughout the world with respect to cap italization, punctuation, and other matters. For further guidance, consult the extensive resources available from the Real Academia Espanola, in cluding such essential guides as the Diccionario panhispanico de dudas and the Ortografia de la lengua espafiola (bibliog. 5).
11.61 Spanish capitalization. In Spanish, a title preceding a proper name is normally lowercased. When abbreviated, however, titles are capitalized.
el senor Jaime Lopez but la senora Lucia Moyado de Barba el Sr. Lopez dona Perfecta Nouns as well as adjectives denoting membership in nations are lower cased, but names of countries are capitalized.
los mexicanos la lengua espanola Inglaterra Names of organizations and institutions, historical events, buildings, streets, and the like are usually capitalized (see also 8.159).
Real Academia Espanol Universidad Nacional Autonoma de Mexico Plaza del Dos de Mayo See also 11.4,11.6,11.18.
11.62 LANGUAGES OTHER THAN ENGLISH 11.62 Spanish question marks and exclamation points. A question or an ex clamation in Spanish is preceded by an inverted question mark or excla mation point and followed by a regular mark.
<Que pasa, amigo? jOlvidalo en ese caso!
If a vocative or dependent construction precedes a question or exclama tion, it is written as follows: Amigo, ique pasa? En ese caso, iolvidalo!
Because the opening marks are integral to Spanish punctuation, they should be retained even when Spanish is being quoted in an English con text (see 11.19).
11.63 Spanish guillemets and quotation marks. Spanish traditionally uses guillemets (« ») as quotation marks. Only punctuation belonging to the quoted matter is placed within the closing guillemets; other punctua tion follows them. Within a quotation, em dashes may be used to set off words identifying the speaker. In Spanish publications, the opening dash is usually preceded by a space; the closing dash is then followed by a space unless immediately followed by punctuation. In English contexts, such spaces need not be used (see also 11.19). (For quotations within quota tions, regular quotation marks are used, as in French; see 11.30.) «Vino el negocio a tanto—comenta Suarez—, que ya andaban muchos tornados por el diablo».
In lexical studies, it is typical to see single quotation marks used for glosses, with no punctuation preceding the gloss (cf. 11.5).
Muchos adverbios se forman anadiendo -ly al adjetivo: courteous ‘cortes’, courte ously ‘cortesmente’, bold ‘atrevido’, boldly ‘atrevidamente’.
Increasingly, Spanish-language publications use regular quotation marks rather than guillemets for all quotations. Where this is the case, the rules for punctuation marks relative to the quotation marks are the same as they are for guillemets (but see 11.11).
11.64 Spanish dialogue. In dialogue, an em dash (or, less frequently, a guille met) introduces each successive speech. Any other matter that follows the quoted speech in the same paragraph is generally preceded by a dash or a comma. See also 11.63.
Languages Using the Latin Alphabet 11.67 —Esto es el area de Noe, afirmo el estanciero.
—K (0416), xc (0436) j k zh z 3 (0417), 3 (0437) 3, 3 z W (0418), M (0438) U i M (0419), a (0439) W, U y i (012D) j K (041A), K (043A) x, K k JI (041B), JI (043B) b, JI 1 M (041C), M (0430 M, M m H (041D), H (043D) H, H n O (041E), 0 (043E) o, 0 0 n (041F), n (043F) n, n P p (0420), P (0440) p, P r c (0421), c (0441) c, c s T (0422), T (0442) T, m t y (0423), y (0443) y, y u (0444) 0, f X (0425), X (0445) X, X kh X U (0426), U (0446) II, K ts ts5 c H (0427), H (0447) H ch c HI (0428), III (0448) IB, Ul sh s m (0429), m (0449) B[, shch sc i.
(042A), ■ b6 (044A) B, b ” (201D)7 " (02BA)8 " (02BA)8 w (042B), w6 (044B) BI, bl y b (0420, b6 (0440 B, b ’ (2019)9 ' (02B9)10 ' (02B9)10 3 (042D), 3 (044D) 3, 9 e e (0117) e (00E8) IO (042E), K) (044E) /0, 10 yu iu5 ju 51 (042F), a (044F) B, a ya la5 n o t e : The Library of Congress and linguistic systems employ the same characters as the US Board system except where noted.
2The Unicode numbers are the same for the upright and cursive characters; the differences in appear ance depend on the italic version of a given typeface.
2 The term linguistic describes a system generally preferred by journals of Slavic studies (see 11.98).
3 Initially and after a vowel or or &.
4 Not considered a separate letter; usually represented in Russian by e.
5 Character tie, sometimes omitted, may be produced by using the combining double inverted breve (U+0361).
6 Does not occur initially. 7 Right double quotation mark.
8 Modifier letter double prime (hard sign). 9 Right single quotation mark.
10 Modifier letter prime (soft sign).
11.100 LANGUAGES OTHER THAN ENGLISH 11.100 Titles of Russian works. Only the first word and any proper nouns are capitalized in titles.
N. A. Kurakin, Lenin i Trotskiy O. I. Skorokhodova, Kak ya vosprinimayu i predstavlyayu okruzhayushchiy mir [How I perceive and imagine the external world] Note that in the original Cyrillic, titles are set in ordinary type; the Cyril lic kursiv is used more sparingly than our italic and never for book titles. In transliterations, however, italic should be used.
11.101 Russian quotations and dialogue. Russian generally resembles French in its use of guillemets (« ») for dialogue and quoted material and of dashes for dialogue (see 11.29,11.31).
«Bozhe, bozhe, bozhe!» govorit Boris.
—S kem ya rabotayu?
—S tovarishchem.
—Kak my rabotayem?
—S interesom.
To set off a quotation within a speech, guillemets may be used, as in French. For an example, see 11.31.
11.102 Russian ellipses. Ellipses are used as in French (see 11.32) to indicate interruptions or breaks in thought.
Ya... vy... my tol’ko chto priyekhali.
In Russian, an exclamation point or a question mark often takes the place of one of the dots; this convention may be regularized to three dots in English publications.
Mitya!... Gde vy byli?...
11.103 Russian uses of the dash. A dash is sometimes inserted, with a space on either side, between subject and complement when the equivalent of is or are is omitted.
Moskva — stolitsa Rossii.
Languages Usually Transliterated (or Romanized) 11.108 Similarly, a dash, preceded and followed by a space, is used in place of a verb omitted because it would be identical to the preceding verb.
Ivan i Sonya poyedut v Moskvu poyezdom, Lev i Lyuba — avtobusom.
11.104 Russian word division—general. Transliterated Russian should be di vided according to the rules governing word division in the Cyrillic orig inal. The guidelines in this section are adapted from the transliteration system of the United States Board on Geographic Names.
11.105 Combinations not to be divided in Cyrillic transliteration. Combina tions representing single Cyrillic letters—ch, kh, sh, shch, ts,ya,ye,ye,yu, zh—should never be divided, nor should combinations of a vowel plus short i (or yod, transliterated^/): ay, ey,yey, and so on.
11.106 Division between Russian consonants. Words may be divided between single consonants or between a consonant and a consonant combination.
ubor-ku chudes-nym mol-cha sred-stvo mor-skoy The following consonant combinations are not normally divided: bl, br, dr, dv,fl,fr,gl,gr, kl, kr, ml,pl,pr, sk, skr, skv, st, str, stv, tr, tv, vl, vr, zhd. They may, however, be divided if they fall across the boundary of a prefix and a root or other such units (e.g., ob-lech’, ras-kol).
11.107 Division of Russian words after prefixes or between parts. Words may be divided after a prefix, but generally the prefix itself should not be di vided.
bes-poryadok pere-stroyka za-dat’ pred-lozhit’ pro-vesti obo-gnat’ Compound words should be divided between parts.
radio-priyemnik gor-sovet kino-teatr 11.108 Division of Russian words after vowel or diphthong. Words may be di vided after a vowel or a diphthong before a single (Cyrillic) consonant.
Si-bir’ voy-na Gorba-chev da-zhe Division after a vowel may also be made before a consonant combina tion.
puteshe-stvennik khi-trit’ pro-stak ru-brika 11.109 LANGUAGES OTHER THAN ENGLISH South Asian Languages 11.109 South Asian special characters. Transliteration of the principal South Asian languages may require the following special characters (see also table 11.2): Aa, Dd, Ee, Hh, Ii, LI, LI, Mm, Mm, Nn, Nri, Nn, Nn, Od, Rr, Rr, Ss, Ss, Tt, Uu, Yy Many writers using South Asian languages, however, employ a simplified style that does not use diacritics at all—for example, substituting sh for various s’s, ignoring subscript dots for dental consonants, and omitting macrons altogether.
Classical Greek 11.110 Transliterating Greek. Isolated Greek words and phrases in works not focusing on ancient Greece are usually transliterated. Table 11.4 shows the Greek alphabet (with Unicode numbers) and corresponding letters of the Latin alphabet. In transliteration, all Greek accents are omitted. The macron is used to distinguish the long vowels eta (e) and omega (o) from the short vowels epsilon (e) and omicron (o). The iota subscript is transliterated by an i on the line, following the vowel it is associated with (avOpdmq), anthrdpdi). The rough breathing is transliterated by h, which precedes a vowel or diphthong and follows the letter r (as in the En glish word rhythm). The smooth breathing is ignored, since it represents merely the absence of the h sound. If a diaeresis appears in the Greek, it also appears in transliteration. Transliterated Greek words or phrases are usually italicized unless the same words occur frequently, in which case they may be italicized at first mention and then set in roman.
11.111 Typesetting Greek. Authors who need to present Greek should use a Unicode-enabled font if at all possible (see 11.2). Publishers need to make sure that a Greek font is available for publication; Greek may need to be set in a slightly different size to make it visually match the surrounding type. Greek is normally not set in italics. Extra white space must occasionally be added where more than one diacritic appears over a vowel.
Classical Greek 11.113 t a b l e 11.4. Greek alphabet (and Unicode numbers) and romanization Greek alphabet Name of letter (and Unicode numbers) Transliteration alpha A (0391), a (03B1) a beta B (0392), p (O3B2) b gamma r (0393), Y1 (O3B3) g delta A (0394), 62 (03B4) d epsilon E (0395), e (O3B5) e zeta Z (0396), (O3B6) z eta H (0397), n (O3B7) e (0113) theta 0 (0398), o3 (O3B8) th iota I (0399), i (O3B9) i kappa K (039A), K (03BA) k lambda A (O39B), X (03BB) 1 mu M (O39C), P (03BC) m nu N (039D), V (03BD) n xi E (039E), ij (03BE) X omicron 0 (039F), 0 (03BF) 0 Pi n (03A0), 7t (03C0) P rho p (03A1), P (03C1) r; initially, rh; double, rrh sigma z (O3A3), a (O3C3), <;♦ (O3C2) s tau T (03A4), T (03C4) t upsilon Y (O3A5), V (O3C5) u; often y, exc. after a, e, e, i phi <D (O3A6), (pS (03C6) ph chi X (O3A7), X (O3C7) kh, ch psi T (O3A8), Y (O3C8) ps omega Q (O3A9), 0) (O3C9) 6 (014D) 1Note that yy becomes ng, and yx becomes nk.
2Sometimes incorrectly appears as d (U+2202, partial differential).
3Also 9 (U+03D1). 4Final letter. 5Also <|> (U+O3D5).
Breathings and Accents 11.112 Greek breathing marks. When Greek is set in the Greek alphabet, every initial vowel or diphthong or rho must be marked with a breathing, either rough (, dasia) or smooth (’, psili). The breathing mark is placed over the initial lowercase vowel (or the second vowel of a diphthong). It is po sitioned to the left of capital letters. Note that a single quotation mark cannot function as a breathing because it is the wrong size and does not sit close enough to the letter.
c u jt e ETepai "EXArp q0r| Tpic; u t t ex^ w k u c ; paSioc; 11.113 Greek accent marks. There are three Greek accent marks: acute, or oxia ('); circumflex, or perispomeni, either tilde-shaped or rounded (~ or ~), 11.114 LANGUAGES OTHER THAN ENGLISH depending on the typeface; and grave, or varia ('). Accents in Greek oc cur only over vowels. The circumflex occurs only on the two final syl lables of a word. The grave accent occurs only on the last syllable. Like breathings, accents are placed over lowercase vowels, over the second vowel of a diphthong, and to the left of capital vowels. A diaeresis is used to indicate that two successive vowels do not form a diphthong but are voiced separately (as in French naif).
11.114 Unaccented Greek words. With two exceptions, all Greek words are marked with accents—usually one, occasionally two (see below). The first exception is a group of monosyllabic words called proclitics, which are closely connected with the words following them. The proclitics are the forms of the definite article d, q, oi, al; the prepositions e k ;, ev, e k (e£); the conjunctions el, cbc;; and the adverb ov (o v k , o v %). The second excep tion is a group called enclitics, short words pronounced as if part of the word preceding them. Enclitics usually lose their accents (Apra^ep^qc; t e), and in certain circumstances the word preceding them gains a sec ond accent ((po0£rrai t ic;).
11.115 Greek vowels. Vowels complete with breathing marks and accents, in all combinations, are an integral part of every Greek font used in publishing. Each font, for example, should be able to provide, for lowercase eta, q, q, q, q, q, q, q, q, q, q, q, q, and, for uppercase eta, H/H/H/H/H/H/H/H/H. Additional symbols are needed for scholarly works treating ancient man uscripts or papyri. Consult the latest Unicode character charts for Greek alphabets.
Punctuation and Numbers 11.116 Greek punctuation. In Greek the period and comma are the same as in English; the colon and semicolon are both represented by a midlevel dot (•); the question mark is represented by a semicolon. The apostrophe (which looks almost like a smooth breathing mark) is used as an elision mark when the final vowel of one word is elided before a second word beginning with a vowel. In English texts, quoted words or passages in the Greek alphabet, of whatever length, should not be enclosed in quotation marks.
11.117 Greek numbers. Numbers, when not written out, are represented in or dinary Greek text by the letters of the alphabet, supplemented by three additional, obsolete Greek letters—stigma, koppa, and sampi: c/ = 6, 9' = 90, = 900. The diacritical mark resembling a prime (and defined for Classical Greek 11.120 t a b l e 11.5. Greek numerals 1 a' 13 iy' 30 X' 600 X' 2 r 14 16' 40 P' 700 V' 3 Y 15 l e' 50 v' 800 co' 4 6' 16 60 e 900 Y3 5 £' 17 IC 70 0' 1,000 ,a 6 18 in' 80 Tt' 2,000 <P 7 c 19 10' 90 9'2 3,000 'Y 8 n' 20 k ' 100 p' 4,000 9 0' 21 Ka' 200 a' 10,000 ,1 10 1' 22 K0' 300 t ' 100,000 'P 11 ia' 23 KY' 400 v' 12 iP' 24 k 8' 500 9' 1 Stigma (U+03DB); also represented with digamma (U+03DD): jr'. 2Archaic koppa (U+O3D9); also represented with koppa (U+03DF): y. 3Sampi (U+03E1); formerly disigma (double sigma).
Unicode as the Greek numeral sign, U+0374) distinguishes the letters as numerals and is added to such a letter standing alone or to the last sign in a series. For example, pia' means 111. For thousands, the foregoing let ters are used with a different diacritical mark (the Greek lower numeral sign, U+0375): ,a = 1,000, ,apia' = 1,111, ,0o k P' = 2,222. See table 11.5.
Word Division 11.118 Greek word division—consecutive vowels. Diphthongs (ai, av, el , ev, qv, o l , ov, vi, (vv) are never divided. But two consecutive vowels that do not form a diphthong are divided.
0e-d-o-|iai xn-oc; t k x v-e-t e v e-o k ; 11.119 Greek word division—single consonants. When a single consonant oc curs between two vowels, the word is divided before the consonant.
cpo)-vq KE-cpa-Xic; ps-ya 8e-8o)-k e v pq-rqp 11.120 Greek word division—two or more consonants. If a consonant is doubled, or if a mute is followed by its corresponding aspirate (iKp, 0T^, 80), the word is divided after the first consonant.
11.121 LANGUAGES OTHER THAN ENGLISH 0d-Xao-oa cnry-xcd-pco If the combination of two or more consonants begins with a liquid (X, p) or a nasal (p, v), division is made after the liquid or nasal.
£|i-7tpo-u0£v (but before pv: |i£-pvr||iai) All other combinations of two or more consonants follow the division.
npd-ypa T£-\vr] (3a-0pd(; ai-uxpoc; Pt-pXt-ov Sa-KTv-Xoc; UKfj-irTpov pd-KTpov 11.121 Greek word division—compounds. Compound words are divided be tween parts; within each part the rules detailed elsewhere in this section apply. The commonest type of compound word begins with a preposition or a prefix.
dpcp-dcp-V71-£^-ePaXov dv-£ b thorn Both edh and thorn represent voiced or unvoiced th, as in them or three.
3 3 Yogh; occurs in Old English representing^ as in good,y as inyear, or gh as in light and thought. Yogh sometimes occurs in Middle English representing^ as inyear andg/i as in light and thought, but normally not g as in good.
JE ae Ligature; should not be printed as two letters in Old English names and text (iElfric).
Authors should use the correct Unicode characters for the ligature and for edh, thorn, and yogh, and should provide their publisher with a list of these and any other special characters (see 11.2). For the long 5 (f), see 13.7.
American Sign Language (ASL) 11.128 11.123 Ampersand and wynn. In Old English and Middle English texts a sort of stylized seven (the Tironian et) may be found for and, but the modern ampersand may be substituted for this. In Old English texts p or p (wynn) is found for w; the modern w is often substituted for this.
11.124 Old English vowels. Modern editors of Old English sometimes distin guish between long and short vowels and diphthongs by means of a ma cron over the long versions (e.g., a, x, e, ea, eo, i, io, d, u,y). Note that, with the exception of the x ligature, diphthongs are usually marked with a macron over only the first vowel.
American Sign Language (ASL) 11.125 Signed languages. The visual-gestural languages used by deaf people in different parts of the world are called signed languages. Signed lan guages are quite different from spoken languages (although there may be regional effects of language contact), and a particular signed lan guage may or may not share the same national or geographic boundaries as spoken languages in the same locations. The individual elements of these languages are known as signs.
11.126 Components of signs. Signs have five major articulatory components— handshape, location, orientation, movement, and (in some cases) dis tinctive nonmanual signals.
11.127 Writing ASL. Many formal systems for writing signed languages exist; however, none has been adopted for widespread use by deaf signers. This section offers an overview of some of the most frequently em ployed conventions for written transcription of signing. For additional resources, see Charlotte Baker-Shenk and Dennis Cokely, American Sign Language: A Teacher’s Resource Texton Grammar and Culture; and Clayton Valli, Ceil Lucas, Kristin J. Mulrooney, and Miako Villanueva, Linguistics of American Sign Language: An Introduction (bibliog. 5).
11.128 Glosses in ASL. The written-language transcription of a sign is called a gloss. Glosses are words from the spoken language written in small capi tal letters: w o ma n , s c h o o l , c a t . (Alternatively, regular capital letters may be used.) When two or more written words are used to gloss a single sign, the glosses are separated by hyphens. The translation is enclosed in double quotation marks.
The sign for “a car drove by” is written as v e h ic l e-d r iv e-b y .
11.129 LANGUAGES OTHER THAN ENGLISH One obvious limitation of the use of glosses from the spoken/written language to represent signs is that there is no one-to-one correspon dence between the words or signs in any two languages.
11.129 Compound signs. Some combinations of signs have taken on a meaning separate from the meaning of the individual signs. Various typograph ical conventions are used to indicate these compounds, including a “close-up” mark or a plus sign. Depending on the transcription system, the sign for “parents” might be glossed as follows: MOTHERFATHER or MOTHER+FATHER 11.130 Fingerspelling. For proper nouns and other words borrowed from the spoken language, the signer may fingerspell the word, using the hand shapes from a manual alphabet. (There are numerous fingerspelling al phabets used by different signed languages, among them the American Manual Alphabet.) Fingerspelled words may be transcribed in any of the following ways: fs-jOHN or j-o -h -n or j-o-h-n 11.131 Lexicalized signs. Over time, some fingerspelled words have taken on the quality of distinct signs, either by omission of some of the individual letter signs or by a change in the orientation or movement of the letter signs. These lexicalized signs are represented by the “pound” symbol (#): #WHAT, #BACK, #DO.
11.132 Handshapes. Most of the handshapes of American Sign Language are described by the corresponding alphabetic or numerical handshape or a variation thereof. For example, a ppl e is made with an X handshape; c r e a t e is made with a 4 handshape; a n y is made with an Open A handshape; y e l l is made with a Bent 5 handshape. Handshapes without a clear relative in the fingerspelling or number system are labeled idio- syncratically according to the transcription system in use. For example, s a r c a s t ic is made with the h o r n s handshape; a ir pl a n e is made with the il y handshape. Handshapes for signed languages that do not use the American Manual Alphabet are often described in relation to the ASL handshapes.
11.133 Transcriptions of signed sentences. Signed sentences are written as a sequence of glosses, often with the spoken/written-language trans lation underneath in italics or quotation marks or both. (For examples, see 11.134,11.135.) Punctuation is generally omitted from sentence tran-American Sign Language (ASL) 11.135 scriptions (though not from the translations). Some writers, however, add question marks and exclamation points, and a comma may be used to indicate a short pause in the sentence.
11.134 Pronouns, possessives, and reference. Pronouns are commonly tran scribed either as ix (since these are frequently produced with the “in dex” finger) or as pr o . Either of these is followed by indication of person and sometimes number. A similar convention is used with the posses sive marker, sometimes glossed as po s s . There are varying conventions about how to indicate person and number. Thus, a third-person singu lar pronoun in ASL (equivalent to English “he,” “she,” or “it”) might be glossed as ix3p, ix-3p, or pr o .3. A second-person plural pronoun could be glossed as ix2p-pl. Subscript indices are often used to show signs ar ticulated in the same location or to indicate coreferential noun phrases. The following example indicates that he and his refer back to the same person: IX3pj LOSE POSS3pi HOUSE He lost his house.
11.135 Nonmanual signals. Nonmanual gestures may be labeled based on ana tomical behavior or grammatical interpretive function. These gestures, indicated by various abbreviations and terms, are typeset in a smaller font followed by a half-point rule above the ASL sentence. For example, the label whq is commonly used to refer to the facial expression that marks questions involving “who,” “what,” “when,” “where,” “how,” or “why.” This expression consists of a cluster of features that include furrowed brows and slightly squinted eyes. In the example below, whq occurs over the entire question (i.e., the expression is articulated simul taneously with all of the manual signs over which the line extends). In the same example, the label t indicates a topic marker that occurs simul taneously with the sign y e s t e r d a y . Correct alignment is critical to an accurate transcription.
t whq YESTERDAY, fs-JOHN SEE WHO Whom did John see yesterday?
12 • Mathematics in Type OVERVIEW 12.1 STYLE OF MATHEMATICAL EXPRESSIONS 12.3 General Usage 12.3 Signs and Symbols 12.9 Punctuation 12.18 Mathematical Expressions in Display 12.21 Numeration 12.24 Delimiters 12.26 Subscripts and Superscripts 12.36 Summations and Integrals 12.39 Radicals 12.43 Fractions 12.45 Matrices and Determinants 12.50 Scalars, Vectors, and Tensors 12.52 Definitions, Theorems, and Other Formal Statements 12.56 Probability and Statistics 12.57 PREPARATION AND EDITING OF PAPER MANUSCRIPTS 12.60 12.1 MATHEMATICS IN TYPE Overview 12.1 Additional resources for math. This chapter is mainly intended to pro vide guidance to authors and editors working in the sciences who have occasional need to compose or edit mathematical expressions. Those who work extensively with mathematics should consult Ellen Swanson’s Mathematics into Type, among the other sources listed in bibliog. 2.4.
12.2 Tools for math. Many authors in mathematics and related quantitative fields prepare their manuscripts in LaTeX, a freely available, device independent document markup and preparation system developed in the 1980s. In.LaTeX, which is designed to work with the TeX typesetting system developed by Donald Knuth, a properly coded manuscript will generate equation numbers, cross-references, and many other elements automatically. Manuscript editors working with LaTeX documents (even on paper) should have some understanding of how the markup works to avoid, for example, marking unnecessary changes or instructions and to know when a particular change can be indicated globally. A good place to start for more information is the LaTeX website. Manuscripts that in clude only the occasional in-line or displayed expression, on the other hand, are usually prepared using a word processor’s equation editor. For the role of MathML, a markup language that complements HTML by de scribing the content and structure of mathematical notation for publica tion in electronic formats, consult the latest version of the standard from the World Wide Web Consortium.1 For marking mathematical copy on paper manuscripts, see 12.61-68.
Style of Mathematical Expressions General Usage 12.3 Standards for mathematical copy. The author and editor should give careful attention to matters of style, usage, sense, meaning, clarity, ac curacy, and consistency. Authors should use correct terminology and 1. MathML, together with alternative text, can enhance accessibility for readers who use text-to-speech and related tools. Publishers are encouraged to consult the guidelines of fered by the Web Accessibility Initiative (WAI) of the World Wide Web Consortium (W3C) and, for EPUB (a standard format for e-books), the Accessibility Guidelines available from the International Digital Publishing Forum.
Style of Mathematical Expressions 12.5 notation and should carefully follow the conventions of their special fields. In most cases, an author’s preferences should prevail, though edi tors should query any apparent typographical or grammatical violations. As a general rule, mathematical copy, including displayed expressions, should “read” as clearly and grammatically as any other kind of copy. The signs for simple mathematical operations and relations have direct verbal translations: ab reads “a is greater than b”; a + b = c reads “a plus b equals c.” The translation is not always straightforward, however, as is the case with df(x)/dxy which means “the derivative of the function/of x with respect to x” and is not the quo tient of two numbers df(x) and dx. Moreover, mathematical notation is often abbreviated: the pair of inequalities a<b and b < c is usually written a < b < c. In mathematics it is also standard to read terms with indices, such as Xi (“x sub i”), as plural or singular depending on the context (e.g., “for a unique xf’ and “for all the xf’ are both grammatically correct). Use this convention wherever possible to avoid ugly mixtures of italic mathematics and roman “s” in forming a plural.
12.4 Consistency of mathematical notation. The letters and other symbols used to denote mathematical objects should be consistent and unambig uous: the same symbol should denote the same thing whenever it occurs and not be used for more than one thing. Typographical distinctions should also be made consistently; for example, if uppercase italic letters A, B, and C are used to denote sets and lowercase italic letters x,y, and z to denote the elements of sets, then a, by and c should not be used for sets at another place without good reason.
12.5 Words versus mathematical symbols in text. In general, mathematical symbols may be used in text in lieu of words, and such statements as “x > 0” should not be rewritten as “x is greater than or equal to zero.” Nonetheless, symbols should not be used as a shorthand for words if the result is awkward or ungrammatical. In the phrase the vectors ri,..., rn, 0, the condition 0” is better expressed in words: the nonzero vectors fi,..., rn or the vectors rly..., rny all nonzero, 12.6 MATHEMATICS IN TYPE depending on the emphasis desired. Moreover, logical symbols should generally not appear in text: 3 a minimum value of the function/on the interval [a, b] should be replaced by there exists a minimum value of the function/on the interval [a, b] or the function/has a minimum value on the interval [a, b].
See also 12.7.
12.6 Concise mathematical expression. Mathematical symbols should not be used superfluously. For example, in the first statement the symbol n is extraneous, as are the parentheses in the second statement: There is no integer n between 0 and 1.
This quantity is bounded above by the sum (a + b).
As a general rule, no letter standing for a mathematical object should be used only once. Symbols that appear to be redundant may be qualified later in the same discussion, however, and editors should never delete a symbol without explicit instruction from the author.
12.7 Sentence beginning with a mathematical symbol. Mathematical sym bols should not begin a sentence, especially if the preceding sentence ended with a symbol, since it may be difficult to tell where one sentence ends and another begins. For example, it is difficult to read Assume that x e S. S is countable.
If a sentence starting with a symbol cannot easily be rephrased, the ap propriate term for the symbol can be inserted in apposition at the begin ning of the sentence: Assume that xeS. The set S is countable.
If the sentences are closely related, a semicolon may be used to connect them: Style of Mathematical Expressions 12.12 A function/is even if/(-x) =/(%);/is odd if/(-%) = 12.8 Adjacent mathematical symbols. Mathematical symbols in adjacent mathematical expressions should be separated by words or punctuation (or both), for the reasons discussed in 12.7: Suppose that a = bq + r, where 0<r<b.
Signs and Symbols 12.9 Mathematical characters. The smallest units of mathematical writing are mathematical signs and symbols, which include letters and numbers. Table 12.1 lists some of the standard mathematical characters and their verbal translations. Unicode numbers are included, where applicable (see also 11.2), as are the LaTeX commands for producing each charac ter. Old-style figures [like this: 1938] should be avoided in mathemati cal contexts because their height and position relative to the baseline is inconsistent and because, in some typefaces, a zero might be mistaken for an oh; lining figures [like this: 1938] should be used instead (see also 3.86, item 4).
12.10 Diacritical and other marks in mathematical notation. Ordinary italic letters are used to represent various kinds of mathematical objects. The set of distinct symbols can be greatly extended by the use of dia critics (including accents), such as tf, J, J, J, a, dy a, and a. (Note that when an i or a j appears with a diacritical mark—e.g., with an overbar or a circumflex—the dotless i or j should be used.) Double diacritics may also be used; for example, a, and a, Marks over or beneath several letters or groups of letters—for example, overlines, underlines, overbraces, and underbraces—are frequently encountered in mathematics, as are other types of stacked expressions.
12.11 Italic letters and kerning in mathematical expressions. Contiguous italic letters that represent separate objects must never be kerned. To this end fonts intended for mathematics are typically designed so that italic letters are less slanted than they might otherwise be (and, therefore, unlikely to crowd or overlap other typeset elements). Contiguous let ters that form an abbreviation—for example, “Aut” for “automorphism group”—should be set roman; see also 12.17.
12.12 Letters and fonts in mathematical notation. The number of symbols can be extended by using letters from other alphabets, most often the t a b l e 12.1. Common mathematical signs and symbols (with Unicode numbers and LaTeX commands) Sign/symbol Name Unicode LaTeX OPERATIONS + x / o U n + + ® e e ® o U Plus sign 002B + Minus sign 2212 -Multiplication sign 00D7 \times Dot operator (multiplication) 22C5 \cdot Division sign 00F7 \div Division slash 2215 /a Ring operator (composition) 2218 \circ Union 222A \cup Intersection 2229 \cap Plus or minus 00B1 \pm Minus or plus 2213 \mp Asterisk operator (convolution) 2217 \ast Circled asterisk operator (convolution) 229B \circledast Circled plus (direct sum, various) 2295 \oplus Circled minus (various) 2296 \ominus Circled times (various) 2297 \otimes Circled dot operator (various) 2299 \odot Ratio 2236 Coproduct or amalgamation 2210 \amalg RELATIONS < > Ill IV IA I Equals sign 003D = Not equal to 2260 \neq Almost equal to, asymptotic to 2248 \approx Approximately equal to, isomorphic to 2245 \approxeq Less than 003C < Much less than 226A \11 Greater than 003E > Much greater than 226B \gg Less than or equal to 2264 \leq Greater than or equal to 2265 \geq Identical to, congruent to 2261 \equiv Not identical to, not congruent to 2262 \nequiv Divides, divisible by 2223 \divides Tilde operator (similar to, asymptoti 223C \sim cally equal to) Colon equals (assignment) 2254 \coloneqq Element of 2208 \in Not an element of 2209 \notin Subset of 2282 \subset Subset of or equal to 2286 \subseteq Superset of 2283 \supset Superset of or equal to 2287 \supseteq Proportional to 221D \propto Approaches the limit, definition 2250 \doteq Tends to, maps to 2192 \rightarrow t a b l e 12.1. (continued) Sign/symbol Name Unicode LaTeX -Maps from 2190 \leftarrow -Maps to 21A6 \mapsto Maps into 21AA \hookrightarrow -Maps into 21A9 \hookleftarrow OPERATORS E Summation 2211 \sum n Product 220F \prod f Integral 222B \int Contour integral 222E \oint LOGIC A And, conjunction 2227 \wedge V Or, disjunction 2228 \vee -Not sign (negation) 00AC \neg => Implies 21D2 \implies -Implies 2192 \rightarrow <=> If and only if 21D4 \iff — If and only if 2194 \leftrightarrow 3 There exists (existential quantifier) 2203 \exists V For all (universal quantifier) 2200 \forall F Assertion 22A6 \vdash Hence, therefore 2234 \therefore Because 2235 \because RADIAL UNITS Minute (prime) 2032 \prime Second (double prime) 2033 \second o Degree 00B0 \degree CONSTANTS Tt Pi («3.14159265) 03C0 \pi e Base of natural logarithms 0065 e («2.71828183) GEOMETRY ± Perpendicular to (up tack) 22A5 \perp II Parallel to 2225 \parallel Not parallel to 2226 \nparallel z Angle 2220 \angle < Spherical angle 2222 \sphericalangle V Equiangular to 225A \veedoublebar MISCELLANEOUS i Square root of-1 0069 i Prime 2032 \prime Double prime 2033 \second Triple prime 2034 \third J Square root, radical 221A \sqrt Cube root 221B \sqrt (continued) 12.13 MATHEMATICS IN TYPE t a b l e 12.1. (continued) Sign/symbol Name Unicode LaTeX !
Factorial 0021 i ii Double factorial 2O3C n 0 Empty set, null set 2205 \varnothingb oo Infinity 221E \infty a Partial differential 2202 \partial A Increment, Laplace operator 2206 \triangle V Nabla, del; also Laplace operator (with superscript 2) 2207 \nabla □ d’Alembert operator (white square) 25A1 \square n o t e : Though there are generic characters that appear similar to the ones listed in the table (e.g., asterisk [002A] for asterisk operator), the mathematical versions are preferred because they encode correct intersymbol spacing.
a LaTeX also defines \slash, which permits a line break after the slash. bAlso \emptyset.
Greek alphabet, and by representing letters from the Latin alphabet in other fonts. Examples of characters from four fonts commonly used in mathematics include the following: Greek Script Boldface italic Boldface Greek ABTAapyS WBCDabccC ABCDabcd ABrAapyfi Lowercase script characters are often not available for a given typeface, though they have been defined for the mathematical alphabets in Uni code (see also 11.2). See also 12.65 for marking fonts on paper manu scripts.
12.13 List of unusual mathematical characters. Special characters specific to mathematics will generally not be available in all typefaces. Even some of the more common characters may need to be checked for availability or inadvertent substitution (e.g., letter a for Greek alpha, a). For elec tronic publication formats, special fonts may need to be embedded (and licensed accordingly). It may be advisable, therefore, to prepare a list of unusual mathematical signs, symbols, and special characters used in the manuscript, before editing begins. This is preferably done by the author but may be done by the editor. In preparing an electronic manuscript, the author should make a list of any special, nonstandard fonts. A copy should be given to the publisher, who will check to make sure the nec essary characters are available for the intended publication formats. If some are not, the author may be asked to use more accessible forms; if Style of Mathematical Expressions 12.16 that is impossible, the typesetter must be asked to obtain or generate the characters needed.
12.14 Special mathematical symbols. Many mathematical symbols have a re served meaning: n stands for the number 3.14159265 ..., e for the num ber 2.71828182 ..., and i for the square root of-1. The symbols V, 3, g , c, and 0 are used in all mathematical disciplines. Double-struck (black board) symbols are reserved for familiar systems of numbers: N for the natural numbers, 2 for the integers (2+ is the same as N), Q for the ratio nal numbers, R for the real numbers, and € for the complex numbers.
12.15 Signs for binary operations and relations. Binary operations act as con junctions to combine two mathematical expressions. Examples of binary operation signs are + (plus sign), - (minus sign), • (multiplication dot), x (multiplication cross), 4 (division sign), / (solidus or slash), and o (com position sign). Binary relations act as verbs and express a relationship between two mathematical expressions. Examples of relation signs are = (equals), (does not equal), > (is greater than), and < (is less than).
12.16 Basic spacing in mathematics. Mathematics isn’t simply read left to right in a machine-like manner, and one should be able to see the parts of an equation if it is properly set. Good mathematical spacing helps to indicate grouping: things that are more closely related should be set more tightly than things that are less closely related. Such spacing will vary according to the elements being set. In simple expressions, how ever, absolute spacing may be called for. Signs for binary operations (i.e., conjunctions); symbols of integration, summation, or union; and signs for binary relations (i.e., verbs) are preceded and followed by medium spaces (i.e., one-fourth of an em space): xn + yn = zn, XU0 = X, (a°b)oc = a°(b°c).
No space follows a binary operation or relation sign when it is modifying a symbol (i.e., used as an adjective): -1, +oo, x5, >7.
In subscripts anvd superscripts, no space precedes or follows operation or relation signs: xa+^, yc~2 12.17 MATHEMATICS IN TYPE t a b l e 12.2. Standard abbreviated notations in mathematical copy sin Sine sn Elliptic function, sn cos Cosine cn dn Elliptic function, cn Elliptic function, dn tan Tangent tg Tangent3 cot Cotangent ctg Cotangent3 sec Secant csc Cosecant sinh Hyperbolic sine cosh Hyperbolic cosine tanh Hyperbolic tangent coth Hyperbolic cotangent sin-1 Inverse sine arcsin Inverse sine log Common logarithm (logio) In Natural logarithm lg Binary logarithm (log2) loge Natural logarithm, alternate form sgn Sign arg Argument det or Det Determinant Tr Trace (also Sp, or spur) Re, 91 Real part Im, 3 Imaginary part curl Curl; vector operator, same as V x div Divergence; vector operator, same as V • prob or Pr Probability mod Modulo (as in a mod b) inf Infimum; greatest lower bound sup Supremum; least upper bound isom Isomorphism Hom Homeomorphism min Minimum max Maximum gcd Greatest common divisor 1cm Least common multiple dex Decimal exponent; from IO’1-5 to IO"3 is 1.5 dex norm Norm; norm (a) = ||fl|| dim or Dim Dimension ker Kernel wrt With respect tob iff If and only ifb Var or var Variance Cov or cov Covariance a Frequently used by non-North American authors. bUsed in informal notation.
Commas used between coordinate points or in lists (see 12.19) should be followed by a medium space. See also 6.120.
12.17 Mathematical functions. For a list of abbreviated functions, see table 12.2. These abbreviations are followed by a thin space (about one-fifth of an em space) unless the argument is enclosed in delimiters, or fences (see 12.26), in which case they are usually closed up to the opening de limiter: ln2it, sin(x + j>), min(xi,x2).
Limits are set as subscripts to the right of the abbreviation in text and below the abbreviation in display: Style of Mathematical Expressions 12.19 x->a max^es(^i), max(fli).
Punctuation 12.18 Mathematical expressions and punctuation. Mathematical expres sions, whether run in with the text or displayed on a separate line, are grammatically part of the text in which they appear. Thus, expressions must be edited not only for correct presentation of the mathematical characters but also for correct grammar in the sentence. For example, if several expressions appear in a single display, they should be separated by commas or semicolons. For example, Xi + x2 + 3 = 3, 12 + 23 + 31 = 6, 123 = -1.
Consecutive lines of a single multiline expression, however, should not be punctuated: (|a + Z?|)2 = (a + b)2 = a2 + 2ab + b2 <a2 + 2\a\b+b2 = \a\2 + 2\a\b\ + \b\2 = (W + |/’I)2.
Expressions must carry ending punctuation if they end a sentence. All ending punctuation and the commas and semicolons separating expres sions should be aligned horizontally on the baseline, even when pre ceded by constructs such as subscripts, superscripts, or fractions.
12.19 Elided lists in mathematical expressions. In elided lists, commas should come after each term in the list and after the ellipsis points if the list has a final term. For example, j = 0,1, 2,... not y = 0, 1, 2...; 1, 2, ... ,n not Xi,x2, ...n.
12.20 MATHEMATICS IN TYPE The ellipsis points should be on the baseline when the terms of the list are separated by commas. Use a medium space after each comma (see also 12.16).
12.20 Elided operations and relations. In elided sums or elided relations, the ellipsis points should be vertically centered between the operation or re lation signs. For example, x1 + x2 + -- + xn not Xi + x2 + ... + xn; th < a2 < ••• < an not ch < a2 <... < an Multiplication is often signified by the juxtaposition of the factors with out a multiplication sign between them. That is, abc means a b • c.
When the multiplication sign is not explicit, the elided product may be denoted with ellipsis points either on the baseline or vertically centered: d±d2... dn or ■ &n • The second alternative is commonly used in displays with built-up fac tors: If the multiplication dot is present, then ellipsis points should be on the baseline and not centered. For example, ’ ^2 ‘... not Ui ■ ci2....... Un.
If the multiplication cross is present, then ellipsis points should be cen tered. For example, #i x a2 x • • • x an-Multiplication signs are always used when the factors need to be sepa rated: lx2x---xl0.
Style of Mathematical Expressions 12.22 In same contexts (such as for numbers) the use of the multiplication cross rather than the dot is a matter of preference, but in many other con texts (such as for vectors) multiplication dots and crosses have different meanings and cannot be used interchangeably.
Mathematical Expressions in Display 12.21 Displaying mathematical expressions. Mathematical expressions should be displayed—that is, set on a separate line clear of text—if they are im portant to the exposition, if they are referenced, or if they are difficult to read or typeset in the body of the text. If different mathematical expres sions are displayed on the same line, the expressions should be separated by spacing, together with words or punctuation: If a = b, then for all real numbers x, a + x = b + x, ax = bx, -a = -b.
If different mathematical expressions are displayed on separate consec utive lines, regardless of whether there is an intervening word between two of the equations, each expression can usually be center aligned, as in the first example in 12.18. Some groups of displayed expressions will be easier to read if they are aligned on an equals sign or other relational sign: If a = b, then for all real numbers x, a + x = b + x, ax = bx, -a = -b.
For another example, see 12.22.
12.22 Qualifying clauses for displayed mathematical expressions. Qualifying clauses may be presented in several ways. If the main expression is dis played, the qualifying clause may also be displayed (separated from the expression by an em space or more): If/is a constant function, then f'(a) = 0 for all a G OL 12.23 MATHEMATICS IN TYPE The qualifying clause may instead appear in the text, following the dis played main expression: Suppose that the prime factorization of the integer a is given by « = p^-'-PY, where thepj are distinct prime numbers and ki > 0.
Or it may precede the displayed main expression: For all real numbers a and b, And qualifying clauses may themselves include displayed expressions: Suppose that assumptions 1 and 2 hold. Then a competitive equilib rium satisfies the following three differential equations: = |[(1 - a1)Yr1(f)VeX(t)«1K(r)-«1x(t)-«1 - 6 - p] -= X(r)“iK(r)1-aix(r)-aiT -Xv? ai K(t) _ [1 - K(£)]{A[x(r)/x(t)] + m2- (a2/a1)mi} K(t) ~ (l-eH+AfKW-Mr)] ’ where with initial conditions x(0) and k (0), and also satisfies the transver- sality condition lim exp r->oo r 12.23 Breaking displayed mathematical expressions. Even in displayed form, some long expressions will not fit on one line. In such cases, displayed expressions may be broken before a relation or operation sign. Such signs include the following: Style of Mathematical Expressions 12.24 Operation signs (conjunctions): + - x 4- ± u n Relation signs (verbs): = ^><><^DCG = = See table 12.1 for a more complete list. In displayed expressions, runover lines are aligned on the relation signs, which should be followed by thick spaces (i.e., one-third of an em space): Mx) = (x-a)(x-p)(x-y) = x3- (a + p + y)x2 + (ap + ay + Py)x - aPy.
If a runover line begins with an operation sign, the operation sign should be lined up with the first character to the right of the relation sign in the line above it, followed by a medium space (one-fourth of an em space): 71 1 1 1 1 1 1 4 2 3x2s+5x25 7x2? + 9x29 llx211 + ’" 11111 + 3 3x3s+ 5 x 35 7x3? + 9x39 1 1 1 1 4 3 x 4^ 5 x 4^ 7 x 42 For additional rules on breaking expressions, consult Ellen Swanson, Mathematics into Type (bibliog. 2.4).
Numeration 12.24 Numbering displayed mathematical expressions. Mathematical ex pressions that are referred to elsewhere in the text should be numbered or otherwise labeled. All numbered mathematical expressions must be displayed. Displayed expressions are usually centered on the line (with out regard to the expression number or label). The number or label, en closed in parentheses to prevent misreading, is usually put at the right margin, but it may be placed at the left margin.
Hence it is apparent that 13+23+••• + n3 = (1 +2+--- + n)2. (1.1) In cross-references, display numbers or labels are enclosed in parenthe ses to match the marginal enumerations: 12.25 MATHEMATICS IN TYPE Recalling equation (1.1), we may conclude that...
or Recalling (1.1), we may conclude that...
A range of numbered equations is referred to by giving the first and last numbers, joined by an en dash: From equations (2)-(5) we obtain...
12.25 Methods of numeration for mathematical expressions. Displayed math ematical expressions may be numbered or labeled, as may definitions, theorems, lemmas, and other formal parts of the exposition. A simple numbering system offers a convenient and space-saving method of cross-reference. In texts with many displayed equations, double or triple numeration is usually preferred. In this system, the displayed expressions in each chapter are labeled with the chapter number first, followed by the section number (if any), followed by the statement number, starting with number 1 (1.1.1,1.1.2,..., 1.2.1,1.2.2,..., etc.). If, on the other hand, single numeration is used (e.g., in a text with relatively few displayed equations), the displays are still usually numbered starting over with 1 in each chapter. In a work with many such numbered statements, it is helpful to the reader to number them together, in a single sequence, but apart from displayed equations (e.g., definition 4.1, lemma 4.2, lemma 4.3, proposition 4.4, corollary 4.5).
Delimiters 12.26 Common delimiters in mathematics. Three sorts of symbols are com monly used to group mathematical expressions: parentheses (), brackets [], and braces {}. They are used in pairs, and their normal order is {[()]}. When necessary, the sequence of delimiters can be extended by large parentheses, brackets, and braces as follows: {[({[( )]})]} In text, the braces are sometimes omitted from this sequence. Angle brackets, vertical bars, and double vertical bars carry special mathemat ical significance and should not be used to supplement the sequence of common delimiters.
12.27 Functional notation. In functional notation, nested pairs of parentheses are used instead of brackets or braces to indicate grouping: Style of Mathematical Expressions 12.31 12.28 Set notation. Braces are used to delimit the elements of a set, and other delimiters should not be substituted. For example, {^1, #2, • • •, an} denotes the set consisting of n objects aly a2)...^ an, and {x:xED} denotes the set of all elements % in a set D. In the second example (called “set-builder” notation), the condition that defines the set follows the co lon. A vertical bar is sometimes used instead of the colon to delimit the condition.
12.29 Ordered set notation. In ordered set notation, parentheses are used as delimiters. For example, (a, b) denotes the ordered pair of objects a and by where a is the first element in the pair and b is the second element. More generally, (^1, a2,..., an) denotes the ordered n-tuple of objects a2,..., an- This notation is standard, and other delimiters should not be substituted.
12.30 Interval notation. In interval notation, parentheses are used to delimit an open interval—that is, one that does not include its endpoints; for ex ample, (a, b) denotes the set of all real numbers between a and b, not in cluding either a or b. Brackets are used to delimit a closed interval—that is, an interval that includes its endpoints. The notation (a, b] signifies the interval not including a but including b, while [a, b) denotes the inter val including a but not including b. Parentheses and brackets in interval notation should not be replaced with other delimiters. (According to an alternative convention, ]a,b[ denotes an open interval, [a, b[ an interval that includes a but not b, etc. This should not be changed if the author has used it consistently.) 12.31 Delimiters denoting inner product. Parentheses are sometimes used to denote the inner product of two vectors: (w, w). Angle brackets are also used as notation for the inner product: . See also 12.52-55.
12.32 MATHEMATICS IN TYPE 12.32 Binomial coefficients. The notation (JP, “n choose fc,” is called the bi nomial coefficient and stands for the number of ways k objects can be chosen from among a collection of n objects. It is defined by n\ k(n-k)\' where n and k are positive integers and the notation ! stands for the fac torial function, n\ = n x (n -1) x • • • x 1.
12.33 Vertical bars in mathematical notation. Vertical bars serve several spe cial purposes. The modulus or absolute value of x is denoted |x|. The no tation |u| is used for the “length” of a vector u, also sometimes called the norm of u and written with a double vertical bar, ||u||. Vertical bars are used to denote the cardinal number of a set. The notation |A | can signify the determinant of a matrix A, which is also denoted det A.
12.34 A single vertical bar in mathematical notation. A single vertical bar with limits is used to denote the evaluation of a formula at a particular value of one of its variables. For example, f^/2 . j |X=7l/2 Jo smxax = -cosx|x=0 =-c o s jc/2 - (-cosO) = 1.
12.35 Cases in mathematical expressions. Displayed mathematical expres sions that present a choice between alternatives, or cases, may be grouped using a single brace and are punctuated as follows: -a, a>0; a < 0.
Another acceptable style is /(x) = 1 ifx>0, 0 otherwise.
As a general rule, each alternative is equivalent to a clause in ordinary language and should be punctuated as such. If the alternatives are very long, they may be stated as separate equations: Style of Mathematical Expressions 12.38 I(t) = AeV1^ 1 + £icos[2rc/(t - tp)]} + B{1 + e2c o s[2n/(t - tp)]}, t < tp.
I(t) = Ae r2^ ^{1 + 8i cos[27ty(t - tp)]} + B{1 + e2 cos[2jry(t - tp)]}, t > tp.
(la) (lb) Subscripts and Superscripts 12.36 Simple mathematical subscripts and superscripts. Inferior and supe rior indices, exponents, and other subscript and superscript symbols oc cur frequently in mathematical copy. Examples are Xi, x2, 2, x', Xijyjk, xab, axbyy xj2.
Multiple indices are written without commas between them unless there is a possibility of confusion: xij instead of x/j, but x1>2 if there is a possi bility of confusing the subscripts “1,2” and “12.” Abbreviations or words that serve as labels in subscripts or superscripts are usually set in roman type: #min, wav 12.37 Complex mathematical subscripts and superscripts. Subscripts and superscripts may themselves have subscripts and superscripts. For ex ample, %ak) 2X , 2 i Mathematical expressions may occur as subformulas in the superior or inferior positions. For example, xa+h, a-xb^-z^, a^', 2^ 12.38 Alignment of mathematical subscripts and superscripts. Subscripts and superscripts may be stacked or staggered 12.39 MATHEMATICS IN TYPE Xabcd, Xij».
Because there are standard conventions for raising and lowering indi ces in some branches of mathematics, especially in tensor calculus, the relative position between superior and inferior indices should not be changed. See also 12.54.
Summations and Integrals 12.39 Summation sign. The summation sign E is used to stand for a sum of a finite or infinite sequence of terms. For example, the sums d± + d2 + • • • + dn and d± + + • • • may be written n oo ^at and ^at, i=i i=i respectively. The variable i in the expressions above is called the index of summation. The subformulas below and above the summation sign are called the limits of summation and indicate where the summation begins and, if it is finite, ends. Summation limits are sometimes omitted if it is clear from the context what the limits are; for example, if all vectors are stated to be of size n and all matrices are of size n x n, it is acceptable to write J When a summation sign occurs in text, its limits are placed to the right of the summation sign to avoid spreading the lines of text: X^idi-12.40 Product sign. Product notation follows similar conventions. The pro ducts d± • d2 •... • dn and d± • d2 •...
may be written n oo n and rn i=i i=i Style of Mathematical Expressions 12.43 respectively, and in text the limits are placed to the right of the product symbol to avoid spreading lines: 12.41 Integral sign. The integral sign J is used to denote two sorts of integrals, called definite and indefinite. A definite integral is the integral of a func tion/ on an interval [fl, b]. This integral is denoted jf(x)dx.
The numbers a and h are called the lower and upper limits of integration, and dx is called the element of integration or the differential. (The d in some notations is not italicized; editors can usually follow an author’s preference on this matter.) The limits of integration are usually placed to the right of the integral sign in both text and display. The indefinite integral is denoted J f or j f(x)dx, without limits of integration. For a function of two variables, it is com mon to denote an integral over both variables by a double integral sign: , y) dx dy.
D Here D is called the set of integration.
12.42 Spacing around differentials. Thin spaces (about one-fifth of an em space) are placed before and after differentials: dV = r2 sinO dr dQ d§.
Differential expressions appearing in derivatives must be closed up to the slash: dx/dt.
Radicals 12.43 Radical signs. The radical sign V is used to denote the square root. A horizontal bar extends from the top of the radical sign to the end of the radicand: V2, >/sin2x + cos2%.
12.44 MATHEMATICS IN TYPE In display, the radical sign extends vertically to accommodate a built-up radicand: The radical sign may be used to denote cube and higher-order roots. For these roots, a superscript-sized number or letter is nested within the rad ical sign: 12.44 Radical signs in text. Radical signs can be used in text if the radicand is a simple expression: a = m + n>/3. If the radicand is more complex or if the text design uses tight leading, radical signs can give the page a crowded look or interfere with descending letters in the line above. One remedy is to substitute the appropriate exponent, using delimiters to indicate the extent of the radicand. For example, •Ja2 + b2 may be replaced by (^2 + /?2)1/2.
Fractions 12.45 Fractions in text. Fractions are set in text with a slash to separate the numerator and denominator: 1/2, 2/3, 1/10, 97/100, k /2, 11/5, a/b.
Some common numerical fractions may be set as case fractions (text sized fractions with a horizontal bar): 1 2 1_ 2’ 3’ 1O’ Fractions should be enclosed in parentheses if they are followed by a mathematical symbol or expression: (a/b)x.
Style of Mathematical Expressions 12.48 For simple algebraic fractions in text, the slash should be used rather than the horizontal fraction bar. For example, (ax + b)/(cx + d) not to it. Thus, a + b/c means a + b c which should be written (a + b)/c.
12.46 Fractions in display. In displayed mathematical expressions, all fractions should be built up unless they are part of a numerator or denominator or in a subscript or superscript: not |x2sin||<i.
Fractions that include summation, product, or integral signs should al ways be displayed. For example: jK/2sin2W%dx Jo_ \n/2sin2n+1xdx Jo If there are no built-up fractions in the display, common numerical frac tions may be set as case fractions: \a-b\<^.
12.47 Fractions in subscripts and superscripts. Fractions in subscripts and superscripts should always use the slash, both in text and in display: xa/b, y3/2.
12.48 Multiple and multilevel fractions. If a mathematical expression contains more than one fraction, it should be displayed, and the horizontal bar should be used for the principal fraction sign: ax+ b _ px + q ex+ d rx +s' Fractions should preferably be limited to two levels: 12.49 MATHEMATICS IN TYPE ^ + c not (p/q) + r P + r q Continued fractions—that is, expressions of the form aT + 1/Z?X, where fex = a2 + l/b2) b2 = a3 + l/b3, and so on—are displayed: #x + 1 u2 + 1 a3 + 1 (I4. + • • • 12.49 Rewriting fractions using exponents. There are times when it is desir able to represent the denominator of a fraction without using a fraction rule or a slash. This may be done by using delimiters followed by the ex ponent -1: ab(cd)-1 instead of cd If there is already an exponent in the denominator, it can be changed to its negative: ab(cd)~2 instead of (cd)2 If an exponential expression, particularly in text, is very complex, it may be rewritten in a simpler form. An exponential term such as e(2niXnj)/Jx2+y2 can be rewritten using the abbreviation exp: expp27w’Sny)/(x2 + j?2)1/2j Matrices and Determinants 12.50 Matrices. Matrices are arrays of terms displayed in rectangular arrange ments of rows and columns and enclosed on the left and right by either large brackets or parentheses: #11 #12 • • #in / #11 #12 • • #in #21 #22 • ■• #2M or #21 #22 '' #2« &mi #W2 ’ • ^mn #mi #W2 ‘' amn \ 7 Style of Mathematical Expressions 12.51 The horizontal lists of entries are called the rows of the matrix, and the vertical lists the columns. A matrix with m rows and n columns is called an m x n matrix. A matrix consisting of a single row is called a row matrix or a row vector; a matrix consisting of a single column is a column matrix or a column vector. For example, [a b c] and a b c are row and column matrices, respectively. The transpose of a matrix A, often denoted AT, is the matrix obtained by interchanging the rows and columns of A. For example, #11 #21 ’ ’ • dmi #12 #22 ’ ’ • dm2 #1H #2« ’ • dmn is the transpose of the mn matrix given above. Column matrices such as " a -b c may be represented in text as (a, b, c)T, col. (a, b, c), or the column vector (a, b, c), and a 2 x n (for small ri) matrix may be set, for example, as (? j). Most matrices are displayed, however. In-line matrices cannot be broken on the line, and display matrices cannot be broken across the column or page. Authors should avoid using matrix notation for multiline equa tions.
12.51 Determinants. If A is a square matrix, the determinant of A, denoted |A| or det A, is a function that assigns a specific number to the matrix A. If A is an n x n matrix, the determinant of A is represented by #11 #12 “ #m #21 #22 ’ ’’ dm #m #H2 ‘‘ dnn 12.52 MATHEMATICS IN TYPE Vertical bars are used to distinguish the determinant of A from the ma trix A. The Jacobian matrix has a standard notation: j = d(Xi,X2,...,Xn)‘ Its determinant (usually referred to as the Jacobian) is denoted as d(%i, #2, • • • > Xn) Scalars, Vectors, and Tensors 12.52 Scalars, vectors, and tensors defined. Three basic quantities often en countered in scientific mathematical material are scalars, vectors, and tensors. Scalars, usually denoted by lowercase italic or Greek letters, are ordinary numbers and are treated as such. Vectors are quantities that have direction as well as magnitude, and they are often denoted by bold face letters or by an arrow diacritic to distinguish them from scalars: r or r.
Because authors do not always follow these conventions, editors should be prepared to query. A vector may be written as the sum of its compo nents: i The circumflex over the e is used to denote a vector of length 1, called a unit vector. Tensors are multidimensional quantities that extend the vector concept. A scalar is a tensor of rank 0, and a vector is a tensor of rank 1.
12.53 Vector and tensor multiplication. Vector and tensor multiplication em ploys a special notation that is relatively easy to identify in text. The inner or dot product of two vectors u and w is denoted u • w; the dot product is signified by the boldface multiplication dot. The vector or cross product of two vectors u and w is denoted u x w; the cross product is signified by the boldface multiplication cross. The multiplication dot and multipli cation cross are not interchangeable for vectors as they are for ordinary multiplication. The standard notation for the tensor product of tensors Style of Mathematical Expressions 12.56 T and S is T ® S. Index notation for vectors and tensors usually takes the following form: S = SjkeJek, bl = al jci.
Note the correspondence of the indices in these expressions (see 12.38 for discussion of index positioning). The Einstein convention has been used here, which implies summation over the repeated index. Thus Q = X a^jk the same as Ck = aJbjk j unless otherwise stated.
12.54 Additional tensor notation. Two additional special notations are used to differentiate tensors. One is A\j, where the subscript comma indicates a coordinate (or “ordinary”) derivative. The other is A\j, where the sub script semicolon indicates the covariant derivative. See also 12.38.
12.55 Dirac notation. A special form of the inner product, used especially in physics, is the Dirac bracket notation, , which can also be used in combination with operators, as in (which is not the same as (aTb)) or (Ta\b). The combinations , for an overbar, x. They can mean very different things. The population standard deviation (the most common measure of dispersion) is denoted by a (Greek lowercase sigma), and the sample standard deviation is more commonly denoted by 5; sd or SD may be used to distinguish it from se or SE, for standard error. The arithmetic mean is most frequently written in physical sciences literature as \ or A and the cumulant as «A». Several abbreviations are used in stochastic theory and probability theory with out special definition: a.e., almost everywhere; a.c., almost certainly; a.s., almost surely. See table 12.3 for statistical notation.
GREEK ALPHABET t a b l e 12.3. Statistical notation a Probability of rejecting a p(X) Mean of the population true null hypothesis E Sum of (type I error) 0 Population standard p Probability of accepting a deviation false null hypothesis a2 Population variance (type II error) X2 Value for the chi-squared K Cumulant; also kappa statistic distribution LATIN ALPHABET df, DF, dof Degrees of freedom 5, sd, SD Sample standard deviation F F-ratio se, SE Standard error H Value from the Kruskal-sem, SEM Standard error of the mean Wallis test t Value from Student’s t-test Ho Null hypothesis T Value from the Wilcoxon In Natural logarithm matched-pairs signed-log Logarithm to base 10 rank test mse, MSE Mean squared error U Value from the Mann-p,P,Pr Probability Whitney test rp Pearson correlation ws Value from the Wilcoxon coefficient rank sum test Value from the Spearman z Value from the normal rank-order test distribution 12.60 MATHEMATICS IN TYPE 12.60 Uncertainties. Uncertainties in quantities are usually written with a plus or minus sign (±): 2.501 ± 0.002 or, if there is an exponent, (6.157 ± 0.07) x 105 or io4-3±°-3. However, there are cases in which the bounds rather than the range are given, and these may be unequal; hence, ... where D/H = 1.651g^ x IO"5....
Uncertainties may also be specified as se for standard error, 1 o (or a larger multiple) or sd for standard deviation. Finally, separation into ran dom and systematic uncertainties is written as 71.0 ± 5.0 (random) ± 2.5 (sys) or 71.01J? (random) (sys) for asymmetric bounds. When such expres sions occur in an exponent, it is preferable to write a separate expression for the exponent (see 12.49).
Preparation and Editing of Paper Manuscripts 12.61 Format of paper manuscripts for mathematics. Manuscripts for math ematical articles and books should be printed out one-sided and double spaced, on 8x/2 x 11-inch white paper, with UA-inch margins for text and 2-inch margins for display work. The print quality should be 300 dots per inch or better. If handwritten equations or symbols are to be inserted in the printout, allow generous space for them. Since the editor will need to provide instructions to the typesetter, there should be ample margins.
12.62 Setting mathematics from the author's hard copy. In the event that the publisher cannot use the electronic files prepared by the author, and to the extent that a manuscript shows all the necessary characters, symbols, and signs as they should appear, the typesetter may simply be instructed to follow the author’s hard copy. Any unusual characters not achieved in the manuscript must be marked or identified. Authors should supply the highest-quality printout possible.
12.63 Marking italic type for mathematics. The editor of a mathematical text should either underline all copy that is to be set in italics or give general instructions to the typesetter to set all Latin single-letter mathematical objects in italics unless they are marked otherwise. The general instruc tions to the typesetter should also specify italic type for such letters used in subscripts or superscripts. If italics have been used in the manuscript, the editor can instruct the typesetter to follow the copy.
Preparation and Editingof Paper Manuscripts 12.67 12.64 Marking common mathematical abbreviations. Abbreviations for com mon functions, geometric points, units of measurement, and chemical elements, which are set in roman type, should be marked as roman by the editor only where ambiguity could occur. For a list of some frequently used abbreviations, see table 12.2.
12.65 Marking single mathematical letters in other type styles. Special mark ing must be used when single letters representing mathematical objects are to be set in any typeface other than italics. A mathematical text may require the use of some roman letters, usually to indicate properties dif ferent from those expressed by the same letters in italics. Underlining is the standard method of indicating italics, but it can be used instead, with instructions to the typesetter, to indicate letters that are to be in roman. If, however, the editor does not use general instructions but underlines all letters to be set italic, then letters not underlined will be set, as im plied, in roman type. Double underlining is used to indicate small capi tals. Wavy underlining is used for boldface. Color codes are often used to indicate other typefaces. For example, red underlining or circling can be used for Fraktur, blue for script, green for sans serif, and so forth. The general instructions to the typesetter must clearly explain the marking and coding system used. If a photocopy must be made of the edited man uscript for estimating by the typesetter or for querying the author, avoid color coding.
12.66 Mathematical fonts to mark on a paper manuscript. Boldface, script, Fraktur, and sans serif are frequently used in mathematical expressions and should be indicated on a paper manuscript as suggested in 12.65.
Boldface Script Fraktur Sans serif ABCD c/EBCT) TOST ABCD Double-struck, or blackboard, characters are often used for special mathematical symbols—for example, N, Z, Q, 0&, C (see also 12.14). These should be clearly indicated on the manuscript.
12.67 Marking mathematical subscripts and superscripts. As long as inferior and superior characters have been marked in a few places by the symbols v and a (see examples below), and new characters or symbols are iden tified when they first appear, a typesetter should have no difficulty inter preting the manuscript. If the spatial relationship of terms is not clearly shown in a typed or handwritten expression, the terms should be marked to avoid ambiguity. For example, given the copy 12.68 MATHEMATICS IN TYPE it may not be clear from the manuscript whether this means or X or X^.
The expression should therefore be marked in one of the following ways for complete clarity: $ or 4 or X^.
tl The examples above show the subscripts and superscripts aligned, or stacked. See 12.38 for discussion and examples of staggered subscripts and superscripts.
12.68 Examples of marked mathematical copy. Figure 12.1 shows a page of a paper manuscript as marked initially by the author and then by the editor before being sent to the typesetter. The author’s marks merely identify ambiguous symbols. Figure 12.2 shows that same page set in type. Figure 12.3 shows the LaTeX source code that would generate the first part of figure 12.2. Signs and symbols that could be misread by the typesetter should be clearly identified on a paper manuscript by marginal notations or in a separate list. For lists of symbols and special characters commonly used in mathematics, see table 12.1. Illegible handwriting and uniden tifiable signs and symbols can reduce composition speed and result in time-consuming and costly corrections. Certain letters, numbers, and symbols can easily be misread, especially when Greek, Fraktur, script, and sans serif letters are handwritten rather than typed. Some of the characters that cause the most difficulty are shown in table 12.4.
for n f w, since b £ G.
u(x, t) is a rectangular array of and n columns: An m / n matrix A over a field F The temperature function is A «X1 2 al ‘2X 1 a n 2 l2 2 a n ‘1 The modulus of the correlation (2.4), where M m i n coefficient of X^ and X2 is a2 From equation ,(a + b) the sum being x+4x,y , (i, + a + b - kh - 1 for || xJI > 0 AX-»0 b extended over all sets (i To summarize our findings: f ig u r e 12.1. An example of typewritten and hand-marked mathematical copy. (Note that this page is not intended to make mathematical sense but is merely meant to illustrate some of the issues that may arise in preparing mathematical copy.) Therefore Fxn c GC\B„ and Fxn C\Bm =0 for n m, since b & G. The temperature function is 2 / u2zc2kt\ . nnx J .
f(x') sin —~—dx. (3.1) An m x n matrix A over a field F is a rectangular array of mn elements af in F, arranged in m rows and n columns: A = ’«I1 «2 • .. an} ~ «I2 «22 ■ • an2 aflm_ The modulus of the correlation coefficient of A) and Xz is A^l/yrill |(.X2|| for ||A/|| > 0, 2 = 1,2.
Hence 9F a?
a f 1 f rx+^y lim —= lim — / Pdx + Qdy Ax-»0 AX Ax~»0 AX I Jafi -I^PA + Qdy^+P + Q.
From equation (2.4), where M = [(a + b - 1)/(A 4-1)1, we obtain / , n i nV V'/ -------Af)> av{a + b~ (-1)1 > —~---- r~-— the sum being extended over all sets (/1,..., fjw). To summarize our findings: v(z;tn) > HSi [v(x) + 0(1)1 - > v(z) + o(l) + ??i/2O(l). f ig u r e 12.2. The page of manuscript shown in figure 12.1 set in type.
\noindent Therefore $F_{x}{}"n\subset G\cap B_n$ and $F_{x}{}~n\cap B_m=\O$ for $n\ne m$, since $b\in G$. The temperature function is \begin{multline} u(x,t)=\frac{2}{L}\sum_l"\infty \exp \left (-\frac{uA2\pi"2 kt} {L-'2}\right) \sin \frac{n\pi x}{L}\ \times \int\nolimits_0~L f(x')\sin \frac{n\pi x'}{L}dx'.
\end{multline} An $m\times n$ matrix {\bf A} over a field $F$ is a rectangular array of $mn$“ele-ments~$a_{j}{}~i$ in $F$, arranged in $m$ rows and $n$ columns: \begin{align} {\bf A}=\left[ {\arraycolsep5pt\begin{array}{@{}cccc@{}} a{l}{}"l &a_{2}{}',l &Udots &a_{n}{}~l\[4pt] a_{l}{}~2 &a_{2}{}~2 &\ldots &a_{n}{p2\[4pt] .&.&\ldots &.\[4pt] a_{l}{}"m &a_{2}{}"m &\ldots &a__{n}{}“mW \end{array}} \end{align} The modulus of the correlation coefficient of $X_1$ and $X_2$ is \begin{align} \rho=|Mangle X_l,\ X_2\rangle|/\|X_l\I\ \|X_2\| \quad{\rm for}\quad \|X_l\|>0,\ 1=1,2.
\end{align} f ig u r e 12.3. LaTeX source listing that would generate a portion of the mathe matical copy shown in figure 12.2.
t a b l e 12.4. Potentially ambiguous mathematical symbols other operation signs (+, 4, etc.). Do not add space when such signs as +, or ± are used to modify symbols or expressions (-3, ±1, etc.). Do not add space when operations appear as subscripts or superscripts.
Symbols set in type3 Marginal notation to operator15 Remarks and suggestions for manuscript preparation a lc “aye” a lc Gr. alpha oc proportional to Leave medium space before and after oc and all 00 infinity binary operation signs (=, <, g , n, c, etc.).
B cap “bee” p lc Gr. beta X lc Gr. chi X cap “ex” X lc “ex” X “times” or “mult” Leave medium space before and after x and all 6 lc Gr. delta d partial differential d lc “dee” Simpler to use printer’s term “round dee.” e lc Gr. epsilon G “element of” r| lc Gr. eta n lc “en” y lc Gr. gamma t lc Gr. tau r lc “ar” t lc “tee” i i lc Gr. iota Avoid using i and i together because of similarity lc “eye” in print.
K k K lc Gr. kappa lc “kay” cap Gr. kappa cap “kay” lc “el” script “el” numeral 1 / 1 In some fonts, 1 and 1 look identical; note “el” but leave numeral unmarked; / should not be used if I is available.
t a b l e 12.4. (continued) Symbols set in typea Marginal notation to operator15 Remarks and suggestions for manuscript preparation V lc Gr. nu Avoid using v and v together because of similar V lc “vee” ity in print.
O cap “oh” Asymptotic upper bounds 0(x) and o(x) may 0 lc “oh” occur together.
0 zero 0 cap Gr. omicron o lc Gr. omicron 0 degree sign A cap Gr. lambda A wedge lc Gr. phi Preference for form <p should be specified by 0 empty or null set author; <|> more commonly used.
n product n cap Gr. pi 7T lc Gr. pi P lc Gr. rho P lc “pee” summation E cap Gr. sigma 0,9 lc Gr. theta Preference for form 9 should be specified by 0 cap Gr. theta author; 0 more commonly used.
U cap “you” u union symbol u lc Gr. upsilon p lc Gr. mu V lc Gr. nu u lc “you” U) lc Gr. omega d) round lc Gr. pi w lc “double-u” Z cap “zee” z lc “zee” 2 numeral 2 (continued) t a b l e 12.4. (continued) groups.
a Latin letters in mathematical expressions will automatically be set in italics unless marked otherwise.
bOnly if symbols, letters, or numbers are badly written or rendered in the manuscript is it necessary to identify them for the typesetter.
Symbols set in type3 Marginal notation to operator15 Remarks and suggestions for manuscript preparation prime Use apostrophe for prime if no prime available.
superscript 1 In handwritten formulas, take care to distin guish prime from superscript 1 and comma from subscript 1.
comma subscript 1 — em dash Use two hyphens for em dash; no space on either side (except in LaTeX, where a double hyphen produces an en dash).
-minus sign To indicate subtraction, leave medium space en dash on each side of sign; omit space after sign if negative quantity is represented.
multiplication dot Use centered period for multiplication dot, allowing medium space on each side; do not show space around a center dot in a chemical formula (CO3-H2).
n o t e : Symbols and letters that are commonly mistaken for each other are arranged in 13 • Quotations and Dialogue OVERVIEW 13.1 PERMISSIBLE CHANGES TO QUOTATIONS 13.7 QUOTATIONS IN RELATION TO TEXT 13.9 Run In or Set Off 13.9 Assimilation into the Surrounding Text 13.11 Quotations and Punctuation 13.13 Initial Capital or Lowercase Letter 13.18 Block Quotations 13.22 Poetry Extracts 13.25 QUOTATION MARKS 13.30 Double or Single Quotation Marks 13.30 Quotations of More than One Paragraph 13.32 Quotation Marks Omitted 13.36 SPEECH, DIALOGUE, AND CONVERSATION 13.39 DRAMA, DISCUSSIONS AND INTERVIEWS, AND FIELD NOTES 13.46 ELLIPSES 13.50 INTERPOLATIONS AND CLARIFICATIONS 13.59 ATTRIBUTING QUOTATIONS IN TEXT 13.64 Sources Following Run-In Quotations 13.68 Sources Following Block Quotations and Poetry Extracts 13.70 13.1 QUOTATIONS AND DIALOGUE Overview 13.1 Scope of this chapter—and where else to look. This chapter offers rec ommendations for incorporating words quoted from other sources—and, to a lesser degree, for presenting speech and other forms of dialogue—in text. For the use of quotation marks for purposes other than direct quo tation, see the discussions throughout 7.48-69. For quotation marks with titles of works, see the discussions in 8.156-201. For citing the sources of quotations, discussed only peripherally here, see chapters 14 and 15. For formatting block quotations in a manuscript, see 2.19,2.20; for the man uscript editor’s responsibilities regarding quoted material, see 2.61. For quotation marks in relation to surrounding text and punctuation, see 6.6, 6.9-11. For quotations of material from languages other than English, see 11.11-17.
13.2 Quotations and modern scholarship. Scholarship has always depended at least in part on the words and ideas of others. Incorporating those words and ideas is central to the act of writing and publishing. The choice between quoting, on the one hand, and merely copying, on the other, can mean the difference between properly acknowledging and crediting the ideas of others and falsely representing them as your own, thus making the conventions outlined in this chapter and in chapters 14 and 15 essen tial to modern scholarship. These conventions extend beyond scholar ship to encompass journalism and other categories of nonfiction.
13.3 Giving credit and seeking permission. Whether quoting, paraphrasing, or using others’ words or ideas to advance their own arguments, authors should give explicit credit to the source of those words or ideas. This credit often takes the form of a formal citation incorporated into a note or parenthetical reference. For a full discussion of documentation, see chapters 14 and 15. In addition, written permission may be needed, es pecially for direct quotations, as follows: for more than a line or two of a poem or a song lyric in copyright; for prose quotations of, say, more than three paragraphs or for many short passages from a work in copyright; or for any excerpt from certain unpublished materials (letters, email mes sages, interviews, and so forth). For more information about permis sions, consult chapter 4, especially 4.75-94.
13.4 When to paraphrase rather than quote. Authors drawing on the work of others to illustrate their arguments should first decide whether direct quotation or paraphrase will be more effective. Too many quotations with Overview 13.6 too little commentary can pose a distraction, and readers may choose to skip over long or frequent quotations. In some cases, authors who notice an error in a passage they wish to quote should paraphrase the original, eliminating the error. For “silent correction,” see 13.7 (item 6); for sic, see 13.61.
13.5 When quotation and attribution is unnecessary. Commonly known or readily verifiable facts can be stated without quotation or attribution unless the wording is taken directly from another source. Authors, of course, must be absolutely sure of any unattributed facts, and editors should flag anything that seems suspicious (see 13.6). Likewise, proverbs and other familiar expressions can usually be reproduced without quo tation or attribution. Of the following statements, only the last—a direct quotation—requires quotation marks and attribution (as well as a source citation, either in the text or in a note; see 13.64-72).
Until July 20,1969, no one had set foot on the moon.
The chemical symbol for gold, Au, derives from the Latin word aurum, Ithaca, New York, is located at the southern end of Cayuga Lake. No one can convince the young that practice makes perfect. If reading maketh a full man, Henry is half-empty.
but It was Shakespeare’s contemporary Francis Bacon who wrote that “reading maketh a ful man, conference a ready man, and writing an exact man.” For the treatment of unconventional spellings in quoted material (as in the quotation from Bacon), see 13.7, item 6.
13.6 Ensuring accuracy of quotations. It is impossible to overemphasize the importance of meticulous accuracy in quoting from the works of others. Authors should check every direct quotation against the original or, if the original is unavailable, against a careful transcription of the passage. This should be done before the manuscript is submitted to the publisher. Manuscript editors can help by spot-checking quotations against avail able resources to get an idea of how accurate the transcriptions are and by querying apparent errors; they may not, however, have ready access to an author’s sources, nor is it typically assumed to be an editor’s respon sibility to confirm the accuracy of quotations. Moreover, it takes far less time for authors to accurately transcribe quotations during the writing stage than for authors or editors to go back to the original sources once a work is submitted for publication. See also 2.33,2.136.
13.7 QUOTATIONS AND DIALOGUE Permissible Changes to Quotations 13.7 Permissible changes to punctuation, capitalization, and spelling. Al though in a direct quotation the wording should be reproduced exactly, the following changes are generally permissible to make a passage fit into the syntax and typography of the surrounding text. See also 13.8.
1. Single quotation marks may be changed to double, and double to single (see 13.30); punctuation relative to quotation marks should be adjusted accord ingly (see 6.9). “Smart” quotation marks and apostrophes may be imposed (see 6.115,6.117), and guillemets and other types of quotation marks from lan guages other than English may be changed to regular double or single quota tion marks (see 11.11).
2. En dashes or hyphens used as em dashes may be changed to em dashes, with any space before or after the dash or hyphen eliminated as needed (see 6.85- 92). Hyphens may be changed to en dashes in number ranges or other contexts where an en dash may be appropriate (for use of the en dash, see 6.78-84).
3. The initial letter may be changed to a capital or a lowercase letter (see 13.18-21).
4. At the end of the quotation, a period or other mark of punctuation in the original may be omitted or changed to a period or comma as required by the surrounding text; a question mark or exclamation point may be retained if it continues to apply to the word or words as quoted. For punctuation with el lipses, see 13.50-58.
5. Original note reference marks (and the notes to which they refer) may be omitted unless omission would affect the meaning of the quotation. If an orig inal note is included, the quotation may best be set off as a block quotation (see 13.9), with the note in smaller type at the end, or the note may be summa rized in the accompanying text. Note references added to a quotation must be distinguished from any note references in the original (see also 14.51). On the other hand, parenthetical text references in the original should be retained; if a parenthetical text reference is added to the original, it must be placed in square brackets (see 13.59-63).
6. Obvious typographic errors may be corrected silently (without comment or sic; see 13.61), unless the passage quoted is from an older work or a manuscript or other unpublished source where idiosyncrasies of spelling are generally preserved. If spelling and punctuation are modernized or altered for clarity, readers must be so informed in a note, in a preface, or elsewhere.
7. In quoting from early printed documents, the archaic long s (f, Unicode char acter U+017F), used to represent a lowercase s at the beginning or in the middle but never at the end of a word (“Such goodnefs of your juftice, that our foul...”), may be changed to a modern s. Similarly, a title like Vanitie and Quotations in Relation to Text 13.10 Vncertaintie may be changed to Vanitie and Uncertainties but writers or editors without a strong background in classical or Renaissance studies should gener ally be wary of changing u to v, i to j, or vice versa. See also 11.56,11.122-23.
13.8 Permissible changes to typography and layout. The following elements of typography and layout may be changed to assimilate a quotation to the surrounding text (see also 13.7): 1. The typeface or font should be changed to agree with the surrounding text.
2. Words in full capitals in the original may be set in small caps, if that is the preferred style for the surrounding text. (See also 10.8.) 3. In drama or dialogue, names of speakers may be moved from a centered posi tion to flush left.
4. Underlined words in a quoted manuscript may be printed as italics, unless the underlining itself is considered integral to the source or otherwise worthy of reproducing.
5. In quoting correspondence, such matters as paragraph indents and the posi tion of the salutation and signature may be adjusted.
For paragraph indents in block quotations, see 13.22. For reproducing poetry extracts, see 13.25-29. For permissible changes to titles of books, articles, poems, and other works, see 8.165.
Quotations in Relation to Text Run In or Set Off 13.9 Run-in and block quotations defined. Quoted text may be either run in to the surrounding text and enclosed in quotation marks, “like this,” or set off as a block quotation, or extract. Block quotations, which are not enclosed in quotation marks, always start a new line. They are further distinguished from the surrounding text by being indented (from the left and sometimes from the right) or set in smaller type or a different font from the text. These matters are normally decided by the publisher’s designer or by journal style. Authors preparing block quotations in their manuscripts can simply indent them from the left margin (see 2.19, 2.20) . See also 13.22-24. For poetry, see 13.25-29.
13.10 Choosing between run-in and block quotations. In deciding whether to run in or set off a quotation, length is usually the deciding factor. In general, a short quotation, especially one that is not a full sentence, should be run in. A hundred words or more can generally be set off as 13.11 QUOTATIONS AND DIALOGUE a block quotation. Other criteria apply, however. A quotation of two or more paragraphs is best set off (see 13.22-24), as are quoted correspon dence (if salutations, signatures, and such are included), lists (see 2.21), and any material that requires special formatting. If many quotations of varying length occur close together, running them all in may make for easier reading. But where quotations are being compared or otherwise used as entities in themselves, it may be better to set them all as block quotations, however short. For setting off poetry, see 13.25-29.
Assimilation into the Surrounding Text 13.11 Logical and grammatical assimilation of quoted text. In incorporat ing fragmentary quotations into a text, phrase the surrounding sentence in such a way that the quoted words fit into it logically and grammati cally—as if there were no quotation marks—and quoting only as much of the original as is necessary. For the incorporation of full sentences (as in the last part of the example below), see 13.13-17.
The narrator’s constant references to “malicious code and obsolete data” detract from a more fundamental issue—that we are dumping “the burden of human his tory” onto computer hard drives. It is this vision of the future that is most alarm ing: “If (when?) we run out of sources of electricity,” she asks, “will we forget who we are?” 13.12 Integrating tenses and pronouns from quoted text. In quoting verba tim, writers need to integrate tenses and pronouns into the new context.
[Original] Mr. Moll took particular pains to say to you, gentlemen, that these eleven people here are guilty of murder; he calls this a cold-blooded, deliberate and premeditated murder.
[As quoted] According to Darrow, Moll had told the jury that the eleven defen dants were “guilty of murder” and had described the murder as “cold-blooded, deliberate and premeditated.” Occasional adjustments to the original may be bracketed. This device should be used sparingly, however.
Mr. Graham has resolutely ducked the issue, saying he won’t play the game of rumormongering, even though he has “learned from [his] mistakes.” Quotations in Relation to Text 13.14 Quotations and Punctuation 13.13 Punctuation relative to closing quotation marks. For a full discussion of the use of periods, commas, and other marks of punctuation relative to closing quotation marks, see 6.9-11. See also 13.7, item 4. The rest of this section is primarily concerned with punctuation relative to the beginning of quoted material.
13.14 Comma to introduce a quotation. When it is simply a matter of iden tifying a speaker, a comma is used after said, replied, asked, and similar verbs to introduce a quotation. Such usage is more traditional than log ical, recognizing the syntactical independence of the quoted material from the surrounding text (even as the surrounding text often becomes dependent on the quoted material). A colon, though never wrong in such instances, should be used sparingly (see 13.16).
Garrett replied, “I hope you are not referring to me.” Fish writes, “What [the students] did was move the words out of a context (the faculty club door) in which they had a literal and obvious meaning into another context (my classroom) in which the meaning was no less obvious and literal and yet was different.” When the sentence is inverted and the quotation comes first (a common arrangement), a comma is usually required at the end of the quotation unless the quotation ends with a question mark or an exclamation point. See also 6.9, 6.10.
“I hope you are not referring to me,” Garrett replied.
When the quotation is interrupted, two commas are required.
“I hope,” Garrett replied, “you are not referring to me.” When the quotation is subordinated to or otherwise integrated into the surrounding text, no comma is needed (see also 13.15,13.19).
Fish observed that “what [the students] did was move the words out of a con text ...” See also 6.40, 6.42.
13.15 QUOTATIONS AND DIALOGUE 13.15 No comma to introduce a quotation. Many writers mistakenly use a comma to introduce any direct quotation, regardless of its relationship to the surrounding text. But when a quotation introduced midsentence forms a syntactical part of the surrounding sentence, no comma or other mark of punctuation is needed to introduce it, though punctuation may be required for other reasons.
Donovan made a slight bow and said he was “very glad.” One of the protesters scrawled “Long live opera!” in huge red letters.
According to one critic, Copland’s style could be called “American urban pastoral, with a touch of jazz and more than a hint of Stravinsky.” She said she would “prefer not to comment.” but Copland’s style—“American urban pastoral, with a touch of jazz and more than a hint of Stravinsky”—owes a debt to several genres and more than one continent.
She said that she would, in short, “prefer not to comment.” She said, “I prefer not to comment.” (See 13.14.) See also 6.41.
13.16 Colon to introduce a quotation. When a quotation is introduced by an independent clause (i.e., a grammatically complete sentence), a colon should be used. Such introductions may include a formal introductory phrase such as the following or as follows (see also 6.64).
The role of the author has been variously described. Henry Fielding, at the begin ning of his History of Tom Jones, defines it as follows: “An author ought to consider himself, not as a gentleman who gives a private or eleemosynary treat, but rather as one who keeps a public ordinary, at which all persons are welcome for their money.” Faraday’s conclusion was alarming: “Without significant intervention, your home town will have gone the way of Atlantis by century’s end.” A colon may also be used in place of a comma to introduce a quotation. Such a colon, never wrong, should be used consistently—for example, to Quotations in Relation to Text 13.19 introduce quotations of more than one sentence or, occasionally, to add emphasis. See also 13.14.
Garrett replied: “I hope you are not referring to me. Because if you are ...” 13.17 Period rather than colon to introduce a block quotation. Unless intro duced by as follows or other wording that requires a colon (see 6.64), a block quotation may be preceded by a period rather than a colon. Such usage should be applied consistently. See also 13.22-24.
He then took a clearly hostile position toward Poland, having characterized it as a Fascist state that oppressed the Ukrainians, the Belorussians, and others.
Under present conditions, suppression of that state will mean that there will be one less Fascist state. It will not be a bad thing if Poland suffers a defeat and thus enables us to include new territories and new populations in the socialist system.
Initial Capital or Lowercase Letter 13.18 Changing capitalization to suit syntax—an overview. Aside from proper nouns and some of the words derived from them (see 8.1), most words are normally lowercased unless they begin a sentence (or, often, a line of poetry). To suit this requirement, the first word in a quoted passage must often be adjusted to conform to the surrounding text. In most types of works, this adjustment may be done silently, as such capitalization does not normally affect the significance of the quoted matter, which is assumed to have been taken from another context (see 13.7, item 3). In some types of works, however, it may be obligatory to indicate the change by bracketing the initial quoted letter; for examples of this prac tice, appropriate to legal writing and some types of textual commentary, see 13.21.
13.19 Initial capital or lowercase—run-in quotations. When a quotation in troduced midsentence forms a syntactical part of the sentence (see also 13.15) , it begins with a lowercase letter even if the original begins with a capital.
Benjamin Franklin admonishes us to “plough deep while sluggards sleep.” With another aphorism he reminded his readers that “experience keeps a dear school, but fools will learn in no other”—an observation as true today as then.
13.20 QUOTATIONS AND DIALOGUE When the quotation has a more remote syntactic relation to the rest of the sentence, the initial letter remains capitalized.
As Franklin advised, “Plough deep while sluggards sleep.” His aphorism “Experience keeps a dear school, but fools will learn in no other” is a cogent warning to people of all ages. (See also 6.41.) On the other hand, for a quotation that is only a part of a sentence in the original but forms a complete sentence as quoted, a lowercase letter may be changed to a capital if appropriate. In the example that follows, “those” begins midsentence in the original (see 13.20).
Aristotle put it this way: “Those who are eminent in virtue usually do not stir up insurrections, always a minority.” but Aristotle believed that “those who are eminent in virtue usually do not stir up insurrections, always a minority.” 13.20 Initial capital or lowercase—block quotations. The consideration of whether to lowercase a capital letter beginning a block quotation is ex actly the same as it is for run-in quotations (see 13.19): the initial letter of a block quotation that is capitalized in the original may be lowercased if the syntax demands it. In the following example, the quotation from Aristotle in the Jowett translation (Modern Library) begins in the original with a capital letter and a paragraph indent. See also 13.22.
In discussing the reasons for political disturbances, Aristotle observes that revolutions also break out when opposite parties, e.g. the rich and the people, are equally balanced, and there is little or no middle class; for, if either party were manifestly superior, the other would not risk an attack upon them. And, for this reason, those who are eminent in virtue usually do not stir up insurrections, always a minority. Such are the beginnings and causes of the disturbances and revolutions to which every form of govern ment is liable. (Politics 5.4) On the other hand, the capital should be retained—or a lowercase letter should be changed to a capital—if the syntax requires it. See also 13.16.
In discussing the reasons for political disturbances, Aristotle makes the following observations: Quotations in Relation to Text 13.22 Revolutions also break out when opposite parties, e.g. the rich and the people, are equally balanced, and there is little or no middle class;...
13.21 Brackets to indicate a change in capitalization. In some legal writing, close textual analysis or commentary, and other contexts, it is consid ered obligatory to indicate any change in capitalization by brackets. Al though this practice is unnecessary in most writing, in contexts where it is considered appropriate it should be employed consistently throughout a work.
According to article 6, section 6, she is given the power “ [t]o extend or renew any existing indebtedness.” “[Rjeal estates may be conveyed by lease and release, or bargain and sale,” ac cording to section 2 of the Northwest Ordinance.
Let us compare Aristotle’s contention that “[inferiors revolt in order that they may be equal, and equals that they may be superior” (Politics 5.2), with his later observation that “ [r] evolutions also break out when opposite parties, e.g. the rich and the people, are equally balanced” (5.4).
Block Quotations 13.22 Block quotations of more than one paragraph. Quoted material of more than a paragraph, even if very brief, is best set off as a block quotation. (For a less desirable alternative, see 13.32.) A multiparagraph block quo tation should generally reflect the paragraph breaks of the original. But if the first paragraph quoted includes the beginning of that paragraph, it need not start with a first-line paragraph indent. Subsequent paragraphs in the quotation should be indicated either by first-line paragraph in dents or (less desirably) by extra line space between the paragraphs (see also 13.24). The following example, from Jane Austen’s Pride and Preju dice, includes four full paragraphs: He began to wish to know more of her, and as a step towards conversing with her himself, attended to her conversation with others. His doing so drew her notice. It was at Sir William Lucas’s, where a large party were assembled.
“What does Mr. Darcy mean,” said she to Charlotte, “by listening to my con versation with Colonel Forster?” “That is a question which Mr. Darcy only can answer.” “But if he does it any more I shall certainly let him know that I see what he 13.23 QUOTATIONS AND DIALOGUE is about. He has a very satirical eye, and if I do not begin by being impertinent myself, I shall soon grow afraid of him.” If the first part of the opening paragraph were to be omitted, it would still begin flush left. For ellipses at the beginning of paragraphs, see 13.56.
13.23 Block quotations beginning in text. A long quotation may begin with a few words run in to the text. This device should be used only when text intervenes between the quoted matter in the text and its continua tion.
“There is no safe trusting to dictionaries and definitions,” observed Charles Lamb.
We should more willingly fall in with this popular language, if we did not find brutality sometimes awkwardly coupled with valour in the same vo cabulary. The comic writers ... have contributed not a little to mislead us upon this point. To see a hectoring fellow exposed and beaten upon the stage, has something in it wonderfully diverting. (“Popular Fallacies,” Es says of Elia, 277) “In short,” says Crane, summarizing Gordon’s philosophy, there has been “almost a continual improvement” in all branches of hu man knowledge;...
A permissible alternative is to set off the entire quotation, enclosing the intervening words of text in brackets.
There is no safe trusting to dictionaries and definitions [observed Charles Lamb]. We should more willingly...
13.24 Text following a block quotation or extract. If the text following a block quotation or extract (whether prose or poetry) is a continuation of the paragraph that introduces the quotation or extract, it begins flush left. If, on the other hand, the resuming text constitutes a new paragraph, it receives a paragraph indent. The decision in each case is usually made by the author or, failing that, may be left to the editor (see 2.19, 2.20). In works where each new paragraph is to appear flush left, distinguished only by extra line space, such a distinction may have to be ignored (or it can be signaled by imposing more line space before new paragraphs than before continued text).
Quotations in Relation to Text 13.27 Poetry Extracts 13.25 Setting off poetry. In a published work, two or more lines of verse are best set off as an extract. (In a note, set off three or more; see 14.38.) A poetry extract, if isolated, is often visually centered on the page between the left and right margins (usually relative to the longest line), but if two or more stanzas of the same poem appear on the same page, a uniform indent from the left may work better (see 13.26). A half line to a full line of space should appear between stanzas. Within each piece or stanza, the indentation pattern of the original should be reproduced (but indents should be distinguished from runover lines; see 13.27). For placement of the source, see 13.71. For advice on formatting poetry extracts in a man uscript, see 2.20.
Sure there was wine Before my sighs did drie it: there was corn Before my tears did drown it.
Is the yeare onely lost to me?
Have I no bayes to crown it?
No flowers, no garlands gay? all blasted?
All wasted?
(George Herbert, “The Collar”) If the quotation does not begin with a full line, space approximating the omitted part should be left.
there was corn Before my tears did drown it.
For text that follows an extract, see 13.24.
13.26 Uniform indents for poetry. Where all or most poetry extracts consist of blank verse (as in studies of Shakespeare) or are very long, uniform indents from the left margin usually work best (e.g., a left indent that matches the one, if any, used for prose extracts).
I have full cause of weeping, but this heart Shall break into a hundred thousand flaws Or ere I’ll weep. O Fool! I shall go mad.
13.27 Long lines and runovers in poetry. Runover lines (the remainder of lines too long to appear as a single line) are usually indented one em from the 13.28 QUOTATIONS AND DIALOGUE line above, as in the following quotation from Walt Whitman’s “Song of Myself”: My tongue, every atom of my blood, form’d from this soil, this air, Born here of parents born here from parents the same, and their parents the same, I, now thirty-seven years old in perfect health begin, Hoping to cease not till death.
Runover lines, although indented, should be distinct from new lines de liberately indented by the poet (as in the Herbert poem quoted in 13.25). Generally, a unique and uniform indent for runovers will be enough to accomplish this. See also 2.20.
13.28 Quotation marks in poems. Quotation marks at the start of a line can usually be aligned with the other lines in the excerpt.
He holds him with his skinny hand.
“There was a ship,” quoth he.
“Hold off! unhand me, grey-beard loon!” Eftsoons his hand dropt he.
Some publishers prefer instead to place quotation marks at the start of a line of poetry outside the alignment of the poem, with lines left-aligned as if the quotation marks were not there. This practice, not followed by Chicago, may be impractical in certain electronic publication formats.
13.29 Run-in poetry quotations. If space or context in the text or in a note requires that two or more lines be run in, the lines are separated by a slash, with one space on either side (in printed works, a thin space to an en space).
Andrew Marvell’s praise of John Milton, “Thou has not missed one thought that could be fit, / And all that was improper does omit” (“On Paradise Lost”), might well serve as our motto.
For running in more than one stanza (to be avoided if at all possible), see 13.34.
Quotation Marks 13.31 Quotation Marks Double or Single Quotation Marks 13.30 Quotations and "quotes within quotes." Quoted words, phrases, and sentences run into the text are enclosed in double quotation marks. Single quotation marks enclose quotations within quotations; double marks, quotations within these; and so on. (The practice in the United Kingdom and elsewhere is often the reverse: single marks are used first, then double, and so on.) When the material quoted consists entirely of a quotation within a quotation, only one set of quotation marks need be employed (usually double quotation marks). For permissible changes from single to double quotation marks and vice versa, see 13.7 (item 1); see also 13.63. For dialogue, see 13.39.
“Don’t be absurd!” said Henry. “To say that ‘I mean what I say’ is the same as ‘I say what I mean’ is to be as confused as Alice at the Mad Hatter’s tea party. You remember what the Hatter said to her: ‘Not the same thing a bit! Why you might just as well say that “I see what I eat” is the same thing as “I eat what I see”!”’ Note carefully not only the placement of the single and double closing quotation marks but also that of the exclamation points in relation to those marks in the example above. Exclamation points, like question marks, are placed just within the set of quotation marks ending the ele ment to which such terminal punctuation belongs. See also 6.9-11.
13.31 Quotation marks in block quotations. Although material set off as a block quotation is not enclosed in quotation marks, quoted matter within the block quotation is enclosed in double quotation marks—in other words, treated as it would be in otherwise unquoted text (see 13.30). An author or editor who changes a run-in quotation to a block quotation must delete the opening and closing quotation marks and change any internal ones. The following examples illustrate the same material first in run-in form and then as a block quotation: The narrator then breaks in: “Imagine Bart’s surprise, dear reader, when Emma turned to him and said, contemptuously, ‘What “promise”?”’ The narrator then breaks in: Imagine Bart’s surprise, dear reader, when Emma turned to him and said, contemptuously, “What ‘promise’?” 13.32 QUOTATIONS AND DIALOGUE Similarly, converting a block quotation to a run-in quotation requires adding and altering quotation marks. For interpolations that include quoted matter, see 13.63.
Quotations of More than One Paragraph 13.32 Quotation marks across paragraphs. Quoted material of more than one paragraph should be presented as a block quotation if at all possible (see 13.10). If for some reason such a passage must be run in to the surround ing text, a quotation mark is needed at the beginning of the quotation and at the beginning of each new paragraph but at the end of only the final paragraph. (Note that each successive paragraph must begin on a new line, as in the original.) The same practice is followed in dialogue when one speaker’s remarks extend over more than one paragraph (see 13.39). See also 13.33.
13.33 Quotations within quotations across paragraphs. Quoted material that contains an interior quotation that runs for more than one paragraph (as in extended dialogue from one speaker quoted in a work of fiction) should be presented as a block quotation if it all possible; original quota tion marks should be preserved, except as described in 13.7. If for some reason such a passage must be run in to the surrounding text, quotation marks should be used as described in 13.32, with the following excep tions: a single quotation mark appears at the beginning and at the end of the interior quotation as a whole, and both double and single marks appear before each new paragraph belonging to the interior quotation. See also 13.30.
13.34 Quoting more than one stanza of poetry. A poetry quotation that spans more than one stanza should be presented as an extract, if at all possible (see 13.25). If it must be run in to the text (set off by opening and closing quotation marks), two slashes (//), with a space before and after, should appear between stanzas. For the use of the slash between run-in lines of poetry, see 13.29.
13.35 Quoting letters in their entirety. A letter quoted in its entirety should be set off as a block quotation. In the undesirable event that it must be run in, it should carry an opening quotation mark before the first line (includ ing the salutation) and before each new paragraph (each of which must begin on a new line, as in the original). A closing quotation mark appears only after the last line (often the signature). See also 13.32.
Quotation Marks 13.38 Quotation Marks Omitted 13.36 Epigraphs. Quotation marks are not used around epigraphs (quotations used as ornaments preceding a text, usually to set the tone for what fol lows, rather than as illustration or documentation). Like block quota tions, epigraphs receive a distinctive typographic treatment—often being set in a smaller typeface and indented from the right or left, and sometimes italicized. Treatment of sources, which are usually set on a separate line, also varies, though more than one epigraph used in the same work should receive consistent treatment. For more on sources, see 13.70-72. See also 1.37.
Oh, what a tangled web we weave, When first we practice to deceive! —Sir Walter Scott It is a truth universally acknowledged, that a single man in possession of a good fortune, must be in want of a wife.
Jane Austen, Pride and Prejudice 13.37 Decorative initials ("drop caps" and raised initials). When the first word of a chapter or section opens with a large raised or dropped initial letter, and this letter belongs to the beginning of a run-in quotation, the open ing quotation mark is often omitted.
O F THE MAKING o f ma n y b o o k s there is no end,” de clared an ancient Hebrew sage, who had himself magnifi cently aggravated the situation he was decrying.
If the opening quotation mark is included, it should appear in the same size and with the same vertical alignment as the regular text.
13.38 Maxims, questions, and the like. Maxims, mottoes, rules, and other familiar expressions, sometimes enclosed in quotation marks, are dis cussed in 6.41 and 7.62. Questions that do not require quotation marks are discussed in 6.42 and 6.69.
13.39 QUOTATIONS AND DIALOGUE Speech, Dialogue, and Conversation 13.39 Direct discourse. Direct discourse or dialogue is traditionally enclosed in quotation marks. A change in speaker is usually indicated by a new paragraph, as in the following excerpt from Huckleberry Finn: “Ransomed? What’s that?” “I don’t know. But that’s what they do. I’ve seen it in books; and so of course that’s what we’ve got to do.” “But how can we do it if we don’t know what it is?” “Why, blame it all, we’ve got to do it. Don’t I tell you it’s in the books? Do you want to go to doing different from what’s in the books, and get things all muddled up?” If one speech (usually a particularly long one) occupies more than a para graph, opening quotation marks are needed at the beginning of each new paragraph, with a closing quotation mark placed at the end of only the final paragraph (see also 13.32).
13.40 Single-word speech. Words such asjyes, no, where, how, and why, when used singly, are not enclosed in quotation marks except in direct dis course. See also 6.69.
Ezra always answered yes; he could never say no to a friend.
Please stop asking why.
but “Yes,” he replied weakly.
Again she repeated, “Why?” 13.41 Faltering speech or incomplete thoughts. An ellipsis may be used to suggest faltering or fragmented speech accompanied by confusion or in security. In the examples below, note the relative positions of the ellipses and other punctuation. (For the use of ellipses to indicate editorial omis sions, see 13.50-58.) “I... I... that is, we... yes, we have made an awful blunder!” “The ship... oh my God!... it’s sinking!” cried Henrietta.
“But... but...,” said Tom.
Interruptions or abrupt changes in thought are usually indicated by em dashes. See 6.87.
Speech, Dialogue, and Conversation 13.44 13.42 Alternatives to quotation marks. In some languages, em dashes are used to present dialogue; for examples, see 11.31,11.47,11.64,11.101. For the use of guillemets (« »), see 11.29,11.41,11.47,11.63,11.101.
13.43 Unspoken discourse. Thought, imagined dialogue, and other internal discourse (also called interior discourse) may be enclosed in quotation marks or not, according to the context or the writer’s preference. If a thought begins midsentence, it normally begins with a capital letter (as in the third example). See also 6.42.
“I don’t care if we have offended Morgenstern,” thought Vera. “Besides,” she told herself, “they’re all fools.” Why, we wondered, did we choose this route?
She thought, If there’s an app for that, I’ll need to program it myself.
The following passage from James Joyce’s Ulysses illustrates interior monologue and stream of consciousness without need of quotation marks: Reading two pages apiece of seven books every night, eh? I was young. You bowed to yourself in the mirror, stepping forward to applause earnestly, striking face. Hurray for the Goddamned idiot! Hray! No-one saw: tell no-one. Books you were going to write with letters for titles. Have you read his F?
13.44 Numerals in direct discourse. In quoting directly from spoken sources (e.g., interviews, speeches, or dialogue from a film or a play), or when writing direct discourse for a drama or a work of fiction, numbers that might otherwise be rendered as numerals can often be spelled out. This practice requires editorial discretion. Years can usually be rendered as numerals, as can trade names that include numerals. And for dialogue that includes more than a few large numbers, it may be more practical to use numerals. See also 9.2, 9.7.
Jarred’s answer was a mix of rage and humiliation: “For the last time, I do not have seven hundred thirty-seven dollars and eleven cents! I don’t even have a quarter for the parking meter, for that matter.” Like most proofreaders, she is a perfectionist. “I’m never happy with a mere ninety-nine and forty-four one-hundredths percent.” 13.45 QUOTATIONS AND DIALOGUE but “Do you prefer shopping at 7-Eleven or Circle K?” “I didn’t get around to reading Nineteen Eighty-Four until 1985,” he finally ad mitted.
“My mobile number is 555-0133.” 13.45 Indirect discourse. Indirect discourse, which paraphrases dialogue, takes no quotation marks. See also 6.42.
Tom told Huck they had to do it that way because the books said so. Very well, you say, but is there no choice?
Drama, Discussions and Interviews, and Field Notes 13.46 Drama. In plays, the speaker’s name is usually set in a font distinct from the dialogue—caps and small caps, for example, or all small caps. The dialogue is not enclosed in quotation marks and is usually set with hang ing indents (a style often used for bibliographies and indexes and illus trated in the following examples; see also 2.11).
R. Ro is t e r Do is t e r . Except I have her to my wife, I shall run mad. M. Me r y g r e e k e . Nay, “unwise” perhaps, but I warrant you for “mad.” Stage directions are usually italicized.
Al g e r n o n . That is quite a different matter. She is my aunt. (Takes plate from be low.) Have some bread and butter. The bread and butter is for Gwendolen. Gwendolen is devoted to bread and butter.
ja c k , advancing to table and helping himself. And very good bread and butter it is too.
13.47 Shared lines and runover lines in verse drama. In quoted excerpts from drama in verse, a single line of verse shared between two speakers in a play should be presented such that the second line continues where the first has left off (as in the example below, where the line begun by Barnardo is finished by Marcellus). Runover lines may be indicated as Drama, Discussions and Interviews, and Field Notes 13.49 in poetry, by an indent of one em or more from the line above (see also 13.27).
Ba r n a r d o .
It would be spoke to.
Ma r c e l l u s . Speak to it, Horatio.
Ho r a t io .
What art thou that usurp’st this time of night, Together with that fair and warlike form In which the majesty of buried Denmark Did sometimes march? By heaven, I charge thee, speak.
13.48 Discussions and interviews. The transcription of a discussion or an in terview is treated in much the same way as drama (see 13.46). Interjec tions such as “laughter” are italicized and enclosed in brackets (rather than parentheses, as in drama; see also 13.59-63). Paragraph indents are usually preferred to hanging indents (though hanging indents, which al low easier identification of the speaker, may work better if several speak ers’ names appear and the comments are relatively brief). Although speakers’ names are usually followed by a period, a colon may be used instead. To save space, names may be abbreviated after their first appear ance.
In t e r v ie w e r . Yo u weren’t thinking that this technology would be some thing you could use to connect to the Office of Tibet in New York or to different Tibet support groups in Europe?
Re s po n d e n t . No . Nobody seemed to have anything to do with GreenNet in the Tibet world at that time. That came much later. That’s not really right. I spe cifically wasn’t interested in connecting to the community of Tibet martyrs and fellow sufferers [laughs] and the emotional pathological there-but-for-the-grace- of-god-go-I people.
An author’s previously unpublished transcriptions of interviews or dis cussions can usually be edited for such matters as capitalization, spelling, and minor grammatical slips or elisions. If an author has imposed more significant alterations, these should be explained in a note, a preface, or elsewhere. See also 4.77. Previously published transcriptions should be quoted as they appear in the original source.
13.49 Case studies and ethnographic field notes. An author’s transcriptions of unpublished ethnographic field notes or material from case studies 13.50 QUOTATIONS AND DIALOGUE (the author’s own or those of a colleague or assistant) pose a special case. Unlike quotations from published sources or transcriptions of in terviews, such material need not be presented verbatim—whether pre sented as quotations or woven into the text. Rather, it should be edited for consistency—with related material and with the surrounding text—in matters of spelling, capitalization, punctuation, treatment of numbers, and so forth. And even if the author is in possession of signed releases, any otherwise anonymous subjects or informants should generally be presented under pseudonyms; a note should be appended to the text to indicate that this is the case. Whereas modifications intended to main tain participants’ anonymity are acceptable, authors must take care to ensure that any changes do not lead to misrepresentation. Editors should query authors if it is not clear that appropriate provisions have been made. See also 4.77.
Ellipses 13.50 Ellipses defined. An ellipsis is a series of three dots used to signal the omission of a word, phrase, line, paragraph, or more from a quoted pas sage. Such omissions are made of material that is considered irrelevant to the discussion at hand (or, occasionally, to adjust for the grammar of the surrounding text). Chicago style is to use three spaced periods (but see 13.53) rather than another device such as asterisks. These dots (which are sometimes referred to as suspension points) may also be used to indicate faltering speech or incomplete thoughts (see 13.41). The dots in an ellipsis must always appear together on the same line (through the use of nonbreaking spaces; see 6.121), along with any punctuation that immediately follows; if an ellipsis appears at the beginning of a new line, any punctuation that immediately precedes it (including a period) will appear at the end of the line above. If they prefer, authors may prepare their manuscripts using their word processor’s nonbreaking three-dot ellipsis character (Unicode 2026), usually with a space on either side; editors following Chicago style will replace these with spaced periods. For bracketed ellipses, see 13.58. See also 11.2.
13.51 Danger of skewing meaning with ellipses. Since quotations from an other source have been separated from their original context, particular care needs to be exercised when eliding text to ensure that the sense of the original is not lost or misrepresented. A deletion must not result in a statement alien to the original material. And in general, ellipses should not be used to join two statements that are far apart in the original. Accu racy of sense and emphasis must accompany accuracy of transcription.
Ellipses 13.54 13.52 When not to use an ellipsis. Ellipses are normally not used (1) before the first word of a quotation, even if the beginning of the original sentence has been omitted; or (2) after the last word of a quotation, even if the end of the original sentence has been omitted, unless the sentence as quoted is deliberately incomplete (see 13.55).
13.53 Ellipses with periods. A period is added before an ellipsis to indicate the omission of the end of a sentence, unless the sentence is deliberately in complete (see 13.55). Similarly, a period at the end of a sentence in the original is retained before an ellipsis indicating the omission of material immediately following the period. What precedes and, normally, what follows the four dots should be grammatically complete sentences as quoted, even if part of either sentence has been omitted. A complete passage from Emerson’s essay “Politics” reads: The spirit of our American radicalism is destructive and aimless: it is not loving; it has no ulterior and divine ends; but is destructive only out of hatred and self ishness. On the other side, the conservative party, composed of the most mod erate, able, and cultivated part of the population, is timid, and merely defensive of property. It vindicates no right, it aspires to no real good, it brands no crime, it proposes no generous policy, it does not build, nor write, nor cherish the arts, nor foster religion, nor establish schools, nor encourage science, nor emancipate the slave, nor befriend the poor, or the Indian, or the immigrant. From neither party, when in power, has the world any benefit to expect in science, art, or humanity, at all commensurate with the resources of the nation.
The passage might be shortened as follows: The spirit of our American radicalism is destructive and aimless.... On the other side, the conservative party... is timid, and merely defensive of property.... It does not build, nor write, nor cherish the arts, nor foster religion, nor establish schools.
Note that the first word after an ellipsis is capitalized if it begins a new grammatical sentence. Some types of works require that such changes to capitalization be bracketed; see 13.21. See also 13.58.
13.54 Ellipses with other punctuation. Other punctuation appearing in the original text—a comma, a colon, a semicolon, a question mark, or an ex clamation point—may precede or follow an ellipsis (except when a period precedes the ellipsis; see 13.53). Whether to include the additional mark of punctuation depends on whether keeping it aids comprehension or is required for the grammar of the sentence. Placement of the other punc 13.55 QUOTATIONS AND DIALOGUE tuation depends on whether the omission precedes or follows the mark; when the omission precedes it, a nonbreaking space should be used be tween the ellipsis and the mark of punctuation to prevent the mark from carrying over to the beginning of a new line (see 13.50). Note that this before-or-after distinction is usually not made with periods, where— without the aid of brackets (see 13.58)—it is likely to go unnoticed (see 13.53) .
It does not build,... nor cherish the arts, nor foster religion.
As to Endymion, was it a poem ... to be treated contemptuously by those who had celebrated, with various degrees of complacency and panegyric, Paris, and Woman, and A Syrian Tale ... ? Are these the men who ... presumed to draw a parallel between the Rev. Mr. Milman and Lord Byron?
When a species ... increases inordinately in numbers in a small tract, epidem ics... often ensue: and here we have a limiting check independent of the struggle for life. But even some of these so-called epidemics appear to be due to parasitic worms...: and here comes in a sort of struggle between the parasite and its prey.
13.55 Ellipses at the ends of deliberately incomplete sentences. An ellipsis alone (i.e., three dots with no additional period) is used at the end of a quoted sentence that is deliberately left grammatically incomplete.
Everyone knows that the Declaration of Independence begins with the sentence “When, in the course of human events...” But how many people can recite more than the first few lines of the document?
Have you had a chance to look at the example beginning “The spirit of our Amer ican radicalism ...”?
Note that no space intervenes between a final ellipsis point and a closing quotation mark.
13.56 Ellipses for the omission of whole or partial paragraphs. The omission of one or more paragraphs within a quotation is indicated by a period followed by an ellipsis at the end of the paragraph preceding the omit ted part (see also 13.53). (If that paragraph ends with an incomplete sen tence, only the three-dot ellipsis is used; see 13.55.) If the first part of a paragraph is omitted within a quotation, a paragraph indent and an el lipsis appear before the first quoted word. It is thus possible to use an ellipsis both at the end of one paragraph and at the beginning of the next, Ellipses 13.58 as illustrated in the following excerpt from Alexander Pope’s “Letter to a Noble Lord”: I should be obliged indeed to lessen this respect, if all the nobility... are but so many hereditary fools, if the privilege of lords be to want brains, if noblemen can hardly write or read....
Were it the mere excess of your Lordship’s wit, that carried you thus trium phantly over all the bounds of decency, I might consider your Lordship on your Pegasus, as a sprightly hunter on a mettled horse....
... Unrivalled as you are, in making a figure, and in making a speech, methinks, my Lord, you may well give up the poor talent of making a distich.
13.57 Ellipses in poetry and verse drama. Omission of the end of a line of verse is indicated by a period followed by an ellipsis if what precedes them is a complete grammatical sentence (see 13.53); otherwise, only the three-dot ellipsis is used (as in the Poe example; see also 13.55). The omission of a full line or of several consecutive lines within a quoted poem or drama in verse is indicated by one line of widely spaced dots approximately the length of the line above (or of the missing line, if that is determinable). See also 13.25-29.
Type of the antique Rome! Rich reliquary Of lofty contemplation...
(Edgar Allan Poe, “The Coliseum”) She would dwell on such dead themes, not as one who remembers, But rather as one who sees.
Past things retold were to her as things existent, Things present but as a tale.
(Thomas Hardy, “One We Knew”) This royal throne of kings, this sceptred isle, This blessed plot, this earth, this realm, this England.
(Richard 11,2.1.40-50) 13.58 Bracketed ellipses. Especially in languages that make liberal use of el lipses for faltering speech or incomplete thoughts, it is a common prac tice to bracket ellipses that are inserted to indicate an omission in quoted text (see, e.g., 11.32; see also 11.19). In an English context where ellipses are needed for a quotation that includes ellipses in the original text, the 13.59 QUOTATIONS AND DIALOGUE latter may be explained at each instance in a note (e.g., “ellipsis in orig inal”; see also 13.62); for more than a few such instances, authors may choose instead to bracket their own ellipses, but only after explaining such a decision in a note, a preface, or elsewhere. The rules for bracketed ellipses are the same as the rules outlined in the rest of this section, with one exception—a period is placed before or after the ellipsis depending on its placement in the original. Compare the passage that follows to the passages in 13.53.
The spirit of our American radicalism is destructive and aimless [...]. On the other side, the conservative party [...] is timid, and merely defensive of property. [...] It does not build, nor write, nor cherish the arts, nor foster religion, nor es tablish schools.
Note that a space appears before an opening bracket; a space appears after a closing bracket except when a period, comma, or other mark of punctuation follows. Within brackets, the sequence is bracket-period- space-period-space-period-bracket. Nonbreaking spaces are needed only for the two spaces between the periods within the brackets (see also 6.121). Bracketed ellipses may also be used in source citations to shorten very long titles; see 14.97.
Interpolations and Clarifications 13.59 Missing or illegible words. In reproducing or quoting from a document in which certain words are missing or illegible, an author may use ellipses (see 13.50-58), a bracketed comment or guess (sometimes followed by a question mark), or both. If ellipses alone are used (useful for a passage with more than a few lacunae), their function as a stand-in for missing or illegible words must be explained in the text or in a note. If a bracketed gloss comes from a different source, the source must be cited in a note or elsewhere. See also 6.99.
If you will assure me of your .. . [illegible], I shall dedicate my life to your en deavor.
She marched out the door, headed for the [president’s?] office.
A 2-em dash (see 6.93), sometimes in combination with an interpolated guess, may also be used for missing material. As with ellipses, this de vice should be used consistently and should be explained (in prefatory material or a note).
Interpolations and Clarifications 13.62 I have great marvel that ye will so soon incline to every man his device and [coun sel and---- ] specially in matters of small importance---- ] yea, and as [it is] re ported [unto me ] causes as meseemeth th [a ] nothing to [ ]ne gentle women.
13.60 Bracketed clarifications. Insertions may be made in quoted material to clarify an ambiguity, to provide a missing word or letters (see 13.59), to correct an error, or, in a translation, to give the original word or phrase where the English fails to convey the exact sense. Such interpolations, which should be kept to a minimum lest they irritate or distract readers, are enclosed in brackets (never in parentheses). See also 6.99,13.12.
Marcellus, doubtless in anxious suspense, asks Barnardo, “What, has this thing [the ghost of Hamlet’s father] appear’d again tonight?” “Well,” said she, “if Mr. L[owell] won’t go, then neither will I.” Saha once remarked of Nehru that “his position in this country can be described by a phrase which Americans use with respect to Abraham Lincoln [read: George Washington], ‘first in war, first in peace.’” 13.61 "Sic." Literally meaning “so,” “thus,” “in this manner,” and traditionally set in italics, sic may be inserted in brackets following a word misspelled or wrongly used in the original. This device should be used only where it is relevant to call attention to such matters (and especially where readers might otherwise assume the mistake is in the transcription rather than the original) or where paraphrase or silent correction is inappropriate (see 13.4,13.7, item 6).
In September 1862, J. W. Chaffin, president of the Miami Conference of Wesleyan Methodist Connection, urged Lincoln that “the confiscation law past [sic] at the last session of Congress should be faithfully executed” and that “to neglect this national righteousness” would prove “disastrous to the American people.” Sic should not be used merely to call attention to unconventional spell ings, which should be explained (if at all) in a note or in prefatory ma terial. Similarly, where material with many errors and variant spellings (such as a collection of informal letters) is reproduced as written, a pref atory comment or a note to that effect will make a succession of sics un necessary.
13.62 "Italics added." An author wishing to call particular attention to a word or phrase in quoted material may italicize it but must tell readers what 13.63 QUOTATIONS AND DIALOGUE has been done, by means of such formulas as “italics mine,” “italics added,” “emphasis added,” or “emphasis mine.” Occasionally it may be important to point out that italics in a quotation were indeed in the origi nal. Here the usual phrase is “italics in the original” or, for example, “De Quincey’s italics.” This information appears either in parentheses fol lowing the quotation or in a source note to the quotation. If there are ital ics in the original of the passage quoted, the information is best enclosed in brackets and placed directly after the added italics. Consistency in method throughout a work is essential.
You have watched the conduct of Ireland in the difficult circumstances of the last nine months, and that conduct I do not hesitate to risk saying on your behalf has evoked in every breast a responsive voice of sympathy, and an increased convic tion that we may deal freely and yet deal prudently with our fellow-subjects be yond the Channel. Such is your conviction. (William Ewart Gladstone, October 1891; italics added) In reality not one didactic poet has ever yet attempted to use any parts or pro cesses of the particular art which he made his theme, unless in so far as they seemed susceptible of poetic treatment, and only because they seemed so. Look at the poem of Cyder by Philips, of the Fleece by Dyer, or (which is a still weightier example) at the Georgies of Virgil,—does any of these poets show the least anxiety for the correctness of your principles [my italics], or the delicacy of your manipu lations, in the worshipful arts they affect to teach? (Thomas De Quincey, “Essay on Pope”) 13.63 Interpolations requiring quotation marks. Occasionally a bracketed or parenthetical interpolation that includes quotation marks appears in material already enclosed in quotation marks. In such cases, the double/ single rule (see 13.30) does not apply; the quotation marks within the brackets may remain double.
“Do you mean that a double-headed calf [“two-headed calf” in an earlier version] has greater value than two normal calves? That a freak of nature, even though it cannot survive, is to be more highly treasured for its rarity than run-of-the-mill creatures are for their potential use?” Attributing Quotations in Text 13.64 Use of parentheses with in-text citations. If the source of a direct quo tation is not given in a note, it is usually placed in the text in parentheses. Although the source normally follows a quotation, it may come earlier if Attributing Quotations in Text 13.67 it fits more smoothly into the introductory text (as in the second example in 13.65). The examples in this section focus on full and short forms of parenthetical citation that may be needed in shorter works with no notes or bibliography or to provide in-text citations to a frequently quoted work. The advice in this section on placement relative to surrounding text is intended to supplement the system of notes and bibliography cov ered in chapter 14. For author-date references, see chapter 15.
13.65 Full in-text citation. An entire source may be given in parentheses im mediately following a quotation (as in the first example below), or some of the data may be worked into the text (as in the second example), with details confined to parentheses. See also 6.101. For more on the proper form for full citations, see 14.23.
“If an astronaut falls into a black hole, its mass will increase, but eventually the energy equivalent of that extra mass will be returned to the universe in the form of radiation. Thus, in a sense, the astronaut will be ‘recycled’” (Stephen W. Haw king, A Brief History of Time: From the Big Bang to Black Holes [New York: Bantam Books, 19881,112).
In their introduction to Democracy in America (University of Chicago Press, 1999), translators Harvey Mansfield and Delba Winthrop write that Tocqueville “shows that the people are sovereign, whether through the Constitution or despite it, and he warns of the tyranny of the majority” (xvii).
13.66 Shortened citations or "ibid." with subsequent in-text citations. If a second passage from the same source is quoted close to the first and there is no intervening quotation from a different source, the author’s name or ibid, (set in roman) may be used in the second parenthetical reference (e.g., “Hawking, 114” or “ibid., 114”). In a departure from previous edi tions of this manual, Chicago now prefers the first form as being the less ambiguous option, though ibid, may be appropriate if used consistently. Avoid overusing either form: for more than the occasional repeated ref erence to the same source—as in an extended discussion of a work of fiction—only a parenthetical page number is necessary. Whichever form is used, if a quotation from another source has intervened, a shortened reference that includes the title may be necessary (e.g., “Hawking, Brief History of Time, 114”). For more on shortened citations, see 14.29-36; for the use of shortened citations versus ibid, in notes, see 14.34.
13.67 Frequent reference to a single source cited in a note. In a work con taining notes, the full citation of a source may be given in a note at first mention, with subsequent citations made parenthetically in the text.
13.68 QUOTATIONS AND DIALOGUE This method is especially suited to literary studies that use frequent quotations from a single source. In a study of Much Ado about Nothing, for example, the note would list the edition and include wording such as “Text references are to act, scene, and line of this edition.” A parenthet ical reference to act 3, scene 4, lines 46-47, would then appear as in the example below. In references to a work of fiction, page numbers alone may be given.
“Ye light o’ love with your heels! then, if your husband have stables enough, you’ll see he shall lack no barns,” says Beatrice (3.4.46-47).
Where a number of such sources (or different editions of a single source) are used in the same work, the title (or edition) may need to be indi cated in the parenthetical references; it may be advisable to devise an abbreviation for each and to include a list of the abbreviations at the be ginning or end of the work (see 14.59,14.60). See also 14.246,14.253, 14.48.
Sources Following Run-In Quotations 13.68 Punctuation following source of run-in quotation. After a run-in quo tation, the source is usually given after the closing quotation mark, fol lowed by the rest of the surrounding sentence (including any comma, semicolon, colon, or dash; but see 13.69) or the final punctuation of that sentence.
With his “Nothing will come of nothing; speak again” (1.1.92), Lear tries to draw from his youngest daughter an expression of filial devotion.
It has been more than a century since Henry Adams said: “Fifty years ago, science took for granted that the rate of acceleration could not last. The world forgets quickly, but even today the habit remains of founding statistics on the faith that consumption will continue nearly stationary” (Education, 493).
Has it been more than a century since Henry Adams observed that “fifty years ago, science took for granted that the rate of acceleration could not last” (Edu cation, 493)?
A parenthetical reference need not immediately follow the quotation as long as it is clear what it belongs to. For examples, see 13.65 (second ex ample), 13.67. See also 15.26.
Attributing Quotations in Text 13.71 13.69 Punctuation preceding source of run-in quotation. When a quotation comes at the end of a sentence and is itself a question or an exclamation, that punctuation is retained within the quotation marks, and a period is still added after the closing parentheses. (Compare the third example in 13.68.
) And finally, in the frenzy of grief that kills him, Lear rails, “Why should a dog, a horse, a rat, have life, / And thou no breath at all?” (5.3.306-7).
Sources Following Block Quotations and Poetry Extracts 13.70 Parenthetical source following a block quotation. The source of a block quotation is given in parentheses at the end of the quotation and in the same type size. The opening parenthesis appears after the final punctu ation mark of the quoted material. No period either precedes or follows the closing parenthesis. See also 6.101,15.26.
If you happen to be fishing, and you get a strike, and whatever it is starts off with the preliminaries of a vigorous fight; and by and by, looking down over the side through the glassy water, you see a rosy golden gleam, the mere specter of a fish, shining below in the clear depths; and when you look again a sort of glory of golden light flashes and dazzles as it circles nearer beneath and around and under the boat;. . . and you land a slim and graceful and impossibly beautiful three-foot goldfish, whose fierce and vivid yellow is touched around the edges with a violent red—when all these things happen to you, fortunate but bewildered fisherman, then you may know you have been fishing in the Galapagos Islands and have taken a Golden Grouper. (Gifford Pinchot, To the South Seas [Philadel phia: John Winston, 1930], 123) Shortened references are treated in the same way as full ones. If a qual ifier such as line, vol., or p. is required at the beginning of the shortened reference (a p. may often be omitted, especially with repeated refer ences), it should be lowercased as with sources to run-in quotations.
At last the fish came into sight—at first a mere gleam in the water, and then his full side. This was not even a distant cousin to the fish I thought I was fighting, but something else again entirely, (p. 142) or (142) 13.71 Parenthetical citations with poetry extracts. In order not to interfere with a poem’s layout and overall presentation, parenthetical citations following poetry extracts are dropped to the line below the last line of 13.72 QUOTATIONS AND DIALOGUE the quotation. They may be centered on the last letter of the longest line of the quotation or set flush with the left margin of the poem; an addi tional line space may be added. Other positions are also possible (as in the examples in 13.25 and 13.72), as long as consistency and clarity are preserved.
Now more than ever seems it rich to die, To cease upon the midnight with no pain, While thou art pouring forth thy soul abroad In such an ecstasy!
(Keats, “Ode to a Nightingale,” stanza 6) 13.72 Shortened references to poetry extracts. Shortened references to po etry are treated the same way as full ones. A quotation from Edmund Spenser’s The Faerie Queene, once the reader knows that reference is to book, canto, and stanza, might appear as follows: Who will not mercie unto others shew, How can he mercy ever hope to have?
(6.1.42) Index References are to paragraph numbers except where specified as table, figure (fig.), or page number (p.). Page numbers in the online edition link directly to terms in the glossary.
a and an and abbreviations, 5.250,7.33,10.9 choice between, 5.74,5.250,7.32,7.33 and coordinate nouns, 5.75 as indefinite articles, 5.70,5.72,5.73 meaning affected by, 5.75,5.76,5.77 omitted from titles in running text, 8.169 in titles, disregarded in alphabetizing, 14.71,14.79,14.80,16.51,16.52, 16.56,16.68 See also articles (definite and indefinite) AAs (author’s alterations), 2.70,2.135, 2.136, p. 975 abbreviations, general alphabetizing of, 1.44,15.37,16.64, 16.93, fig. 1.8 a or an with, 5.250,7.33,10.9 articles with, 5.250,7.33,10.9 in chart labels, 3.45 in compound terms, 6.80 definitions and uses, 10.2,10.3 derived from proper names, 10.6,10.49, 10.52 at end of sentence, 6.14 indexing of, 16.46,16.49,16.64 lists of: alphabetizing, 1.44, fig. 1.8; cross-checking, 2.32; format and placement, 1.4,1.44,2.23,14.60, fig. 1.8, fig. 14.3; for frequently cited works, 14.60, fig. 14.7; when to create, 10.3 in notes, 10.3,10.24,10.26,10.27,10.42, 10.44 numbers with, 7.89 (sec. 1), 9.16-17 plurals of, 7.15, 8.121,10.49,10.52,10.53, 10.65,14.150 possessives of, 6.43, 6.44,7.17,8.189 punctuation with: ampersands, 6.21, 10.10; commas (see commas: with abbreviations, general (continued) abbreviations); hyphens, 6.80; peri ods (seeperiods [punctuation]: with abbreviations); slashes, 6.109,10.4; and surrounding text, 6.123 spelled out: capitalization, 2.23,10.6, 10.21,10.52; marking manuscript for, 2.93; marking proofs for, 2.129; parentheses with, 7.17,10.3 vs. spelled-out forms: after first occur rence, 1.44,10.26; repeated after long interval, 10.3 (see also under abbreviations, specific) in tables, 3.56, 3.62,3.67,3.81, fig. 3.12 typographic considerations: capitaliza tion, 10.6,10.25-26,10.49,10.52, 10.63; italic vs. roman type, 2.23, 10.7; line breaks, 7.44; small caps vs. lowercase, 10.41; small vs. full-size capitals, 10.8,10.38; spacing, 9.16, 9.55-56,10.5,10.10,10.12,10.49, 10.58,10.66 word division, 7.44 See also abbreviations, specific; acro nyms; contractions; initialisms abbreviations, specific Bible, 10.42,10.44-48,14.238-39 business and commerce, 10.69 (see also company names below) commas with: academic and profes sional designations, 10.21,10.22; addresses, mailing, 10.27,10.29; e.g. and i.e., 5.250; etc. and et al., 6.20; Inc., Ltd., and such, 6.44; Jr., Sr., and such, 6.43; o.s., n.s. (old series, new series), 14.126,14.184 company names, 8.189,10.23-25,14.133, 14.134,14.136 computer terminology, 7.77,7.80, 9.11, 10.39,10.40,10.49 INDEX abbreviations, specific (continued) eras, 9.34, 9.64,10.38,13.67,14.59-60, fig. 14.7 geographical terms: compass points, 10.34,10.35; latitude and longitude, 10.36; postal vs. standard abbrevia tions, 10.33 for illustrative matter in indexes, 16.116 of italicized terms, 2.23,10.7 journal titles, 14.170,15.13,15.46,16.49 Latin, 7.55,10.7,10.42 (see also et al.; ibid.) in legal-style citations (see under legal and public documents, source cita tions for) in mailing addresses, 10.27,10.28, 10.29,10.33,10.34 mathematical, 12.3,12.11,12.17,12.36, 12.49,12.60,12.64, table 12.2 musical works, 8.195,8.196 names, personal, 10.11-22; civil titles, 10.13,10.14; given names, 10.11; /r., Sr., Ill, and such, 6.43,7.42,10.19, 16.41; list of, 14.60; punctuation and spacing, 10.12; with Saint, 16.74; speakers in interviews or discus sions, 13.48 (see also initials: in and as names; titles and offices of people below) organization names, 10.23-26,14.274, 15.37,16.46,16.64 place-names: Canadian provinces and territories, 10.28; city plus state, comma with, 10.29; country names, 10.31,10.32,14.290; with Fort, Saint, and such, 10.30,11.26,16.93; US, when to use, 10.32; US states and territories, 10.4,10.8,10.27, 14.130 in publication details, 1.21 resources on, 10.1 scholarly, 10.42 scientific and technical (see under scien tific and technical terminology) in source citations: classical Greek and Latin works, 14.244-47,14.250; clas sic English works, 14.254; editions, 14.113,15.6; editor, translator, and such, 14.20-21,14.23,14.68-69, 14.103-4,15.6,15.9,15.15; frequently cited works, 13.67,14.59-60, fig.
abbreviations, specific (continued) 14.7; interviews and personal communications, 15.53; legal-style citations (see under legal and public documents, source citations for); non-English bibliographic terms, 14.102; organization names, 14.274, 15.37; pseud, 14.80,15.35; publish ers’ names, 14.133-34,14.136; recto and verso, 14.154-55; reference lists, generally, 15.15; self-published materials, 14.137; states, provinces, and countries, 14.130; s.v. (sub verbo), 14.232; titles of works, 14.59-60, 14.170.15.13.15.46, fig. 14.7 vs. spelled-out form: Bible, 10.44; civil titles, 10.13,10.14; compass points in place-names, 10.34; country names, 10.31,10.32,14.290; edited by, 14.20-21,14.103-4,14.122,15.6, 15.9,15.15; given names, 10.11; latitude and longitude, 10.36; legal and public documents, 14.274; orga nization names, 10.24,10.26; Saint, St., 10.20; social titles, 10.16,10.17; states and territories, 10.27,10.28; United States vs. US, 10.32 time designations (see under time des ignations) titles and offices of people: academic and professional designations, 7.15, 10.4,10.16,10.21,10.22; civil titles, 10.13,10.14; honorifics, 10.18; mili tary, 8.24,10.13,10.15; social titles, 10.4,10.16,10.17; St., 10.20 titles of works: italics for, 10.7; resources on, 15.46; vs. short titles, 14.59; in source citations, 14.59-60,14.170, 15.13.15.46, fig. 14.7 for typescripts, 14.226 units of measurement (see under units of measurement) in vessel names, 8.116 See also postal codes; and specific abbre viations about the author. See biographical notes abridgements, 2.61,4.14,4.75 absolute (independent) possessive pro nouns, 5.49 absolute (positive) adjectives, 5.84 absolute (positive) adverbs, 5.162 INDEX absolute constructions, 5.35,5.111 abstractions, 8.37, 8.94 abstract nouns, 5.5, 5.7 abstracts of books, 1.76,2.25 and copyright issues, 4.64 of dissertations and theses, 14.215 of journal articles, 1.87,1.92-95,2.25, 14.186 manuscript preparation guidelines, 2.3, 2.25 and metadata, 1.92,2.25 proofreading of, 2.100 source citations for, 14.186,14.215 Abu, Abd, Ibn, 16.75 academic concerns course names, 8.86 degrees and affiliations: abbreviations, 7.15,10.4,10.16,10.21,10.22; of au thor, 1.19,1.50,1.64,1.66,1.92,1.95, 1.100; capitalization, 8.29,10.21, 10.22; in contributors’ list, 1.64,2.41, fig. 1.10; omitted in source citations, 14.73 honors and awards, 8.31, 8.83 institutions and departments, 6.81,8.68, 8.85,14.215,14.217 (see also institu tions; universities) lectures and lecture series, 8.87,14.217, 14.264,14.267 letter grades, 7.65 student status terms, 8.29 subjects and disciplines, 7.2,8.30,8.85, 8.86 titles and offices, 8.28-31,16.40 See also universities academic year, 1.80. See also publication date accents. See special characters access dates author-date reference system, 15.50 generally, 14.12 journal articles, 1.83,14.176 maps, 14.237 reference works, 14.233 scientific databases, 14.257 websites and web pages, 14.12,14.207, 15.50 accessibility illustrations, 3.28 resources for, 2.82nl, 3.28,12.2nl accessibility (continued) subsidiary rights for, 4.64 tables, 3.88 text-to-speech tools, 2.82nl, 3.28,7.49, 12.2nl accession numbers for databases, 2.34, 14.215,14.257,14.270 access rights and abstracts, 1.93 and database searches, 1.111 as metadata, 1.92 open-access publishing models, 4.52, 4.60,4.61,4.66 in table of contents, 1.87 accordingly, 6.57 accusative case. See objective (accusative) case ache, compound terms with, 7.89 (sec. 3) acknowledgments on copyright page, 1.20,1.30,1.31, figs.
1.3-4 of data sources, 3.77 format of term, 8.179 of grants and subventions, 1.20,1.31 illustration credits in, 3.30 in journals, 1.89,1.95 material appropriate for, 1.43 omitted in indexing, 16.109 and permissions, 4.102,4.103 placement in book, 1.4,1.41,1.42 separate section for, 1.42,4.103 submission of, 2.3 in table notes, 3.77 in unnumbered notes, 14.55 acquisition numbers, for musical record ings, 14.263 acronyms as adjectives, 10.9 articles with, 5.250,7.33,10.9 capitalization, 10.6,10.26 definition and use, 10.2 disease and medical terms, 8.144 indexing of, 16.49,16.64 journal titles, 16.49 small vs. full-size capitals for, 10.8 space omitted in, 10.5 See also abbreviations, general; abbrevia tions, specific; initialisms active voice, 5.19,5.115, 5.118,5.119,5.135, 5.192 INDEX acts and treaties source citations for, 14.275,14.290 treatment in text, 8.66,8.80,8.81 See also legal and public documents AD (anno Domini), and such, 9.34,9.64,10.38 additions and insertions marking manuscript for, 2.94,2.95, fig. 2.6 marking proofs for, 2.125, fig. 2.6 stetting or reversal of, 2.130 address, spoken. See dialogue; direct ad dress; speech; speeches addresses, email hyperlinks to, 1.121 line breaks in, 7.46 omitted in source citations, 14.214 punctuation with, 6.8, 6.104 treatment in text, 14.17 See also addresses, mailing; URLs addresses, mailing abbreviations in, 10.4,10.8,10.27-29, 10.33-34 compass points in, 10.34 numbers in, 9.50-52,9.54 publishers’, 1.19,1.20,1.21,1.84,1.95 treatment in text, 6.39,10.27,10.29 See also addresses, email; compass points and directions; geographical terminology adhesive binding, p. 975 adjective clusters, 5.167 adjectives, 5.68-96 acronyms as, 10.9 attributive, 5.79 avoiding bias in uses, 5.260 fee-verbs as part of, 5.154 clusters, 5.167 coinage of (see derived from proper names below) compound, 6.80 dates as, 5.83 defined, 5.68 degrees, 5.84-89; comparative, 5.85, 5.87-88,5.201; equal and unequal comparisons, 5.88; noncomparable, 5.89; positive (absolute), 5.84; super lative, 5.86,5.87,7.89 (sec. 1) dependent clauses as, 5.219,5.225 derived from proper names: in non English languages, 11.26,11.39, 11.70; from organization names, 8.62; from personal names, 8.60, adjectives (continued) 8.61; from place-names, 5.69,8.45; proper adjectives, defined, 5.69; religious terms, 8.96, 8.97 (see also proper adjectives) hyphens with, 7.89 (sec. 2) infinitives as, 5.107 as interjections, 5.208 irregular, 5.87 as nouns (adnouns), 5.94 nouns as, 5.24, 5.259,7.27,7.89 (sec. 2) and parenthetical elements in sen tences, 6.48 participles as, 5.90,5.111 position, 5.79-83; after possessive pronouns, 5.80; basic rules, 5.79; and dates, 5.83; meaning affected by, 5.76,5.77; predicate, 5.79, 5.82; when modifying pronoun, 5.81 postpositive, 5.79 predicate, 5.79,5.82 prepositional phrases as, 5.176,5.178 and pronouns, 5.29,5.49,5.78,5.80,5.81 punctuation: commas, 5.91, 6.36-37; dates in text, 5.83; hyphenation, 5.92, 5.93,6.80 relative, 5.56 repeated, 6.37 sex-specific labels as, 5.259 special types: coordinate, 5.91, 6.36; participial, 5.90,5.111; phrasal (com pound modifiers), 5.79, 5.92, 5.93, 7.85.7.89 (secs. 2,3) as verbs, 5.95 See also articles (definite and indefinite) adjuncts, 5.68. See also adjectives administrative bodies, 8.63. See also busi ness and commerce; governmental entities; institutions; organization names adnouns, 5.94 adverbial clauses, 5.225 adverbial objectives, 5.26 adverbs, 5.156-71 adverbial conjunctions, 5.202 adverbial phrases, 5.158,5.161, 6.31, 6.32.7.89 (sec. 3) compound, 5.161,7.86,7.89 in compound modifiers, 7.85 conjunctive (however, therefore, and such), 5.204, 6.49, 6.57 INDEX adverbs (continued) defined, 5.156 degrees, 5.162-66; comparative, 5.163, 5.201; irregular, 5.165; noncompara ble, 5.166; positive (absolute), 5.162; superlative, 5.86,5.87, 5.164,7.89 (sec. 1) dependent clauses as, 5.225 flat (bare), 5.160,5.250 formation of, 5.158 formed from nouns, 5.158 hyphens with, 7.89 (sec. 2) infinitives as, 5.107 as interjections, 5.208 in introductory phrases, 6.31 -ly ending, 5.93, 5.158,5.160,5.163, 5.164.7.85.7.86.7.89 (sec. 2) for negation, 5.230,5.233 nouns as (adverbial objectives), 5.26 phrasal, 5.161 phrasal verbs as, 5.102 position, 5.167-71; and linking verbs, 5.170; meaning affected by, 5.167; modifying intransitive verbs, 5.169, 5.170; modifying participles, 5.114; modifying words other than verbs, 5.168.7.89 (sec. 2); replacing preposi tional phrases, 5.190; in verb phrases, 5.104,5.171 prepositional phrases as, 5.176,5.178 vs. prepositions, 5.184 punctuation with, 6.31, 6.49, 6.57 sentence, 5.157 simple, 5.160 in split infinitives, 5.108 and suffixes, 5.158-59,5.163-64 See also infinitives; participles advertising. See promotion affixes. See prefixes; suffixes African languages, 11.23-24. See also Arabic language after, 5.202, 5.250 afterwords, 1.4,1.26,1.54,14.110 agents, literary, 4.18 age terms, hyphenation of, 7.89 (sec. 1) aggregators, 1.105, 4.63,4.64,4.65 agreement of pronoun with antecedent, 5.42 of verb with subject, 5.138,5.139, 5.141, 5.143, 5.205, 5.250 ah, 6.35 AH (anno Hegirae or anno Hebraico), and such, 9.34,9.64,10.38 aircraft, 8.2,8.116-18. See also vehicles and vessels a.k.a., 10.69 al, el (Arabic definite article), 8.14,11.79, 11.80,16.75 Albanian language, 11.70 albeit, 5.201 albums, 8.197 alignment of facing pages, 2.117 marking manuscript for, 2.98 marking proofs for, 2.127 in mathematical expressions, 12.9,12.18, 12.20,12.21,12.23,12.38,12.67 in outlines, 6.132 of quotation marks in poetry, 13.28 in tables: cells, 3.70-74, figs. 3.19-21; decimal points, 3.72,3.86, figs. 3.14- 15, fig. 3.20 See also lists in text; margins all, compound terms with, 7.89 (sec. 3) all caps acronyms and initialisms, 10.6,10.25- 26 computer terminology, 7.77,8.155 for emphasis, 7.52 gene symbols, 8.132 newspaper headlines, 14.192 postal codes, 10.33 See also abbreviations, general; abbrevia tions, specific; acronyms; initialisms all rights reserved, 1.20,1.22,4.41, figs. 1.1-4 along with, 5.142 alphabetizing, 16.56-93 abbreviations and abbreviations lists, 1.44,15.37,16.46,16.64,16.93, fig. 1.8 articles disregarded in, 14.71,14.79-80, 15.34,16.48-49,16.51-52,16.56, 16.68,16.88,16.91 basic rule for indexing, 16.56 bibliographies, 2.63,14.62,14.65-66, 14.71, fig. 14.8 checking, in indexes, 16.133 vs. chronological order, 16.70 compound terms, 16.58,16.60,16.72, 16.84 computerized sorting, 14.65,14.67,15.17, 16.5,16.57,16.104,16.123 dates, 16.65 INDEX alphabetizing (continued) dedicated indexing software for, 16.104, 16.123 glossaries, 1.61 index cross-references, 16.17,16.20 indexes of first lines, 16.144,16.145 index main headings, 16.56 index subentries, 16.68-70 index subentries, generally, 16.68-70 letter-by-letter: basics of, 16.59; for bibliographies, 14.65; for indexes of first lines, 16.144,16.145; preferred, 16.58; word-by-word compared, 16.57.16.58.16.61.16.123 list of contributors, 1.64, fig. 1.10 names, non-English personal, 16.75-87; Arabic, 16.75; Asian, other, 16.87; Burmese, 16.76; Chinese, 16.77; Hungarian, 8.13,16.78; Indian, 16.79; Indonesian, 16.80; Japanese, 16.81; Korean, 16.82; Portuguese, 16.83; Spanish, 16.83,16.84; Thai, 16.85; Vietnamese, 16.86 names, personal: basic rules, 16.71- 75; compound, 16.72; as corporate names, 16.89; with initials, 8.13, 16.63,16.78; with Mac, Me, or O’, 16.73; with numbers, 16.66; with par ticles, 16.71,16.84; with Saint, 16.74; sovereigns and other rulers, 16.37; titled persons, 16.38 (see also names, non-English personal above) non-English words, 16.67,16.92 numerals, 16.65,16.66,16.103 vs. numerical order for index subentries, 16.69 organization names, 16.46,16.64, 16.88-89 as part of indexing process, 16.122, 16.123 person, place, and thing with same name, 16.62 place-names, 16.90-93 and prepositions, 16.53,16.68 publisher’s preferences, 16.103 and punctuation, 16.59-61 reference lists, 2.63,15.11, fig. 15.1 resources on, 16.58 special characters, 16.67 titles of works, 16.51-54,16.56 (see also articles disregarded in above) alphabetizing (continued) word-by-word, 14.65,16.57,16.58,16.60, 16.61,16.123 alphabets dictionary tables, 11.76,11.91 modernizing archaic letters, 13.7 non-Latin, 11.2,11.72-73,11.90,11.110, 11.115,11.122,12.12, tables 11.2-5 See also Arabic language; Cyrillic alphabet; Greek alphabet; Hebrew language; International Phonetic Alphabet; Latin alphabet; letters (alphabet); special characters; Unicode standard; and names of other languages alterations author’s (AAs), 2.70,2.135,2.136, p. 975 defined, p. 975 designer’s (DAs), 2.135, p. 978 editor’s (EAs), 2.135,2.136,14.47, p. 979 and printer’s errors, 2.135, pp. 984, 985 and proofreading, 2.110 alternative questions, 5.212 alternative text, 2.82nl, 3.28, 3.88 although, 5.201 ALWD Guide to Legal Citation (Associa tion of Legal Writing Directors and Barger), 14.269. See also legal-style citations a.m.,p.m., 9.37,9.38,10.41 ambitransitive (ergative) verbs, 5.99 American Indians, 7.10,8.5,8.38 American Medical Association (AMA), 14.3, 14.76 American National Standards Institute (ANSI), 1.35 American Psychological Association (APA), 14.3 American Sign Language (ASL), 11.125-35 and Deaf, deaf terms, 8.43 fingerspelling in, 6.77,11.130 resources on, 11.127 signed languages, defined, 11.125 signs: components of, 11.126; com pound, 11.129; glosses of, 11.128-35; handshapes, 11.132; lexicalized, 11.131; nonmanual, 11.135; pronouns, possessives, and reference, 11.134 ampersands alphabetizing, 16.64 in company names, 6.21,10.24,14.135 INDEX ampersands (continued) initialisms with, 10.10 in legal-style citations, 14.300 for Old and Middle English, 11.123 in publishers’ names, 14.135 serial comma omitted before, 6.21 spacing with, 10.10 in titles of works, 8.165 in URLs and email addresses, 7.46 See also and; conjunctions analytic vs. synthetic languages, 5.221 anaphoric pronouns, 5.28. See also pro nouns: antecedents and ampersand for, 6.21 between with, 6.78, 9.60 both with, 5.199,5.244,5.250 coordinate adjectives separated by, 5.91 in generic cross-references of indexes, 16.23 in lists, 6.131 pronoun and antecedent with, 5.32, 5.33,5.34 in publishers’ names, 14.135 punctuation with, 6.19, 6.22, 6.23 sentences beginning with, 5.203,5.250 and serial commas, 6.19 slash instead of, 6.106 in spelled out numbers, 9.5 as subordinating conjunction, 5.201 in titles of works, 8.165 for two or more authors or editors in source citations, 14.76 and verb number, 5.205 See also ampersands; conjunctions and if, 6.26 and/or, 5.250 and other stories, 14.92 and so forth or and the like, 6.20,11.32 and then, 6.23, 6.57 angle brackets for callouts, 2.30 in generic markup, 2.16,2.30,2.81, 6.104 for less than and more than, 3.83,12.15 in mathematical expressions, 6.104, 12.26,12.31,12.55,12.59 for special characters, 2.16 with URLs, 6.8,14.17 in XML, 6.104 See also brackets animals illustrations, fig. 3.7 resources on, 8.119, 8.128 scientific names, 8.119, 8.120-24, 8.126-27 vernacular names, 8.128-29 animations, 1.107,2.4. See also multimedia content Annals of the Congress of the United States, 14.287 annotated bibliographies, 14.64, fig. 14.10 annotations as copyrightable, 4.5 keyed to line or page numbers, 14.53, figs. 14.5-6 in PDF files, 1.118,2.71,2.119,2.133 See also notes; source citations announcements in journals, 1.86,1.87,1.89, 1.99 anonymity of research subjects, 13.49 anonymous works copyright of, 4.25 source citations for, 14.79,14.212, 15.34 use of Anonymous and Anon., 10.42, 14.79.15.34 ANSI (American National Standards Insti tute), 1.35 ante, as prefix, 7.89 (sec. 4) antecedents. See under pronouns anthologies author-and-title indexes for, 16.6 as collective works, 4.8 and copyright issues, 4.55,4.64 editorial additions bracketed in, 6.99 material copyrightable in, 4.5 permissions and fees for, 4.105 unnumbered source notes in, 14.54 See also compilations of previously pub lished material; derivative works anti, as prefix, 7.89 (sec. 4) any, 5.7 any, as prefix, 5.238, 5.250 APA (American Psychological Association), 14.3 aphorisms. See figures of speech; maxims Apocrypha, 8.106,10.46. See also Bible apodosis, 5.228 apostrophes, 6.116-18 directional or “smart,” 2.80,2.85, 6.115, 6.117 INDEX apostrophes (continued) and electronic file cleanup, 2.80 marking on manuscript, 2.94 marking on proofs, 2.132 in non-English languages: African lan guages, 11.24; Chinese, 11.84; Czech, 11.70; French, 11.35; German, 11.40; Greek, 11.116; Hebrew, 11.93; Italian, 11.48,11.52-53; Japanese, 11.87 other punctuation with, 6.118 in plurals, 6.116,7.13,7.15 in possessives: attributive nouns, 7.27; basic use, 5.50, 6.116; compounds, 7.24; for... sake expressions, 7.21; general rule, 7.16; genitive case, 7.25; gerunds, 7.28; italicized or quoted terms, 7.29; joint (group) possessives, 5.22; nouns plural in form, singular in meaning, 7.20; with of 7.26; proper names, 7.17,7.19,7.28; two nouns as unit, 7.23; words ending in unpro nounced s, 7.18 vs. prime symbol, 9.16 vs. single closing quotation mark, 2.80, 6.9 uses, other: abbreviated decades (e.g., ’70s), 9.33; abbreviated years, 9.30; contractions, 5.50, 6.116,7.30; vs. hamza, 11.77; inappropriate, 5.49, 5.50,5.250, 6.117,7.9,7.65; legal-style citations, 14.274; nouns, genitive, 5.9, 5.20,5.21 appendixes chronologies as, 1.60- discographies in, 1.63 as electronic supplementary data, 1.59, 1.78 figures or illustrations in, numbering of, 3.11 format of term, 8.179 glossaries as, 1.61 indexing of, 16.109 multiple, 1.48 notes to, 1.62 numbering of, 1.59,8.180 parenthetical references to, 14.56 placement in book, 1.4,1.59 running heads for, 1.14 submission of, 2.3 as work made for hire, 4.10 appositives clauses as, 5.227 commas with, 5.23, 6.28, 6.41 defined, 5.23 place-names, 8.54 and pronouns, 5.12,5.36,5.51 vs. serial commas, 6.19 subordinating conjunctions in, 5.201 titles and offices of people as, 8.21,8.30, 8.36 apps defined, p. 975 functional features in, 1.119 generally, 1.118 indexes in, 1.125 for magazine articles, 14.189 markup for, 2.83 multimedia content, 14.268 names of, 8.154, 8.155 navigation in, 1.120,1.123 PDF format for, 1.118 proofing and testing of, 2.137, 2.138, 2.139,2.140 required to read electronic publications (see file formats and devices) site map for, 2.137 specified in source citations, 14.137, 14.159,14.160,14.189,14.268 terminology specific to, 7.76 video games as, 8.190 See also electronic publications; multiple formats, publications in Arabic language, 11.76-81 articles, 8.14,11.79,11.80,16.75 capitalization, 8.14,11.80 names, 8.14,16.75 resources on, 11.76 special characters, 11.76,11.77,11.81, 11.92, table 11.2 spelling, 11.78 word division, 11.81 arabic numerals building and apartment numbers, 9.52 Chicago’s preference for, 9.67 defined, p. 975 divisions in publications and docu ments, 9.26, 9.27, 9.28 enzyme names, 8.133 illustrations, 3.12 and line breaks, 7.43 INDEX arabic numerals (continued) parts of book, 8.180 ranges (inclusive), 9.61,9.62 roads and thoroughfares, 9.50,9.51 vs. roman numerals, 14.147, table 9.1 sequels, 8.189 in source citations: classical Greek and Latin works, 14.243,14.244,14.249; journal volume numbers, 14.171; parts of books, 8.180,14.147; parts of poems and plays, 8.184,14.254; vs. roman numerals, 14.147; scriptural references, 14.241; volume numbers of multivolume works, 14.116 spacecraft, 9.44 table columns, 3.56 text page numbers, 1.4,1.9,1.46,1.47, 2.38 unions and lodges, 9.49 See also inclusive (continuing) numbers; numbers; roman numerals archaeology, 7.2,9.34 archival practices, 1.114,2.78,2.87. See also backup copies archives, 4.83,14.292,14.296. See also legal and public documents; letters (correspondence); manuscript col lections; unpublished and informally published materials articles (definite and indefinite), 5.70-78 and abbreviations, 5.250,7.33,10.9 appropriate use of, 5.74,5.250,7.32,7.33 in Arabic, 8.14,11.79,11.80,16.75 astronomical names with, 8.138, 8.140 a vs. an, 5.74,5.250,7.32,7.33 in blog titles, 8.192 collective nouns with, 5.7 common nouns with, 5.5 coordinate nouns with, 5.75 defined, 5.70 definite, generally, 5.71,5.73,5.75 disregarded in alphabetizing, 14.71, 14.79-80,15.34,16.48-49,16.51-52, 16.56,16.68,16.88,16.91 with epithets, 8.35 and gender, 5.11, 5.255 in headline-style capitalization, 8.159 with honorifics, 10.18 indefinite, generally, 5.72-74 in indexes of first lines, 16.144 articles (definite and indefinite) (continued) in index subentries, 16.51,16.52,16.68, 16.129 meaning affected by, 5.75,5.76, 5.77 omission of, 5.77,8.169,8.170,14.33, 14.170,14.193 organization names with, 8.68, 8.70, 14.134,16.88,16.89 in personal names, 8.11,16.75 place-names with, 8.59,16.91,16.92 as pronoun substitute, 5.78 proper names with, generally, 5.6 with titles of people used in apposition, 8.21,8.30 in titles of works, 8.169-70,8.192,14.98, 14.170,14.193-94,16.48-49,16.51-52,16.56 zero (implicit), 5.77 See also a and an; the articles, periodical. See journal articles; magazines; newspapers art styles and movements, 8.79. See also cultural movements and styles artwork author’s inventory of, 3.17 commissioned, 3.33 continuous-tone, 3.3, p. 978 copyright issues, 4.15,4.98 covers and jackets, 1.30,1.73,4.77,4.90, 4.100, 4.101 creators’ names, indexing of, 16.50 credit lines for, 1.30,1.73,3.33, 8.199 cropping, scaling, and shading, 3.19 defined,p. 975 identification for publisher, 3.16 inventory of, 3.17-18 original dimensions noted for, 3.27 redrawn by publisher, 3.20 scans, 2.27,3.15 source citations for, 14.235-37 submission to publisher, 2.2,2.4,2.27, 3.3, 3.4, 3.15-20 titles of, 8.198-201; in captions, 3.22; cartoons, 8.200; exhibition catalogs, 8.201,14.236; generic, 3.22; index ing, 16.50; maps, 8.199,14.237; paint ings, statues, and such, 8.57,8.198; in source citations, 14.86,14.235 use of term, 3.1 See also captions; illustrations INDEX as, 5.196,5.201,5.202 as, as if, 5.185,5.250 as, like, 5.185, 5.250 as. .. as, 5.46,5.88,5.199 ascenders, p. 975 ASCII (plain-text) files, p. 985 as far as, 5.201, 5.250 as follows, 6.64,13.16. See also introductory words and phrases Asian names Burmese, 16.76 Chinese, 8.15,11.83,11.85,11.88,16.77 Indian, 16.79 Indonesian, 8.18,16.80 Japanese, 8.16,11.88,16.81 Korean, 8.17,16.82 other, 8.18,16.87 Thai, 16.85 Vietnamese, 16.86 See also South Asian languages as if, 5.201, 5.250 ASL. See American Sign Language as much as, 5.201 as so, so as, 5.199,5.244 associations, 8.70,10.26. See also organiza tion names as soon as, 5.201 asterisks for footnotes, 14.25,14.27,14.49,14.51, fig. 14.4 in legal-style citations, 14.276 in tables, 3.79, 3.80, fig. 3.22 for text breaks, 1.58,2.8 as though, 5.201 astronomical and astrophysical terminol ogy, 8.137-42 abbreviations in, 9.10,10.60-62 celestial bodies, 8.138-42,10.61 descriptive terms, 8.142 Myr and Gyr, 9.10 resources on, 8.137,10.60,10.62 sun and moon, 8.141 See also metric system; scientific and technical terminology as well as, 5.142,5.201, 6.19 at, in legal-style citations, 14.273,14.276, 14.285 atlases, 4.10 at sign (@), 16.64 attribute pronouns, 5.45 attribution and fair-use doctrine, 4.75,4.84,4.92 importance of, 4.15,4.75,4.92 unnecessary, 13.5 See also illustration credits and credit lines; permissions; source citations; text citations attributive adjectives, 5.79 attributive nouns, 5.24,7.27 audiobooks, 1.75,14.264. See also multime dia content; sound recordings audio rights, 4.64 audiovisual materials. See movies and film; multimedia content; musical works; sound recordings; television and television programs Australia, 1.34 author-and-title indexes, 16.6 author-date reference system authors’ names: anonymous works, 15.34; authors with same last name, 15.22; basic form, 15.6,15.12; editor in place of author, 15.9,15.19,15.36; examples and variations, 15.9; initials vs. full names, 15.12,15.33,15.53; more than three authors, 15.9,15.29; multimedia content, 15.57; orga nizations as authors, 15.37; proper form of name to use, 15.12,15.33; pseudonyms, 15.35; same author, same year, 15.20,15.29; single vs. several, order for, 15.16; 3-em dashes for, 15.17-20 basic format: example, fig. 15.1; ex amples and variations, 15.9; page numbers and other locators, 15.8; reference list entry structure, 15.5, 15.6,15.22; text citation structure, 15.5,15.7,15.22 blogs and blog posts, 15.51 cross-checking, 2.32,15.21 cross-references, 15.42 disciplines using, 14.2 and indexing, 16.114,16.115 interviews and personal communica tions, 15.53 in journals, 1.108 legal and public documents, 14.271, 15.58-59 multiauthor books, 15.10 INDEX author-date reference system (continued) multimedia content, 15.57 and notes and bibliography system, 14.2, 15.1,15.2,15.3 notes used with, 14.39,14.50,14.271, 15.11,15.31, fig. 15.2 patents, 15.55 periodicals, 15.9,15.46-49; initials for authors’ given names, 15.33; issue number in parentheses, 15.47; news papers and magazines, 15.49; titles of, 15.13,15.38,15.46 publication date: basic structure and punctuation, 15.5,15.6,15.9,15.25; blog posts, 15.51; editions with mul tiple, 15.40;forthcoming, 15.18,15.45; letters in published collections, 15.43; manuscript collections, 15.54; mul timedia content, 15.57; multivolume works published over several years, 15.41; newspapers and magazines, 15.49; “no date,” 15.44,15.50; vs. notes and bibliography system, 15.2, 15.3; placement of, 15.14; references arranged by, 15.18,15.19; reprint and modern editions, more than one date, 15.40; social media content, 15.52 secondary sources, quotations from, 15.56 social media content, 15.52 syntactic considerations with, 15.28 websites and web pages, 15.4,15.50 See also reference lists; source citations; text citations; titles of works in source citations; and specific materials to document author queries and ambiguity in tracking changes, 2.84 bias-free language issues, 5.254 comment style for, 2.69 on cross-references, 2.60 example of, fig. 2.4 and fact-checking, 2.56 generally, 2.2 inserting in electronic files, 2.87 marking, 2.92 placement in margins of, 2.10,2.43, 2.91,2.92-93 author queries (continued) on quotations and previously published material, 2.61 on running heads, 2.76 on silent changes, 2.85 on source citations, 2.62,2.63,2.64, 2.69 sticky notes for, 2.92 on subheads in table of contents, 2.57 on table totals, 3.81 authors academic degrees and affiliations, 1.19, 1.50,1.64,1.66,1.92,1.95,1.100 accuracy of, 2.7,2.56,13.6 alterations on proofs (AAs), 2.70,2.135, 2.136, p. 975 approval before editing, 2.50 backup copies of manuscript as respon sibility of, 2.4, 2.40, 2.71 biographical notes (see biographical notes) books published by (see self-published materials) collected works of single, 8.178 communications with manuscript editors, 2.68-72; checking author’s changes, 2.72; cover letter and instructions, 2.70,2.88,2.92; early contact, 2.68; editing sample, 2.52, 2.68; permissions double-checked, 4.79; sample of editing, 2.52,2.68; sending edited manuscript, 2.71,2.88 (see also author queries) field notes of, 13.49 financial assistance for, 1.31 indexing, role in: list of terms for indexer, 16.30,16.108; manuscript preparation, 16.24; vs. professional indexers, 16.3,16.4; responsibility for, 2.2,2.70; and software, 16.104 interaction with readers, 1.118 interviews by, 13.48 (see also interviews and discussions) list of previous publications on copyright page, 1.18 multiple (see multiauthor volumes) own work used by, 3.31,4.66,4.78 papers in collection of, 14.231 permissions as responsibility of (see permissions) INDEX authors (continued) proofreading as responsibility of, 2.101 publication process and schedule for, 2.2, figs. 2.1-2 review of copyedited manuscript, 2.2, 2.70-72,2.82,2.88,2.101 royalties, 4.37,4.52,4.58,4.64,4.69, 4.104 translations by, 11.14 warranties by, 4.52,4.72,4.73 websites of, 1.113,1.121,4.66,4.67 See also author queries; authors’ names; contributors to multiauthor volumes and journals; copyright; manuscript preparation guidelines for authors; proofreading; publishing agreements author’s alterations (AAs), 2.70,2.135, 2.136, p. 975 authorship, in copyright. See copyright authors’ names in books: chapter display, 1.50,2.32; copyright notice, 1.22,4.42; covers and jackets, 1.69,1.70; epigraphs, 1.37; forewords and introductions, 1.40,14.105; multiauthor volumes, 1.38,1.50,1.64,2.32, fig. 1.10; run ning heads, 1.83; table of contents, 1.38; title page, 1.19,14.73,14.82, 14.101,15.12 as copyright management information, 4.15 in electronic file names, 2.37 given names, 10.11,14.83,15.33 hyperlinks from, 1.121 indexing: in separate index, 16.6,16.115, 16.145; with titles of works, 16.50, 16.97 in journals and journal articles: first page of article, 1.95,1.100; as metadata, 1.92,1.100; retractions, 1.91; running heads, 1.83; source citations, 14.168, 14.185,15.33; table of contents, 1.87, fig. 1.11 as metadata, 1.75 in production checklist, 2.77 proofreading of, 2.108,2.134 screen names, 14.208,14.209,15.52 in source citations (see authors’ names in source citations) in species names, 8.124 See also anonymous works; contributors authors’ names (continued) to multiauthor volumes and journals; pseudonyms and pseudonymous works authors’ names in source citations author-date reference system: alpha betization, 15.11,15.16,15.18,15.20; anonymous works, 15.34; authors with same last name, 15.22; basic form, 15.6,15.7,15.12; editor in place of author, 15.9,15.19,15.36; examples and variations, 15.9; and indexing, 16.114,16.115; initials vs. full names, 15.12,15.33,15.53; more than three authors, 15.9,15.29; organizations as authors, 15.37; proper form of name to use, 15.12,15.33; pseudonyms, 15.35; same author, same year, 15.20, 15.29; single vs. several authors, order for, 15.16; 3-em dashes for, 15.17-20 legal-style citations, 14.272,14.285, 14.305 notes and bibliography system, 14.72- 84; alphabetization, 14.62,14.65-66, 14.71, fig. 14.8; alternative forms of, 14.82; anonymous works, 14.79; authors known by given name, 14.83; basic note and bibliography entry structure, 14.20-21; chronological order for single author with multiple titles, 14.71; classical Greek and Latin works, 14.244,14.245; different coauthors with, 14.66; editor in place of author, 14.103; editor vs. author, 14.104; examples and variations, 14.23; footnotes vs. endnotes, 14.48, 14.57; forewords, prefaces, and such, 14.110; form of, generally, 14.19,14.23,14.73,14.101; initials vs. full names, 14.73-74; mononyms, 14.83; in multivolume works, 14.117, 14.122; organizations as authors, 14.70,14.84; pseudonyms, 14.80- 81; punctuation of, 6.20; repeated names, 1.124, 6.94,14.65,14.67-71, 14.79; short forms, 14.30-32,14.108, 14.185; single author, 14.75; 3-em dashes in, 1.124,2.14, 6.94,14.65, 14.67-71,14.79; in title of work, 14.78; two or more authors or editors, 14.23, INDEX authors’ names in source citations (continued) 14.32,14.76; two or more authors or editors with same family name, 14.77 automobiles, 8.117. See also vehicles and vessels auxiliary (helping) verbs, 5.144-53 can, could, 5.146,5.250 in contractions, 5.105 defined, 5.103,5.144 do, 5.152,5.215, 5.231 have, 5.132,5.133,5.153 with infinitives, 5.106 may, might, 5.147,5.250 modal, generally, 5.145 must, 5.148 ought, should, 5.149,5.150,5.250 and parallel structure, 5.245 in verb phrases, 5.104 will, would, 5.131,5.134,5.151,5.231 averages, in tables, 3.74 awards and prizes, 1.99,8.31, 8.83,8.115, 8.172 axes, x andj>, 3.43, 3.52, fig. 3.8 Azeri language, 11.70 back counters (subtrahends), 9.66 back issues, 1.84 back margin, p. 975 back matter, 1.59-68 acknowledgments, 1.42 added later in production process, 1.9 afterwords, 1.4,1.26,1.54,14.110 appendixes, 1.59 bibliography or reference list, 1.63 biographical notes, 1.66 chronologies, 1.60, fig. 1.9 colophons (logos), 1.67 editor’s notes, 1.45 in electronic publications, 1.122,1.125 endnotes, 1.4,1.61,1.62 glossaries, 1.61 illustration credits, 1.4, 3.30 indexable material in, 16.109 indexes, 1.65 in journals, 1.88,1.90,2.134 list of abbreviations, 1.44, fig. 1.8 list of contributors, 1.50,1.64,2.41, 14.55, fig. 1.10 overview, 1.3 pagination of, 1.4,1.8,1.9 in production checklist, 2.77 back matter (continued) recto vs. verso for elements, 1.4,1.53, 1.65 running heads for, 1.14,1.15 submission of, 2.3 in table of contents, 1.38, fig. 1.5 See also many of the above elements under their own headings backslashes, 6.105,6.112 backup copies of citation data, 14.5 electronic files, 2.4,2.40,2.78,2.80,14.5 paper-only manuscripts, 2.6,2.71,2.90 See also archival practices bar charts, 3.43, fig. 3.8 bar codes, 1.69,1.74,1.75,1.84 bare (flat) adverbs, 5.160,5.250 barely, 5.90,5.230 baseline, p. 975 battles, campaigns, wars, and such, 8.113- 14 BC, AD, and such, 9.34,9.64,10.38 because, 5.201,5.250, 6.24 before, 5.202 Berne Convention, 4.2,4.30,4.39 besides, 6.57 beta (Greek character), 11.45 beta testing, 2.138, p. 975 between... and, 6.78,9.60 be-verbs (be, is, are, was, etc.) in cleft sentences, 5.246 contractions with, 5.105 forms and tenses of, 5.100,5.154,5.155 it as subject of, 5.47,5.240 as linking verbs, 5.45,5.101 with not, 5.231 noun or pronoun following, 5.18 in passive voice, 5.118,5.154 with present participles, 5.118 in progressive tenses, 5.135 bi, as prefix, 5.250,7.89 (sec. 4) bias-free language, 5.251-60 alternatives for, 5.255 avoiding biases other than gender, 5.253,5.260 avoiding gender bias, 5.47-48,5.250, 5.252,5.255-59 editor’s attention to, 2.51,5.254 and gender-specific suffixes, 5.257 necessary gender-specific language, 5.258 INDEX bias-free language (continued) noun forms in, 5.11 purposes of, 5.251 sex-specific labels, 5.259 Bible abbreviations, 10.44-48; of Bible and biblical, 10.42; books of Bible, 10.44-47; chapter and verse, 14.239; resources on, 10.44,14.238; versions and sections of Bible, 10.48,14.240 books of, 8.105,10.44-47 capitalization of, 8.103, 8.104 in Hebrew language, 11.97 named parts of (e.g., Sermon on the Mount), 8.107 resources on, 10.44,14.238 sections of, 8.106,10.48,14.240 versions of, 8.104,10.48 See also biblical citations biblical citations abbreviations in, 10.44-48 chapter and verse, 14.239 punctuation of, 6.62, 9.26 resources on, 14.238 versions of Bible in, 10.48,14.240 See also Bible bibliographic essays, 14.64, fig. 14.11 bibliographies, 14.61-71 abbreviations in, 10.3,10.27 alphabetizing: authors with different co authors, 14.66; authors with multiple works, 14.68-69,14.71; example, fig. 14.8; letter-by-letter vs. word-by-word, 14.65; multiple authors, 14.66,15.16, 15.18,15.20; readability of, 14.62 authors’ names in (see authors’ names in source citations) chronological order for, 2.63,14.71 for each chapter in multiauthor vol umes, 1.63,14.62 editing and checking, 2.32,2.56,2.63, 2.64,2.80 format: basic structure, 14.21,14.62; ex amples, 14.23, figs. 14.8-11; hanging (flush-and-hang) indentation, 1.63, 2.11,2.24, fig. 14.8; headnotes, 14.63, 14.64, fig. 14.9; manuscript prepa ration guidelines, 2.24,14.67,15.17; sections in, 14.63,14.263, fig. 14.9; 3-em dashes, 1.124,2.14, 6.94,14.65, 14.67-71,14.79; type size, 1.63 bibliographies (continued) format of term, 8.179 as guide to further reading, 14.63 and indexing, 16.109 notes in relation to, 14.61 omission of, 14.31 placement in book, 1.4,1.63,14.62 vs. reference lists, 14.71,15.3 special types: annotated, 14.64, fig. 14.10; bibliographic essays, 14.64, fig. 14.11; discographies, 1.63,14.262, 14.263,15.57, fig. 14.12; selected, 14.64, fig. 14.9; single-author, 14.64, 14.71 subheads in, 14.63,14.263 submission of, 2.3 symbols in, 10.43 titles of works in (see titles of works in source citations) as work made for hire, 4.10 works appropriate for inclusion, 2.63 See also notes and bibliography system; reference lists; source citations; and specific types of sources billion, 5.250 bills, legislative, 14.283. See also legislation binary operations and relations, 12.15, 12.16,12.20,12.23 binary systems, 9.11 binding defined, p. 975 types: adhesive, p. 975; burst, p. 976; case or hardcover, p. 976-77; limp (flexi-), p. 980; notch, p. 983; perfect, p. 984; saddle stitched, p. 987; sewn, pp. 987,988 bio, as prefix, 7.89 (sec. 4) biographical notes components of, 1.18,1.66 on covers or jackets, 1.66,1.70,1.71 editing of, 2.57 of editors, compilers, and such, 1.66 in journals, 1.100,14.55 as metadata, 1.75,1.100 in multiauthor volumes, 1.50,1.64,2.41, 14.55, fig. 1.10 placement in book, 1.4 proofreading of, 2.108 in unnumbered notes, 1.50,14.55 biographies, indexing of, 16.70 BISAC, 1.75 INDEX bitmapped images, 2.137, p. 976 blackboard (double-struck) symbols, 12.14, 12.66 blanket, p. 976 blank lines, 1.58,2.8,2.98 bleed, p. 976 blind (unexpressed) folios, 1.6,1.7,1.8,1.39, 1.46,1.49,3.6, p. 980 blind stamping, p. 988 block quotations beginning in text, 13.23 capitalization in, 13.20 converting to run-in, 13.31 defined, 13.9, p. 976 generic markup of, 2.74,2.80,2.81 indentation in, 2.11,2.19 manuscript preparation guidelines for, 2.11,2.19,2.20,13.9 from non-English materials, 11.11 note references or notes within, 13.7 paragraphs within, 2.19,13.22,13.32-33, 13.35,13.56 permissions for, 4.95,13.3 of poetry (see poetry: quotations from) proofreading of, 2.113 punctuation in: colons, 6.63, 6.65; edit ing of, 2.61; ellipses, 13.56; periods, 13.17; quotation marks, 13.31 quotation beginning in text, 13.23 vs. run-in, 13.9-10 text citations for, 13.70-72,15.26 text following, 2.12,2.21,13.24 within notes, 14.38 See also quotations; source citations; text citations blogs and blog posts defined, 14.205 source citations for, 14.205-6,14.208, 15.51 treatment in text, 8.192 See also online publications; social media content; URLs; websites and web pages Bluebook, The (Harvard Law Review Asso ciation), 14.269. See also legal-style citations bluelines, p. 976 blurb, author’s. See biographical notes boards, p. 976 boats. See vehicles and vessels body text, p. 976 boldface defined, p. 976 in manuscript, 2.15 marking manuscript for, 2.97 marking proofs for, 2.131 proofreading of, 2.113 and punctuation, 6.3 uses: computer terminology, 7.79; dynamics in music writing, 7.75; em phasis, 7.51; highlighting key terms, 7.56; indexes, 16.116,16.140,16.141, 16.143; mathematical expressions, 12.52,12.56; notes, 14.53, fig. 14.6 See also typographic considerations book, compound terms with, 7.89 (sec. 3) book-club rights, 4.64 book design, 1.30,2.73 Book Industry Study Group, 1.75 book notes section, 1.94,1.98. See also reviews book reviews, 1.87,1.94,1.98,4.10,14.153, 14.202 books abstracts for, 1.76,2.25 divisions of: for letters and diaries, 1.52; outline of, 1.4; overview, 1.3; pagination of, 1.7,1.8; for poetry, 1.51; running heads in, 1.12; in table of contents, 1.38, fig. 1.5 (see also parts of below) electronic supplementary data, 1.59, 14.112 errata, 1.68 interactive, 14.268 (see also multimedia content) ISBN (see ISBN) metadata for, 1.75,1.76,2.25 multiple formats, 1.2,1.26,14.159 pages, definitions, 1.5 pagination in, 1.6-9,1.46,1.49,1.51 parts of, 1.4; back matter (see back mat ter); chapter display, 1.8,1.12,1.16, 1.46,1.49-50,2.32,13.37; chapters, 1.49; conclusion, 1.53; divisions into, 1.48; epilogue, 1.4,1.54; front matter (see front matter); galleries, 1.6; generic terms for, 3.5,8.179,14.110; illustrations and tables, 1.8,1.39; introductions, 1.43,1.47; markup of, 2.73; running heads, 1.10-16; second half title, 1.4,1.8,1.46; subheads, INDEX books (continued) 1.55-57; text breaks, 1.58,2.8 (see also divisions of above, and many of the above elements under their own headings) publication process and schedule, 2.2, fig. 2.1 reviews of, 1.87,1.94,1.98,4.10,14.153, 14.202 signatures in, 1.41, 3.6,14.154, p. 988 simultaneous print/electronic publish ing model, 1.2,14.159 source citations for, author-date refer ence system, 15.9,15.40-45 source citations for, notes and bibliog raphy system, 14.100-163; author’s names, generally, 14.19,14.23,14.73, 14.101 (see also authors’ names in source citations: notes and bibliog raphy system); basic format, 14.19; book-length work within, 14.109; on CD-ROMs and fixed media, 14.163; chapters, 14.23,14.54,14.106-8, 14.120,14.161,14.181,15.9; consulted online, 14.161-62; e-books, 14.137, 14.159-60; editions (see editions: source citations for); editor, trans lator, and such, 14.103-5; electronic supplementary data, 14.112; elements to include, 14.100; examples and variations, 14.23; ghostwritten books, 14.105; introductions, prefaces, etc., 14.110; letters in published collec tions, 14.111; locators, 14.53,14.106, 14.153-58, fig. 14.5 (see also page numbers in source citations); multi volume works, 14.116-22,14.124-25, 14.144; non-English bibliographic terms and abbreviations, 14.102; publication details (see publication details); self-published materials, 14.132,14.137; series, 14.123-26; titles (see titles of works in source citations: notes and bibliography system) See also authors’ names; covers of jour nals and paperback books; e-books; electronic publications; hardcover books; multiple formats, publica tions in; online publications; source citations; titles of works Books, in publishers’ names, 14.134 borne, compound terms with, 5.250,7.89 (sec. 3) Bosnian language, 11.70 botanical terminology, 8.119-30 illustrations, figs. 3.3-4, fig. 3.6 resources on, 8.119,8.128 scientific names: authors’ names in, 8.124; divisions higher than genus, 8.126; English derivatives from taxo nomic system, 8.127; genus, species, and subspecies, 8.2, 8.119,8.120-25, 8.127, 8.158-59,8.173; hybrids, 8.125; resources on, 8.119 vernacular names, 8.128-30 See also scientific and technical termi nology both... and, 5.199,5.244,5.250 braces in mathematical expressions, 6.104,12.26,12.28,12.35. See also delimiters brackets, 6.99-104 angle brackets: for callouts, 2.30; in ge neric markup, 2.16,2.30,2.81, 6.104; for less than and more than, 3.83, 12.15; in mathematical expressions, 6.104,12.26,12.31,12.55,12.59; in XML, 6.104 Dirac bracket notation, 12.55 font for, 6.5 other punctuation with: commas, 6.18; ellipses, 11.32,11.49,11.65,13.53, 13.58,14.97; exclamation points, 6.73, 6.74; generally, 6.103; parenthe ses, 6.97, 6.101,14.38,15.28,15.40; periods, 6.13; question marks, 6.70; quotation marks, 13.63 in quotations (see under quotations) in source citations: annotated bibliog raphies, 14.64; for author’s name, 14.74,14.79,15.34; for counted page numbers, 14.155; for editor’s or translator’s notes, 14.51; with ellipses in very long titles, 14.97; legal-style citations, 14.294,14.298; for non-English titles, 14.99; within parentheses, 14.38,15.28,15.40; for pseudonyms, 14.80-81,14.208, 15.35; for publication details, sur mised, 14.132,14.145,15.34,15.44 uses, other: editorial interpolations and clarifications, 6.99; mathematical ex-INDEX brackets (continued) pressions, 12.50 (see also delimiters); music dynamics, 7.75; phonetics, 6.102; with sic, 13.61; transcriptions of discussions or interviews, 13.48; translated text, 6.96, 6.100,11.15; URLs, 6.8,14.17 See also parentheses; punctuation Braille, 4.64 brand names, 7.80,8.69, 8.146,8.153-54.
See also trademarks breathing marks, 2.80,11.110,11.112 British Library, 1.34 British style acronyms, 10.6 contractions, 10.4 dashes, 6.83, 6.85 dates, 6.38 double consonants in past indicative, 5.130 millions, billions, and trillions, 5.250, 9.8 parentheses within parentheses, 6.97 past tense and past participle, 5.100 period as full stop, 6.12 plural-form proper names, 5.15 pronounced h, 7.32 quotation marks, 6.9,13.30 shan't in, 5.231 singular they, 5.48n5 spelling, 7.3,16.29 than with 1,5.183 titles of nobility, 8.32 which and that, 5.250, 6.27 See also European style; United Kingdom broadcast networks, channels, and such, 8.189,10.25 broadside layout defined, p. 976 for tables, 2.115, 3.51,3.86,3.87, fig. 3.16, fig. 3.25 brochures. See pamphlets, brochures, and reports buildings names of, 8.57,8.68,8.172,11.26,11.61 numbers of, 9.52 religious, 8.101, 9.48 bulk, p. 976 bullets, 2.21, 6.130, 6.131 Burmese names, 16.76 burst binding, p. 976 business and commerce abbreviations used in, 10.23-25,10.69, 14.133,14.134,14.136 generic terms vs. proper names, 7.80 in source citations: corporate features omitted from names in, 14.134, 14.136; imprints, parent companies, and such, 14.138-41; news services, 14.200; private documents, 14.219; reports, brochures, and such, 14.220 titles and offices in, 8.27 See also company names; governmental entities; institutions; organization names but for negation, 5.237 punctuation with, 6.22, 6.23, 6.46 sentences beginning with, 5.203,5.250 but if, 6.26 by, in long passive, 5.192 bylaws, 9.28. See also legal and public documents ca. (circa), 7.55,10.42 calendar designations. See dates; time designations calendar year, 1.80. See also publication date calibers, firearm, 9.19 call letters (broadcasting), 10.25 callouts (illustrations and tables) cross-checking of, 2.32,2.65,2.66 definition and format, 2.26,2.30, 3.8 figure, table spelled out in, 8.180 markup of, 2.87 and placement of feature, 3.8, 3.51 vs. text references, 2.30, 3.8,3.51 working numbers in, 3.13 calls for papers, 1.89,1.99 campaigns, military, 8.114 campus locations, 6.81 can, could, 5.146,5.250 can, may, 5.250 can, not, 5.231 Canada CIP (Cataloging-in-Publication) data, 1.34 currency, 9.21 First Peoples, 8.38 legal and public documents, 14.269, 14.293-96 INDEX Canada (continued) numbers, 5.250, 9.55 provinces and territories, 10.28,14.130, 14.193 capitalization abbreviations, 10.6,10.25-26,10.49, 10.52,10.63 academic degrees and affiliations, 8.29, 10.21,10.22 academic subjects and courses of study, 8.85-86 brand names and trademarks, 8.69, 8.146,8.153-54,10.6 calendar and time designations, 8.88- 90 with colons, 6.63 and common nouns, 5.5 company names, 8.68, 8.69 computer terminology, 7.77,7.79,7.80, 8.155 consistency in text and illustrations, 2.65 dialogue, 7.52,13.43 editor’s note on, 1.45 and ellipses, 13.53 for emphasis, 7.52 generic terms for parts of books, 8.179, 8.180 in glossaries, 2.23 groups of people: ethnic and national groups, 8.38; generations, 8.42; physical characteristics, 8.43; sexual orientation and gender identity, 8.41; socioeconomic classes, 8.40 historical and cultural terminology, 8.71-79; acts, treaties, and govern ment programs, 8.66,8.75,8.80-81; awards, 8.31,8.83,8.115; cultural movements and styles, 8.60, 8.61, 8.79; events, 8.75-78, 8.89, 8.108, 8.113-14; oaths and pledges, 8.84; periods, 8.71-74 I (pronoun), 5.40 in illustration labels and keys, 3.20 in indexes: checking, 16.133; cross- references, 16.17,16.18,16.20; main heading and subentries, 16.11 initial the in periodical titles, 8.170 intercaps (midcaps), 8.154 interjections, 7.31 lectures and lecture series, 8.87 capitalization (continued) letters as shapes, 7.67 in lists, 2.23, 6.130, 6.131 marking manuscript for, 2.97 marking proofs for, 2.131 military terminology, 8.112-15 names, personal (see names, personal, capitalization of) non-English languages: African lan guages, 11.23; Arabic, 8.14,11.80; Asian, other, 8.18; Azeri, 11.70; Chinese, 8.15,11.88,11.89; Danish, 11.70; Dutch, 8.10,11.70; in English context, 11.3,11.4; English language compared, 11.18; English transla tions, 11.9-10,11.18,14.99; French, 8.7,11.26,11.27; German, 7.54,8.8, 11.39; Hebrew, 11.93; Hungarian, 8.13; Indonesian, 8.18,16.80; Italian, 8.9,11.46; Japanese, 8.16,11.88, 11.89; Korean, 8.17; Latin, 11.54; Norwegian, 11.70; Polish, 11.70; Portuguese, 8.8,11.70; Russian, 8.12, 11.99,11.100; Spanish, 8.11,11.61; Swedish, 11.70; titles of works (see under titles of works: capitalization); Turkish, 11.70 organization names, 8.62-70; abbre viations, 10.6,10.26; associations, 8.70; companies, 8.68; governmental entities, 8.52,8.62-65; institutions, 8.68, 8.85; political and economic organizations and movements, 8.66- 67,8.70 part and chapter titles, 2.58 permissible changes to, 8.165,13.7,13.18, 13.19,14.88 personifications, 8.37 physical characteristics, 8.43 place-names: cities and towns, 5.69, 8.45,8.53; compass points and direc tions, 8.46, 8.47; continents, coun tries, oceans, 8.45,11.61; political divisions, 8.51,8.52; popular names and epithets, 8.48; public places and structures, 8.56-58; real vs. metaphorical names, 8.50; regions, 8.47,8.54; topographical divisions, 8.53-55; urban areas, 8.49 pronouns, 5.40 proper nouns, generally, 5.6,8.1 INDEX capitalization (continued) questions within sentence, 6.42 quotations, 8.162,13.18-21,13.53 religious terms, 8.91-111; deities, 8.91, 8.92; events and concepts, 8.89,8.94, 8.99,8.108-10; groups, 8.96-102; names and titles, 8.19,8.26, 8.33, 8.36; objects, 8.111; places of wor ship, 8.101; religious figures, 8.93, 8.95; religious works, 8.103-7 scientific and technical terminology: abbreviations, 10.6,10.49; astro nomical and astrophysical, 8.138, 8.140-42; botanical and zoological, 8.120,8.126,8.127,8.128,8.158; ge netic, 8.132; geological, 8.135,8.136; medical, 8.144,8.145,8.146; physical and chemical, 8.148, 8.149; titles of works, 8.168,15.13,15.38 seasons as issue numbers, 1.80 in source citations: and other stories and such, 14.92; author-date reference system, 15.6,15.9,15.13,15.38; blog comments, 14.208; classical Greek and Latin works, 14.247; examples and variations, 14.23; legal-style cita tions, 14.272,14.274,14.280,14.284, 14.290; magazine departments, 14.190; newspaper headlines, 14.192, 14.195; news services, 14.200; pub lishers’ names, 14.136; titles of works (see under titles of works in source citations) sparing use of (“down style”), 8.1 spelled-out forms of abbreviations, 10.6, 10.21,10.52 subheads, 1.56,2.18,2.59,8.158 tables, 3.54, 3.55, 3.56,3.59 text references to illustrations and tables, 3.50 titles and offices (see under titles and offices of people) titles of works (see under titles of works) unusual, 8.4, 8.69,8.154 URLs, DOIs, and such in text, 14.17 vessels and vehicles, 8.116-17 words derived from proper names, 8.1, 8.60-62, 8.96-97 words for Platonic ideas (e.g., Truth), 8.94 See also all caps; capital letters; headline capitalization (continued) style capitalization; historical and cultural terminology; lowercase let ters; names, personal; names, proper; sentence-style capitalization capital letters for compass points in addresses, 10.34 drop caps, 13.37, p. 979 full-size: for emphasis, 7.52; preferred to small caps, 9.34,10.8,10.38; small caps compared to, 7.52 for generations, 8.42 initial, decorative, 13.37 for keyboard combinations and short cuts, 7.78 for musical elements, 7.72-74 plurals of, 7.15,7.65 for scholastic letter grades, 7.65 standing for names, 7.66 used as words, 7.15 See also caps and small caps format; letters (alphabet); small caps caps, p. 976. See also capital letters caps and small caps format in indexes, 16.140,16.145 in legal-style citations, 14.272 for mathematical copy, 12.56 for speakers’ names in drama, 13.46 captions, 3.21-28 accessibility issues, 3.28 capitalization, 3.21,3.22 for charts, 3.44, fig. 3.9 credit lines in (see illustration credits and credit lines) defined, 3.7,3.21 electronic supplementary data noted in, 3.26 examples of, 3.21-27, fig. 3.1, figs. 3.4-9 for illustrations with parts, 3.12, fig. 3.4, figs. 3.6-7 journal cover, inside, 1.84 in lists of illustrations, 1.39, 3.40, fig. 1.7 locators for parts of illustration in (e.g., above, left), 3.24 manuscript and editorial concerns: cross-checking, 2.32; editing, 2.65; guidelines for authors, 2.3,2.26,2.37; proofreading, 2.115; separate file for, 2.26,2.37, 3.18; submission of, 2.3; working numbers, 2.28, 3.13 for maps, 8.199 INDEX captions (continued) for musical examples, fig. 3.5 omission of, fig. 3.3 original dimensions noted in, 3.27 proofreading of, 2.113 punctuation in, 3.21, 3.23,3.24, 6.14 separating illustration number from, 3.23, 6.120 syntax of, 3.21 titles of works in, 3.21, 3.22 carets, 2.94,2.125,2.132 cartoons, 8.200 cascading style sheets (CSS), p. 978 case binding, p. 976-77 case fractions (text-sized with horizontal bar), 12.45 case of nouns and pronouns, 5.17-22 and conjunctions, 5.196 defined, 5.9,5.17 errors in, 5.37 genitive, overview, 5.20 joint and separate genitives, 5.22 nominative (subjective), overview, 5.18 nouns, generally, 5.9 objective (accusative), overview, 5.19 of-genitive, 5.21, 5.191 personal pronouns, 5.44-46 pronouns, generally, 5.35 pronouns, rules for, 5.41 pronouns in apposition, 5.36 relative pronouns, 5.57 word order in place of, 5.221 See also genitive case; nominative (sub jective) case; objective (accusative) case case studies, 13.49 castoff, p. 977 catachresis, 5.249 Cataloging-in-Publication (CIP) data, 1.20, 1.34,1.75, fig. 1.1 catalogs archival (UK), 14.297,14.303 astronomical, 8.139 exhibitions, 8.201,14.236 musical, 8.196 See also library catalogs ccy 10.69 CCC (Copyright Clearance Center), 1.79, 1.103,4.97 CD-ROMs, source citations for, 14.163 CE, BCE, ADy and such, 9.34,9.64,10.38 celestial bodies, 8.138-42,10.61 cells. See tables centuries first vs. later decades of, 9.33 hyphenation of, 7.87,7.89 (sec. 3) numerical designations for, 8.71 plurals for, 7.8 spelled out and lowercased, 7.89 (sec.
3), 9.32 See also historical and cultural termi nology cf. (“compare”), 10.42,14.42 changes from earlier practice. See depar tures from earlier practice chapter display, 1.8,1.12,1.16,1.37,1.46, 1.49- 50,2.32,13.37 chapter numbers as alternative to page numbers in elec tronic publications, 14.160,14.161 arabic numerals for, 8.180,9.26,14.147 in chapter display, 1.49 chap, with, 14.150 for conclusions, 1.53 consecutive, across parts of book, 1.48 cross-references to, 2.35 in double and multiple numeration, 1.57, 2.28,3.11,3.50 in endnote subheads, 1.62,14.46 generic markup of, 2.81 in index locators for endnotes, 16.111 proofreading of, 2.113 in running heads, 1.12,1.15,14.47 in source citations, 14.106,14.153 chapters abbreviation of term, 10.42,14.150 abstracts for, 1.76,2.25 appendixes at end of, 1.59 bibliographies or reference lists for each, 1.63,14.62,15.10 books offered as individual, 1.59,1.62, 1.63,14.62 conclusions as, 1.53 cross-references to, 2.35, 8.180 decorative initials for first word, 13.37 division into, 1.49 endnotes after each, 1.62,14.43,14.46, fig. 14.3 multiple authors of, 1.50 openers of (chapter display), 1.8,1.12, 1.16.1.37.1.46.1.49- 50,2.32,13.37 pagination of, 1.8,1.46,1.49 INDEX chapters (continued) publishing agreements for, 4.58, fig. 4.2 running heads for, 1.12,1.15,14.47 source citations for: chapter in mul tivolume work, 14.120; chapter in single-author book, 14.106; chapters in multiauthor volumes, 14.107-8, 15.42; DOIs for, 14.161; examples of, 14.23,15.9; note referring to entire, 1.49,2.22,14.27; previously published, 14.54,14.181; shortened citations, 14.108; titles, 14.23,14.86, 14.106- 7,14.272,15.9 unnumbered source notes for, 1.49,1.50, 14.27,14.54 as works made for hire, 4.58 See also chapter numbers; chapter titles chapter titles conclusions, 1.53 editing of, 2.58 in endnote subheads, 1.62,14.46 generic markup of, 2.80,2.81 introductions, 1.47 in legal-style citations, 14.272 manuscript preparation guidelines, 2.17 no punctuation following, 6.14 and note reference numbers, 1.49,2.22, 14.27 proofreading of, 2.134 quotation marks for, 8.163,14.23,14.86, 14.106- 7,15.9 in running heads, 1.12,2.76 in table of contents, 1.38,2.17,2.32,2.58, fig. 1.5 wording, 1.49,1.55 character count, p. 977 character encoding, p. 977 character-level formatting, 2.73,2.81,2.82, 2.82nl, 7.49 character references, p. 977 characters, p. 977 characters, fictional, 5.129,8.35. See also dialogue; literary works characters, special. See special characters character sets. See fonts and typefaces; special characters charts and graphs, 3.41-46 as alternative to tables, 3.48 appropriate use and definition of, 3.41 axes and curves in, 3.43, 3.52, fig. 3.8 consistency in, 2.65, 3.42 charts and graphs (continued) credit line for commissioned, 3.33 electronic files, preparation of, 3.43 examples of, figs. 3.8-10 fair use of, 4.91 genealogical and pedigree, 3.46, fig. 3.10 indexing of, 16.116 labels on, 3.44,3.45 as line art, 3.4,3.41 symbols or patterns used in, 3.25, figs.
3.8-9 titles of, 3.44 See also illustrations; tables chemical terminology, 8.147-52 abbreviations, 10.63 laws and theories, 8.148 mass number, 8.150 metric units, 8.152 names and symbols, 7.89 (sec. 1), 8.149, 10.63 resources on, 8.147,10.63 Chinese language, 11.82-85,11.88-90 characters for, 11.90 names, 8.15,11.83,11.85,11.88,16.77 romanization systems (Pinyin, Wade-Giles), 11.82-85,16.77 titles of works, 11.89 chronological order for bibliographies and reference lists, 2.63,14.71,15.18,15.19 for index subentries, 16.70,16.133 for letters and diaries, 1.52 chronologies, 1.4,1.60, fig. 1.9 chronology systems, 9.34,9.64,10.38 Churchy churchy 8.98,8.101. See also reli gions and religious concerns CIP (Cataloging-in-Publication) data, 1.20, 1.34,1.75, fig. 1.1 circay ca, 7.55,10.42 circling editorial marking, 2.93,2.95 proofreader’s marking, 2.121,2.124, 2.127,2.128,2.129,2.131 ciSy as prefix, 7.89 (sec. 4) citation. See source citations citation management software, 14.5,14.12 cited-by statistics, 1.120 cities and towns administrative agencies and depart ments, 8.63 judicial bodies, 8.64 INDEX cities and towns (continued) laws and ordinances, 14.288,14.300 legislative bodies, 8.62 names: in addresses, punctuation of, 6.39,10.27,10.29; alphabetiz ing, 16.90-93; areas within, 8.49; capitalization, 5.69,8.45, 8.53; English forms of non-English cities, 14.131; with Fort, Mount, Saint, and such, 10.30,11.26,16.90,16.93; as locations of manuscript collections, 14.227; in newspaper titles, 14.193, 14.194,16.48; in place of publication, 14.129; on title page, 1.19 numbered divisions of, 9.46 See also civil and civic concerns; govern mental entities; place of publication civil and civic concerns awards and honors, 8.31,8.83 committee minutes and such, 14.229 holidays, 8.89 titles and offices, 8.19,8.22,10.13,10.14, 14.284 See also cities and towns; governmental entities; public places and structures class, socioeconomic, 8.40. See also groups of people classical Greek and Latin works, 14.242-51 artworks, 8.198 capitalization, 11.54,14.247 resources on, 14.244 source citations, 14.242-52; abbrevia tions, 14.244-47,14.250; authors’ names, 14.244,14.245; collections of inscriptions, 14.249; editions, 14.246, 14.248,14.251; editors’ names, 14.260,14.263-65; fragments of texts, 14.250; identifying numbers or letters, 14.243,14.245,14.249; punctuation in, 14.245,14.250; superscripts in, 14.248; text citations, 14.242; titles, 14.245,14.247,14.248; translators’ names, 14.246 See also Greek language; Latin language classic English works, 1.47,14.162,14.253-54,14.267-68 clauses, 5.225-28 adjectival, 5.219,5.225 adverbial, 5.225 appositive, 5.227 case of, 5.66 clauses (continued) conditional (protases), 5.228 contact, 5.226 coordinate, 5.218,5.220 defined, 5.225 dependent (subordinate) (see dependent [subordinate] clauses) elliptical (see ellipses, grammatical) independent (see independent clauses) parallel structure in, 5.242 personal pronoun as subject of, 5.44 punctuation with, 5.23,6.22-27, 6.56-59,6.89 relative, 3.54,5.56,5.59,5.60,5.225, 5.226,5.227, 6.27 restrictive and nonrestrictive, 5.23, 6.27, 6.28 See also conjunctions; phrases cleft sentences, 5.246-48 clerical titles, 8.26,8.33,10.18,16.39. See also religions and religious concerns climate, named events of, 8.77 closed (solid) compounds, 7.82,7.83,7.88. See also compound terms close-up mark, 2.125,11.130, fig. 2.6 clothbound, p. 977 CMYK mode, 3.3, p. 977 co, as prefix, 7.89 (sec. 4) Code of Federal Regulations, 14.289 codes and coding. See markup for manu script editing; markup languages, formal coinage of words, 7.14. See also words derived from proper names collected works. See compilations of previ ously published material; derivative works; multiauthor volumes collected works of author, 8.178 collections. See legal and public documents; manuscript collections; unpublished and informally published materials collections of classical inscriptions, 14.249 collective nouns, 5.5,5.7, 5.15,5.138 collective works, 4.8,4.10,4.58, fig. 4.2. See also multiauthor volumes colloquial speech adverb suffixes in, 5.158 conjunctions in, 5.196 contractions in, 7.30 flat adverbs in, 5.160 vs. grammar, 5.249 INDEX colloquial speech (continued) grammatical ellipses in, 5.229 interjections as, 5.207 like in, 5.185, 5.250 singular they in, 5.48 they without antecedent in, 5.30 See also common expressions; dialect; figures of speech; maxims; slang colons, 6.61-67 and capitalization, 6.63 common misuses of, 6.67 em dashes instead of, 6.85 marking manuscript for, 2.93 marking proofs for, 2.132 other punctuation with: ellipses, 13.54; parentheses, 6.98; quotation marks, 6.10, 6.65 spacing with, 2.9, 6.62,14.116,14.152, 14.177 uses: captions, 3.24; with dates in titles of works, 14.93; direct address, 6.53, 6.66; double or multiple numeration, 1.57; glossaries, 1.61; indexes, 16.19, 16.20,16.25,16.95; introducing lists, 6.129, 6.130; non-English languages, 11.116; overview, 6.61; with page numbers for journal articles, 15.9; pathnames, 6.112; with publication details, 14.127,14.177; with quotations or speech, 6.10, 6.63, 6.65,13.16; ratios, 6.62,9.58; scriptural citations, 9.26,14.239, 14.241; tables, 3.60,3.78; table source notes, 2.31, 3.77; between titles and subtitles, 1.19,8.164-65,14.89-91, 14.96,14.151,14.169; transcriptions of discussions or interviews, 13.48; twenty-four-hour system of time, 9.39,9.40; between volume and page numbers, 14.23,14.116,14.152, 14.177,15.23,15.48 See also punctuation; semicolons colophons (logos), 1.19,1.69,1.70, p. 977 colophons (production details), 1.67, p. 977 color in charts and graphs, 3.43 continuous tone vs. halftone images, 3.3, 3.6, p. 978 in editing paper-only manuscripts, 2.91, 12.65 in electronic journals, 1.107 color (continued) in electronic supplementary data, 1.78 for ethnic groups, 8.38 hyphenation of, 7.89 in illustrations, 1.6,2.37, 3.3, 3.15, 3.19, 3.43 for journal covers, 1.84 in line art, 3.19 names for, 7.89 (sec. 1) process printing, 8.11 for proofreading marks, 2.123 and punctuation, 6.3 for tracking changes, 2.84 color proofs, p. 977. See also prepress color separation, p. 977-78 columns avoiding multiple, 2.23 in indexes, 1.65,16.24,16.135,16.138 in vertical lists, 6.130 See also magazines; newspapers; tables; tabular material combining forms, 7.82,7.89 (secs. 2, 4) command papers (UK), 14.302 commands, 5.214. See also imperative sentences commas, 6.16-55 with abbreviations: academic and pro fessional designations, 10.21,10.22; addresses, mailing, 10.27,10.29; e.g. and i.e., 5.250; etc. and etal., 6.20; Inc., Ltd., and such, 6.44; Jr., Sr., and such, 6.43; old series and new series, 14.126,14.184 with clauses: in compound predicates, 6.23; dependent, 6.24-27; indepen dent, 6.22-23, 6.32; relative, 6.27; restrictive and nonrestrictive, 5.23, 6.27, 6.28 em dashes instead of, 6.85 in indexes: and alphabetizing, 16.59- 61; cross-references, 16.19-20; generally, 16.94; inverted names (see indexing: names, personal); locators, 16.25,16.94; titles ending with question mark or exclamation point, 16.54 marking manuscript for, 2.93,2.94 marking proofs for, 2.132 other punctuation with: apostrophes, 6.118; brackets, 6.18; ellipses, 13.54; exclamation points, 6.125; paren-INDEX commas (continued) theses, 6.18, 6.98; question marks, 6.42,6.125; quotation marks, 2.80, 6.9, 6.40 pairs of, 6.17, 6.26, 6.32 vs. parentheses, 6.95 serial (Oxford), 6.19-21, 6.23,8.165, 14.135 in source citations: with access dates, 14.176; and other stories and such, 14.92; between authors’ names, 14.76; classical Greek and Latin works, 14.245; interviews and personal communications, 15.53; legal-style citations, 14.273; maga zine article page numbers, 14.188; nonconsecutive locators, 14.148; notes, 14.19,14.20; or with double titles, 14.91; page and issue numbers, 14.177,15.47; periodical citations, 14.167,14.171,14.176; with publica tion details, 14.127-28; publishers’ names, 14.134-35; text citations, 15.7, 15.9,15.23,15.29,15.30,15.44,15.45; titles of works, 14.91,14.93,14.229; with volume numbers, 14.152 uses, other: addresses, 6.39,10.27, 10.29; adverbial phrases, 6.31, 6.32; appositives, 5.23, 6.28, 6.41; with but, 5.250; captions, 3.24; com pound predicates, 6.23; conjunctive adverbs, 6.49, 6.57; and consecutive conjunctions, 6.26; coordinate adjec tives, 5.91, 6.36; with coordinating conjunctions, 6.22, 6.32; dates, 5.83, 6.17, 6.38, 9.54; direct address, 6.53; editorial interpolations, 6.18; global positioning coordinates, 10.36; glossed American Sign Language, 11.133; grammatical ellipses, 6.54; homonyms, 6.55; with including, 6.50; interjections, 5.206, 6.35; introductory phrases, 6.30-31, 6.33-35,13.14; with Jr., Sr., and such, 6.43,16.41; lists, 6.129, 6.131,12.19; mathematical expressions, 12.16, 12.18,12.19,12.28,12.36,12.38; with not phrases, 6.45, 6.46; numbers, 9.54-56, 9.63; parenthetical elements in sentences, 6.48, 6.51; participial commas (continued) phrases, 6.30, 6.32; place-names, 5.69, 6.39; preceding main clause, 6.24; questions, 6.42; quotations, 6.40,13.14; quoted titles or expres sions, 6.41; repeated adjectives, 6.37; such as, 6.50; suspended hyphens, 7.88; that is, namely, for example, and such, 6.51; the more, the less, and such, 6.47; titles of works, 6.17,8.165, 8.167,16.48,16.49; with too and either, 6.52; twenty-four-hour system of time, 9.40; and which vs. that, 6.27 when to omit, 6.125 See also punctuation commercial terms and abbreviations, 10.69 common expressions, 7.60,7.64. See also colloquial speech; figures of speech; maxims common-law copyright, 4.2, 4.15,4.19,4.23 common nouns (concrete, abstract, and collective), 5.5 communications. See interviews and discus sions; personal communications comp (comprehensive layout; compositor), p. 978 companies. See business and commerce; company names company names abbreviations, 8.189,10.23-25,14.133, 14.136 alphabetizing of, 16.89 ampersands in, 6.21,10.24,14.135 articles in, 8.68,14.134,16.88 capitalization, 8.68, 8.69 corporate features in, 14.134,14.136 Inc., Ltd., and such with, 6.44, 6.123, 10.23,10.24 indexing of, 16.89 non-English names, 10.23,14.135, 14.136 omissible parts of, 14.134 possessives of, 7.17 spelled out in running text, 10.24 with unusual capitalization (e.g., eBay), 8.69 comparative adjectives, 5.85, 5.87-88,5.201 comparative adverbs, 5.163,5.201 comparison or degree, 5.201. See also de grees under adjectives and adverbs INDEX compass points and directions abbreviations, 10.34,10.35 capitalization, 8.46, 8.47 hyphenation of, 7.89 (sec. 1) latitude and longitude, 10.36 competing publications, 4.52 compilations of previously published mate rial, 2.43-47 and copyright, 2.46,4.5 illustrations in, 2.47 notes in, 2.45 permissible changes to, 2.44,2.61 permissions for, 4.105 source notes in, 1.50,2.46,14.54 as work made for hire, 4.10 See also anthologies; compilers; deriv ative works; previously published materials compilers abbreviation of term in source citations, 10.42,14.103-4,15.36 in addition to author, 14.104 as authors of forewords, 1.40 biographical notes on, 1.66 and copyright, 4.8 notes on changes, 2.44 permissions as responsibility of, 4.105 in place of author, 14.103,15.36 See also compilations of previously published material; editors complex (phrasal) prepositions, 5.174 complex sentences, 5.219,5.225 composition sign, 12.15 compositors and composition. See type setters and typesetting; typographic considerations compound adverbs, 5.161,7.86,7.89 compound-complex sentences, 5.220 compound predicates, 6.23 compound sentences, 5.218 compound terms abbreviated, 6.80 adjectives as, 6.80 adverbs as, 5.161,7.86,7.89 alphabetizing of, 16.58,16.60,16.72, 16.84 inASL, 11.129 and bias-free language, 5.257 closed compounds, trend toward, 7.83 conjunctions as, 5.197 compound terms (continued) defined, 7.82 en dashes in, 6.80-81 genitives of, 5.20 hyphenation, 7.81-89; adverbs ending in -/y, 7.86,7.89; with compound modifiers, 5.92,5.93,7.8,7.85; en dash vs. hyphen, 6.80; ethnic groups and nationalities, 7.89 (sec. 2), 8.39; guide to, 7.89; headline-style titles, 8.161,11.26; and line breaks, 2.112; personal names, 8.6; with prefixes or suffixes, 6.80,7.40,7.81,8.161 names, personal, 8.6,16.72 names, proper, 5.6,7.85,7.89 non-English languages: French, 11.26; German, 11.44; Greek, 11.121; Latin, 11.58; Spanish, 11.68 open, generally, 6.106,7.82,16.58 permanent, 7.82 person pronouns (-self forms), 5.48,5.51, 5.250 in phrasal adjectives, 5.79, 5.92,5.93, 7.85.7.89 (secs. 2, 3) plurals of, 7.7 possessives of, 7.24 prepositional phrases with, 5.179 prepositions as, 5.173 pronoun case errors in, 5.37 pronouns, relative, 5.65 readability of, 7.84 slashes with, 6.106 suspended (second part omitted), 7.88, 7.89 (sec. 1) temporary, 7.82 types of, 7.82,7.83 and word division, 2.112,7.40,7.42,11.44 See also hyphens and hyphenation; pre fixes; suffixes; word division computer software. See software computer terminology, 7.76-80 abbreviations in, 7.77,7.80, 9.11,10.39, 10.40,10.49 application-specific variations in, 7.76 apps, devices, and operating systems, 7.78,8.155 binary systems, 9.11 capitalization, 7.77,7.79,7.80,8.155 fonts for, 7.79 resources for, 7.76 INDEX computer-to-plate (CTP) technology, p. 978 concert reviews, 14.201,14.203 conclusions, 1.4,1.48,1.53,1.76 concordances, 16.5 concrete nouns, 5.5,5.7 conditional clauses (protases), 5.228 conferences announcements of, 1.86,1.89,1.99 names of, 8.70 papers and posters presented at, 14.217 proceedings of, 1.18,1.39,4.8,4.59, 14.217 See also speeches; unpublished and informally published materials; working papers conflicts of interest, 1.95 Congress, US. See US Congress Congressional Globe, 14.287 Congressional Record, 14.286 conjugation. See verbs conjunctions, 5.196-205 adverbial, 5.202 as and as if, 5.185 beginning a sentence with, 5.203, 5.204, 5.250 compound predicates joined by, 6.23 consecutive, 6.26 defined, 5.196 in imperative sentences, 6.22 independent clauses with, 6.22, 6.32, 6.59 in index subentries, 16.68 as interjections, 5.208 vs. prepositions, 5.184 punctuation with, 6.19, 6.22, 6.23, 6.26, 6.32, 6.59 and quotations in text, 6.40 than as, 5.183 types: coordinating, 5.198, 5.230, 6.22, 6.26, 6.32, 8.159; correlative, 5.198, 5.199,5.230,5.244,5.250; simple vs. compound, 5.197; subordinating, 5.175,5.200-201,5.225, 6.24, 6.26 and verb number, 5.205 in vertical lists, 6.131 See also and; but; or conjunctive adverbs (however, therefore, and such), 5.204, 6.49, 6.57 connecting (linking) verbs, 5.45,5.82,5.93, 5.101,5.154,5.170, 6.30 connectives, misleading, 5.142 constitutions and amendments, 8.80, 9.28, 14.272,14.280 contact clauses, 5.226 content proof, 2.140 continents, 8.45,8.53 continued lines in indexes, 16.139 in tables, 3.86-87, fig. 3.25 continuous (imperfect, progressive) tenses, 5.119, 5.128, 5.135 continuous publishing model, 1.82,1.113, 14.174 continuous-tone artwork, 3.3, p. 978 contractions apostrophes in, 5.50, 6.116,7.30 defined, 5.105,10.2 in non-English languages, 11.40,11.48 non-US-style, 10.4,10.23 with not, 5.231 vs. possessives, 5.50 in verb phrases, 5.104 contract proof, p. 978 contracts, private, 14.219 contracts, publication. See publishing agreements contributed volumes. See collective works; multiauthor volumes contributors to multiauthor volumes and journals biographical notes, 1.100,14.55 communications with, 2.68 lists of: vs. biographical notes, 14.55; cross-checking, 2.32; format and placement of, 1.4,1.50,1.64,1.100, fig. 1.10; volume editor’s responsibil ity for, 2.41 manuscript preparation guidelines, 2.36 other (ghostwriters, authors of fore words, etc.), 14.105 publishing agreements for, 2.42, 4.55- 57,4.58,4.59, fig. 4.1 See also authors; authors’ names; journal articles; manuscript preparation guidelines for authors; multiauthor volumes controlled vocabularies, 1.93 conversation. See dialogue; direct address; speech coordinate adjectives, 5.91, 6.36 coordinate clauses, 5.218, 5.220 coordinated universal time (UTC), 10.41 INDEX coordinate nouns, 5.75 coordinating conjunctions, 5.198, 5.230, 6.22, 6.26, 6.32,8.159 coordination, 5.242 copula (linking) verbs, 5.45,5.82,5.93, 5.101, 5.154,5.170, 6.30 copy, types of, 2.110 copyediting. See manuscript editing; manu script editors copyright, 4.1-38 accuracy and candor in process, 4.49 alternatives to, 4.62 assignment or licensing of, 4.34-38 authorship rights, 4.6,4.13-18 authorship varieties, 4.7-12; collective works, 4.8; individual and joint authors, 4.7,4.12; “life plus seventy” rule, 4.23,4.25,4.26; original owner, 1.22,1.24,4.6,4.11,4.42; works made for hire, 4.9-12,4.24,4.38 benefits of registering, 4.50 changes in, 1.24 date of, 1.22,1.23,1.24,14.142 deposit requirements, 4.47 and derivative works, 4.14,4.31,4.45 of dissertations and theses, 4.60 dual system of, 4.2,4.19,4.23 duration: jointly authored works, 4.23; lengthened in 1978,4.20; overview, table 4.1; for works created after 1977, 4.23-25,4.98; for works created before 1978,4.19,4.26-30,4.98 electronic publications, 4.13,4.47,4.63- 66,14.14 extensive paraphrasing under, 4.89 importance of, 4.1 law relevant to, 4.2,4.10, 4.19,4.26 material covered by, 4.3,4.5 of material derived from public-domain works, 4.22 for new editions, 4.28 of non-US publications, 4.29-30,4.47 of online publications, 4.13,4.47,14.14, 14.267 and open-access publishing models, 4.61 and photocopying, 4.17,4.35,4.55,4.64, 4.66 preregistration, 4.50 of previously published materials, 2.46, 4.5 copyright (continued) and public display, 4.13,4.14,4.34 publisher’s responsibilities for, 4.32 “reasonable effort” to correct mistakes, 4.45 registration of, 4.4,4.19,4.46,4.48-50 renewal of, 1.24,4.19,4.27,4.31-33,4.41, 14.143 subdivision of, 4.34 subsidiary rights, 4.64-69; author’s electronic use of own works, 4.66; author’s retention of, 4.18; vs. basic rights, 4.17; categories of, 4.64; dis tribution outside the US, 4.34,4.64, 4.76; economic considerations, 4.65, 4.66,4.69; electronic rights, 4.63, 4.64,4.65; granting permissions for, 4.70,4.71; moral, 4.15; non-US pub lications, 4.29-30; and public-access policies, 4.68; translation rights, 4.34, 4.64; university licenses, 4.67 symbol for, 1.22,4.41,10.43 termination of transfers under, 4.38 word forms for, 5.250 works ineligible for, 4.11 See also copyright notice; copyright page, contents of; fair-use doctrine; illustration credits and credit lines; intellectual property rights; licenses for copyrighted works; permissions; publishing agreements Copyright Act (1976) (and amendments), 1.20,4.2,4.20,4.84 Copyright Clearance Center (CCC), 1.79, 1.103,4.97 copyright lines, journals, 1.79,1.84,1.86, 1.95,1.103,2.134 copyright management information, 4.15 copyright notice, 4.39-46 components, 1.22,4.41-46, figs. 1.1-2 as copyright management information, 4.15 different regimes of, 4.2,4.40 mistakes in, 4.39,4.46 necessity for, 4.27 in notes, unnumbered, 14.54 old rules removed, 4.4,4.39 placement of, 1.20,1.22, 4.43, figs. 1.1-4 renewal, 1.24 types of material: derivative works, 4.45; electronic publications, 1.122; INDEX copyright notice (continued) government-produced works, 4.44; journals, 1.79,1.84,1.86,1.95,1.103, 2.109,2.134, 4.43 copyright page, contents of acknowledgments, 1.20,1.30,1.31, figs. 1.3-4 author’s previous publications, 1.18 changes and renewals, 1.24 CIP data, 1.20,1.34, fig. 1.1 copyright notice, 1.20,1.22,4.43, figs.
1.1-4 country of printing, 1.20,1.27 DOIs, 1.20,1.28,1.33, fig. 1.1, figs. 1.3-4 editions, 1.20,1.23,1.25,1.26,14.113, fig-1.2 examples, figs. 1.1-4 grant information, 1.31 illustration credits, 1.30,1.73, 3.30,4.78 impression line, 1.20,1.28, figs. 1.1-4 ISBN, 1.20,1.32, fig. 1.1 pagination omitted from, 1.7 paper durability statement, 1.20,1.35, fig. 1.1 permissions, 1.20,1.30,4.78, fig. 1.4 place of publication, 14.129 proofreading of, 2.134 publication date, 1.22,1.25,14.142 publisher’s address, 1.20,1.21 publisher’s responsibility for, 2.3,2.57 publishing history, 1.25 running heads omitted from, 1.11 translation information, 1.29, fig. 1.3 See also copyright notice corporations. See business and commerce; organization names correlative conjunctions, 5.198-99, 5.230, 5.244,5.250. See also either ...or; neither... nor; not only... but also correspondence. See letters could, can, 5.146, 5.250 counter, as prefix, 7.89 (sec. 4) counties, judicial bodies of, 8.64 countries adjectives from names, 5.69, 8.45 capitalization, 11.61 in newspaper titles, 14.193 in place of publication, 14.130 regions of, 8.47,8.54 when to abbreviate names, 10.31,10.32, 14.290 countries (continued) See also cities and towns; governmental entities; states (US); and specific countries courses of study, 8.86 courtesy, in illustration credit lines, 3.34, 4.102 courts, 14.276-79 basic elements in source citations, 14.276 Canadian, 14.294 international, 14.304 names and terms, treatment in text, 8.64 United Kingdom, 14.298 US federal, 14.278 US state and local, 14.279 US Supreme Court, 4.93,14.277 See also legal and public documents; legal-style citations cover image, 1.75,1.84 covers of journals and paperback books artwork, 1.30,1.73,4.77,4.90,4.100, 4.101 author’s previous publications, 1.18 biographical notes, 1.66,1.70,1.71 colophons (logos), 1.69,1.70 country of printing, 1.27 defined, p. 978 design of, 1.115 DOIs, 1.33 endpapers, 1.72 and front matter, 1.85 gatefolds (French flaps), 1.70 as historical context, 1.105 illustration credit lines, 1.73 image of, 1.75,1.84 ISBNs and bar codes, 1.32,1.69,1.74, 1.75,1.84 metadata on, 1.75 promotional copy on, 1.70 proofreading of, 2.103,2.108,2.109, 2.134 table of contents, 1.87,2.109 See also hardcover books: jackets Creative Commons (CC) licenses, 3.32, 3.34,4.52,4.60,4.62 credits and credit lines. See illustration credits and credit lines criminal cases. See legal cases Croatian language, 11.70 cropping, 3.19, p. 978 INDEX cross, compound terms with, 7.89 (sec. 3) CrossMark, 1.112 Crossref (DOI registration agency), 1.33, 1.79,1.112,14.8 cross-references, in indexes, 16.15-23 accuracy, 16.130 alphabetizing in, 16.17,16.20 and used with generic, 16.23 blind, 16.16 capitalization of, 16.17,16.18,16.20 checking and editing, 16.126,16.130, 16.133 correspondence with entry wording, 16.21 dedicated indexing software for, 16.104 vs. double posting, 16.16 following subentries, 16.10,16.18,16.20, 16.26 generally, 16.15 generic, 16.23 in indented style, 16.20,16.26,16.28 italics in, 16.15,16.20,16.22,16.23 for organization names, 16.46,16.89 overuse of, 16.15 for personal names: Arabic names, 16.75; Chinese names, 16.77; as corporate names, 16.89; married women’s, 16.36; with numbers, 9.42; with particles, 16.71; Portuguese and Spanish names, 16.83,16.84; saints, 16.42; titled persons, 16.38; Vietnam ese, 16.86 placement and punctuation, 16.17-20, 16.23,16.26,16.95-97 for place-names, 16.93 in run-in style, 16.20,16.25,16.96 “see also” and “see also under” 16.20, 16.26,16.96 “see” and “see under” 16.16-19,16.96 shortened, 16.21 to subentries, 16.19,16.20 and term choice, 16.29,16.30,16.126 typing and modifying, 16.122 cross-references, in source citations alternative real names, 14.82 authors preferring initials, 14.74 legal-style citations, 14.272 newspapers, 14.196 pseudonyms, 14.81 several chapters from same book, 15.42 in text citations, 15.30 cross-references, in text alterations in, 2.136 cf. and see, 14.42 checking and editing, 2.32,2.35,2.60 electronic publications, 1.120,1.121,2.35 for illustrations, 2.28 mathematical copy, 12.24,12.25 mathematical enunciations, 12.56 notes and citations, 14.28,14.31 in previously published materials, 2.44 proofreading, 2.134 specific page numbers, 2.35,2.60 to whole chapters, 2.35, 8.180 See also hyperlinks CSS (cascading style sheets), p. 978 cultivated varieties (cultivars), 8.130 cultural movements and styles descriptive designations, 8.72 period names, 8.73,8.74 treatment in text, 8.60,8.61, 8.79,11.88 See also historical and cultural termi nology curly braces in mathematical expressions. See delimiters currency, 9.20-25 dates with, 9.25 K abbreviation in, 9.24 large amounts of, 9.24 non-US, 9.21-23, 9.25 resources on, 9.21 words vs. symbols and numerals for, 9.20 curves, in graphs, 3.43 cut-in heads (tables), 3.53, 3.58, fig. 3.14 cyan, p. 978 cyber, as prefix, 7.89 (sec. 4) Cyrillic alphabet and capitalization, 11.99 languages using, 11.70 titles of works, 11.100 transliteration, 11.98, table 11.3 and word division, 11.104-8 See also Russian language Czech language, 11.70 da (Portuguese particle), 8.8 daggers and double daggers (symbols), 3.79,14.25,14.49 dangling terms gerunds, 5.116 infinitives, 5.109 participles, 5.115 INDEX Danish language, 11.70 DAs (designer’s alterations), 2.135, p. 978 dashes length of, compared, 6.75 marking manuscript for, 2.96 vs. parentheses, 6.95 preceding epigraph source, 1.37 2-em dashes, 2.96, 6.75, 6.93,7.66,13.59 typing, in manuscript preparation, 2.14 See also em dashes; en dashes; hyphens and hyphenation; punctuation; 3-em dashes data, as singular or plural, 5.14,5.250 databases accession numbers for, 2.34,14.215, 14.257,14.270 bibliographic, 14.67 and copyright issues, 4.55,4.56 dissertations and theses in, 4.60 and electronic rights, 4.65 hyperlinks to, 1.86,1.96,1.109 institutional repositories, 4.60,4.66, 4.67,14.215 legal, 14.270,14.276,14.300 metadata in, 1.75,1.92,1.121 in place of index, 1.110,1.111 for proofreading, 2.140 publisher’s rights database, 4.71 search for copyright owner via, 4.82 in source citations: dissertations and theses, 14.215; formatting hiding publications from, 15.17; legal-style citations, 14.270,14.276; magazine articles, 14.189; names vs. URLs or DOIs for, 14.9,14.11,14.161,14.164, 14.175; newspaper articles, 14.191; permalinks for, 14.9; scientific, 14.257; shortened citations, 14.276 data cells. See tables data falsification, 1.91 data sets, as electronic supplementary data, 1.78 date of download. See access dates date of publication. See publication date dates abbreviations: months, 10.39,14.171, 14.224,15.15; years, 9.30, 9.64 adjectival use of, 5.83 alphabetizing of, 16.65 avoiding ambiguity in, 6.108,9.35 dates (continued) avoiding o/in, 5.250 centuries (see centuries) of copyright, 1.22,1.23,1.24,14.142 currency with, 9.25 decades, 9.33 in diaries, 1.52 editorial doubt, 6.68 fiscal year (FY), 9.64 in foreword, 1.40 formats for: all-numeral styles, 6.108, 9.35, 9.36; day-month-year, 6.38, 6.108, 9.35,14.224, fig. 1.9; month- day-year, 6.38, 6.108, 9.35,14.224; year-month-day (ISO), 6.108,9.36, 9.40 holidays, 8.89 inclusive, 6.78-79, 9.34, 9.60,9.64, 14.117,14.144,14.171 incomplete, 6.79 in legal-style citations: acts and treaties, 14.290; bills and resolutions, 14.283; cases and court decisions, 14.276, 14.279,14.294,14.298; hearings, 14.284; international entities, 14.305; legislation, 14.288,14.295,14.300; legislative debates and hearings, 14.285-87,14.301 months (see months) numbers for (see under numbers) in prefaces, 1.41 punctuation: commas, 5.83, 6.17, 6.38, 9.54; en dashes, 6.78-79,14.117, 14.144,14.171,15.41; hyphens, 6.108, 9.36; slashes, 6.107, 6.108, 6.113 in source citations: access dates, 14.12, 14.176,14.207,14.233,14.237,14.257, 15.50; interviews, 14.211,15.53; manuscript collections, 14.224; multimedia content (recording or performance date), 14.261,14.263; patents, 14.258; unpublished mate rials, 14.215-17,15.53 (see also in legal-style citations above) in titles of works, 14.93 unspecified date of death, 6.79 work period known (fl.), 10.42 years (see years) See also periods of time; publication date; time designations INDEX day-month-year date format, 6.38, 6.108, 9.35,14.224, fig. 1.9 days of the week, 8.88,10.40. See also dates; time designations de, as prefix, 7.89 (sec. 4) de, d’, de la, du, 8.7,8.9,16.84 dead (foul) copy or proofs, 2.105,2.110 deadlines indexing, 2.2,16.4,16.101 proofreading, 2.102 publication schedule, 2.2, figs. 2.1-2 Deaf, deaf, 8.43 debates, source citations for, 14.286-87, 14.301 decades, 9.33 decimal points in celestial coordinates, 10.61 dex (decimal exponent), 9.12 vs. double and multiple numeration, 1.57nl in global positioning coordinates, 10.36 with millions, billions, and such, 9.8, 9.24 in monetary amounts, 9.20, 9.23 in non-US style, 9.55 in SI style, 9.55, 9.56 in tables, alignment of, 3.72,3.86, figs.
3.14-15, fig. 3.20 and use of zero, 9.19 See also fractions; periods (punctuation) decked heads (spanner heads), 3.53,3.57, fig. 3.13, fig. 3.16 declarative sentences (statements), 5.211, 5.246, 6.12, 6.68 dedications format and placement of, 1.4,1.7,1.36 omitted in indexing, 16.109 running heads omitted from page, 1.11 submission of, 2.3 defamation, 4.70 defective words, 5.146 definite articles. See articles (definite and indefinite) definitions, in text highlighted in index, 16.143 mathematical, 12.25,12.56 numbering of, 12.25,12.56 punctuation of, 7.58 See also glossaries degrees (academic). See academic con cerns: degrees and affiliations degrees (measurement), abbreviations for, 10.49,10.52,10.58. See also International System of Units; metric system; units of measurement degrees (positive, comparative, super lative). See under adjectives and adverbs degree symbol, 9.16,10.36,10.49,10.58, 10.61 deictic (demonstrative) pronouns, 5.52 deities, 8.91,8.92. See also religions and religious concerns; religious works deletions marking manuscript for, 2.94,2.95, 2.124, fig. 2.6 marking proofs for, 2.124 stetting or reversal of, 2.130 whiz-deletions, 5.226,5.247 delimiters, 12.26-35 alternative cases, 12.35 binomial coefficients, 12.32 common, 12.26 functional notation, 12.27 functions enclosed by, 12.17 inner product denoted by, 12.31 intervals, 12.30 for radicals in text, 12.44 rewriting fractions using exponents, 12.49 set notation, 12.28-29 vertical bars, 12.26,12.28,12.33-34 demonstrative (deictic) pronouns, 5.52 denominal verbs, 5.25 departments administrative type, 8.63 capitalization, 14.190 in magazines, 8.177,14.190 See also academic concerns: institutions and departments departures from earlier practice biographical notes placement, 1.66 commas with etc. and et al., 6.20 commas with questions in sentences, 6.42 double titles connected by or, 8.167, 14.91 email not hyphenated, 7.89 (sec. 3) short form vs. ibid., 13.66,14.34 INDEX departures from earlier practice {continued) titles of websites, 8.191,14.206 US as noun, 10.32 dependent (subordinate) clauses commas with, 6.24-27 in complex sentences, 5.219 in compound-complex sentences, 5.220 conditional (protases), 5.228 defined, 5.225 with different, 5.195 em dashes with, 6.89 parallel structure in, 5.242 quotations as, 13.14 relative, 3.54,5.56,5.60, 5.225,5.226, 5.227, 6.27 remote, 5.60 subordinating conjunctions with, 5.200- 201, 5.225, 6.24, 6.26 See also relative pronouns derivative works and copyright issues, 4.14,4.31,4.45 and Creative Commons licenses, 4.62 and enhanced electronic rights, 4.64 license to make, 4.34 public-domain work in, 4.22 See also compilations of previously pub lished material; previously published materials descenders, p. 978 describing words. See adjectives design book, 1.30,2.73 and footnotes vs. endnotes, 14.44 of journals, 1.84,1.115' of tables, 3.86 templates for, 2.2 testing, 2.139 designer’s alterations (DAs), 2.135, p. 978 design mockups (wire frames), 2.139 determinants, 12.51 determiners, 5.5, 5.6,5.7. See also articles (definite and indefinite) developmental editing, 2.48,2.50 devices. See file formats and devices dex {decimal exponent), 9.12 diacritical marks. See special characters diaeresis (Greek), 11.110,11.113 diagrams genealogical, 3.46, fig. 3.10 text references and callouts for, 2.30 diagrams {continued) wording of, 2.65 See also charts and graphs; illustrations dialect, 5.25,5.185, 5.236,10.42. See also colloquial speech dialogue capitalization in, 7.52,13.43 direct discourse, 13.39,13.44 in drama, 13.8,13.46-47 and fair-use doctrine, 4.87 faltering or interrupted, 6.87, 6.89,13.41, 13.50,13.58 in field notes, 13.49 hanging (flush-and-hang) indentation in, 13.46 indirect discourse, 13.45 interjections in, 5.207 in interviews and discussions, 13.48 multiple paragraphs in, 13.32,13.39 in non-English materials: French, 11.31; Italian, 11.47; Russian, 11.101; Span ish, 11.64 numerals in, 13.44 permissible changes to, 13.8 punctuation: colon preceding, 6.63, 6.65; ellipses, 13.41; em dashes, 6.91,11.11,11.19,13.41-42; quotation marks, 13.32,13.39-45 single-word speech, 13.40 titles and offices of people in, 8.20 unspoken discourse, 13.43 See also direct address; quotations; speech diaries, 1.52. See also letters (correspon dence); manuscript collections dictionaries for abbreviations, 10.1,10.3,10.6,10.23, 10.31,14.244 alphabetizing in, 16.58 alphabet tables in, 11.76,11.91 biographical, 8.5,16.32,16.71,16.84 contractions in, 7.30 electronic versions, 1.118,1.120,1.121, 14.233 for forms of address, 8.33 for generation terms, 8.42 for geographical terminology, 8.47,16.32 HTML format for, 1.118 for hyphenation of compound terms, 7.81,7.82,7.89 INDEX dictionaries (continued) for indexing of proper names, 16.32 interjections in, 7.31 for irregular adjectives, 5.87 and language settings, 2.80 legal, 8.82 matter not found in, 7.4 music, 7.70, 8.193 for named natural phenomena, 8.77 for names of letters, 7.68 for non-English words and phrases, 7.53, 7.54,11.3,11.75 organizing principle of, 1.123 for personal names, 8.3, 8.5,8.6,16.71, 16.84 for place-names, 8.44 for plant and animal names, 8.119, 8.128, 8.129 for plural forms, 5.13,7.5,7.6,7.7,7.9, 7.10,7.14 for problematic words, 5.250 for pronounced A, 7.32 recommended, 2.54,2.112,5.250,7.1,7.2 running heads in, 1.13 source citations for, 14.232-33 for special characters, 11.76 for spelling, generally, 7.1 for technical abbreviations, 10.49 for trademarks, 8.153 for transliteration, 11.74,11.83,11.86, 11.91,11.98 for word division, 7.36,7.37,7.38,7.41, 7.42 for words derived from proper names, 8.60, 8.61 See also reference works die copy, 1.69,2.108 Digital Library Federation, 1.114 Digital Object Identifiers. See DOIs digital printing, 1.28, p. 978 digital proofs, p. 978 digital rights, 4.63,4.64,4.65 digital rights management (DRM), 4.65, p. 978-79 dimensions, 3.27 diphthongs capitalization, 11.70 never divided, 7.39,11.33,11.42,11.50, 11.66,11.118 non-English languages: Dutch, 11.70; diphthongs (continued) French, 11.33; German, 11.42; Greek, 11.118; Italian, 11.50; Latin, 11.55; Old and Middle English, 11.124; Russian, 11.108; Spanish, 11.66 Dirac bracket notation, 12.55 direct address in imperative mood, 5.122 punctuation, 6.53, 6.66 titles and offices of people in, 8.20,8.32, 10.16,10.17 See also dialogue; quotations; speech directional (“smart”) quotation marks and apostrophes, 2.85,6.115,13.7 directions. See compass points and direc tions directives. See imperative sentences directors’ names, 14.203 direct quotations. See block quotations; quotations; run-in quotations disabilities, people with. See accessibility disciplines and subjects, academic, 7.2, 8.30,8.85, 8.86. See also academic concerns; scholarly works, generally discographies, 1.63,14.262,14.263,15.57, fig. 14.12 discussions. See dialogue; direct address; interviews and discussions; personal communications; speech; transcrip tions diseases, infections, procedures, and such, 8.144, 8.145 display (mathematical expressions). See under mathematical expressions display pages, 1.7,1.11,1.12,1.16 display type, p. 978 dissertations and theses copyright for, 4.60 institutional repositories, 4.60,4.66, 4.67,14.215 microform editions, 14.115 source citations for, 14.215 titles of, in text, 8.188 distance, 7.89 (sec. 1), 10.49. See also metric system; physical quantities; units of measurement distribution copyright owner’s right of, 4.13 license for, 4.34 publishing agreements on, 4.51 INDEX distribution (continued) source citations for, 14.141 subsidiary rights to, 4.34,4.64,4.76 distributive pronouns, 5.67 ditto marks, 3.62,10.42 division of words. See word division division sign, 12.15 divisions of books. See books: divisions of do, 5.152,5.215, 5.231 docket numbers, 14.276 Doctor, Dr., 10.16 documentation. See source citations document comparison software, 2.86 document-delivery companies, 4.64 document type definitions (DTDs), p. 979 DOIs (Digital Object Identifiers) and continuous publishing model, 1.82 on copyright page, 1.20,1.28,1.33, fig.
1.1, figs. 1.3-4 defined, 1.33,1.79,14.8, p. 978 for journal articles, 1.79,1.82,1.92,1.103 in metadata, 1.33,1.79,1.92 resolvers for, 14.8 resources on, 14.8n2 in source citations: book reviews, 14.202, books, 14.100,14.161; vs. database names, 14.11,14.161,14.175; electronic supplementary data, 14.187; journal articles, 14.23,14.175, 14.178-79,14.186,15.9; periodicals, generally, 14.164; vs. permalinks, 14.9; publications in multiple formats, 14.16; reference entries, 14.234; short forms for, 14.10 treatment in text, 14.17-18 in version tracking, 1.28 See also URLs dollars. See currency dots. See periods (punctuation) double comparative or superlative error, 5.87 double genitives (double possessives), 5.21, 5.49,7.26 double negatives, 5.236 double or multiple numbering, 1.57,1.57nl, 2.28,2.29,3.11, 3.50,12.25 double possessives (double genitives), 5.21, 5.49,7.26 double posting, in indexes, 16.16,16.30 double-struck (blackboard) symbols, 12.14, 12.66 “down style” (capitalization), 8.1. See also sentence-style capitalization drama divisions of, 8.184 format of, 3.8,13.46 quotation of, 13.47,13.57 source citations for: classic English works, 14.253-54; frequently cited works, 13.67; reviews, 14.201,14.203; short forms for, 14.254 titles of, 8.183 See also classical Greek and Latin works; dialogue; speech dramatic rights, 4.64 drawings. See artwork; charts and graphs; illustrations; line art DRM (digital rights management), 4.65, pp. 978-79 drop caps, 13.37, p. 979 drop folios, 1.6,1.7,1.8,1.49, p. 980 drug names, 8.146 DTDs (document type definitions), p. 979 Dublin Core Metadata Initiative, 1.92 dummy auxiliaries, 5.152. See also do “dummy” folios, 1.81 dust jackets, 1.71, p. 979. See also covers of journals and paperback books; hardcover books: jackets Dutch language, 8.10,8.45,11.70 DVDs. See multimedia content dynasties, 9.45,11.85 e, compound terms with (e.g., email), 7.89 (sec. 3) each, 5.33, 5.67,5.250 each other, one another, 5.53 earth, 8.140 EAs (editor’s alterations), 2.135,2.136, 14.47, p. 979 East, eastern, 8.46, 8.47 e-books biographical notes in, 1.50 CIP (Cataloging-in-Publication) data, 1.34 as common publishing format, 1.2 cross-references in, 2.35 defined,p. 979 front matter in, 1.122 functional features in, 1.119 hyperlinks in, 1.118,1.120,1.121 impression numbering, 1.28 INDEX e-books (continued) indexes in, 1.125,16.7,16.13 ISBNs for, 1.28,1.32, fig. 1.1, fig. 1.4 markup for, 2.83 navigation in, 1.120,1.123,2.35 parts of, 1.4 PDF as basis of, 1.118 percentage of work as orientation method, 1.123 proofing and testing of, 2.139, fig. 2.8 as reprints, 14.159 source citations for, 14.100,14.137, 14.159-60 See also electronic publications; EPUB; multiple formats, publications in ECF (elemental chlorine-free), p. 979 edited by on title page, 1.19 when to abbreviate or spell out, 14.20- 21,14.103-4,14.122,15.6,15.9,15.15 edited works. See compilations of previ ously published material; derivative works; editors; multiauthor volumes editing, types of, 1.116,2.48-50. See also manuscript editing editions abbreviation of term, 10.42,14.20-21, 14.113,15.6 capitalization of term, 8.176 citations to forthcoming works in new, 14.146 copyright issues, 4.28 on copyright page, 1.20,1.23,1.25,1.26, 14.113, fig. 1.2 definition and preferences concerning, 1.26, p. 979 facsimile, 1.47 folio, 14.155,14.225 vs. impressions, 1.23,1.26 as metadata, 1.75 microform, 4.64,14.115 number of, 1.19,1.70 permissions for, 4.101 preface for new, 1.41 publications in multiple formats as, 1.26 publishing agreements on subsequent, 4.52 source citations for, 14.113-15; classical Greek and Latin works, 14.246, 14.248,14.251; classic English works, 14.253; copublications, 14.140; dic-editions (continued) tionaries and encyclopedias, 14.232, 14.233; folio, page citation for, 14.155, 14.225; microform, 14.115; multi media content, 14.265; newspapers, 14.191; other than first, 14.113; pub lication date, 14.142,15.40; reprint and modern, 14.54,14.114,14.159, 15.40; standards, 14.259 titles, 8.176 editorial board information, 1.84,1.85, 2.109,2.134 editorial interpolations and clarifications, 13.59-63 editorial doubt, 6.68,14.132,15.34, 15.44 and errors, 13.7 italics for emphasis, 13.62 missing or illegible words, 13.59 notes, 13.7,13.58 punctuation for: brackets, generally, 6.99, 6.103; commas, 6.18; exclama tion points, to be avoided, 6.73; 2-em dashes, 6.93,13.59 quotation marks in, 13.63 sic, 6.73,7.55,13.61 in transcriptions of discussions or inter views (e.g., laughter), 13.48 See also permissible changes to quoted and referenced materials editorials, 1.87,1.102,14.195. See also letters to the editor editorial style. See house style editorial we, 5.47 editors and anthology permissions, 4.105 biographical notes on, 1.66 copyright ownership of, 4.8,4.57,4.58, 4.59 forewords by, 1.40 of journals, 1.84,1.87,1.89,1.102,2.134, 4.57 manuscript preparation guidelines for, 2.41-42 notes of, 1.45,1.95,14.51 prefaces by, 1.41 and publishing agreements, 4.57 responsibility for bias-free language, 2.51,5.254 See also authors; editorial interpolations and clarifications; editors’ names; INDEX editors (continued) manuscript editors; multiauthor volumes editor’s alterations (EAs), 2.135,2.136, 14.47, p. 979 editors’ names abbreviation of term, 10.42,14.20-21, 14.23.14.68- 69,14.103-4,15.6,15.9, 15.15 on covers or jackets, 1.69,1.70 form of, 1.19 in multiauthor volumes, 1.64 proofreading of, 2.109,2.134 in source citations, 14.23,15.9; abbre viation of term, 14.20-21,14.23, 14.68- 69,14.103-4,15.6,15.9,15.15; vs. authors’ names, 14.104; of book reviews, 14.202; classical Greek and Latin works, 14.246,14.249,14.250; of journal articles, 14.183; of journal supplements, 14.179; legal-style cita tions, 14.305; of multivolume works, 14.117,14.119,14.122; omission of term in text citations, 15.7,15.9,15.22, 15.36; in place of author, 14.103,15.9, 15.19,15.36; repeated names, 14.68- 69; series of works, 14.123; shortened citations, 14.32; of special issues of journals, 14.178 special issues of journals, 1.87,14.178 See also authors’ names editor’s notes, 1.45,1.95,14.51 educational use of materials, 4.64,4.66 e.g., 5.250, 6.51,10.7,10.42 either, 5.67, 6.52 either... or, 5.33, 5.143,5.199, 5.205,5.244 elect, compound terms with, 7.89 (sec. 3) electronic enhancements. See supplemen tary data, electronic electronic files archival practices for, 1.114 for author’s review, 2.71 backups of, 2.4,2.40,2.78,2.80,14.5 cleaning up, 2.39,2.53,2.59,2.61,2.63, 2.79-80,2.85,14.5 conversion of, 2.7,2.79,2.80, 2.113,2.137 as copyrightable, 4.3 date style for sorting in, 9.36 document comparison software, 2.86 formats for (see file formats and devices) illustrations and tables, preparation electronic files (continued) of: artwork, prints vs. scans, 2.27; charts, 3.43; digital artwork, 3.15; file formats, pp. 981,982,985,989; identified for publisher, 3.16; notes, 2.31; numbering of, 2.28,2.29; prepa ration of tables, 3.47; separate files for each component, 2.26,2.36,2.80, 3.1, 3.16; submission to publisher, 3.3, 3.4,3.15-20; text references and callouts, 2.30 indexer’s use of, 16.102,16.105 index submitted as, 16.131 list of, in production checklist, 2.77 for manuscript submission, 2.4,2.7, 2.36-40 MP3 audio files, 10.49 naming of (see file names) numbering pages in, 2.38 password protection of, 2.88 prepress proofs generated from, 2.107 updating, 2.72 See also manuscript editing; PDF files; software; tracking changes; version control; word-processing software electronic mailing lists, 14.210. See also email; social media content electronic publications accessibility issues, 2.82nl, 3.28, 3.88, 7.49,12.2nl aggregators, 1.105,4.64, 4.65 and alignment of quotation marks in poetry, 13.28 authority and permanence of, 14.14 common use of, 1.2 components: biographical notes, 1.50, 14.55; cross-references, 1.120,1.121, 2.35; front and back matter, 1.122, 1.125; hyperlinks, generally, 1.121; illustrations, 1.107,3.8, 3.28, 3.38; indexes, 1.125,16.1,16.7,16.13,16.15, 16.102; running heads, 1.12; site map, 2.137; table of contents, 1.87,1.122, 1.123, fig. 1.11; tables, 1.106,3.48,3.76 copyright issues, 4.13,4.47,4.63-66, 14.14 country of printing, 1.27 devices required to read (see file formats and devices) educational use of, 4.64 enhanced electronic rights, 4.64 INDEX electronic publications (continued) formats for, 1.77,1.118,14.175 (see also file formats and devices; multiple formats, publications in) functional features in, 1.119,1.120,1.121 institutional repositories, 4.60,4.66, 4.67,14.215 markup for, 2.83 (see also markup lan guages, formal) navigation as organizing principle for, 1.120 page number markers in, 1.123n2,1.125, 14.160,16.13 periodicals as, generally, 14.164-65 prepress, 2.2 print elements not always used in: fixed page numbers, 1.6,1.81,1.123, 2.35,14.47,14.49,14.160; footnotes, 14.27,14.49,14.52; paper durability statement, 1.35; recto and verso, 1.5; running heads, 1.10,1.83,1.123,14.47 proofing and testing of, 1.92,2.103, 2.137-40, fig. 2.8 published before print version, 1.78, 1.81,1.86,1.104,1.112-13,14.172 punctuation in, 6.2, 6.4, 6.8,6.84 reflowable formats: illustration place ment in, 3.8; and indexes, 16.13; line breaks in, 6.82,6.90, 6.113,7.36, 7.47,14.18; location information, 1.6,1.123; navigation in, 1.123,2.35, 14.22,14.160,14.174,15.23; and text breaks, 1.58 scanned versions, 14.162 searchable nature of, 14.22,16.2 source citations for: access dates, 14.12, 14.176,14.207,14.233,14.237,14.257, 15.50; alternate versions, 14.16; books consulted online, 14.161-62; CD-ROMs and fixed media, 14.163; e-books, 14.137,14.159-60; editions of older works, 14.162; examples of, 14.23,15.9; and id. (idem), 14.279; identifiers for (see DOIs; URLs); and navigation, 14.22,14.160,14.174, 15.23; permanent records of chang ing sources, 14.15,14.191; revision dates, 14.13,14.207,14.233,15.50 spaces and spacing, 6.11, 6.120, 9.55, 9.56 unpublished manuscripts as, 14.216 electronic publications (continued) See also apps; databases; DOIs; e-books; hyperlinks; hyperlinks for source citations; online publications; simul taneous print/electronic publishing model; URLs; websites and web pages electronic rights, 4.63,4.64,4.65 electronic workflow checklists for, 2.139 and formal markup languages, 1.117, 2.83,2.137 metadata for, 1.92 proofing and testing, 1.92,2.137-40, fig. 2.8 as standard, 1.2,1.77 See also electronic files; manuscript edit ing; markup for manuscript editing; markup languages, formal elided elements in mathematical expres sions, 12.19,12.20 ellipses, 13.50-58 danger of skewed meaning with, 13.51 defined, 6.15,13.50 editing of, 2.61 and electronic file cleanup, 2.80 non-English uses: French, 11.32; Italian, 11.49; Russian, 11.102; Spanish, 11.65 other punctuation with: brackets, 11.32, 11.49,11.65,13.53,13.58,14.97; colons, 13.54; exclamation points, 11.102,13.54; generally, 13.50,13.54; periods, 13.53,13.56-58,14.97 spacing of, 6.121,11.49,11.65,13.50, 13.54,13.55,13.58 uses, other: in deliberately incomplete sentences, 13.55; in drama, 13.57; in mathematical expressions, 12.19,12.20; missing or illegible words, 13.59; for partial or whole paragraphs, 13.56; in poetry, 13.57; shortening very long titles, 13.58, 14.97; in table cells, 3.67,3.68, fig. 3.15, fig. 3.18 when not to use, 13.52 See also ellipses, grammatical ellipses, grammatical clauses with, 5.226, 6.54 colons with, in lists, 6.67 commas with, 6.54 INDEX ellipses, grammatical (continued) and conjunctions beginning sentences, 5.203 defined, 5.229 exclamations, 5.216 in one-word sentences, 5.97 and relative pronouns, 5.226,5.247 else, 5.67, 5.201 em, p. 979 email addresses: hyperlinks to, 1.121; line breaks in, 7.46; omitted in source citations, 14.214; punctuation with, 6.8,6.104; treatment in text, 14.17 as copyrightable, 4.3 journal table of contents sent by, 1.86 non-English materials in, 11.30 source citations for, 14.210,14.214 term not hyphenated, 7.89 (sec. 3) embedded elements comments, 2.39 fonts, 12.13 indexes, 16.7 URLs, 1.121 em dashes, 6.85-92 in British style, 6.83, 6.85 defined, p. 979 and electronic file cleanup, 2.80 keyboarding for manuscript, 2.14 length of, 6.75 and line breaks, 6.90 marking manuscript for, 2.96 marking proofs for, 2.132 in non-English languages: with dia logue, 6.91; French, 11.28,11.31; Italian, 11.47; for quotations, 11.11, 11.19,13.42; Russian, 11.101,11.103; Spanish, 11.63,11.64 other punctuation with, 6.87, 6.89, 6.90 and permissible changes, 13.7 uses: before that is, namely, for example, and such, 6.51, 6.88; dialogue, 6.91, 11.11,11.19,13.41-42; glossaries, 1.61; indexes, 6.92,16.27,16.99,16.141; lists, 6.92; parenthetical elements in sentences, 6.48, 6.85; to set off not phrase, 6.46; between subject and pronoun, 6.86; sudden breaks, 6.87, 13.41; table cells with no data, 3.67, 3.68, 6.92; titles of works, 8.164, 8.165,14.90 em dashes (continued) See also dashes; en dashes; hyphens and hyphenation; 3-em dashes; 2-em dashes emojis and emoticons, 5.250 emphasis added to original quotation, 13.62 adjective position for, 6.79 boldface or underlining for, 7.51 capitals for, 7.52 colons for, with quotations, 13.16 commas for, 6.46 exclamation points for, 6.71 and hyphenation, 9.14 italics for, 7.48,7.50,13.62,14.37 quotation marks (scare quotes) for, 7.57, 7.60,11.47,14.94 and so-called, 7.59 word order for, 5.224 employment opportunities, 1.99 em spaces, 2.126, 3.23,3.54, 6.120 en, p. 979 enclitics (Greek), 11.114 encyclopedias for geographical terminology, 8.47 for named natural phenomena, 8.77 organizing principle of, 1.123 for personal names, 8.3,16.71 source citations for, 14.232-33 See also reference works en dashes, 6.78-84 in British style, 6.83, 6.85 defined,p. 979 length of, 6.75 and line breaks, 6.82 in manuscript, 2.14 marking manuscript for, 2.96 marking proofs for, 2.132 permissible changes to, 13.7 punctuation and spacing with, 6.79, 14.144 slash instead of, 6.107 uses: classical Greek and Latin works, 14.245; in compound terms, 6.80-81; in directions, 7.89 (sec. 1); with ex as prefix, 7.89 (sec. 3); in harmonic progressions, 7.73; in inclusive dates, 6.78-79,14.117,14.144,14.171,15.41; in inclusive numbers, 2.80, 6.78-79, 9.60,16.100,16.134; in indexes, 16.100,16.111,16.134; as minus sign, INDEX en dashes (continued) 6.84; in musical pitch series, 7.70; in source citations for journal articles, 14.171; for unfinished number ranges, 6.79,15.41 See also dashes; em dashes; hyphens and hyphenation; inclusive (continuing) numbers end matter. See back matter endnotes after each chapter, 1.62,14.43,14.46, fig. 14.3 editing of, 2.62 examples, figs. 14.6-7 vs. footnotes, 1.124,2.77,2.80,14.43-48, 14.57,14.59 footnotes used with, 14.49-51, fig. 14.4 generic markup of, 2.81 glossary terms in, 1.61 index entries for, 16.111 in journal articles, 1.108,14.43,14.46 keyed to line or page numbers, 14.53, fig. 14.6 length considerations, 14.56 lists of abbreviations placed with, 14.60, fig. 14.3 manuscript preparation guidelines, 2.22 multiple paragraphs in, 14.40 numbering of (see note numbers) overview, 14.43 placement in book, 1.4,1.61,1.62,14.46 in previously published materials, 2.45 pros and cons of, 14.45 running heads for: checking proofs for, 2.114; description of, 1.15,1.62,14.47; example, fig. 14.2; importance of, 14.45; page numbers in, 1.15,1.62, 2.76,2.114,2.136,14.47, fig. 14.2; placement of, 1.10; preparation of, 2.76 special needs of, 14.48 subheads in, 1.62,2.81,14.146, fig. 14.2 See also footnotes; notes; shortened citations endpapers, 1.72, p. 979 English language abbreviation of English, 10.42 Old and Middle, 7.35,11.70,11.122-24 See also British style enhanced electronic rights, 4.64 enhancements. See supplementary data, electronic en spaces, 6.120,13.29 Enter key (hard return), 2.12,2.19,2.80 enunciations, mathematical, 12.25,12.56 environmental impact, paper standards for, 1.35, pp. 979,984, 988 enzyme names, 8.133 epigraphs defined, 13.36 and fair-use doctrine, 4.87 format and placement of, 1.37,13.36 generic markup of, 2.80 notes to, 1.49,14.52 omitted in indexing, 16.109 pagination omitted, 1.7 as part of chapter displays, 1.49 placement in book, 1.4 quotation marks omitted from, 13.36 running heads omitted from page, 1.11 submission of, 2.3 epilogues, 1.4,1.54 epithets, 8.34, 8.35, 8.48. See also genus and species names eponyms, 5.6 EPS (encapsulated PostScript), p. 979 EPUB and accessibility, 7.49 defined,pp. 979-80 and impression numbering, 1.28 and indexes, 16.13 journals, 1.77 and navigation, 1.120 overview, 1.118 page number markers in, 1.123n2,1.125 proofing and testing of, fig. 2.8 Unicode standard required by, 11.2 and XML, 1.117 See also e-books equal-space mark, 2.125, fig. 2.6 equals sign, 7.46,10.43,10.45-47,14.250, fig. 3.10. See also mathematical expressions equations. See mathematical expressions eras, 9.34,9.64,10.38. See also periods of time ergative (ambitransitive) verbs, 5.99 errata books, 1.68 journals, 1.90,1.94,1.112 errors in grammar and punctuation apostrophes, 5.49,5.50,5.250, 6.117, 7.9,7.65 INDEX errors in grammar and punctuation (continued) colons, 6.67 dangling gerunds, 5.116 dangling infinitives, 5.109 dangling participles, 5.115 double comparative or superlative, 5.87 misleading connectives, 5.142 number of predicate nouns, 5.141 overuse of compound adverbs, 5.161 overuse of prepositions, 5.187-92 possessive pronoun with apostrophe, 5.50 in pronoun antecedents, 5.28 pronoun case, 5.37 remote clauses, 5.60 See also errors in text errors in text checking for, 2.56,2.108,13.6 correction of, 1.27,1.28,2.69,2.70 elimination of, 2.2 errata pages, 1.68,1.90,1.94,1.112 and file conversion, 2.80,2.113,2.137 indexer’s notation of, 16.125,16.131 in original quotation, 13.7,13.61 paraphrasing to avoid, 13.4 in prepress proofs, 2.107 regression testing for, 2.138 responsibility assigned for (see alter ations) silent correction, 13.7 spelling, 2.111 See also errata; errors in grammar and punctuation; permissible changes to quoted and referenced materials; proofreading Esq., 10.16 eszett, 11.45 et al.
appropriate use of, 5.250,15.53 and indexing authors’ names, 16.115 meaning of, 10.42 punctuation with, 6.20,6.123,10.4 roman type for, 7.55 in source citations: author-date ref erence system, 15.9,15.29; four or more authors, 14.23,14.32,14.76; not italicized in text citations, 15.29; shortened citations, 14.32 etc. (etcetera), 5.250, 6.20,10.7,10.42 ethnic groups, 7.10,7.89 (sec. 2), 8.38,8.39. See also countries; groups of people ethnographic field notes, 2.61,13.49 etseq., 10.42,16.12 Eucharist, 8.110 Europe currency, 9.20,9.23 resources on European Union, 14.304 See also British style; European style; and specific countries and languages European style all-numeral dates, 9.35 decimal markers, 9.55 metric units, spelling of, 8.152 numerals with sovereigns, 9.41 period between title and subtitle, 14.89 space between digits, 9.55,9.56 twenty-four-hour system of time, 9.39, 9.40 See also British style events historical, 8.75-76,11.61 natural phenomena, 8.77 religious, 8.89,8.108,8.110 speeches, 8.76 sporting, 8.78 ever, as prefix, 7.89 (sec. 3) ever, as suffix, 5.65,5.66 every, 5.33 ex, as prefix, 7.89 (sec. 3) except, 5.201,5.237,5.250 exclamation points, 6.71-74 marking proofs for, 2.132 other punctuation with: brackets, 6.73, 6.74; commas, 6.125; ellipses, 11.102, 13.54; em dashes, 6.89; parentheses, 6.74, 6.98; periods, 6.124; question marks, 6.126; quotation marks, 6.10, 6.74,13.30,13.69 uses: editorial interpolation, to be avoided, 6.73; with ellipses, in Rus sian, 11.102; exclamatory questions, 5.213, 6.72; generally, 6.71; with introductory oh or ah, 6.35; inverted, in Spanish works, 11.19,11.62; in mathematical expressions, 12.32; in titles of works, 6.124-26,14.92, 14.96,16.54 See also exclamations; interjections; punctuation INDEX exclamations, 5.216, 6.35. See also exclama tion points; interjections exclamatory questions, 5.213 exhibitions, 8.201,14.236 expletives, 5.30, 5.47, 5.239-41, 6.93 exponents abbreviations with, 10.66,12.49 dex (decimal exponent), 9.12 rewriting fractions using, 12.49 uncertainties in, 12.60 See also mathematical expressions expressed folios, p. 980 expression, defined in copyright, 4.5 extra, as prefix, 7.89 (sec. 4) extracts. See block quotations fables, 8.185 Facebook, 8.191. See also social media content facing pages alignment of, 2.117 tables on (broadside), 2.115, 3.51, 3.86, 3.87, fig. 3.16, fig. 3.25 facsimile editions, 1.47 fact-checking, 2.56,13.5 facts of publication. See publication details fair-use doctrine, 4.84-94 and attribution, 4.75,4.84,4.92 Chicago’s guidelines for, 4.94 and Creative Commons licenses, 4.62 and dissertations and theses, 4.60 general rules, 4.86 legal overview, 4.84 and photocopying, 1.103 “rules of thumb,” 4.85 and specific materials: charts, tables, and graphs, 4.91; data in tables, 3.77; epigraphs, 4.87; illustrations, 3.32; interior monologues, 4.87; interviews, 4.77; paraphrasing, 4.89; pictorial and graphic, 4.90; unpublished works, 4.88 statements on, 1.84 and unnecessary permissions, 4.93 See also acknowledgments; illustration credits and credit lines; permissible changes to quoted and referenced materials; source notes fairy tales, 8.185 F&Gs (folded-and-gathered sheets), 2.107, p. 980 fan diagrams, 3.46. See also charts and graphs feature definition documents, 2.139 federal court decisions. See legal and public documents; legal cases; legal-style citations Federal Register, 14.289 female and male, 5.259 few, 5.230 fewer, less, 5.250 ffi, 10.42,14.148-49,16.12 fiction. See dialogue; literary works field notes, 2.61,13.49 figures abbreviation of term, 3.9,10.42,14.150, 14.158,16.116 defined, p. 980 source citations for, 14.158 text references to, 3.9 use of term, 3.1, 3.5, 3.9, 3.23 See also artwork; captions; illustrations figures of speech, 2.51,7.60,7.62 file extensions, 2.37,7.77 file formats and devices for e-books, 14.100,14.137,14.159 for electronic supplementary data, 14.112,14.187 functional features in, 1.119 and hyperlinks, 1.121 and illustration quality, 1.107 for illustrations, pp. 981, 982, 985, 989 for journal articles, 14.175 for magazine articles, 14.189 in metadata, 1.75 for multimedia content, 2.4,14.187, 14.261,14.267,14.268 names of, 8.155 for publications in multiple formats, 1.77,14.16 terminology and punctuation specific to, 7.76.7.78 testing of, 2.138,2.139,2.140 for unpublished documents, 14.225 See also apps; electronic publications; software; and specific formats file-hosting services, for version control, 2.78 file names for artwork, 3.16 extensions, 2.37,7.77 INDEX file names (continued) format for, 7.79 in production checklist, 2.77 for proofreading, 2.103 slashes in, 6.112 typographic considerations, 7.79 for version control, 2.78 files, p. 980. See also electronic files file transfer protocol (FTP), p. 980. See also URLs film. See movies and film; multimedia content financial terminology, 10.69. See also busi ness and commerce; currency fingerspelling, 6.77,11.130 finite verbs, 5.35,5.138 Finnish language, 11.70 firearm calibers, 9.19 first-line indentation. See paragraph (first- line) indentation first lines indexes of, 16.6,16.144,16.145 references to, 8.182 fiscal year (FY), 9.64 fl. (floruit), 10.42 fl (flush left), 2.127 flat (bare) adverbs, 5.160,5.250 flexibinding (limp binding), p. 980 flush, p. 980 flush-and-hang style. See hanging (flush- and-hang) indentation flush left (ragged right) justification, 2.10, 3.73,7.47,16.136, fig. 3.19, p. 986. See also justification flyleaf, 1.72 fold, compound terms with, 7.89 (sec. 3) folded-and-gathered sheets (F&Gs), 2.107, p. 980 folios defined, 1.5, p. 980 folio editions, 14.155,14.225 manuscript collections, 14.225 See also page numbers folktales, 8.185 fonts and typefaces archaic, 13.7 for computer terminology, 7.79 defined, pp. 980, 989 for digital artwork, 3.15 and electronic file cleanup, 2.80 embedded, 12.13 fonts and typefaces (continued) Gothic (Fraktur), 11.45,12.65,12.66, 12.68 for letters as shapes, 7.67 for mathematical copy, 12.11-13,12.66 for old-style numbers, 3.86 permissible changes to, 13.8 proofreading of, 2.113,2.131,10.8 and punctuation, 6.4-6, 6.129 sans serif, p. 987 serif, p. 987 specialized, for non-English materials, 11.21,11.72,11.90,11.110 See also boldface; italics; roman (type); typographic considerations; Unicode standard foot folios, p. 980 footnotes continued across pages, 14.41,16.113, fig. 14.1 editing of, 2.62 vs. endnotes, 1.124,2.77, 2.80,14.43-48, 14.57,14.59 endnotes used with, 14.49-51, fig. 14.4 examples, fig. 14.1, figs. 14.4-5 index entries for, 16.112 in journals, 1.108 keyed to line or page numbers, 14.53, fig. 14.5 length considerations, 14.40,14.56 manuscript preparation guidelines, 2.22 not always used in electronic publica tions, 14.27,14.49,14.52 numbering of (see note numbers) overview, 14.43 in previously published materials, 2.45 proofreading, 2.134 pros and cons of, 14.44 for tables (see tables: notes to) unnumbered, 1.50,2.46,14.52,14.54 See also endnotes; notes; shortened citations for, 5.201 foreign-language materials. See non English materials foreign names. See non-English names foreign rights. See rights to distribution outside the US foreign words and phrases. See non-English words and phrases INDEX forewords format of, 1.40 format of term, 8.179,14.110 indexing of, 16.109 material appropriate for, 1.43 mentioned on covers or jackets, 1.70 pagination, 1.7 placement in book, 1.4 vs. prefaces, 5.250 source citations for, 14.105,14.110 as work made for hire, 4.10 for example, 6.51, 6.58, 6.64, 6.88 format information. See file formats and devices formatting manuscript for submission. See manuscript preparation guidelines for authors forms, titles of, 8.187 forms of address, 6.66, 8.33,10.16,10.17.
See also dialogue; direct address; speech for... sake expressions, 7.21 Fort, 10.30,11.26,16.90 forthcoming generally, 8.188,14.146 for journal articles, 1.78,1.104,1.113, 14.172 in reference lists, 15.18,15.45 forums, electronic, 14.210 forward slash. See slashes foster, compound terms with, 7.89 (sec. 3) foul (dead) copy or proofs, 2.105,2.110 fr (flush right), 2.127 fractions case (text-sized with horizontal bar), 12.45 hyphenation, 7.89, 9.14 and line breaks, 6.113 in mathematical copy, 12.45-49; in display, 12.46,12.47; multiple and multilevel, 12.48; rewritten with exponents, 12.49; and subscripts and superscripts, 12.47,12.49; in text, 12.45,12.47 simple, 9.14 slashes in, 6.110, 6.113,12.45,12.47 spelled out, hyphenation of, 7.89 (sec. 1) symbols for, 6.110, 9.15 whole numbers with, 9.15 See also decimal points Fraktur (Gothic) type, 11.45,12.65,12.66, 12.68 France, currency, 9.23. See also French language fraud. See data falsification; plagiarism free, compound terms with, 7.89 (sec. 3) French flaps (gatefolds), 1.70 French language, 11.25-37 abbreviation of French, 10.42 abbreviations, 10.17 capitalization, 8.7,11.26,11.27 decimal markers, 9.55 dialogue, 11.31 ligatures, 7.34,7.35 proper names, 8.7,11.26,16.93 punctuation: ellipses, 11.32; em dashes, 11.28,11.31; guillemets, 11.28-32; periods omitted from contractions, 10.4; quotation marks, 11.30; spacing with, 11.28-29,11.31-32 resources on, 11.25,11.27 Saint in place-names, 11.26,16.93 social titles, 10.17 special characters, 11.37 titles of works, 11.27,16.48,16.52 word division, 11.33-36 from, 6.78, 9.60 frontispieces, 1.18,1.39,3.22 front matter, 1.17-45 acknowledgments, 1.4,1.41,1.42,1.89 author’s signature, 1.41 biographical notes, 1.18,1.66 chronologies, 1.4,1.60 copyright information (see copyright notice; copyright page, contents of) dedication, 1.4,1.7,1.11,1.36,2.3, 16.109 editing of, 2.57 editorials, 1.102 in electronic publications, 1.122 epigraph and epigraph source, 1.4,1.37 examples, figs. 1.1-4 forewords, 1.40 frontispieces, 1.18,1.39, 3.22 half-title page, 1.17,2.3 as historical context, 1.105 indexing of, 16.109 information for contributors, 1.88 introduction appropriate to, 1.43 journals, 1.84-85,1.87-91,2.102,2.105, 2.134 list of contributors, 1.64 lists of abbreviations, 1.44, fig. 1.8 INDEX front matter (continued) lists of illustrations and tables, 1.39, 3.38, figs. 1.6-7 online publications, 1.122 overview, 1.3 pagination of, 1.4,1.7,1.9,1.18,2.38,2.57 prefaces, 1.41 in production checklist, 2.77 publisher’s, translator’s, and editor’s notes, 1.45,1.95,14.51 recto vs. verso for elements, 1.4 running heads, 1.11,1.16 series title, 1.4,1.18 special issue noted, 1.84,1.87,1.89 submission of, 2.3 table of contents, 1.38,1.87, fig. 1.5 title page, 1.17-19 See also many of the above elements under their own headings FTP (file transfer protocol), p. 981. See also URLs full, fill, as suffixes, 7.89 (sec. 3) full bibliographies, 14.64. See also bibliog raphies full measure, p. 983 full-size caps (capitals) for emphasis, 7.52 preferred to small caps, 9.34,10.8,10.38 small caps compared to, 7.52 full stops. See periods (punctuation) funding sources, 1.92 fused participles, 5.114 future perfect tense, 5.134 future tense, 5.131 FY (fiscal year), 9.64 galleries callouts not needed for, 3.8 defined, p. 981 in list of illustrations, 3.39 manuscript preparation guidelines, 2.28 numbering illustrations in, 2.28, 3.14 placement of, 3.6 plates in, 3.5 in table of contents, 1.38,1.39,3.39, fig.
1.5, fig. 1.7 unpaginated, 1.6, 3.6, fig. 1.7 galley proofs, 2.104, p. 981 games, 8.190. See also video games gatefolds (French flaps), 1.70 gay, lesbian, 8.41 gender avoiding bias, 5.47,5.48,5.250,5.252, 5.255-59 capitalization for terms, 8.41 for named natural phenomena, 8.77 necessary gender-specific language, 5.258 and non-English personal names, 8.11, 8.12 in nouns, 5.11 and personal identification, 5.256 pronouns, 5.43,5.47,5.57 See also bias-free language gender identity or expression, 5.256, 8.41 genealogical charts, 3.46, fig. 3.10. See also charts and graphs general, compound terms with, 7.89 (sec. 3) generic markup. See markup for manuscript editing genetics, 8.131-33 genitive case definition and uses, 5.9,5.20-22 form of, 5.41 joint (group), 5.22 o/-genitive, 5.21,5.191 possessives, 5.20,7.25 pronoun and gender in, 5.43 reciprocal pronouns, 5.53 See also possessives genus and species names, 8.120-25 abbreviation of genus name, 8.121 and author names, 8.124 divisions higher than genus, 8.126 English derivatives from, 8.127 italics for, 8.2,8.120,8.122, 8.123,8.173 plant hybrids, 8.125 resources on, 8.119 in sentence- and headline-style capital ization, 8.158, 8.159 subspecies and varieties, 8.122 unspecified species and varieties, 8.123 geographical terminology abbreviation of geography, 10.42 abbreviations: Canadian provinces and territories, 10.28; city plus state, comma with, 10.29; compass points, 10.34,10.35; country names, 10.31, 10.32,14.290; latitude and longi tude, 10.36; place-names with Fort, INDEX geographical terminology (continued) Saint, and such, 10.30,11.26,16.90, 16.93; postal vs. standard abbrevia tions, 10.33; US states and territories, 10.4,10.27,14.130; US vs. United States, 10.32 capitalization: continents, countries, cities, oceans, and such, 8.45; topo graphical divisions, 8.53-55 compass points and directions, 7.89 (sec.
1), 8.46-47,10.34-36 generic terms for, 8.53,8.54, 8.55,8.56, 8.57,10.30,11.26 non-English terms for, 8.55,11.26,11.99, 16.93 real vs. metaphorical names, 8.50 regions, 8.47,8.54 resources on, 8.47,16.32 urban areas, 8.49 See also place-names geological terminology, 8.134-36,10.42 German language, 11.38-45 articles (parts of speech), 16.52 billion, 5.250 capitalization, 7.54, 8.8,11.39 names, 8.8 nouns, 7.54,11.39 orthographic reform, 11.38,11.43,11.45 punctuation, 11.40-41 resources on, 11.38 special characters, 11.45 titles of works, indexing of, 16.48,16.52 word division, 11.42-44 Germany currency, 9.23 formerly GDR (German Democratic Republic) and FRG (Federal Republic of Germany), 10.31 See also German language gerunds in compound terms, hyphenation of, 7.89 (sec. 2) dangling, 5.116 defined, 5.112 vs. participles, 5.113 possessives with, 7.28 word division, 7.41 See also participles ghostwritten books, source citations for, 14.105 GIF files, p. 981 global changes in manuscript editing, 2.69,2.79,2.80, 2.85 in proofreading, 2.111,2.113,2.122 global positioning coordinates, 10.36 glossaries boldface for terms appearing in, 7.56 format and alphabetizing, 1.61,2.23 format of term, 8.179 hyperlinks to, 1.121 and indexing, 1.61,16.109 placement in book, 1.4 punctuation, 1.61,2.23 submission of, 2.3 glosses ASL signs, 11.128-35 in index entries: abbreviated organiza tion names, 16.46; authors’ names after titles, 16.50,16.97; newspaper titles, 16.48; parentheses for, gener ally, 16.98 (see also personal names in indexes below) parentheses for, generally, 6.96,16.98 personal names in indexes: clerical titles, 16.39; entities with same name, 16.45,16.62; married women, 16.36; people with same name, 16.35; persons with unknown full names, 16.43; pseudonyms, 16.34; saints, 16.42; sovereigns and other rulers, 16.37,16.97; titled persons, 16.38 single quotation marks for, 11.63 translations of non-English terms, 6.96, 6.100,11.5 See also editorial interpolations and clar ifications; translations and translated works GMT (Greenwich mean time), 10.41 “gold” open access, 4.61 Google Books, 14.10,14.162 Google Play Books, 14.159. See also file formats and devices Gothic (Fraktur) type, 11.45,12.65,12.66, 12.68 governmental entities abbreviations, 10.26,14.281 acts and treaties of, 8.66,8.80, 8.81, 14.275,14.290 administrative, 8.63 INDEX governmental entities (continued) capitalization of, 8.52, 8.62-65 courts, 14.276-79; basic elements in source citations, 14.276; Canadian, 14.294; international, 14.304; names and terms, treatment in text, 8.64; United Kingdom, 14.298; US federal, 14.278; US state and local, 14.279; US Supreme Court, 4.93,14.277 forms published by, 8.187 legislative and deliberative, 8.62, 14.281-88 loose-leaf services of, 14.220 numbered divisions, 9.46 numbers for successive governing bodies, 9.45 programs of, 8.66, 8.75,8.80 public-domain works produced by, 4.21, 4.44, 4.68, 4.75 titles and offices of people, 8.19, 8.22 See also cities and towns; civil and civic concerns; countries; institutions; legal and public documents; legal cases; legislation; presidential docu ments; states (US); unpublished and informally published materials grades, scholastic, 7.65 graduate student publications. See disserta tions and theses grammar correction of, in previously published materials, 2.44 defined, 5.1 parts of speech, 5.3, i0.42 preventive, 5.57 schools of thought, 5.2,5.218 vs. usage, 5.249 See also adjectives; adverbs; conjunc tions; errors in grammar and punc tuation; exclamations; interjections; nouns; prepositions; pronouns; syntax; verbs; word usage grants and subventions, 1.20,1.31 graphs. See charts and graphs great,great-grand, 7.89 (sec. 3) Great Britain. See British style; United Kingdom Greek alphabet beta (0) vs. eszett (E), 11.45 for mathematical copy, 12.12,12.13, 12.68, table 12.3 Greek alphabet (continued) numbers in, 11.117, table 11.5 transliteration, 11.110-15, table 11.4 typesetting, 11.111 See also Greek language Greek language, 11.110-21 abbreviation of Greek, 10.42 possessives in, 7.19 punctuation, 11.116 resources on, 11.110 unaccented words, 11.114 word division, 11.118-21 See also classical Greek and Latin works; Greek alphabet “green” open access, 4.61 Greenwich mean time (GMT), 10.41 group (joint) genitive, 5.22 groups of people, 8.38-43 colors designating, 8.38 compound nationalities, 8.39 ethnic and national groups, 7.10,7.89 (sec. 2), 8.38, 8.39 generations, 8.42 kinship terms, 8.36 physical characteristics, 8.43 socioeconomic classes, 8.40 See also bias-free language; names, per sonal; titles and offices of people guillemets permissible changes to, 11.7,11.11,11.19, 13.7 spacing with, 11.28-29,11.47 uses: French, 11.28-32; German, 11.41; Italian, 11.47; Russian, 11.101; Span ish, 11.63,11.64 gutter, p. 981 habitual actions, 5.129 hairline rule, p. 981 hair spaces, 6.5, 6.11, 6.120, p. 989 half, compound terms with, 7.89 (sec. 3) half titles content of, 1.17 editing of, 2.57 pagination of, 1.4,1.7,1.8,1.17 proofreading of, 2.134 publisher’s responsibility for, 2.3 running heads omitted from, 1.11 second half title, 1.4,1.8,1.46 halftones, 3.3, 3.6, figs. 3.1-2, p. 981 halftone screen, p. 981 INDEX Handle System, 14.8 handshapes (ASL), 11.132 hanging (flush-and-hang) indentation bibliographies, 1.63,2.11,2.24, fig. 14.8 defined, 2.11, pp. 980,982 dialogue and drama, 13.46 glossaries, 2.23 indexes, 16.24,16.106,16.131 lists of abbreviations, 2.23 reference lists, 1.63,2.24 runover lines in, 2.24,16.137 transcriptions of interviews or discus sions, 13.48 vertical lists, 2.11, 6.130 See also indexes: indented style; indexes: run-in style Hansard Parliamentary Debates (UK), 14.301 Hanyu Pinyin system. See Pinyin system hard copy, p. 981. See also manuscript editing: paper-only; manuscripts: paper-only hardcover books binding, p. 977 clothbound covers: defined spine copy, 1.69; description of, 1.69; endpapers, 1.72; proofreading of, 2.108 jackets (dust jackets), 1.71; artwork, 1.73, 4.100,4.101; author’s previous publications, 1.18; biographical in formation, 1.66; country of printing, 1.27; defined, p. 978; DOIs, 1.33; illustration credit lines, 1.73; ISBN and bar codes, 1.32,1.69,1.74; meta data on, 1.75; proofreading of, 2.100, 2.103, 2.108 paper-over-board format (lithocase), 1.71 See also covers of journals and paperback books hardly, 5.230 hard return (Enter key), 2.12,2.19,2.80 hard vs. soft hyphens, 2.96 Hausa language, 11.23-24 have, has, had, 5.132,5.133,5.153 Hawaiian language, 11.70,11.70nl he, him, his, 5.47, 5.48,5.49, 5.50,5.252 headings chronologies, fig. 1.9 endnotes, 14.46 indexes, 16.131 headings (continued) letters and diaries, 1.52 lists of abbreviations, 14.60 lists of illustrations and tables, fig. 1.6 for navigation, 14.160,15.23 with note numbers, 14.27 proofreading of, 2.113,2.134 punctuation in, 6.14 reference lists, 15.5,15.10 See also index entries; subheads headlines, 7.52,14.191,14.192,14.195 headline-style capitalization in author-date reference system, 15.6 chapter and part titles, 2.58 examples of, 8.160 generally, 8.1 hyphenated compounds in, 8.161,11.26 journal article titles, 8.168,14.169 magazine article titles, 14.188 magazine departments, 14.190 mottoes, 7.62 newspapers, 14.192,14.195 non-English words and phrases, 11.4, 11.10,11.27,11.54 and permissible changes, 8.165 principles of, 8.159 publishers’ names, 14.136 series titles, 14.123 signs and notices, 7.61 subheads, 2.18,2.59,8.158 table titles, 3.54,3.55 titles containing quotations, 8.162 titles in captions, 3.21, 3.22 titles of websites, 14.206 titles of works, generally, 8.168,14.87 unpublished titles, 8.188 website titles, 8.191 See also sentence-style capitalization head margin, p. 981 headnotes bibliographies with sections, 14.63, 14.64, fig. 14.9 indexes, 16.116,16.140,16.141,16.143, 16.145 notes to tables, 3.78 previously published materials, 2.46 Hebrew language, 11.91-97 capitalization and italics, 11.93 prefixes, 11.93 resources on, 11.91 romanization systems, 11.91 INDEX Hebrew language (continued) special characters, 11.92, table 11.2 unromanized phrases in, 11.96 vowels, 11.97 word division, 11.95,11.96 height, 3.27,7.89 (sec. 1), 10.66. See also units of measurement helping (auxiliary) verbs. See auxiliary (helping) verbs help menus, 1.122, 8.155 hence, 6.57 he or she, him or her, 5.250, 5.255 Hepburn system, 11.86-87 her, hers, 5.49,5.50 herself, 5.51 hidden text, 2.39,2.80,2.113 highways and interstates, 8.56, 9.50,10.33, 11.26. See also addresses, mailing himself, 5.51 historic, historical, 5.250,7.33 historical and cultural terminology abbreviations, 10.42 acts, treaties, and government pro grams, 8.66,8.75,8.80-81,14.275, 14.290 awards, 8.31,8.83, 8.115 events, 8.75-78; battles and campaigns, 8.114; natural phenomena, 8.77; religious, 8.89,8.108, 8.110; Spanish language, 11.61; speeches, 8.76; sporting, 8.78; wars and revolutions, 8.113 movements and styles: descriptive des ignations, 8.72; period names, 8.73, 8.74; treatment in text, 8.60,8.61, 8.79,11.88 oaths and pledges, 8.84 periods of time: capitalization, 8.71-74; centuries, 7.8,7.87,7.89,7.89 (sec. 3), 8.71,9.32,9.33; decades, 9.33; descriptive designations, 8.72; eras, 8.135,9.34,9.64,10.38; numerical designations, 8.71; traditional names, 8.73 See also academic concerns; legal cases; political terminology holidays, 8.89 home pages, 1.84,1.86,1.87,1.122. See also websites and web pages homonyms, 6.55,7.38 Honorable, Hon., 8.32,8.33,10.18 honorifics abbreviations, 10.18 alphabetizing, 16.67,16.80,16.87 capitalization of, 8.26,8.33 in index entries, 16.39 non-English, 16.76,16.80,16.87 pronouns in, 5.40 See also titles and offices of people honors. See awards and prizes horticultural terminology. See botanical terminology house, compound terms with, 7.89 (sec. 3) house style author communications, 2.68,2.70 editing for, 2.49,2.55 generally, 1.116 for indexes, 16.103,16.106,16.118, 16.123,16.129 proofreading for, 2.113 for source citations, 1.108-9,2.64, 14.2-4 for tables, 3.86 See also style sheets (editorial) how, 5.202, 6.69 however, 5.157,5.204, 6.49, 6.57 HTML (hypertext markup language), 1.77, 1.117,1.118,1.119,2.139, p. 981. See also online publications; websites and web pages HTTP (hypertext transfer protocol), 7.80, 14.17, p. 981. See also URLs HTTPS, p. 981 humanities style. See notes and bibliogra phy system; source citations hundreds, thousands, and hundred thou sands, 9.3,9.4,9.54 Hungarian language, 8.13,11.70,16.78 hybrids, plant, 8.125 hyper, as prefix, 7.89 (sec. 4) hyperlinks added to printed work, 4.64 cross-checking, 2.32,2.34 for DOIs, 1.33,1.121 in e-book formats, 1.118,1.120,1.121 for errata, 1.68,1.90 generally, 1.121 in HTML format, 1.118 for illustrations, 1.96,1.107, 3.8,3.51 in indexes, 1.125,16.7,16.13,16.15, 16.102,16.108 INDEX hyperlinks (continued) in journals: for errata, 1.90; home pages, 1.86; illustrations, 1.107; to related works, 1.86; for retractions, 1.91; for source citations, 1.96,1.106,1.109, 14.6; tables, 1.106; tables of contents, 1.86,1.87,1.90, fig. 1.11 markup for, 2.73,2.83 metadata for, 1.92,1.121 in online publications, 1.120,1.121,1.122 in PDF files, 1.118,1.121 and tables, 1.96,1.106, 3.51 testing of, 2.137,2.138 See also DOIs; hyperlinks for source citations; URLs hyperlinks for source citations advantages of, 1.124 annotations, 14.53 and author-date reference system, 15.5, 15.9 checking, 2.34 consistency in, 2.64 and footnotes vs. endnotes, 14.43,14.45 generally, 14.19,15.5 in journals, 1.96,1.106,1.109,14.6 and metadata, 1.92 multiauthor volumes, 1.50 and shortened citations vs. ibid., 1.124, 14.34 source notes for previously published material, 14.54 tables, 1.106 and 3-em dashes, 1.124,14.67,15.17 unnumbered notes, 14.27,14.52 and URLs in citations, 14.6,14.23 to works cited, 1.109,14.61 hypertext, p. 981. See also HTTP; hyperlinks hypertext markup language (HTML), 1.77, 1.117,1.118,1.119,2.139, p. 981. See also online publications; websites and web pages hyphenated compounds, 7.82. See also compound terms hyphens and hyphenation capitalization issues, 8.161,11.26 in compound terms, 7.81-89; adverbs ending in -ly, 7.86,7.89; before or after noun, 7.85; with compound modifiers, 5.92, 5.93,7.8,7.85; en dash vs. hyphen, 6.80; ethnic groups and nationalities, 7.89 (sec. 2), 8.39; hyphens and hyphenation (continued) guide to, 7.89; headline-style titles, 8.161,11.26; and line breaks, 2.112; multiple hyphens, 7.87; personal names, 8.6; with prefixes or suffixes, 6.80,7.40,7.81,7.87-89, 8.161; sus pended hyphens, 7.88,7.89 (sec. 1); trend toward closed, 7.83 vs. dashes, 6.75, 6.80 guide to, 7.89 manuscript and editorial concerns: electronic file cleanup, 2.61,2.80; guidelines for authors, 2.13; marking manuscript for, 2.96; marking proofs for, 2.132; proofreading, 2.105,2.112, 2.116; soft vs. hard hyphens, 2.96; stacks of, 2.112,2.116,7.47; typo graphic considerations, 7.47 non-English languages: Arabic, 11.79; Chinese, 11.84,11.88; compound terms from, 7.89; French, 11.26; glossed American Sign Language, 11.128; Hebrew, 11.93; Japanese, 11.87,11.88 personal names with, 8.6, 8.7, 8.14, 11.88,16.36,16.72 readability as key to, 7.84 software settings for, 2.13,2.80,7.36,7.38 URLs, DOIs, and such in text, 2.13,7.46, 14.18 uses, other: all-numeral dates, 6.108, 9.36; double or multiple numera tion, 1.57; fractions, 7.89, 9.14; gene names, 8.132; global positioning coordinates, 10.36; keyboard com binations and shortcuts, 7.78; mass, in chemical terms, 8.150; music writing, 7.71; noun plus numeral or enumerator, 7.89, 9.13; phrasal adjec tives, 5.92,5.93; separators (numbers and letters), 6.77; in source citations for journal articles, 14.171; telephone numbers, 9.57 See also compound terms; dashes; em dashes; en dashes; punctuation; 3-em dashes; 2-em dashes I (pronoun) antecedent absent, 5.30 appropriate use of, 5.250 capitalization of, 5.40 INDEX I (pronoun) (continued) misuse of, 5.37 with personally, 5.250 than with, 5.183 ibid.
meaning of, 10.42 roman type for, 7.55,10.7 uses: notes, 1.124,14.34,14.48; text citations, 13.66,14.58 Ibn, Abu, Abd, 16.75 iBooks, 14.159. See also file formats and devices ice age, 8.74,8.135 Icelandic language, 11.70 id. (idem), 10.42,14.35,14.275,14.279 idioms earth in, 8.140 and editorial discretion, 2.51 and ergative verbs, 5.99 and fused participles, 5.114 and genitive case, 5.20 and grammatical ellipses, 5.229 negative, 5.211 only in, 5.186 prepositional, 5.193-95 and problematic words and phrases, 5.250 and transitive and intransitive verbs, 5.98 zero article in, 5.77 See also colloquial speech; dialect; slang i.e., 5.250, 6.51,10.7,10.42 if, 5.124, 5.201, 6.24, 6.40. See also con junctions if, whether, 5.250 if... then, 5.199,5.244 illustration credits and credit lines, 3.29-37 contents of, 3.29,4.102 editing of, 2.65 format of, 4.102 importance of, 4.75,13.3 placement of, 3.30; in back matter, 1.4, 3.30; on copyright page, 1.30,1.73, 3.30,4.78; on cover or jacket, 1.73; list of illustrations and tables, 3.38 proofreading of, 2.108 types of material: adapted material, 3.37; author’s own work, 3.31, 4.78; commercial agency material, 3.36; commissioned material, 3.33; cover or jacket artwork, 1.30,1.73; interior illustration credits and credit lines (continued) book design, 1.30; maps, 3.33, 8.199; obtained free of charge, 3.34; public-domain work, 3.35; requiring permission, 3.32; translations of quoted materials, 11.14 See also permissions; source notes illustrations, 3.3-46 accessibility issues, 3.28 acquisition of: adapted material, 3.37; commercial agency material, 3.36, 4.99,4.101; commissioned material, 3.33; material obtained free of charge, 3.34; previously published material, 2.47; public-domain ma terial, 3.35; published work needing permission, 2.2,2.3, 3.32; works made for hire, 3.33,4.10,4.75 blind page numbers for, 1.7,1.39 credits for (see illustration credits and credit lines) defined, 3.1 electronic files, preparation of: artwork, prints vs. scans, 2.27; charts, 3.43; digital artwork, 3.15; file formats, pp. 981,982, 985,989; identified for publisher, 3.16; numbering of, 2.28; separate files for each component, 2.26,2.36,2.80,3.16; submission to publisher, 3.3,3.4, 3.15-20; text references and callouts, 2.30 as electronic supplementary data, 3.9, 3.26 examples of, figs. 3.1-10 fair use of, 3.32,4.90 frontispieces, 1.18,1.39,3.22 galleries: callouts not needed for, 3.8; in list of illustrations, 3.39; manu script preparation guidelines, 2.28; numbering illustrations in, 2.28, 3.14; placement of, 3.6; plates in, 3.5; in table of contents, 1.38,1.39, fig. 1.5, fig. 1.7; unpaginated, 1.6, 3.6, fig. 1.7 hyperlinks to, 1.96,1.107, 3.8, 3.51 indexing of, 16.116,16.140,16.141 on journal cover, 1.84 keys to (legends), 3.7, 3.12,3.19-21,3.24, fig. 3.4, fig. 3.6 in letters to the editor, 1.101 manuscript and editorial concerns: INDEX illustrations (continued) cross-checking, 2.32; editing of, 2.65; electronic file cleanup, 2.80; guide lines for authors (see manuscript preparation guidelines for authors: illustrations and tables); inventory of artwork, 3.17-18; placement, 2.62, 3.8; proofreading, 2.107,2.115,2.137 numbering, 3.9-14; arabic numerals for, 3.12; continuous vs. separate, 3.10; conventions of, 2.28; double or multiple numeration, 1.57,2.28, 3.11; in galleries, 2.28,3.14; in index locators, 16.116; separating captions from, 3.23; text references to, 3.9, 3.50, 8.180; working (temporary) numbers, 2.28,3.13 original dimensions noted for, 3.27 pagination of, 1.8,1.39,2.115 parts, identification of, 3.12,3.24, figs. 3.6-7 permissions for, 2.2,2.3, 3.18,3.29,3.30, 3.32,4.95,4.98-101 photo releases for, 4.77 plates: defined, 3.5, p. 985; indexing of, 16.116; lists of, 1.39, fig. 1.7; number ing of, 2.28; text references to, 3.9; use of term, 3.23 reproduction of: cropping, scaling, and shading, 3.19; halftones, 3.3,3.6, figs. 3.1-2, p. 981; previously published material, 2.47; publisher’s redrawing of, 3.20; scans, 2.27,2.47, 3.15 and running heads, 1.8,1.16,2.115 source citations for, 14.158 submission to publisher, 2.2,2.3,2.4, 2.27,3.3,3.4,3.15-20 symbols or patterns used in, 3.7, 3.25, 3.45, fig. 3.4, fig. 3.6, figs. 3.8-9 vs. tables, 3.2 text references and callouts (see callouts; text references to illustrations and tables) thumbnails, 1.96,1.107,3.8,3.51,4.90, p.989 types: color, 1.6,2.37,3.3, 3.15,3.19,3.43; continuous-tone, 3.3, p. 978; line art, 3.4, 3.19-20,3.41, figs. 3.3-4, fig. 3.6; musical examples, 3.4, fig. 3.5; text figures and plates as, 3.5 unnumbered, 3.13, fig. 3.3 illustrations (continued) as works made for hire, 3.33,4.10,4.75 See also artwork; captions; charts and graphs; lists of illustrations and tables; tables image agencies, 3.36,4.99,4.101 images. See artwork; illustrations imperative mood, 5.122, 5.155, 5.160,5.214, 5.255. See also imperative sentences imperative sentences (directives) as in, 5.250 conjunctions in, 6.22 conjunctive adverbs in, 6.57 defined, 5.214 exceptional, 5.215 for gender neutrality, 5.255 indirect questions as, 6.69 infinitives in, 5.109 negative, 5.152,5.235 periods with, 6.12 question marks with, 6.68 understood you in, 5.51, 5.122, 6.22 imperfect (progressive, continuous) tenses, 5.119,5.128, 5.135 implicit (zero) articles, 5.77 imposition, pp. 981-82 impressions defined, 1.26, p. 982 vs. editions, 1.23,1.26 impression lines, 1.20,1.28, figs. 1.1-4 and publication date, 14.143 See also reprints imprints, 14.138-39, p. 982. See also publish ers’ names in, after chapter titles in source citations, 14.106,15.9 inasmuch as, 5.201,5.250 Inc., 6.44,10.23,10.24,14.134 including, 6.50,10.42 inclusive (continuing) numbers, 9.60-64 abbreviating and condensing, 9.61, 9.62, 9.64,16.14 with abbreviations and symbols, 9.17 commas with, 9.63 dates, 6.78-79,9.34, 9.60, 9.64,14.117, 14.144,14.171 en dashes in, 2.80, 6.78-79, 9.60, 16.100,16.134 full style for, 9.62, 9.64 in index entries, 16.12,16.14,16.29, 16.100,16.103,16.113,16.121 INDEX inclusive (continuing) numbers (continued) with percent, 7.89 (sec. 1) and repeated quantities, 9.17 roman numerals, 9.61,16.14 in tables, 3.83 unfinished ranges of, 6.79,15.41 words between, 9.60 See also numbers; page ranges inclusive language. See bias-free language indeed, 6.49, 6.57 indefinite articles. See articles (definite and indefinite) indefinite pronouns, 5.47,5.67,5.139,5.256 indentation bibliographies, 1.63,2.11,2.24,14.64, fig. 14.8 definitions, 2.11, pp. 980,982 dialogue and drama, 13.8,13.46 and electronic file cleanup, 2.80 glossary entries, 2.23 indexes, 16.24,16.106,16.131,16.137 marking manuscript for, 2.98 paragraphs, 2.11,2.12 proofreading of, 2.113,2.126 quotations: block quotations, 2.11, 2.19; drama excerpts, 13.8,13.47; interviews and discussions, 13.48; paragraphs within, 2.19,13.22; poetry extracts, 2.11,2.20,13.25-27 runover lines (see runover lines) in tables, 3.60, 3.61, 3.63,3.73,3.74, fig.
3.13, fig. 3.21 text following block quotations, 2.21, 13.24 types of, 2.11 vertical lists, 2.11,2.21, 6.130 word-processing function, 2.11,2.12, 2.19,2.20,2.21,2.24,2.80,16.131 See also block quotations; indexes: in dented style; margins; quotations indented style (indexes). See under indexes independent (absolute) possessive pro nouns, 5.49 independent clauses adverbs modifying, 5.156 antecedents in, 5.57,5.59 colons with, 6.61 commas with, 6.22-23, 6.32 in complex sentences, 5.219 in compound-complex sentences, 5.220 in compound sentences, 5.218 independent clauses (continued) conjunctions with, 6.22, 6.32, 6.59 defined, 5.225 introducing quotations, 13.16 passive voice in, 5.115 quotations of, 6.40, 8.162 semicolons with, 6.56-59 and sentence types, 5.217-20 in simple sentences, 5.217 See also dependent (subordinate) clauses independent possessives, 5.49 independent professionals book designers, 2.73 editors, 2.48,2.73 indexers, 2.70,16.3,16.4 for obtaining permissions, 4.76nl proofreaders, 2.101 index cards, 16.107 index entries cross-references (see cross-references, in indexes) double posting, 16.16,16.30 examples, 16.141-45 inclusive (continuing) numbers in, 16.12, 16.14,16.29,16.100,16.103,16.113, 16.121 locators: appropriate use of, 16.12; chap ter numbers with, 16.111; checking, 16.133; ffi and et sec. not appropriate with, 16.12; italics and boldface for, 16.116,16.140,16.141,16.143; long strings of, 16.120,16.128,16.133; with main headings, 16.9; across mul tiple volumes, 1.9; for notes, 14.157, 16.111-13,16.117; number per entry, 16.10,16.120,16.128,16.133; number perpage, 16.118; page ranges, 16.12, 16.14,16.100,16.113,16.121; pagina tion final for, 2.104-6,2.113,2.118, 16.108; paragraph numbers as, 1.125, 2.106,16.12,16.13,16.108; punc tuation with, 16.25,16.94,16.100; for scattered entries, 16.12; for subentries, 16.10; undifferentiated, after main headings with subentries, 16.129 main headings: alphabetizing, 16.56 (see also alphabetizing); checking of, 16.133; cross-references following, 16.17,16.26; definition and use, 16.9; headings beginning with numerals, INDEX index entries (continued) 16.65; initial lowercase letters, 16.11; inversion in, 16.9,16.56,16.94; noun or noun phrase for, 16.9; refining terms for, 16.126; vs. subentries, 16.127; typing and modifying, 16.122; undifferentiated locators after, 16.129 punctuation, 16.94-100; and alpha betizing, 16.59; checking, 16.133; colons, 16.19,16.20,16.25,16.95; cross-references, 16.17-20,16.26, 16.95- 97; em dashes, 6.92,16.27, 16.99,16.141; en dashes, 16.100, 16.111,16.134; in indented style, 16.26,16.28,16.95; inversions, 9.42, 16.94; with locators, 16.94,16.100; in run-in style, 16.25,16.27,16.95, 16.96; subentries, 6.92,16.95-96; sub-subentries, 16.27,16.28,16.99, 16.142 subentries: alphabetizing, 16.68-70 (see also alphabetizing); articles in, 16.51, 16.52,16.68,16.129; checking of, 16.133; chronological order, 16.70, 16.133; cross-references in, 16.10, 16.18,16.20,16.26; cross-references to, 16.19,16.20; definition and use, 16.10; excessive number of, 16.133; indentation, 16.137; initial lowercase letters in, 16.11; vs. main entries, 16.127; marking proofs for, 16.120; numerical order, 16.69; punctuation, 6.92.16.95- 96; run-in vs. indented- style, 16.25-26,16.68,16.103,16.141- 43; syntax for, 16.10,16.68,16.129; typing and modifying, 16.122; when to use, 16.120,16.128,16.133 sub-subentries: avoiding, 16.27,16.133; and continued lines, 16.139; indented style for, 16.143; punctuation, 16.27, 16.28,16.99,16.142; run-in style for, 16.27,16.142; syntax for, 16.129 sub-sub-subentries, 16.28,16.143 vanity entries, 16.31 See also alphabetizing; indexes; indexing indexes automatic “generation” not possible, 16.105 continued lines, 16.139 editing of, 2.67,16.132-34 indexes (continued) electronic publications, 1.125,16.1,16.7, 16.13,16.15,16.102 examples, 16.141-45 format of term, 8.179 hanging (flush-and-hang) indentation, 16.24.16.106.16.131 headnotes for, 16.116,16.140,16.141, 16.143,16.145 indented style: continued lines, 16.139; cross-references, 16.20,16.26,16.28; description, 16.26; examples, 16.26, 16.28.16.142- 43; hanging (flush-and- hang) indentation, 16.24,16.106, 16.131; punctuation in, 16.26,16.28, 16.95; runover lines, 16.26,16.28, 16.137; subentries in, 16.26,16.68, 16.142- 43; sub-subentries in, 16.28, 16.143 length of, 16.103,16.118,16.133 order of multiple indexes, 1.65 placement in book, 1.4,1.65 proofreading of, 2.2,2.105 publisher’s preferences, 16.103,16.106, 16.118,16.123,16.129 quality of, 16.132 repagination issues, 2.118 run-in style: chronological order for subentries in, 16.70; continued lines, 16.139; cross-references, 16.20, 16.25,16.96; description, 16.25; ex amples, 16.141; hanging (flush-and- hang) indentation, 16.24,16.106; indentation guidelines, 16.137; punc tuation in, 16.25,16.27,16.95,16.96; for sub-subentries, 16.28,16.142; sub-subentries in, 16.27,16.28,16.141 running heads for, 1.14,16.6 vs. searching, 1.110,1.111,16.2,16.7 single vs. multiple indexes, 16.6,16.115, 16.131 submission format of, 16.131 in table of contents, 1.38, fig. 1.5 titles for, 16.131 types: author-and-title, for anthologies, 16.6; authors’ names, 16.6,16.115, 16.145; embedded, 16.7; first lines, 16.6,16.144,16.145; hyperlinked, 1.125,16.7,16.13,16.15,16.102, 16.108; journals (see journals: indexes); multivolume works, 1.9; INDEX indexes (continued) online, 1.96,1.109,1.125; past con tributors, 1.100; single vs. multiple, 16.6.16.115.16.131 typographic considerations: bad breaks, 16.138; continued lines, 16.139; dis tinguishing types of entries and lo cators, 16.116,16.140,16.141,16.143; justification, 16.131,16.136; multiple indexes, 16.6; running heads, 1.14, 16.6; separate vs. consecutive pagina tion in multivolume works, 1.9; titles of works, 16.47; typesetting instruc tions, 16.134; type size and column width, 1.65,16.135 value of, 16.2 as works made for hire, 4.10 See also alphabetizing; index entries; indexing indexing, 16.56-93 author’s role in: list of terms for indexer, 16.30,16.108; manuscript prepara tion, 16.24; vs. professional indexers, 16.3,16.4; responsibility for, 2.2, 2.70; and software, 16.104 from first proofs, 2.104,2.106,2.118 glossaries, 1.61,16.109 mechanics: alphabetizing (see alphabet izing); checking cross-references, 16.130; editing and refining entries, 16.124,16.126-30; editing someone else’s index, 16.132-33; index-card method, 16.107; marking proofs, 16.117-21, fig. 16.1; noting errors in text, 16.125,16.131; pagination final for, 2.104-6,2.113,2.118,16.108; parts of work to index, 16.109-16; preliminary work, 16.108; prepa ration, 16.101-7; and publisher’s preferences, 16.103,16.106,16.118, 16.123,16.129; software, 16.5,16.57, 16.104,16.117,16.123; submitting index, 16.131; typesetting instruc tions, 16.134; typing entries, 16.122, 16.123.16.131 for multiauthor volumes, 2.42 names, non-English personal, 16.75-87; Arabic, 16.75; Asian, other, 16.87; Burmese, 16.76; Chinese, 16.77; Hungarian, 8.13,16.78; Indian, 16.79; indexing (continued) Indonesian, 16.80; Japanese, 16.81; Korean, 16.82; Portuguese, 16.83; Spanish, 16.83,16.84; Thai, 16.85; Vietnamese, 16.86 names, personal: academic titles and of fices in, 16.40; basic rules, 16.71-75; clerical titles with, 16.39; compound, 16.72; as corporate names, 16.89; familiar forms of, 16.33,16.71; full name unknown, 16.43; hyphens in, 16.72; with initials, 8.13,16.63, 16.78; Jr., Sr., /I/, and such with, 6.43, 16.41; Mac, Me, or O’ with, 16.73; as main entry, 16.9; married women’s, 16.36,16.84; non-English (see names, non-English personal above); with numbers, 16.37,16.41,16.66; with particles, 16.71,16.84; people with same name, 16.35; pseudonyms, 16.34; saints, 16.42; Saint with, 16.74; sovereigns and other rulers, 16.37, 16.66; titled persons, 16.38 names, proper: abbreviations as, 16.46, 16.64; cross-references for (see cross-references, in indexes); entities with same name, 16.35,16.45,16.62; incomplete or unknown, 16.44; or ganizations, 16.46,16.64,16.88-89; passing mentions of, 16.31; person, place, and thing with same name, 16.62; resources on, 16.32,16.71; variants of, 16.32 (see also names, per sonal above and place-names below) notes, 14.157,16.110-13,16.117 place-names, 16.90-93; definite article with, 16.91,16.92; entities with same name, 16.62; with Fort, Mount, Saint, and such, 16.90,16.93; old vs. pres ent forms, 16.29; from short form in text, 16.44; two places with same name, 16.45 professional indexers, 2.70,16.3,16.4 publication process and schedules, 2.2, 2.67,2.70,16.4,16.101 resources on, 16.8,16.13,16.58,16.84, 16.104,16.107 term choices: editing of, 16.124, 16.126-27,16.129; marking proofs for, 16.117-20, fig. 16.1; noun or noun INDEX indexing (continued) phrase, 16.9; passing mentions omit ted, 16.31; reflective of text, 16.29- 30; and type of material, 16.118 titles of works, 16.47-55; article at beginning of, 16.48-49,16.51-52, 16.56; authors, first lines, and titles, 16.145; with authors’ names in parentheses, 16.50,16.97; double posted under authors’ names, 16.50; non-English, 16.48,16.49,16.52, 16.53; periodicals, 16.48-49; preposi tion at beginning of, 16.53; question mark or exclamation at end of, 16.54; separate index of titles, 16.6; subtitles omitted, 16.54; typographic treatment of, 16.47; works entries, 16.50 (see also alphabetizing: articles disregarded in) See also alphabetizing; index entries; indexes India, names in, 16.79 indicative mood, 5.121, 5.122, 5.124, 5.129, 5.130 indirect questions, 5.201, 6.42, 6.69 Indonesian language, names, 8.18,16.80 infinitives, 5.106-9 dangling, 5.109 defined, 5.106 gerunds as substitute for, 5.112 with have, 5.153 personal pronoun as subject of, 5.44 split, 5.108,5.171 uses of, 5.107 infinity symbol, circled (paper durability), 1.35 inflected genitives, 5.20 informal speech. See colloquial speech; slang infra, as prefix, 7.89 (sec. 4) initialisms ampersands in, 10.10 articles with, 5.250,7.33,10.9 capitalization, 10.6,10.25,10.26 company names, 8.189,10.24,10.25 country names, 10.31,10.32 definition and use, 10.2 disease and medical terms, 8.144 indexing of, 16.49,16.64 journal titles, 14.170,16.49 initialisms (continued) possessives of, 7.17, 8.189 small vs. full-size capitals for, 10.8 space omitted in, 10.5,10.10 versions and sections of Bible, 10.48 See also abbreviations, general; abbrevia tions, specific; acronyms initials decorative, in text, 13.37 in and as names: alphabetizing of, 8.13, 16.63,16.78; in author’s preface, 1.41; for authors with same last name, 15.22,16.35; for hyphenated given name, 8.7; and indexes of authors’ names, 16.115; punctuation of, 7.66, 10.4,10.12; in source citations, 14.73- 74,15.12,15.33,15.53; spacing of, 8.4; vs. spelled-out names, 16.63,16.115; and word division, 7.42 spacing of, 6.121,7.66, 8.4,10.12,14.74, 15.33 See also initialisms in-law, compound terms with, 7.89 (sec. 3) in order that, 5.201 in press, 1.78,1.86,1.113,8.188,14.146. See also forthcoming inscriptions, collections of, 14.249 insertions. See additions and insertions Instagram, 8.191. See also social media content institutions churches as, 8.98 names: academic, 6.81, 8.68; as authors, 14.70; capitalization, 8.68,8.85; in non-English languages, 11.26,11.61, 11.88; place-names within, 6.39, 6.81; possessives of, 7.20; in source citations for dissertations, theses, lectures, and such, 14.215,14.217; unusual capitalization in, 8.69 titles and offices in, 8.25,8.27, 8.28 See also academic concerns; business and commerce; governmental enti ties; organization names; universities integral sign, 12.41 intellectual property rights basic vs. subsidiary, 4.17-18 derivative works and performance, 4.14 originality, 4.3,4.5,4.72 public display, 4.13,4.14, 4.34 INDEX intellectual property rights (continued) reproduction and distribution, 4.13, 4.34, 4.51, 4.64,4.76 subdivision of, 4.34 trademark protection, 4.16 See also copyright; subsidiary rights intensifiers, 5.156 intensive and reflexive pronouns, 5.41, 5.51,5.53 inter, as prefix, 7.89 (sec. 4) interactive books, 14.268. See also multime dia content intercaps (midcaps), 8.69,8.154 interior (internal) discourse, 13.43 interior monologues, 4.87,13.43. See also dialogue; speech interjections, 5.206-9 abbreviation of interjection, 10.42 capitalization, 7.31 as colloquial, 5.207 defined, 5.206 exclamations as, 5.216, 6.35 functional variation in, 5.208 punctuation with, 5.206, 6.35 spelling of, 7.31 words used exclusively as, 5.209 See also exclamation points internal (interior) discourse, 4.87,13.43 International Digital Publishing Forum, 1.28 International ISBN Agency, 1.32,1.74 International Organization for Standardiza tion (ISO) archival practices, 1.114 country name abbreviations, 10.31 currency codes, 9.21,9.23 ISBNs, 1.32 standard date format, 6.108,9.35,9.36 International Phonetic Alphabet (IPA), 11.22 International Standard Book Number. See ISBN International Standard Serial Number. See ISSN International System of Units (Systeme international d’unites, SI), 10.51-59 base quantities and units, 10.52,10.54 binary systems, 9.11 decimal points, 9.55,9.56 derived units, 10.57,10.60 form for, 10.52 International System of Units (Systeme international d’unites, SI) (continued) grams, 10.55 tera-, and such, 9.10, 9.11, 10.49,10.55,10.56 non-SI units, 10.59 overview, 10.51 resources on, 9.56,10.51 SI units referred to as abbreviations, 10.2 technical abbreviations, list of, 10.49 internet abbreviations of related terms, 10.49 defined, p. 982 images from, 3.15 protocols, 14.17 treatment of term, 7.80 See also addresses, email; blogs and blog posts; electronic publications; hy perlinks; online publications; URLs; websites and web pages interpolations and clarifications. See edito rial interpolations and clarifications interrogative pronouns, 5.54-55, 5.58 interrogative sentences. See questions interviews and discussions editing of, 2.61 releases for, 4.77 source citations for, 14.211-13,15.53 transcription of, 13.48 See also dialogue; direct address; personal communications; speech; transcriptions in-text citations. See text citations intra, as prefix, 7.89 (sec. 4) intransitive verbs, 5.35,5.98,5.169, 5.170 introductions abstracts for, 1.76 copyright of, 4.75 format of term, 8.179,14.110 indexing of, 16.109 to journal special issues, 1.102 location and format of, 1.43,1.46,1.47, 1.48 to parts of text, 1.38,1.48 placement in book, 1.4, fig. 1.5 for previously published materials, 2.46 source citations for, 14.105,14.110 submission of, 2.3 introductory words and phrases and alphabetizing, 16.68 colons with, 6.63-65,13.16 INDEX introductory words and phrases (continued) commas with, 6.30-31, 6.33-35,13.14 em dashes with, 6.86 italics for, in legal-style citations, 14.272 no punctuation with, 13.15 inversion of names bibliographies and reference lists: first author’s name only, 4.21,14.23, 14.66,14.76,14.101,15.9; single au thor, 14.75,15.6; titles of manuscript collections, 14.229 personal names in indexes (see indexing: names, personal) inverted word order adverbial phrases with, 6.31 in exclamations, 5.216 indexes, 16.9,16.56,16.68,16.90,16.94 quotations with, 13.14 as syntax variation, 5.224 of titles beginning with articles, 16.48- 49,16.51-52,16.56 See also inversion of names Ireland, 1.34 irregular (strong) verbs, 5.100,5.130,5.144. See also auxiliary (helping) verbs irregular adjectives, 5.87 irregular adverbs, 5.165 irregular nouns, 5.20,7.5,7.16,7.27 ISBN (International Standard Book Num ber) on copyright page, 1.20,1.32, fig. 1.1 on covers and jackets, 1.74 defined, p. 982 for e-books, 1.28,1.32, fig. 1.1, fig. 1.4 hyphenation of, 6.77 as metadata, 1.75 proofreading of, 2.108 for publications in multiple formats, 1.20,1.32,4.64, fig. 1.1, fig. 1.4 resources on, 1.32 ISBN Agency for the United States, 1.32, 1.74 ISO. See International Organization for Standardization ISSN (International Standard Serial Num ber), 1.20,1.32,1.79,1.84,1.86,1.92, 1.103, p. 982 ISSN International Centre, 1.32 issue numbers in copyright line, 1.103 on journal cover, 1.84,2.109 issue numbers (continued) as metadata, 1.92 numerals for, 9.27,14.171 omission of, 15.48 overview, 1.80 proofreading of, 2.134 and retractions, 1.91 in source citations, 14.23,14.165,14.171, 14.177,14.185,15.9,15.47 in table of contents, 1.87 issues (of periodicals), p. 982. See also issue numbers it, 5.47,5.115,5.239,5.240,5.246,8.77,8.118 Italian language, 11.46-53 abbreviation of Italian, 10.42 capitalization, 8.9,11.46 punctuation, 11.47,11.48,11.49,11.52-53 quotations and dialogue, 11.47 resources on, 11.46 special characters, 11.53 word division, 11.50-52 italics defined,p. 982 within italicized titles, reversed, 8.173, 14.95 marking manuscript for, 2.73,2.97,12.63 marking proofs for, 2.131 plurals of words in, 7.12 possessives in, 7.29 proofreading of, 2.113,2.131 and punctuation, 6.2,14.89 and semantic markup, 2.82nl, 7.49 for titles of works (see under titles of works) vs. underlining, 2.15,2.80 uses: abbreviations, 2.23,10.7,10.50, 14.60; added to original quota tion, 13.62; computer terminology, 7.79; in cross-references in source citations, 14.81; emphasis, 7.48,7.50, 13.62,14.37; enzyme names, 8.133; epigraphs, 13.36; gene names, 8.132; genus and species, 8.2,8.120, 8.122, 8.123, 8.173; glossary terms, 2.23; glossed American Sign Language, 11.133; highlighting key terms, 7.56; index cross-references, 16.15,16.20, 16.22,16.23; index references to illustrations, 16.116,16.140,16.141; legal-style citations, 14.272; for let ters as letters (e.g., a), 7.64; locators INDEX italics (continued) (e.g., above, left) in captions, 3.24; mathematical copy, 12.10,12.11, 12.56,12.63,12.65; music dynamics, 7.75; N, n, in tables, 3.85, fig. 3.20; non-English terms in English con text, 7.53,7.54,11.3,11.5,11.75,11.88, 11.94,14.272; notes keyed to text by page numbers, 14.53, fig. 14.6; notes to tables, 3.77,3.78; numbers and letters in lists, 6.129; rhyme schemes, 7.69; run-in subheads, 2.18; SI base quantities, 10.54; sic, 6.73,7.55,13.61; significance (probability) levels (p), 3.80; stage directions, 13.46; totals in tables, 3.63, fig. 3.15; transcriptions of discussions or interviews, 13.48; words and phrases used as words, 6.2,7.63 word-processing styles for, 2.82 See also non-English materials; non English words and phrases; typo graphic considerations Italy, 9.23. See also Italian language it is I, it is me, 5.250 its, it’s, 2.111,5.50,5.250 jackets of books. See covers of journals and paperback books; hardcover books: jackets Jacobian matrix, 12.51 Japan, 9.23. See also Japanese language Japanese language, 11.86-90 characters for, 11.90 names, 8.16,11.88,16.81 resources on, 11.85 romanization system (Hepburn system), 11.86-87 special characters for transliteration, 11.87, table 11.2 titles of works, 11.89 jargon, 5.25,5.95,5.174 JATS (Journal Article Tag Suite), 1.117 Javanese names, 16.80 joint (group) genitive, 5.22 joint (group) possessives, 5.22 joint works, 4.7,4.12,4.52 journal articles abstracts of, 1.87,1.92-95, 2.25,14.186 access dates, 1.83,14.176 journal articles (continued) access rights, 1.87,1.92,1.93 authors’ names: first page of article, 1.95,1.100; as metadata, 1.92,1.100; retractions, 1.91; running heads, 1.83; source citations, 14.168,14.185, 15.33; table of contents, 1.87, fig. 1.11 continuous publishing model, 1.82, 1.113.14.174 copyright lines in, 1.95,1.103 dates of submission and acceptance, 1.95,1.104 editor’s notes in, 1.95 endnotes in, 1.108,14.43,14.46 first-page information, 1.95,1.103 hyperlinks within, 1.96 making changes to after submission, 2.5 menus in, 1.96 metadata for, 1.92,1.100 page ranges for, 1.79,1.81,1.87,1.92, 1.103.14.23.14.174 photocopying of, 1.103 preserving context of, 1.105 publication dates, 1.92,1.95,1.104, 14.171 published before print version, 1.78, 1.81,1.86,1.104,1.112-13,14.172 publisher-assigned numbers for, 1.103 publishing agreements for, 4.55-57, fig- 4.1 retracted, 1.91 source citations for (see journal articles, source citations for) subheads in, 1.97 substantive editing of, 2.50 titles: editing of, 2.58; in first-page information, 1.95; as metadata, 1.92; and note reference numbers, 14.27; proofreading of, 2.134; quotation marks for, 8.163, 8.177,14.86,14.169, 15.9; retractions, 1.91; in running heads, 1.83,2.76; shortened, 14.23, 14.185; in table of contents, 1.87 unique identifiers for, 1.79,1.82 See also abstracts; chapters; journals; offprints journal articles, source citations for, 14.168- 87,15.46-48 abstracts, 14.186 access dates, 14.176 INDEX journal articles, source citations for (continued) anonymous works, 14.79 author-date reference system, 15.9, 15.46-48 authors’ names, 14.168,14.185,15.33 and continuous publishing model, 1.82, 14.174 download options for, 1.87 electronic supplementary data, 14.187 examples and variations, 14.23,15.9 forthcoming articles, 14.172 generally, 14.164-65 house style, 1.108-9,2.64,14.2-4 journals vs. magazines, 14.166 non-English-language considerations, 11.26-27,14.170,14.183 note referring to entire, 14.27 page numbers, 14.174,14.177,15.23 papers presented at meetings as, 14.217 parentheses or comma with issue num ber, 14.177,15.47 place of publication, 14.182 preprints, 14.173 previously published articles, 14.181 published in installments, 14.180 reference lists, 15.10 shortened citations, 14.23,14.185 special issues, 14.178 supplements, 14.179 titles of articles: capitalization, 14.169, 15.38; generally, 14.165,14.169; legal- style citations, 14.272; shortened citations, 14.23,14.185 translated or edited articles, 14.183 URLs and DOIs, 1.79,1.82-83,1.92, 1.103,14.23,14.175,14.187,15.9 volume, issue, and date, 14.165,14.171, 14.177,15.9,15.47 See also journals: titles Journal Article Tag Suite (JATS), 1.117 journals, 1.77-116 advisory boards, 1.85 back issues, 1.84 as collective works, 4.8 components, 1.77-111; abstracts, 1.87, 1.92-95,2.25,14.186; acknowledg ments, 1.89,1.95; announcements, 1.86,1.87,1.89,1.99; articles and other text, 1.94-105; back matter, journals (continued) 1.88,1.90,2.134; biographical notes, 1.100,14.55; book review and book notes sections, 1.87,1.94,1.98, 14.153; copyright notice, 1.79,1.84, 1.86,1.95,1.103,2.109,2.134,4.43; covers (see covers of journals and paperback books); editorial board information, 1.84,1.85,2.109,2.134; editorials, 1.87,1.102; electronic supplementary data, 1.78,1.87,1.96, 1.114, 3.26,14.187; front matter, 1.84-85,1.87-91,2.102,2.105,2.134; generally, 1.94; home page, 1.84, 1.86,1.87; hyperlinks (see under hyperlinks); illustrations, 1.84,1.107; indexes (see indexes); issues, 1.80; letters to the editor, 1.94,1.101; list of upcoming issues, 1.84,1.86,2.109; masthead, 1.84,1.85; pagination, 1.81,1.82, fig. 1.11; retractions, 1.91; running heads or feet, 1.83,2.76; sections, 1.83,1.87,1.95; table of con tents, 1.84,1.86,1.87,1.90,2.109, fig. 1.11; tables, 1.106,3.60, 3.64, 3.86; volumes, 1.80,1.81, fig. 1.11 continuous publishing model, 1.82, 1.113,14.174 contributors (see contributors to multi author volumes and journals; journal articles: authors’ names) copyright of, 1.79,1.84,1.95,1.103, 2.109, 4.43, 4.59, 4.64 and Creative Commons licenses, 4.62 defined, 14.166 design, 1.84,1.115 editorial policy, 1.84,1.86,1.88 editor’s copyright, 4.57 electronic workflow for, 1.77 errata and corrections, 1.90,1.94,1.112 frequency of publication, 1.80,1.84,1.86 history of, on home page, 1.86 house style, 1.108-9,1.116,2.49,2.64, 14.2-4,16.103 indexes: authors’ names, 16.6; informa tion about, 1.84,1.86; keywords, 1.93, 16.7,16.30,16.126; of past contribu tors, 1.100; preparation for, 16.102, 16.103; to printed volumes, 1.110; schedule for, 16.4; vs. searching, INDEX journals (continued) 1.110,1.111,16.7; term choices for, 16.30 ISSN, 1.20,1.79,1.84,1.86,1.92,1.103, p. 982 issue numbers (see issue numbers) new series of, 14.184 open-access publishing models, 4.61 organizing principle, 1.80 peer review for, 2.5 production schedule, 2.102, fig. 2.2 proofreading of, 2.102,2.134 publishing agreements, 4.55-57, fig. 4.1 simultaneous print/electronic pub lishing model for (see simultaneous print/electronic publishing model) special issues, 1.84,1.87,1.89,1.102, 4.59,14.178 sponsorship information, 1.84,1.86, 1.89,1.103,1.105 submission requirements, 1.84,1.86, 1.88,2.3,2.109 subscription information, 1.84,1.86, 2.109,2.134 supplements, 14.179 titles: abbreviations of, 14.170,15.13, 15.46,16.49; in copyright line, 1.103; on cover, 1.84; indexing of, 16.49; initial the in, 8.170,14.170; italicized terms within, 8.173; italics and capi talization, 8.168,14.86,14.170,15.9; magazine journal, review, and such, 8.171; in names of awards and prizes, 8.172; non-English language, 14.98, 14.170; in running heads, 1.83; series, 14.184; shortened, 15.13,15.46; similar names, 14.182; subjects not interchangeable with, 8.174; in table of contents, 1.87 trademark protection, 4.16 treatment of term, 8.171 version control, 1.112 volume editor’s responsibilities, 2.41,2.42 volume numbers (see under volumes and volume numbers) work made for hire in, 4.10 See also abstracts; contributors to multi author volumes and journals; covers of journals and paperback books; journal articles; and many of the above elements under their own headings JPEG files, p. 982 Jr. (“Junior”), 6.43,7.42,10.19,16.41 judicial bodies, 14.276-79 basic elements in source citations, 14.276 Canadian courts, 14.294 international courts, 14.304 names and terms, treatment in text, 8.64 numbered divisions of, 9.46 UK courts, 14.298 US courts, 14.277-79 See also legal and public documents; legal-style citations justification defined, p. 982 flush left (ragged right), 2.10,3.73,7.47, 16.136, fig. 3.19, p. 986 indexes, 16.131,16.136 manuscript preparation guidelines, 2.10 marking manuscript for, 2.98 marking proofs for, 2.127 and spacing, 2.125, 6.120, 6.121 words in table columns, 3.73, fig. 3.19 See also margins K (1,000), 9.24 kern, kerning, 12.11, p. 983 keyboard combinations and shortcuts, 7.78 keys, computer, 7.77 keys, musical, 7.74 keys to illustrations (legends) capitalization and consistency, 3.20 vs. captions, 3.21 defined, 3.7 letters in, 3.7, 3.12,3.24, fig. 3.4, fig. 3.6 scaling of, 3.19 See also labels keywords in abstracts, 1.93 for books, 1.75,1.76,2.25 context-sensitive searching for, 1.111, 1.120 in journal indexes, 1.93,16.7,16.30, 16.126 for journals, 1.92,1.95 manuscript preparation guidelines, 2.25 in shortened running heads, 2.76 kind, kinds, 5.52 Kindle, 14.159. See also file formats and devices kinship terms, 7.89 (sec. 3), 8.36 INDEX Koran (Qur’an), 8.103,11.77,14.241 Korean language, 8.17,16.82 I. (line) and //. (lines), 14.156 labels vs. captions, 3.21 in charts, 3.44-45 in illustrations and keys, 3.7,3.12,3.24, fig. 3.3, fig. 3.4, fig. 3.6 marked on line art to be redrawn, 3.20 scaling of, 3.19 on submitted artwork, 2.27 of x andy axes, 3.43, fig. 3.8 See also keys to illustrations labor unions, 8.70, 9.49 landscape, p. 983 “last modified” dates, 14.13 LaTeX, 12.2,12.9,12.68, fig. 12.3, table 12.1 Latin alphabet IPA notation, 11.22 languages using, generally, 11.18-20, 11.70 special characters, 11.2,11.21,11.22, table 11.1 (see also special characters) See also transliteration and transliterated text; and specific languages Latin language, 11.54-59 abbreviations in, 7.55,10.7,10.42 (see also et al., ibid.) capitalization, 11.54 ligatures for, 7.34 prepositions in, 8.159 for provisional chemical elements, 10.63 roman type for terms in, 7.55,10.7, 10.42,13.66 special characters, 11.59 titles of works in, 11.54 word division, 11.55-58 See also classical Greek and Latin works; genus and species names; scientific and technical terminology latitude, 10.36. See also compass points and directions laws and statutes. See legislation laws and theories mathematical, 12.56 schools of thought, 8.79 scientific, 8.148 See also philosophical terms and movements; scientific and technical terminology layout, p. 983 le, 8.7 leaders (several spaced periods), 3.64, fig.
3.16. See also ellipses leading, 2.125, p. 983. See also line spacing leaf numbers. See folios least, 5.86,5.87,5.88,5.163,7.89 (sec. 2) leaves, book, 1.5. See also folios; page numbers lectures and lecture series, 8.87,14.217, 14.264,14.267. See also speeches left or right indent, 2.11 legal and public documents acts and treaties, 8.66,8.80,8.81, 14.275,14.290 constitutions, 8.80,9.28,14.272,14.280 numbered divisions in, 9.28 presidential, 8.66,8.75-76,8.80-81, 14.275,14.289-90 quotations in, 13.18,13.21 resources and stylebooks on, 14.3, 14.269,14.270,14.287,14.289, 14.290,14.292,14.297 shall in, 5.250 See also legal and public documents, source citations for legal and public documents, source cita tions for, 14.269-305 abbreviations in: cases and court decisions, 14.276-77,14.279,14.294, 14.298; congressional debates and hearings, 14.285-87; constitutions, 14.280; generally, 14.274; interna tional entities, 14.304; legislation, 14.295; legislative debates, 14.301; legislative documents, 14.281, 14.299; UK command papers, 14.302 acts and treaties, 14.275,14.290 arabic vs. roman numerals in, 14.147 author-date reference system, 14.271, 15.58-59 bills and resolutions, 14.283 Canada, 14.269,14.293-96 cases and court decisions, 8.82,14.272, 14.275-79,14.294,14.298,15.59 Chicago- vs. legal-style, 14.291 constitutions, 14.272,14.280 courts, 14.276-79,14.294,14.298, 14.304 id. in, 14.35,14.275,14.279 international entities, 14.304-5 INDEX legal and public documents, source cita tions for (continued) loose-leaf services, 14.220 municipal ordinances, 14.288,14.300 note form, 14.271 online documents, 14.270,14.276, 14.298.14.300.14.305 page numbers and other locators, 14.273,14.276,14.284,14.290, 14.294.14.303.14.305 private (e.g., wills), 14.219 resources and stylebooks on, 14.3, 14.269,14.270 secondary sources, 14.291 shortened citations, 14.275,14.276, 14.279 typefaces, 14.272 United Kingdom, 14.297-303; command papers, 14.302; legal cases, 14.298; parliamentary debates, 14.301; par liamentary publications, 14.299; re sources on, 14.297,14.300; statutes, 14.300; unpublished, 14.303 United States: bills and resolutions, 14.283; congressional reports and documents, 14.285; debates before 1873,14.287; debates since 1873, 14.286; hearings, 14.284; legal cases, 14.277-79; legislation, 14.282, 14.288; municipal ordinances and state laws, 14.288; presidential docu ments, 14.289; secondary sources on, 14.291; unpublished, 14.292 unpublished, 14.292,14.296,14.303 legal blacklining (redlining), p. 986. See also tracking changes legal cases source citations for: basic elements, 14.276; Canadian cases, 14.294; ital ics for, 14.272,14.276; notes and text citations, 8.82,15.59; short forms, 14.275; UK courts, 14.298; US courts, 14.277-79 treatment in text, 8.82 v. in, 8.82,14.276 legal-style citations, 14.269-75 abbreviations in, generally, 14.274 vs. Chicago-style, 14.291 note form, 14.271 page numbers and other locators, legal-style citations (continued) 14.273,14.276,14.284,14.290, 14.294,14.303,14.305 resources on, 14.3,14.269,14.270 shortened citations, 14.275,14.276, 14.279 typefaces, 14.272 See also courts; legal and public docu ments legends, 3.7,3.12,3.19-21, 3.24, fig. 3.4, fig. 3.6. See also captions legislation abbreviations, 14.281,14.299 bills and resolutions compared, 14.283 Canadian, 14.295 debates on, 14.286-87,14.301 formal and generic names for, 8.80,8.81 hearings on, 14.284 published text of, 14.282 secondary sources on, 14.291 short forms for names, 14.275 state, 14.288 United Kingdom, 14.300 US federal, 14.282 US state and municipal, 14.288 See also acts and treaties; copyright; legal and public documents legislative and deliberative bodies abbreviations, 14.281,14.283 hearings of, 14.284 names and terms for, 8.62 See also legal and public documents length (measurement). See International System of Units; metric system; units of measurement length (parts of text) books, 14.153 indexes, 16.103,16.118,16.133 manuscripts, 2.4,2.52, p. 977 notes, 14.40,14.56 quotations, 13.10,13.25 table titles and column heads, 3.54 lesbian, gay, 8.41 less, 5.7,5.85,5.87,5.88,5.163, 6.47,7.89 (sec. 2) less, fewer, 5.250 let, 5.215 letter-by-letter alphabetizing basics of, 16.59 for bibliographies, 14.65 INDEX letter-by-letter alphabetizing (continued) for indexes of first lines, 16.144,16.145 preferred for indexes, 16.58 vs. word-by-word, 16.57,16.58,16.61, 16.123 letters (alphabet) archaic, 13.7 hyphens to separate, 6.77 as letters, 7.15,7.64 marking spacing changes between, 2.125 names of (spelled-out pronunciations), 7.68 ordinal, superscript, 11.27 plurals of, 7.15,7.64,7.65 spellings for, 7.68 uses: in common expressions, 7.64; generations, 8.42; as identifiers in classical Greek and Latin works, 14.243,14.245,14.248; illustration parts, 3.7,3.12,3.24, fig. 3.4, fig. 3.6; lists and outlines, 2.21, 6.127, 6.129, 6.132,7.45; multiple appendixes, 1.59; musical symbols, 7.71-75; notes to parts of tables, 3.79, figs. 3.14-15; parts of book, 8.180; rhyme schemes, 7.69; same-author, same-year works in reference lists, 15.20; scholastic grades, 7.65; shapes (e.g., S curve), 7.67; standing for proper names, 7.66; vessel names, 8.116 See also alphabetizing; alphabets; capital letters; initialisms; initials; special characters letters (correspondence) chronological order, 1.52 copyright issues, 4.19 multiple paragraphs quoted, 13.35 permissible changes, 13.8 punctuation in, 6.14,6.53, 6.66 quotations from, 13.10,13.61 source citations for: examples, 14.229; in manuscript collections, 14.228; private collections, 14.231; published collections, 14.111,15.43; shortened citations, 14.111 See also letters to the editor; manuscript collections; personal communica tions; unpublished and informally published materials letters to the editor, 1.94,1.101,14.196. See also editorials lexicalized signs (ASL), 11.131 libel, 4.73 libraries. See archival practices; databases; library catalogs; manuscript collec tions Library and Archives Canada, 1.34 library catalogs alphabetization in, 16.58 Chinese romanization in, 11.82 non-English bibliographic terms in, 14.102 for non-English titles, 14.98 for personal names, 8.3,16.71 for publishers’ names, 14.133,14.138 Library of Congress, 1.32,1.34,4.47, 11.82,11.84. See also Cataloging- in-Publication (CIP) data; library catalogs licenses for copyrighted works Copyright Clearance Center, 1.79,1.103, 4.97 Creative Commons, 4.62 electronic works, 4.64,4.65,4.66 exclusive vs. nonexclusive, 4.35 goals of parties to, 4.36 handling requests for, 4.70 journals, 1.103 and open-access publishing models, 4.61 payment for, 4.37 rights database as record of, 4.70 and subsidiary rights, 4.17,4.34 termination of, 4.38 universities’ claim to, 4.67 See also copyright; intellectual property rights; permissions; publishing agreements; subsidiary rights “life plus seventy” rule, 4.23,4.25,4.26 ligatures, 7.34,7.35,11.37,11.122, p. 983 like, as, 5.185, 5.250 like, compound terms with, 7.89 (sec. 3) limp (flexi-) binding, p. 980 line art charts and graphs as, 3.4, 3.41 color in, 3.19 defined, p. 983 examples of, figs. 3.3-4, fig. 3.6 publisher’s redrawing of, 3.20 INDEX line art (continued) scaling of, 3.19 See also illustrations line breaks and em dashes, 6.90 and en dashes, 6.82 mathematical expressions, 12.23,12.50 and nonbreaking spaces, 6.11, 6.119, 6.121,7.42,7.44,7.45,11.28 and numbers, 7.42,7.43,7.44 and proper names, 7.42 in reflowable electronic formats, 6.82, 6.90, 6.113,7.36,7.47,14.18 run-in lists, 7.45 and slashes, 6.113 and URLs, 2.13, 6.113,7.46,14.18 See also word division line editing. See manuscript editing line numbers avoiding use of /., //., 14.156 footnotes keyed to, 14.53, fig. 14.5 format of, 14.156 punctuation of, 9.54 in text citations, 13.67 lines (rules) above continued footnotes, 14.41, fig. 14.1 avoided in illustrations, 3.19 to be filled in, 6.93 in tables, 3.53,3.57,3.58,3.74, 3.86, figs. 3.13-14, fig. 3.16 lines (text) blank, 1.58,2.8,2.98 and electronic file cleanup, 2.80 indentation, symbol for, 2.98,2.126 marking manuscript for, 2.98 proofreading of, 2.116,2.117 as text break, 1.58,2.8 widows and orphans, 2.116,2.117, pp. 984,990 See also justification; leading; line spacing line spacing and electronic file cleanup, 2.80 for manuscript editing, 2.91 for manuscript submission, 2.6,2.8, 2.19-20,2.22-24 between paragraphs or stanzas in quota tions, 13.22,13.24,13.25 word-processing styles for, 2.82 linking (copula or connecting) verbs, 5.45, 5.82,5.93,5.101,5.154,5.170, 6.30.
See also be-verbs links. See hyperlinks lists in text, 6.127-32 elided, in mathematical expressions, 12.19 outlines, 2.21,2.36,6.127, 6.132 run-in, 6.128, 6.129, 6.131,7.45 vertical: abbreviations in, 10.29; as alternative to tables, 3.51; as block quotations, 13.10; manuscript prepa ration guidelines, 2.21; in mathe matical expressions, 12.16,12.18; placement of, 2.62; punctuation and format, 2.11,2.21, 6.92, 6.129-31, 6.132; vs. run-in, 6.128, 6.129, 6.131 See also outlines lists of abbreviations alphabetizing, 1.44, fig. 1.8 cross-checking, 2.32 format and placement, 1.4,1.44,2.23, 14.60, fig. 1.8 for frequently cited works, 13.67,14.60, fig. 14.7 when to create, 10.3 lists of donors, 8.20 lists of illustrations and tables, 3.38-40 alterations in, 2.136 appropriate use of, 1.39, 3.38 artwork inventory linked to, 3.17 captions shortened for, 1.39, 3.40, fig. 1.7 cross-checking, 2.32 editing of, 2.57 format and placement, 1.39,3.30, 3.38, figs. 1.6-7 for galleries, 3.39 and indexing, 16.109 manuscript preparation guidelines, 2.26 as navigation aid, 3.38 placement in book, 1.4,2.26 proofreading of, 2.115 submission of, 2.3 See also illustration credits and credit lines lists of special characters, 2.3,2.16,11.21, 11.122,12.13 literary agents, 4.18 literary works characters in, 5.129,8.35 cultural terms for (e.g., deconstruction), 8.79 dramatic rights, 4.64 fair use of, 4.86,4.87 frequent reference to single, 13.67 INDEX literary works (continued) personification in, 8.37 rhyme schemes, 7.69 sound recordings of, 14.264 titles, 9.43,16.50 and trademark protection, 4.16 See also classical Greek and Latin works; plays; poetry; quotations; titles of works literature cited. See source citations lithocase (paper-over-board format), 1.71 little, 5.90, 5.230 live-stage rights, 4.64 location numbers, in e-books, 1.123,14.160 locators in captions, for parts of illustrations (e.g., above, left), 3.24 in classical Greek and Latin works (see classical Greek and Latin works: source citations) in electronic publications: alternatives, 1.123,2.35,14.22,14.160,14.174, 15.23; database accession numbers, 2.34,14.215,14.257,14.270 (see also hyperlinks) ffi, 10.42,14.148-49,16.12 in index entries (see index entries: locators) in legal-style citations, 14.273 paragraph numbers as, 1.125,2.106, 14.148,16.12,16.13,16.108 See also locators in source citations; page numbers; volumes and volume numbers locators in source citations chapter numbers, 14.106,14.153 folio editions, 14.155 generally, 14.22,15.8,15.23 illustration and table numbers, 14.158 line numbers, 14.53,14.156, fig. 14.5 numbered notes, 14.157 signed signatures, 14.154 loc. cit. (loco citato), 10.42,14.36 logo, publisher’s, 1.19,1.69,1.70 longitude, 10.36. See also compass points and directions loose-leaf services, 14.220 Lord, Lady, 8.32,16.38 lowercase letters abbreviations ending with, 10.4 defined, p. 983 lowercase letters (continued) “down style,” 8.1 in index entries, 16.11 marking manuscript for, 2.97 marking proofs for, 2.131 personal names beginning with, 8.4,8.5 proper names beginning with, 8.69, 8.154 See also capitalization; sentence-style capitalization Ltd., 6.44,10.23,14.134 -ly, words ending in adjectives, 5.158 adverbs, 5.93,5.158, 5.160, 5.163, 5.164, 7.85,7.86 hyphenation not needed with, 7.85,7.86, 7.89 (sec. 2) lyrics, 4.87,13.3. See also musical works Mac, Me, 16.73 macro, as prefix, 7.89 (sec. 4) macros, 2.79,2.80, p. 983 Madame, Mme, 10.17 Mademoiselle, Mlle, 10.17 magazines copyright notice, 4.43 defined, 14.166 indexing names of, 16.49 serial rights, 4.64 source citations for, 14.188-90; article titles, 14.188; author-date reference system, 15.49; dates, 14.165,14.188; vs. journals, 14.166; magazine titles, 14.190; newspaper supplements, 14.197; online articles, 14.189; page numbers, 14.188-89; regular depart ments, columns, and features, 8.177, 14.190 treatment of term, 8.171 See also journals; newspapers; period icals mailing addresses abbreviations in, 10.4,10.8,10.27-29, 10.33-34 compass points in, 10.34 numbers in, 9.50-52, 9.54 publisher’s, 1.19,1.20,1.21,1.84,1.95 treatment in text, 6.39,10.27,10.29 See also compass points and directions; email; geographical terminology mailing lists, electronic, 14.210 INDEX mail-to links, 1.86,1.121 main headings. See under index entries male and female, 5.259 manuscript collections noncopyright restrictions on, 4.83 quotations from, 4.83,13.61 source citations for, 14.221-31; author date reference system, 15.54; col lections of letters and such, 14.228; dates for, 14.224; examples, 14.229, 14.230; folios, page numbers, and such, 14.225; lists of abbreviations for, 14.60, fig. 14.7; locations of depositories, 14.227; musical scores, 14.256; note forms vs. bibliographic entries, 14.222; one vs. several items cited from a collection, 14.230; pa pers and manuscripts (terms), 14.226; private collections, 14.231; resources on, 14.221; specific vs. generic collec tion titles, 14.223 titles of, 8.188,14.222,14.223,14.226 See also letters (correspondence); un published and informally published materials manuscript editing, 2.48-99 basics: author availability, 2.52; author queries (see author queries); author’s review, 2.2,2.70-72,2.82,2.88, 2.101; vs. developmental editing, 2.48, 2.50; errors and fact-checking, 2.56,2.108,13.6; estimating time for, 2.52; mechanical editing, 1.116, 2.45,2.49; reference works for, 2.54; stages of editing, 2.53,2.79; style sheets, 2.55,2.83,2.111, fig. 2.3; substantive editing, 2.48,2.50, 2.51; volume editor’s responsibilities, 2.41-42 checking consistency and format: cross-checking, 2.32-35,2.53,2.59; en dash, 2.14; note citations against reference lists and bibliographies, 2.32,2.63,15.21; notes vs. reference numbers in text, 2.32,2.53,2.62; per missions against manuscript, 4.79; style sheets for, 2.55, fig. 2.3; subhead hierarchy, 2.59; text citations against reference lists, 2.32,2.63,15.21; titles and subtitles against table of contents, 2.58 manuscript editing (continued) electronic editing mechanics, 2.78-89; backup copies, 2.40,2.78,2.80; cleaning up files, 2.39,2.53,2.59, 2.61,2.63,2.79-80,2.85,14.5; doc ument comparison software, 2.86; final checks and preparation, 2.89; global changes, 2.69,2.79,2.80,2.85; markup (see markup for manuscript editing; markup languages, formal); in PDF files, 1.118; silent changes, 2.69,2.85; version control, 2.5, 2.78,2.79; word-processing styles, 2.82,2.83, fig. 2.4 (see also tracking changes; word-processing software) independent professionals for, 2.48,2.73 paper-only, 2.90-99; author queries, 2.92; backup copies, 2.6,2.71,2.90; capitalizing and lowercasing, 2.97; checking author’s changes, 2.72; circling, 2.93,2.95; conventions of, 2.91; dashes and hyphens, 2.96; and electronic file cleanup, 2.79; example of, fig. 2.5; and generic markup, 2.99; indentation and spacing, 2.98; insertions, deletions, and substi tutions, 2.94,2.95, fig. 2.6; italics and boldface marks, 2.97; keeping a clean copy, 2.90; manuscript format considerations for, 2.6,2.8; marking manuscript components, 2.99; mathematical copy, 12.61-68, figs. 12.1-2; note renumbering, 2.62; and proofreading, 2.110; punctuation changes, 2.95; style sheets for, 2.55, fig. 2.3; type specifications, 2.74,2.93 (see also author queries) preparing final manuscript for produc tion, 2.73-77; checking markup, 2.73; checklist for, 2.77; document com parison software for, 2.86; markup (see markup for manuscript editing; markup languages, formal); pagi nation, 2.75-76; preparing running heads, 2.76; type specifications, 2.74 resources on, 2.54,2.56 schedule for, 2.2,2.52 specific parts: captions, 2.65; cross- references, 2.60; front matter, 2.57; illustrations, 2.65; indexes, 2.67, 16.132-34; lists and outlines, 2.21; INDEX manuscript editing (continued) mathematical copy, 12.2,12.13, 12.61-68, figs. 12.1-2; quotations and previously published material, 2.61, 2.69,2.91,11.16; source citations, 2.62-64; subheads, 2.59; tables (see below); titles of parts, chapters, and articles, 2.58 tables, 3.81-88; adjusting and checking, 2.32,3.81, 3.86; basic approach, 2.66; continued lines and repeated column heads, 3.86-87, fig. 3.25; electronic file cleanup, 2.80; N, n, 3.85; number ranges, 3.83; percent, percentage, 3.82; signs and symbols, 3.82-84 and word division, 7.36,7.46 See also manuscript editors; manuscript preparation guidelines for authors manuscript editors communications with authors, 2.68-72; checking author’s changes, 2.72; cover letter and instructions, 2.70, 2.88,2.92; early contact, 2.68; edit ing sample, 2.52,2.68; permissions double-checked, 4.79; sending edited manuscript, 2.71,2.88 (see also author queries) discretion and flexibility, 2.48,2.51,2.64 post-proofreading tasks, 2.101 production process role, 2.2 schedule for editing, 2.52 software knowledge, 2.79 See also manuscript editing manuscript preparation guidelines for authors, 2.3-47 basics: accuracy and consistency, 2.7, 2.56,13.6; changes after submis sion, 2.5; checklist, 2.3; indexing as responsibility of authors, 2.2,2.70; letter to publisher, 2.4,2.15,2.39; list of special characters, 2.3,2.16,11.21, 11.122,12.13; proofreading before submission, 2.100; publication pro cess and schedule for, 2.2, figs. 2.1-2; submission requirements, 2.2,2.3-6, 2.36-40 cross-checking tasks: checklist of, 2.32; citation data and format, 14.5; cross- references, 2.32,2.35; quotations, 2.32,2.33,13.6; URLs, 2.3,2.32,2.34 electronic files: accuracy and consis- manuscript preparation guidelines for authors (continued) tency, 2.7; backups, 2.4,2.40,14.5; embedded and hidden comments removed, 2.39; naming, 2.37; for notes in compilations, 2.45; number ing pages in, 2.38; separate vs. single, 2.36,2.80; source citation coding re moved, 14.5 (see also electronic files) formatting, 2.7-25; abbreviations list, 2.23; abstracts and keywords, 2.3, 2.25; bibliographies and reference lists, 2.24,14.67,15.17; block quota tions, 2.11,2.19,2.20,13.9; chapter and part titles, 2.17; dashes, 2.14; glossaries, 2.23; hyphens, 2.13; ital ics, boldface, and roman, 2.15,12.63, 12.65; justification and margins, 2.10; keyboard combinations and shortcuts, 7.78; line spacing, 2.6,2.8, 2.19-20,2.22-24; lists, 2.21; notes, 2.22,14.24,14.41,14.43,14.53; para graphs, 2.12,2.19,2.20; publisher’s guidelines, 2.7; spacing between words, 2.9; subheads, 2.18; word division, 2.13,7.36,7.46,7.47 (see also electronic files) illustrations and tables, 2.26-31, 3.15- 20; artwork, scans for, 2.27; captions, 2.3,2.26,2.37; identification and inventory of artwork, 3.16-18; notes, 2.31; numbering of, 2.28,2.29; preparation of tables, 3.47; separate files for each component, 2.26,2.36; submission to publisher, 3.3,3.4, 3.15; text references and callouts for, 2.26, 2.30,2.32,2.65-66,3.8-9 (see also illustrations; tables) special types of works: compilations and previously published materials, 2.43- 47; multiauthor volumes or journals, 2.41-42 See also mathematical copy: manuscript preparation; permissions; proofread ing; version control manuscripts abbreviation of term, 7.15,7.33,10.42, 14.226 acceptance of, 4.52 file conversion, 2.7,2.79,2.80 length of, 2.4,2.52 INDEX manuscripts (continued) making changes after submission, 2.5 numbering pages of, 2.38 paper-only, 2.6,2.8,2.37,2.38,2.43, 2.45,2.61 version of record, 2.5 See also electronic files; manuscript collections; manuscript editing; man uscript preparation guidelines for authors; word-processing software maps consistency with text, 2.65 credit lines for, 3.33, 8.199 keys to, 3.7 lists of, 1.39 numbering of, 3.10 place-names on, 2.60,3.20,8.45 preparation for reproduction, 3.20 Source citations of, 14.237 symbols or patterns used in, 3.25 titles of, 8.199,14.237 as works made for hire, 4.10 See also illustrations margins adjusting for tables, 3.86 author queries and corrections in, 2.10, 2.43,2.91,2.92-93 definitions, pp. 975, 981, 983 and electronic file cleanup, 2.80 indexes, 16.131 manuscript preparation guidelines, 2.10 marking proofs for, 2.127 proofreading corrections in, 2.120 running heads in, 1.10 screen, in online publications, 1.122 silent changes to, 2.85 See also justification marketing. See promotion markup defined,p. 983 semantic, 2.82nl, 7.49 word-processing styles for, 2.18,2.21, 2.82,2.83, fig. 2.4 See also markup for manuscript editing; markup languages, formal markup for manuscript editing (generic markup) angle brackets in, 2.16,2.30,2.81, 6.104 callouts, 2.87 character-level formatting, 2.73,2.81 and electronic file cleanup, 2.80 markup for manuscript editing (generic markup) (continued) vs. formal markup languages, 2.81 information, in production checklist, 2.77 overview, 2.73 and paper-only markup, 2.99 part and chapter titles, 2.58 semantic, 2.82nl, 7.49 subheads, 2.59,2.80,2.81 tables, 3.86 tags for, 2.81 type specifications in, 2.74,2.93 See also markup languages, formal markup languages, formal abbreviations of names, 10.49 and accessibility, 3.28, 3.88,12.2nl and electronic workflow, 1.117,2.83, 2.137 vs. generic markup, 2.81 HTML, 1.77,1.117,1.118,1.119,2.139, p. 981 and hyperlinks, 1.120 JATS, 1.117 LaTeX, 12.2,12.9,12.68, fig. 12.3, table 12.1 MathML, 12.2,12.2nl, p. 983 and metadata, 1.75,1.92 overview, 2.83 in plain-text files, p. 985 and punctuation, 6.4, 6.8, 6.104 semantic markup in, 2.82nl, 7.49 SGML, p. 987 small caps in, 10.8 Unicode standard required by, 11.2 and word-processing styles, 2.82 XHTML, p. 990 XSL, p. 990 mass (noncount) nouns, 5.7 mass-market paperbacks, 4.64 masthead, journal, 1.84,1.85 mathematical copy abbreviations: in mathematical expres sions, 12.3,12.11,12.17,12.64, table 12.2; of mathematics, 10.42; statistics, 10.50, table 12.3 enunciations, 12.25,12.56 fractions in, 12.45-49 manuscript preparation, 12.61-68; author’s role in, 12.3,12.13,12.62; examples of, 12.68, figs. 12.1-2; INDEX mathematical copy (continued) fonts, 12.11-13,12.66; format, 12.61; kerning, 12.11; potentially ambiguous symbols, 12.68, table 12.4; software for, 12.2,12.2nl, 12.9, table 12.1; subscripts and superscripts, 12.67; typefaces, 12.63,12.65 punctuation in, 12.7,12.18,12.38,12.54 resources on, 12.1-2,12.23 See also decimal points; International System of Units; mathematical expressions; metric system; scientific and technical works mathematical expressions, 12.3-60 abbreviations in, 12.3,12.11,12.17,12.36, 12.49,12.60,12.64, table 12.2 commas in, 12.16,12.18,12.19,12.28, 12.36,12.38 consistency of notation, 12.4 delimiters, 12.26-35; alternative cases, 12.35; binomial coefficients, 12.32; common, 12.26; functional notation, 12.27; functions enclosed by, 12.17; inner product denoted by, 12.31; intervals, 12.30; for radicals in text, 12.44; rewriting fractions using expo nents, 12.49; set notation, 12.28-29; vertical bars, 12.26,12.28,12.33-34 determinants in, 12.51 in display, 12.21-23; alternative cases, 12.35; centering, 12.21,12.24; frac tions, 12.46,12.47; line breaks, 12.23, 12.50; numbering of, 1.57,12.24-25, 12.56; qualifying clauses, 12.22; radicals, 12.43-44 elided elements in, 12.19,12.20 fractions in, 12.46-49 in letters to the editor, 1.101 matrices in, 12.50-51 means in, 12.59 probability, 9.19,10.50,12.57,12.58 punctuation with, 12.15,12.18-20,12.28, 12.35,12.42 qualifying clauses in, 12.22 radicals, 12.43-44 scalars, vectors, and tensors, 12.38, 12.52-55 sentences beginning with mathematical symbols, 12.7 signs and symbols: adjacent, 12.8; angle brackets, 6.104,12.26,12.31, mathematical expressions (continued) 12.55,12.59; binary operations and relations, 12.15; braces, 6.104,12.35 (see also delimiters above); brackets and parentheses (see delimiters above); conciseness of, 12.6; consis tency of, 12.4; delimiters (seeabove); differentials, spacing around, 12.42; division sign, 12.15; double-struck (blackboard) symbols, 12.14,12.66; fonts for, 12.11-13,12.66; functions, 12.17,12.26, table 12.2; generally, 12.9; integral sign, 12.41; list of, table 12.1; means, 12.59; minus sign, 6.84, 10.36,12.15,12.60; multiplication, 12.15,12.20,12.53; potentially ambig uous, 12.68, table 12.4; preferred to generic alternatives, table 12.1; probability, 12.58; product sign, 12.40; proof symbol, 12.56; radical sign, 12.42-44; and sentences, 12.7, 12.18; slashes, 6.110,12.15,12.42, 12.45,12.47; special, 12.14; special characters for, 12.10,12.12-13, 12.68, tables 12.3-4; statistics, table 12.3; summation sign, 12.39; verbal translations of, 12.3; vertical bars, 12.26,12.28,12.33-34,12.46,12.51; vs. words, 12.5 spacing in, 12.16-17,12.19,12.21-22,12.42 standard deviations in, 12.59 standards for, 12.3 statistics, 12.58-60, table 12.3 subscripts and superscripts, 12.36-38; alignment, 12.38,12.67; complex, 12.37; and fractions, 12.47,12.49; ital ics for, 12.63; marking, 12.67; simple, 12.36; spacing, 12.16; uncertainties expressed as, 12.60; vector and tensor notation, 12.53,12.54 summations and integrals, 12.39-42 in tables, 3.84 uncertainties, 12.60 vertical lists in, 12.16,12.18 See also decimal points; mathematical copy MathML, 12.2,12.2nl, p. 983 matrices (mathematical), 12.50-51 matrixes (table), 3.68, figs. 3.17-18 maxims (proverbs), 7.60,7.62,13.5,13.38.
See also figures of speech INDEX may, can, 5.250 may, might, 5.147,5.250 Me, Mac, 16.73 me, 5.37,5.57,5.183,5.250 mean, notation of, 3.74,10.50,12.59, table 12.3. See also tables measure, p. 983 measurement. See decimal points; frac tions; International System of Units; mathematical expressions; metric system; physical quantities; units of measurement mechanical editing, 1.116,2.48,2.49. See also manuscript editing medals, 8.115 media, as singular, 5.14 media companies, 10.25 medical terminology, 8.143-46 medieval works, 7.35,11.122-24,14.252 medium spaces, 12.23 meetings. See conferences; proceedings of conferences and symposia; speeches; unpublished and informally pub lished materials; working papers megrtjgiga, tera, and such, as prefixes, 7.89 (sec. 4), 9.10,9.11,10.49,10.55,10.56 memoranda, 14.111,14.229,14.231. See also pamphlets, brochures, and reports meta, as prefix, 7.89 (sec. 4) metadata and abstracts, 1.92,2.25 for books, 1.75,1.76,2.25 defined, 1.75, p. 983 DOIs, 1.33,1.79,1.92 • for hyperlinks, 1.92,1.121 for journal articles, 1.92,1.100 page number markers as, 1.123n2 publishers’ databases for, 1.75,1.92, 1.121 for running heads, 1.12 search tools for, 1.33 version numbers in, 1.28 See also keywords Metadata Handbook, The, 1.75 metaphorical use of proper names, 8.50, 8.66,8.79 meteorological phenomena, 8.77 metric system, 8.152,10.49. See also Inter national System of Units Mexico, 9.21 micro, as prefix, 7.89 (sec. 4) microform editions, 4.64,14.115 mid, as prefix, 7.87,7.89 (sec. 4) midcaps (intercaps), 8.69,8.154 Middle and Old English, 7.35,11.122-24 might, may, 5.147, 5.250 military terminology battles and campaigns, 8.114 capitalization, 8.112-15 medals and awards, 8.115 numbered military units, 9.47 resources on, 8.116,10.15 ships and other craft, 8.2,8.116 titles and ranks, 8.19,8.24-25,10.13, 10.15 twenty-four-hour system, 9.39 units, 8.112,9.47 wars and revolutions, 8.113 millions, billions, and trillions, 5.250,9.8, 9.24 mini, as prefix, 7.89 (sec. 4) minus sign, 6.84,10.36,12.15,12.60 mobile devices, 1.118. See also apps modal auxiliary verbs, 5.145-51 modern editions, 14.114,15.40 Modern Language Association, 14.3 Moldavian language, 11.70 monarchs. See sovereigns and other rulers money dates with currency, 9.25 hyphenation for, 7.88,7.89 (sec. 1) K abbreviation in, 9.24 large amounts of, 7.89 (sec. 1), 9.24 non-US, 9.21-23,9.25 resources on, 9.21 words vs. symbols and numerals for, 9.20 See also bar codes mononyms, 14.83 Monsieur, M., 10.17 Montenegrin language, 11.70 month-day-year date format, 6.38, 6.108, 9.35,14.224 months abbreviations, 10.39,14.171,14.224, 15.15 capitalization, 8.88 dates in text, 9.31 inclusive span of, 14.171 in non-English-language titles, 14.98 See also dates monuments, 8.57. See also sculpture INDEX mood, of verbs, 5.120-27 defined, 5.120 imperative, 5.122,5.155, 5.160,5.214, 5.255 indicative, 5.121,5.124 subjunctive, 5.123-27 moon, 8.141 moral rights, 4.15 more, 5.85, 5.87-88,5.163, 6.47,7.89 (sec. 2) mosques, 8.57, 8.101. See also religions and religious concerns most, 5.86,5.87,5.88,5.163,7.89 (sec. 2) motion pictures. See movies and film; multi media content mottoes. See figures of speech; maxims Mount, mountains, 8.53,10.30,16.90 move-left/move-right marks, 2.127 movies and film characters in, writing about, 5.129 contributions to, as work made for hire, 4.10 sequels to, 9.43 source citations for, 14.201,14.203, 14.265,14.267,15.57 and subsidiary rights, 4.14,4.64 titles of, 8.189,14.86 See also multimedia content MP3 audio files, 10.49 Mr, Mrs., Ms., 10.16 much, 5.7 much, very, 5.90, 5.250 multi, as prefix, 7.89 (sec. 4) multiauthor volumes authors’ names in, 1.38,1.50,1.64,2.32, fig. 1.10 as collective works, 4.8 components: appendixes, 1.59; bibli ographies and reference lists, 1.63; biographical notes, 1.50,1.64,2.41, 14.55, fig. 1.10; dedication, 1.36; illustrations, 2.28, 3.11,3.31; notes, 1.62,14.43,14.46, fig. 14.3; prefaces, 2.41,2.44; table of contents, 1.38, 1.39,2.32,2.41 consistent style across, 2.49 contributors (see contributors to multi author volumes and journals) editor’s responsibilities, 2.41-42 entries for, in source citations: alpha betical order for, 14.66,15.16,15.18, 15.20; authors with same family multiauthor volumes (continued) name, 14.77; chapters, 14.23,14.107- 8,15.9,15.42; examples, general, 14.23,15.9; indexing of, 16.115; repeated names in bibliography, 14.69; shortened citations, 14.32, 14.108; text citations for, 15.7,15.22, 15.29; word order and punctuation, 14.76,15.7 indexing for, as responsibility of volume editor, 2.42 as joint works, 4.7,4.12,4.52 permissions for, 2.41 proofreading of, 2.42 publishing agreements for, 2.42, 4.58, 4.59, fig. 4.2 source citations in: bibliographies or reference lists for each chapter, 1.63, 14.62,15.10; checking on submission, 2.41; endnotes after each chapter, 1.62,14.43,14.46, fig. 14.3; unnum bered source notes, 1.50,2.46,14.54 symposium proceedings as, 4.59 version control for, 2.41 See also manuscript preparation guidelines for authors; previously published materials multimedia content abbreviations for terms (e.g., DVD), 10.49 animations, 1.107,2.4 archival practices for, 1.114 contributions to, as work made for hire, 4.10 in electronic journals, 1.107 as electronic supplementary data, 1.78, 14.187 file formats and devices specified, 2.4, 14.187,14.261,14.267,14.268 in HTML, 1.118 as illustrations, 3.1 metadata for, 1.75 online, 14.267 in PDF files, 1.118 source citations for, 14.261-68; apps, 14.268; author-date reference system, 15.57; basic elements, 14.261; discog raphies, 1.63,14.262,14.263,15.57, fig. 14.12; musical recordings, 14.263; online, 14.267; performances, 14.261, 14.266; recorded readings, lectures, INDEX multimedia content (continued) audiobooks, and such, 14.264; video and film recordings, 14.265 still images for, 3.15 streaming formats, 14.263 subsidiary rights for, 4.64 video clips and files, 1.107,1.114,2.4 on websites, blogs, and social media, 14.205 multiple formats, publications in books, 1.2,1.26,14.159 chapters as works made for hire, 4.58 CIP (Cataloging-in-Publication) data, 1.34 common practice of, 1.2 copyright issues, 4.34,4.47, 4.64 electronic supplementary data: books, 1.59,14.112; hyperlinks to, 1.96; illustrations, 3.9,3.26; noted in print version, 1.78; in table of contents, 1.78,1.87 electronic version published before print, 1.78,1.81,1.86,1.104,1.112-13, 14.172 embedded indexes for, 16.7 and formal markup languages, 1.117 format choices, 1.77,14.16 illustrations, 1.107, 3.9, 3.10,3.17,3.26 ISBNs for, 1.20,1.32,4.64, fig. 1.1, fig. 1.4 ISSNs for, 1.79,1.86 and letters to the editor, 1.101 noting differences, 1.78,1.81,1.114, 2.137, 3.9, 3.17, 3.26,14.16 and page number markers, 14.160 proofreading for, 2.140 as separate editions, 1.26 source citation considerations, 14.16, 14.172 source citations for, 14.112,14.159,14.187 and version control, 1.112 workflow for, 2.82,2.137,2.138 See also DOIs; electronic files; simulta neous print/electronic publishing model multiple or double numbering, 1.57,1.57nl, 2.28,2.29, 3.11, 3.50,12.25 multiplication crosses (dimensions), 3.27 multiplication signs in mathematical expressions, 12.15, 12.20,12.53 plant hybrids, 8.125 multivolume works components: acknowledgments, 1.42; indexes, 1.9; pagination, 1.9; titles and volume numbers, 8.175 consistent style across, 2.49 source citations for, 14.116-22; authors’ and editors’ names, 14.117,14.119, 14.122; chapters and parts of individ ual volumes, 14.120; one volume in two or more books, 14.121; particular volume, 14.118-19; published over several years, 14.117,14.144,15.41; vs. series, 14.124; specific volume of, 14.118,15.41; volume and page numbers, 14.116,14.118-20,15.41; within series, 14.125; work as whole, 14.117 See also series of works; volumes and volume numbers municipalities. See cities and towns; civil and civic concerns; governmental entities; place of publication museums, 4.98, 8.68,14.140 musical group names, 8.68 musical works and fair-use doctrine, 4.87 indexes of first lines of songs, 16.6, 16.144 letters for, 7.71-75 musical examples in, 3.4, fig. 3.5 resources on, 7.70, 8.193 source citations for: performances, 14.201,14.203,15.57; recordings, 14.263; scores, 14.255,14.256 symbols in, 7.71-75 titles, 8.193-97; indexing of, 16.50; instrumental works, 8.195; operas, songs, and such, 8.194; opus numbers, 8.196; recordings, 8.197; resources on, 8.193 See also multimedia content; sound recordings must, 5.148 my, mine, 5.49, 5.50 myself, 5.250 N, n, in tables, 3.55, 3.72, 3.75, 3.85, fig. 3.20 n., nn. (note, notes), 7.15,14.150,14.157, 15.23,16.111,16.112 n/a (not applicable), 3.67 namely, 6.51, 6.58, 6.64, 6.88 INDEX names, personal abbreviations, 10.11-22; given names, 10.11; honorifics, 10.18; initials (see initials: in and as names); Jr., Sr., Ill, and such, 6.43,7.42,10.19,16.41; list of, 14.60; punctuation and spacing, 10.12; speakers in discussions or interviews, 13.48 alphabetizing, basic rules, 16.71-74 (see also alphabetizing) alternative spellings indicated, 6.106 articles in, 8.11,16.75 capitalization (see names, personal, capitalization of) clerical titles with, 8.26,8.33,10.18, 16.39 compound, 8.6,16.72 consistency in spelling, 8.3 deities, 8.91,8.92 epithets, 8.34,8.35 familiar forms, 11.78,14.74,16.33,16.71 fictional characters, 8.35 honorifics with (see honorifics) hyphenated and extended, 8.6,8.7,8.14, 11.88,16.36,16.72 incomplete or unknown, 16.43-44 indexing (see under indexing) initials (see initials: in and as names) and kinship terms, 8.36 letters standing for, 7.66 in list of contributors, 1.64, fig. 1.10 Mac, Me, or O’ in, 16.73 married women’s, 8.11,16.36,16.84 non-English: alphabetizing of (see under non-English names); in English con text, 8.7-18; particles in, 8.5, 8.7-11 nonliteral use, 8.60 numerals with, 6.43,7.42, 9.42,10.8, 10.19,16.37,16.41,16.66 particles in: alphabetizing, 16.71,16.84; capitalization, 8.4,8.5,8.7-10 patronymics, 8.12 personal usage preferences, 8.3,8.4 as phrasal adjectives, 5.93 plurals of, 7.9 possessive forms of, 7.17,7.19,7.28 pseudonyms, 4.25,13.49,14.80-81, 14.208,15.35,16.34 religious figures, 8.93,8.95,10.20,14.83, 16.42 resources on, 8.3,8.5,8.6,16.71,16.84 names, personal (continued) with Saint, 16.74 saints, 8.93,10.20,14.83,16.42 screen names, 14.208,14.209,15.52, 16.64 shortened, 8.6,8.11,8.14,16.43 sovereigns and other rulers, 9.41,16.37, 16.66,16.98 stage names, 16.34 titled persons, 16.38 2-em dash to indicate omitted, 6.93 word division, 7.42 words derived from, 8.60,8.61 See also authors’ names; contributors to multiauthor volumes and journals; editors’ names; titles and offices of people; translators’ names names, personal, capitalization of, 8.3-18 Asian, other, 16.87 Burmese, 16.76 Chinese, 8.15,11.83,11.85,11.88,16.77 Dutch, 8.10 epithets and nicknames, 8.34, 8.35,8.48 French, 8.7 German, 8.8 Hungarian, 8.13,16.78 hyphenated and extended names, 8.6 Indian, 16.79 Indonesian, 8.18,16.80 Italian, 8.9 Japanese, 8.16,11.88,16.81 Korean, 8.17,16.82 non-English names, 8.7-18 overview, 8.4 particles with, 8.4,8.5,8.7-10 Portuguese, 8.8 resources on, 8.3 Russian, 8.12 Spanish, 8.11,16.84 Thai, 16.85 unusual capitalization, 8.4 Vietnamese, 16.86 names, proper abbreviations, indexing of, 16.46,16.64 abbreviations derived from, 10.6,10.49, 10.52 as academic subjects, 8.85 beginning with lowercase letter, 8.69, 8.154 brand names and trademarks, 4.16,7.80, 8.69,8.130,8.146, 8.153-54,10.6 INDEX names, proper (continued) broadcast networks, channels, and such, 8.189,10.25 capitalization, generally, 5.6, 8.1 compound, 5.6,7.85,7.89 defined, 5.6 generic terms with, 8.53,8.56, 8.57 in headline-style capitalization, 8.159 in historical period designations, 8.71-73 hyphenation of, 7.85,7.89,8.39 indexing (see under indexing) non-English (see under non-English names) ©/-genitive with, 5.21 organizations (see organization names) personal (see names, personal) place-names (see place-names) plurals, 5.15,7.9 possessives of, 7.17,7.19,7.20,7.27,7.28 real vs. metaphorical, 8.50,8.66,8.79 in titles of works, 11.89 used as words, 7.63 variants of, 16.32 vessels and vehicles, 8.2,8.116-18 word division, 7.42 words derived from (see words derived from proper names) See also capitalization narrow measure, p. 983 National Information Standards Organi zation (NISO), 1.35,1.117. See also International Organization for Standardization nationality, terms for, 8.38, 8.39. See also countries; groups of people National Library of Australia, 1.34 nations. See countries; governmental enti ties; nationality, terms for Native Americans, 7.10,8.5, 8.38 natural phenomena, named events, 8.77 navigation, 1.120,3.38. See also hyperlinks n.b., NB (nota bene), 10.42 n.d. (no data), 3.67 n.d. (no date), 10.42,14.145,14.263,15.18, 15.44,15.50,15.54 near, compound terms with, 7.89 (sec. 3) negation, 5.230-38 any- and some- words with, 5.238 and auxiliary verbs, 5.147 but and except for, 5.237 in cleft sentences, 5.246 negation (continued) and comparisons, 5.88 defined, 5.230 double negatives, 5.236 in idioms, 5.211 interrogative and imperative state ments, 5.152,5.235 neither-nor for, 5.230, 5.234 no for, 5.233 not for, 5.231, 6.45 pronouns and adverbs for, 5.230,5.233 in verb phrases, 5.104 negative questions, 5.104,5.235 neither, 5.67 neither... nor, 5.33,5.143,5.199, 5.230, 5.234, 5.244, 5.250 neo, as prefix, 7.89 (sec. 4) Netherlands, 8.45. See also Dutch language networks, broadcast, 8.189,10.25. See also television and television programs neutral citations, for Canadian and UK legal cases, 14.294,14.298 new series (n.s.), 10.42,14.126,14.184 newspapers copyright issues, 4.8,4.10 headlines, 7.52,14.191,14.192,14.195 serial rights, 4.64 source citations for, 14.191-200; author-reference system, 15.49; basic information needed, 14.165,14.191; editorials, 14.195; headlines, 14.191, 14.192,14.195; letters to the editor, 14.196; news services and news releases, 14.200; readers’ comments, 14.196; regular columns, 14.195; in textvs. bibliography, 14.198; time stamps, 14.191; titles of papers, 14.193,14.194; unsigned articles, 14.199,15.49; weekend supplements, magazines, and such, 14.197 titles: in building or award names, 8.172; city names in, 14.193,14.194,16.48; indexing of, 16.48; initial the in, 8.170,14.193,14.194; instead of au thor’s name, 14.199,15.49; italics for, 14.86,14.193; non-English, 14.194; plurals of, 7.12 news releases (press releases), 14.200 news services, 14.200 New Testament, 8.106,10.47. See also Bible New Zealand, 9.21 INDEX nicknames, 8.34,8.35,8.48 9/11 (September 11,2001), 9.35 NISO (National Information Standards Organization), 1.35,1.117. See also International Organization for Stan dardization no, 5.33, 5.230,5.232,5.250, 6.34 no., nos. (musical works), 8.195 nobility, 8.32,16.38. See also sovereigns and other rulers no data (nJ.), 3.67 no date (nJ.), 10.42,14.145,14.263,15.18, 15.44,15.50,15.54 “no derivatives” clause, 4.62 nominalizations (zombie nouns), 5.189 nominative (subjective) case absolute constructions, 5.35 and gender, 5.43 intensive pronouns, 5.51 misuse of, 5.37 nouns, 5.9, 5.18 pronoun after linking verb, 5.45 pronoun as subject of clause, 5.44 pronouns, overview, 5.41 relative pronouns, 5.58,5.66 non, as prefix, 7.89 (sec. 4) nonbreaking spaces with ellipses, 6.121,13.50,13.54,13.58 generally, 6.119 with Jr., Sr., and such, 7.42 in non-English-language materials, 11.28,11.29 in numbers, 6.121,7.44,9.55,9.56 in run-in lists, 7.45 between single and double quotation marks, 6.11 Unicode for, 6.121 in units of measurement, 6.121,7.44 noncomparable adjectives, 5.89 noncomparable adverbs, 5.166 noncount (mass) nouns, 5.7 none, 5.67,5.250 non-English materials capitalization: African languages, 11.23; Arabic, 8.14,11.80; Asian, other, 8.18; Azeri, 11.70; Chinese, 8.15, 11.88,11.89; Danish, 11.70; Dutch, 8.10,11.70; in English context, 11.3, 11.4; English language compared, 11.18; English translations, 11.9-10, 11.18,14.99 (see also translations non-English materials (continued) and translated works); French, 8.7, 11.26,11.27; German, 7.54,8.8,11.39; Hebrew, 11.93; Hungarian, 8.13; Ital ian, 8.9,11.46; Japanese, 8.16,11.88, 11.89; Korean, 8.17; Latin, 11.54; Norwegian, 11.70; Polish, 11.70; Portuguese, 8.8,11.70; Russian, 8.12, 11.99,11.100; Spanish, 8.11,11.61; Swedish, 11.70; titles of works (see under titles of works: capitalization); Turkish, 11.70 English context considered, 8.7-18, 11.3-5,11.19 IPA notation, 11.22 overview, 11.1 permissible changes to, 11.7,11.15-16, 11.19,11.28,13.7 punctuation: apostrophes (see under apostrophes); brackets, 6.96, 6.99, 14.99; colons, 11.116; ellipses, 11.32, 11.49,11.65,11.102; em dashes (see under em dashes); exclamation points, 11.19,11.62,11.102; glossed American Sign Language, 11.133; guillemets, 11.7,11.19,11.28-32, 11.41,11.47,11.63-64,11.101,13.7; permissible changes to, 11.7,11.11; question marks, 11.19,11.62,11.102, 11.116; quotation marks (see under quotation marks); semicolons, 11.7, 11.116; in translations, 6.96, 6.99, 6.100,11.12,11.19 quotations in English context: French, 11.28-32; Greek, 11.116; punctua tion, 11.3,11.19; Russian, 11.101; typographic style of, 11.11 (see also translations and translated works: quotations) running heads, 2.76 in source citations: bibliographic terms, 14.102; English forms of cities (place of publication), 14.131; journal articles, 11.26-27,14.170,14.183; journal titles, 14.98,14.170; musical scores, 14.255; newspaper titles, 14.194; publishers’ names, 14.135, 14.136; series titles, 14.123; titles of works (see titles of works below) special characters, generally, 11.2,11.21, tables 11.1-5 INDEX non-English materials (continued) titles of works, 11.6-10; articles in, 8.170,14.98,14.170; brackets for, 14.99; capitalization (see under titles of works: capitalization); cautions about shortening, 2.76; dates in, 14.93; indexing of, 16.48,16.49, 16.52,16.53; italic vs. roman type, 11.8,11.89,11.100,14.99; permissible changes to, 11.7; short forms, 14.33; subtitles, 14.89; translated, 11.9, 11.10,14.99, fig. 1.3 titles of works, specific languages: Chinese, 11.89; French, 11.27,16.48, 16.52; Japanese, 11.89; Latin, 11.54; Russian, 11.100 See also European style; non-English names; non-English words and phrases; special characters; transla tions and translated works; Unicode standard; and specific languages non-English names personal: alphabetizing of (see under al phabetizing); indexing of (see index ing: names, non-English personal); particles in, 8.5,8.7-10; treatment in English context, 8.7-18 personal, by group: Arabic, 8.14,16.75; Asian, other, 8.18,16.87; Burmese, 16.76; Chinese, 8.15,11.83,11.85, 11.88,16.77; Dutch, 8.10; French, 8.7; German, 8.8; Hungarian, 8.13, 16.78; Indian, 16.79; Indonesian, 8.18,16.80; Italian,’ 8.9; Japanese, 8.16,11.88,16.81; Korean, 8.17,16.82; Portuguese, 8.8,16.83; Russian, 8.12, 11.98; Spanish, 8.11,16.83,16.84; Thai, 16.85; Vietnamese, 16.86 place-names, 8.59,11.61,11.85,11.87, 11.99,14.131,16.92 proper names, other: company names, 10.23,14.135,14.136; in English context, 11.4; French, 11.26; generic terms in geographic names, 8.55; German, 11.39; newspaper titles, 14.194; nouns, roman for, 7.53; publishers’ names, 14.135,14.136; in titles of works, 11.89 non-English words and phrases alphabetizing of, 16.67,16.92 bibliographic terms, 14.102 non-English words and phrases (continued) first use of, 7.53,11.75 generic terms in geographic names, 8.55 glossaries for, 1.61 hyphenation, 7.89 (sec. 1), 11.26,11.79, 11.84,11.87-88,11.93,11.128 italics for, 7.53,7.54,11.3,11.5,11.75, 11.88,11.94,14.272 in legal-style citations, 14.272 ligatures in, 7.35 as mottoes, 7.62 in place-names, 8.55,11.26 plurals of, 7.12,11.3 roman type, 7.53-55 translations of, 6.96, 6.100,11.4-5, 11.18-19 word division (see under word division) nonprofit organizations, 4.64,4.68 non-US publications, copyright of, 4.29-30, 4.47 NOOK, 14.159. See also e-books noon and midnight, 9.38 no place (n.p.), 10.42,14.132 nor, 5.33,5.34,5.138,7.8 North, northern, 8.46,8.47 Norwegian language, 11.70 not, 5.104,5.230,5.231, 6.45 not... but, 6.46 notch binding, p. 984 note numbers, 14.24-28 citations to, 14.157 format of, 14.24 in index entries, 16.111,16.112 keyed to line or page numbers, 14.53, figs. 14.5-6 for multiple citations and references, 14.28 placement of, 14.26,14.27 and punctuation, 14.26 reducing number of, 14.57 vs. reference numbers, 14.24 renumbering of, 2.44,2.62 sequencing of numbers and symbols, 14.25,14.49, fig. 14.4 symbols instead of, 3.79,14.24,14.25, 14.49,14.51,16.112, fig. 14.4 in tables, 2.22,2.31,3.79 with titles and subheads, 14.27 unnumbered notes in relation to, 14.52 See also note reference numbers/sym- bols INDEX note reference numbers/symbols avoiding in chapter display, 1.49,2.22, 14.27 cross-checking, 2.32,2.53,2.62 for editor’s or translator’s notes, 14.51 format for, 14.24 generally, 14.19 multiple, 14.28 and note placement, 2.134 placement of, 14.26 with subheads, 14.27 for table notes, 3.79 See also note numbers notes, 14.24-60 abbreviation of term (n., nn.), 7.15, 14.150,14.157,15.23,16.111,16.112 abbreviations in, 10.3,10.24,10.26, 10.27,10.42,10.44 author-date reference system with, 14.39,14.50,14.271,15.11,15.31, fig. 15.2 authors’ names (see authors’ names) basic structure, 14.20,14.22,14.23 biblical citations in, 10.44,10.45-48, 14.239 bibliographies in relation to, 14.61 for chapters of multiauthor volumes, 1.62,14.43,14.46, fig. 14.3 citations plus commentary in, 14.37 citations to, 14.157,15.23 continued across pages, 14.41,16.113, fig. 14.1 converted in electronic file cleanup, 2.80 cross-checking of, 2.32,2.63,15.21 cross-references to, 14.28 editing of, 2.62,2.63 examples, figs. 14.1-7 footnotes not always used in electronic publications, 14.27,14.49,14.52 footnotes plus endnotes, 2.22,14.49-51, fig. 14.4 footnotes vs. endnotes, 1.124,2.77,2.80, 14.43-48,14.57,14.59 format of, 1.62 for frequently cited works, 13.67,14.59- 60, fig. 14.7 generic markup of, 2.81 glossary terms in, 1.61 hyperlinks to (see hyperlinks for source citations) notes (continued) ibid, in, 1.124,14.34,14.48 indexing of, 14.157,16.110-13,16.117 in journals, 1.108,14.43,14.46 keyed to line or page numbers, 14.53, figs. 14.5-6 legal cases in, 8.82 legal-style citation form, 14.271 length considerations, 14.40,14.56 manuscript preparation guidelines, 2.22, 14.24,14.41,14.43,14.53 multiple citations in single, 14.28,14.35, 14.57 multiple paragraphs in, 14.40 numbering (see note numbers) original, in edited or translated material, 13.7,14.51, fig. 14.4 overview, 14.43 part titles for, 1.48 placement in book, 1.4,1.61,1.62,14.46 in previously published materials, 2.44, 2.45 proofreading, 2.134 publication facts (see publication details) quotations within, 11.13,14.38 reference numbers or symbols for (see note reference numbers/symbols) remedies for excessive, 14.56-60; abbreviations, 14.59-60, fig. 14.7; avoiding overlong, 2.62,14.56; footnotes vs. endnotes, 14.45,14.59; several citations in single, 14.57; text citations, 14.34,14.58 scholarly symbols in, 10.43 see and cf. in, 14.42 shortened citations in, 14.19,14.29-36, 14.48 source notes (see source notes) submission of, 2.3 substantive, 14.37-39,14.49,16.110 for tables (see under tables) titles of works in (see titles of works in source citations) unnumbered: acknowledgments in, 14.55; biographical notes in, 1.50, 14.55; for compilations, 2.46; for en tire chapter, 1.49,1.50,14.27,14.54; for epigraphs, 1.49,14.52; keyed to text by line or page numbers, 14.53, figs. 14.5- 6; numbered notes in relation to, 1.49, 14.25,14.49,14.52, fig. 14.4 INDEX notes (continued) within quotations, 13.7,13.58 See also endnotes; footnotes; note num bers; notes and bibliography system; source citations; source notes; and specific materials to document notes and bibliography system, 14.19-23 and author-date reference system, 14.2, 15.1,15.2,15.3 disciplines that use, 14.2 elements to include, 14.100 examples and variations, 14.23 flexibility of, 14.2 notes, basic structure, 14.20 overview, 14.19 page numbers and other locators, gen erally, 14.22 relationship of notes and bibliographies in, 14.61 See also bibliographies; notes; shortened citations; source citations; titles of works in source citations; and specific materials to document not only ...but also, 5.199, 5.244, 6.46 noun phrases in American Sign Language, 11.134 as antecedents of relative pronouns, 5.59, 5.62 with expletives, 5.239 hyphenation of, 7.89 (sec. 2) for index entries, 16.9 with possessive, 5.80,7.24 See also nouns nouns, 5.4-26 as adjectives, 5.24,5.259,7.27,7.89 (sec. 2) adjectives as (adnouns), 5.94 as adverbs (adverbial objectives), 5.26 adverbs formed from, 5.158 agreement of personal pronoun with, 5.42 attributive, 5.24,7.27 case (see case of nouns and pronouns) coinages, 7.14 collective, 5.5,5.7,5.15, 5.138 in compound terms, hyphenation of, 7.89 concrete, 5.5, 5.7 coordinate, 5.75 definitions: common (concrete, abstract, nouns (continued) and collective), 5.5; generally, 5.4; mass (noncount), 5.7; proper, 5.6 dependent clauses as, 5.219 em dash between pronoun and intro ductory, 6.86 formed from verbs (nominalizations), 5.189 functional variations of, 5.24-26 genitive case (see genitive case) for index entries, 16.9 infinitives as, 5.107 as interjections, 5.208 irregular, 5.20,7.5,7.16,7.27 nominative (subjective) case (see nomi native (subjective) case) in non-English languages: Danish, 11.70; German, 7.54,11.39; Norwegian, 11.70 objective (accusative) case (see objective (accusative) case) possessives, generally, 7.16 (see also possessives) predicate, 5.18,5.141 prepositional phrases as, 5.176 prepositions with, 5.172 proper (see names, personal; names, proper; place-names) properties, 5.8-12; case, 5.9 (see also case of nouns and pronouns); gender, 5.11; number (see number, of nouns and pronouns); person, 5.12, 5.30, 5.39, 5.40 quoted titles or expressions as, 6.41 repeating, for gender neutrality, 5.255 table titles as, 3.54 titles of works as singular, 8.166 as verbs, 5.25,5.250 See also appositives; noun phrases; place-names; plurals; pronouns noun-to-verb transitions, 5.25,5.250 now, 5.202 n.p. (no place, no publisher, no page), 10.42, 14.132 n.s. (new series), 10.42,14.126,14.184 nth degree, 9.6. See also ordinal numbers number, of nouns and pronouns and antecedents, 5.32-33, 5.42,5.57, 5.62,5.255 defined, 5.10, 5.41 INDEX number, of nouns and pronouns (continued) indefinite pronouns, 5.67 numbers, 9.19,9.20 with one in antecedent, 5.62 and verb number, 5.137,5.138,5.139 number, of verbs, 5.137,5.138, 5.139,5.140, 5.141, 5.142,5.205 numbering appendixes, 1.59, 8.180 captions, 2.28,3.13 chapters (see chapter numbers) double or multiple, 1.57,1.57nl, 2.28, 2.29,3.11, 3.50,12.25 illustrations (see illustrations: number ing) impressions, 1.28, figs. 1.1-4 manuscript pages in electronic file, 2.38 mathematical definitions, 12.25,12.56 mathematical expressions, 1.57,12.24-25,12.56 musical works, 8.195, 8.196 notes (see note numbers) record groups, in government archives, 14.292 subhead considerations, 1.57 table columns, 3.56 tables, 1.57,2.29, 3.50, 3.54,16.116 See also issue numbers; numbers; page numbers; pagination; tables numbers abbreviation for number, 8.195,8.196, 14.123,14.171,15.15 (see also issue numbers) abbreviations and symbols with, 7.89 (sec. 1), 9.16-17 addresses and thoroughfares, 9.50-52, 9.54 alignment, in tables, 3.72, 3.86, figs. 3.14-15, fig. 3.20 alphabetic characters in (error), 2.80 alphabetizing in indexes, 16.65,16.66, 16.103 a or an with, 7.33 basic principles, 9.2; alternative rule, 9.3; consistency, flexibility, and read ability, 9.1, 9.7,9.13,9.14,9.19 building and apartment numbers, 9.52 Chicago’s preference for arabic vs.
roman numerals, 9.67 commas between digits, 9.54-56 numbers (continued) in compound terms, 7.89 (sec. 4) dates: centuries, 7.8,8.71, 9.32,9.33; decades, 9.33; eras, 9.34,9.64; ISO all-numeral style, 6.108,9.35, 9.36; month and day, 9.31; periods of time, 8.71; year abbreviated, 9.30; year alone, 9.29 defined, p. 975 editorial uses: column heads in tables, 3.86, fig. 3.24; divisions in publica tions and documents, 9.26-28; file names, 2.37; lists, 2.11,2.21, 6.129, 6.130, 6.131,7.45; notes applying to specific parts of table, 3.79; outlines, 6.127, 6.132; table columns, 3.56 enzyme names, 8.133 fractions (see fractions) Greek, 11.117, table 11.5 hundreds, thousands, and hundred thousands, 9.3,9.4,9.54 hyphenation in: all-numeral dates, 6.108,9.36; double or multiple nu meration, 1.57; fractions, 7.89 (sec. 1), 9.14; noun plus numeral or enumer ator, 7.89 (sec. 2), 9.13; separators, 6.77; telephone numbers, 9.57 inclusive (continuing) (see inclusive (continuing) numbers) index entries beginning with, 16.65 in initialisms, 10.2 large, 5.250, 9.8-12,9.24 and line breaks, 7.42,7.43,7.44 millions, billions, and trillions, 5.250, 9.8,9.24 numbered divisions in publications and documents, 9.26, 9.27,9.28 old-style, 3.86,12.9 ordinals (see ordinal numbers) paragraphs (see paragraph numbers, as locators) parts of book in text, 8.180 physical quantities, 9.13-17; with abbreviations and symbols, 9.16-17; dimensions, 3.27; in general context, 9.13; simple fractions, 9.14; whole numbers plus fractions, 9.15 plurals of, 7.15,9.53 possessive forms of, 7.17 in proper names: and line breaks, 7.42; INDEX numbers (continued) military units, 9.47; personal names, 6.43,7.42,9.42,10.8,10.19,16.37, 16.41,16.66; places of worship, 9.48; political and judicial divisions, 9.46; sequels, 8.189, 9.43; sovereigns and other rulers, 9.41,16.66; successive governments and dynasties, 9.45; unions and lodges, 9.49; vehicles and vessels, 8.116, 9.44 ranges (see inclusive [continuing] num bers; page ranges) ratios, 6.62, 9.58 resources on, 9.1 roads and thoroughfares, 9.50, 9.51 sections (see section numbers, as locators) sentences beginning with, 9.5 sequels, 8.189,9.43 source citations: arabic vs. roman numerals, 14.147; classical Greek and Latin works, 14.243,14.244,14.249; parts of books, 14.147; parts of poems and plays, 8.184,14.254; scriptural references, 14.241 space between digits, 6.121, 9.55,9.56 spacecraft, 9.44 spelled out: alternative rule, 9.3; begin ning a sentence, 9.5; building and apartment numbers, 9.52; Chica go’s general rule, 9.2; consistency, readability, and flexibility, 9.7; dates, 9.29, 9.31-33, 9.35; direct discourse, 13.44; fractions, 7.89 (sec. 1), 9.14, 9.15; general rule, 9.2; in headline style capitalization, 8.161; hundreds, thousands, and hundred thousands (round numbers), 9.4; hyphens with, 7.89 (sec. 1), 9.13; inclusive range of, 9.60; marking manuscript for, 2.93; marking proofs for, 2.129; military units, 9.47; money, 9.20; ordinals, 7.89 (sec. 1), 9.6; places of worship, 9.48; plurals of, 9.53; political and judicial divisions, 9.46; roads and thoroughfares, 9.51; successive governments and dynasties, 9.45; time designations, 9.37, 9.38,9.40; in titles of works, 8.165; to with inclu sive numbers, 9.60 (see also physical quantities above) numbers (continued) telephone numbers, 6.77, 9.57 in titles of works, 8.165,14.88 with units of measurement, 7.44, 9.16- 17,10.49 uses, other: age terms, 7.89 (sec. 1); binary systems, 9.11; celestial bodies, 8.139; chemical elements, 10.63; decimal fractions, 9.19, 9.59; dex with, 9.12; gene and enzyme names, 8.132, 8.133; mass, in chemical terms, 8.150; mega-,giga-, tera-, and such, 9.10,9.11,10.49,10.55,10.56; mon etary amounts, 9.20-25; multiple books within one volume, 14.121; percentages, 9.18; powers of ten, 9.9, 9.11, 9.12 See also arabic numerals; chapter numbers; currency; decimal points; International System of Units; issue numbers; line numbers; mathemati cal expressions; metric system; note numbers; numbering; page numbers; page ranges; roman numerals; subscripts; superscripts; Unicode standard; volumes and volume numbers number sign (pound sign), 2.98,2.125, 3.79, 11.131 numerals. See arabic numerals; numbering; numbers; roman numerals numerical order, index subentries in, 16.69 nursery rhymes, 8.185 O, o/i, 6.35,7.31. See also interjections O’, 16.73 oaths, 8.84 objective (accusative) case and gender, 5.43 and like, 5.185 nouns, 5.9,5.19 in prepositional phrases, 5.44, 5.172, 5.183 pronoun as object of infinitive, 5.44 pronoun as object of verb or preposition, 5.35, 5.37, 5.44 pronouns, overview, 5.41 relative pronouns, 5.57, 5.66 oblique objects, 5.172 oceans, 8.45, 8.53 o’clock, 9.37, 9.38,10.41 INDEX OCR (optical character recognition), 2.43, 10.49, p. 984 odd, compound terms with, 7.89 (sec. 3) Of in genitive case, 5.20,5.21,5.191 limiting use of, 5.250 and possessive, 7.20,7.25,7.26,7.29 offprints, p. 984 offset printing, 3.3, figs. 3.1-2, p. 984 of-genitive, 5.21, 5.191 of whom, of which, 5.63,5.64 oh, O, 6.35,7.31. See also interjections ohm symbol, 10.49,10.56 ‘okina, 11.70,11.70nl old, compound terms with, 7.89 (sec. 3) Old and Middle English, 7.35,11.70,11.122- 24 old series (o.s.), 14.126 old-style numbers, 3.86,12.9 Old Testament, 8.106,10.45. See also Bible on, as prefix, 7.89 (sec. 3) once, 5.201 one, 5.62, 5.255 one another, each other, 5.53 one of, 5.140 ONIX, 1.75 online publications copyright of, 4.13, 4.47,14.14,14.267 formal markup for, 2.83 front and back matter in, 1.122 functional features in, 1.119 hyperlinks in, 1.120,1.121,1.122 indexes in, 1.96,1.109,1.125 navigation in, 1.120,1.123 PDF format for, 1.118 proofing and testing of, 2.137-40 reference works, 1.118,1.120,1.121,14.233 screen margins, 1.122 site maps in, 2.137 source citations for: books, 14.161-62; dictionaries and encyclopedias, 14.233; legal and public documents, 14.270,14.276,14.298,14.300, 14.305; magazine articles, 14.189; multimedia, 14.267 (see also websites and web pages: source citations for) wikis, 14.13, p. 990 See also electronic publications; HTML; multiple formats, publications in; supplementary data, electronic; websites and web pages only, 5.186 op., opp. (musical works), 8.196 opacity, p. 984 op. cit., 10.42,14.36 open-access publishing models, 4.52, 4.60, 4.61,4.66 open compounds, 6.106,7.82,16.58. See also compound terms open-source model, 4.61,7.80. See also open-access publishing models operas, 8.194. See also musical works operating systems, 7.78, 8.155. See also file formats and devices optical character recognition (OCR), 2.43, 10.49, p. 983 option clauses, 4.53 or in dates, 7.8 in double titles of works, 8.167,14.91 in lists, 6.131 pronoun and antecedent with, 5.33, 5.34 punctuation with, 6.22, 6.51 slash instead of, 6.106 as subordinating conjunction, 5.201 and verb agreement, 5.138, 5.143, 5.205, 5.250 See also conjunctions; coordinating conjunctions ordinal numbers day of the month as, 9.31 format of, 9.6 hyphenation, 7.89 (sec. 1) in legal-style citations, 14.274,14.300 military units, 9.47 nd, th, and rd in (e.g., 122nd), 9.6 in non-English languages, 11.27 numbered streets, 9.51 permissible changes to, 14.88 personal names with, 6.43, 9.42,10.19 successive governments and dynasties, 9.45 organization names abbreviations, 10.23-26,14.274,15.37, 16.46,16.64 alphabetizing of, 16.46,16.64,16.88-89 articles in, 8.68,8.70,14.134,16.88, 16.89 associations, 8.70,10.26 as authors, 14.70,14.84,15.37 capitalization (see under capitalization) companies (see company names) INDEX organization names (continued) in non-English languages, 11.26,11.61 numbered branches of, 9.49 periodical titles in, 8.172 place-names within, 6.39, 6.81 possessives of, 7.20,7.27 in source citations, 14.84,14.259, 14.305,15.37 treatment in text, 8.62-70 See also business and commerce; govern mental entities; institutions original expression and originality, 4.3,4.5, 4.72. See also intellectual property rights ornaments for text break, 1.58,2.8 orphans (lines), 2.116, p. 984 “orphan works” problem, 4.82 orthographic reforms, 11.38,11.43,11.45, 11.67,11.70 o. s. (old series), 14.126 otherwise, 5.201 ought, should, 5.149, 5.250 out, in compound terms, 7.89 (sec. 2) outlines, 2.21,2.36, 6.127, 6.132. See also lists in text out-of-print publications, 4.52 over, as prefix, 7.89 (sec. 4) overlapping characters, 6.5 Oxford (serial) commas, 6.19-21,6.23, 8.165,14.135. See also commas p (probability), 3.80,10.50 p. ,pp., 7.15,10.42,14.150-51,14.225 page, 9.27 page count. See length (parts of text) page numbers (folios) alternatives for reflowable electronic publications, 1.123,2.35,14.22, 14.160,14.174,15.23 appropriate use of, 1.6,9.26 blind (unexpressed), 1.6,1.7,1.8,1.39, 1.46,1.49,3.6, p. 980 in books, arabic vs. roman numerals, 9.26 cross-checking text citations and refer ence lists for, 2.32,15.21 cross-references to specific pages within same text, 2.35,2.60 descriptive headers with, 2.38 drop, 1.6,1.7,1.8,1.49, p. 980 “dummy,” 1.81 page numbers (folios) (continued) fixed, not always used in electronic pub lications, 1.6,1.81,1.123,2.35,14.47, 14.49,14.160 inclusive ranges (see page ranges) as index locators (see index entries: locators) in lists of illustrations and tables, 1.39, 3.40, fig. 1.7 location of, 1.6 of manuscript, 2.38 markers for, in electronic publications, 1.123n2,1.125,14.160,16.13 in multivolume works, 1.9,14.116, 14.120,15.41 no comma in, 9.54 for permission requests, 4.95 proofreading of, 2.114,2.134 in running heads for endnotes, 1.15, 1.62,2.76,2.114,2.136,14.47, fig. 14.2 in source citations (see page numbers in source citations) in table of contents, 1.38,2.134, fig. 1.5 three zeros for temporary, 2.35 See also inclusive (continuing) numbers; locators; page numbers in source citations page numbers in source citations alternatives for electronic publications, 14.22.14.160.14.174.15.23 arabic vs. roman numerals, 14.147,14.171 author-date reference system, generally, 15.9.15.23 chapters with, 14.106-7 with citations to notes, 14.157 classical Greek and Latin works, 14.243, 14.251 classic English works, 14.253 e-books, 14.160 examples and variations, 14.23 ff. and passim, 14.148-49 forthcoming works, 14.146 generally, 14.22,15.8 ibid, with, 14.34 journal articles, 14.174,14.177,15.23 legal-style citations, 14.273,14.276, 14.284,14.290,14.294,14.303, 14.305 magazine articles, 14.188-89 multivolume works, 14.116,14.120,15.41 in notes, 14.153 INDEX page numbers in source citations (continued) notes and bibliography system, gener ally, 14.22 notes keyed to, 14.53, fig. 14.6 p. andpp., 14.150-51,14.225 periodicals, generally, 14.165 ranges, author-date reference system, 15.8,15.9,15.21 ranges, notes and bibliography system, 14.23,14.106-7,14.110,14.148-49, 14.167.14.174 reprint and modern editions, 14.114 for sources of quotations, 13.67 text citations, 15.27,15.30 volume numbers with (e.g., 10:122), 14.23,14.116,14.152,14.177,15.23, 15.48 page proof, p. 984. See also proofs page ranges condensing, 9.61,16.14 en dashes in, 6.78,9.60,16.100,16.134 as index locators, 16.12,16.14,16.100, 16.113,16.121 for journal articles, 1.79,1.81,1.87,1.92, 1.103.14.23.14.174 on journal cover or spine, 1.84,2.109, 2.134 in running heads, 1.15,2.76,14.47, fig. 14.2 in source citations: author-date refer ence system, 15.8,15.9,15.21; vs.ffi, 14.148- 49; notes and bibliography system, 14.23,14.106-7,14.110, 14.148- 49,14.167,14.174 pages abbreviation of term (p.,pp.), 7.15,10.42, 14.150-51,14.225 bad breaks in, 16.138 checking overall appearance, 2.116, 2.117 defined, 1.5 facing: alignment of, 2.117; proofreading of, 2.116; tables on, 2.115, 3.51, 3.86, 3.87 See also page numbers; pagination; proofs pagination avoiding repagination, 2.122 basic guidelines for, 1.6-9 in book review and book notes sections, 1.98,14.153 pagination (continued) and continuous publishing model, 1.82, 14.174 determining page 1,1.46 of final manuscript for production, 2.75-76 of folio editions, 14.155 and indexing, 2.104,2.105,2.106,2.113, 2.118,16.108 journals, 1.81,1.82, fig. 1.11 and metadata, 1.75 multivolume works, 1.9 parts of books: back matter, 1.4,1.8,1.9; chapter display, 1.8,1.49; chapters, 1.8,1.46,1.49; conclusions, 1.53; dedication, 1.7,1.36; epigraphs, 1.37; front matter, 1.4,1.7,1.17,2.38,2.57; galleries, illustrations, and tables, 1.8,1.39,2.115; introductions in front matter, 1.43; introductions in main text, 1.47; part-title page with text, 1.8; second half title, 1.4,1.8,1.46 roman numerals in, 1.4,1.7,1.9,2.38, 2.57, 9.26 in signed signatures, 14.154 within volumes, 1.81, fig. 1.11 See also page numbers; page ranges; recto pages; verso pages paintings, 8.198,14.235. See also artwork pamphlets, brochures, and reports art exhibition catalogs, 8.201,14.236 congressional reports and documents, 14.285 in manuscript collections, 14.229 in private collections, 14.231 source citations, generally, 14.220 titles of, 8.186 paper durability statements, 1.20,1.35,1.84, fig-1-1 environmental standards, 1.35, pp. 979, 984, 988 opacity, p. 984 trim size, p. 989 paperback rights, 4.64 paperbacks, 4.64, p. 984. See also covers of journals and paperback books paper durability statements, 1.20,1.35, 1.84, fig. 1.1 paper-only editing. See under manuscript editing INDEX paper-over-board format (lithocase), 1.71 papers, unpublished, 14.217,14.218. See also unpublished and informally published materials paragraph (first-line) indentation defined, 2.11, p. 982 glossary entries, 2.23 paragraph format, 2.11, 2.12 paragraphs instead of lists, 6.130 paragraphs within block quotations, 2.19,13.22 text following block quotations, 2.21, 13.24 in transcriptions of discussions or inter views, 13.48 .
paragraph mark, 2.98,2.126,10.43,14.273 paragraph numbers, as locators in electronic publications, 1.123,1.125, 14.160 in indexes, 1.125,2.106,16.12,16.13, 16.108 legal and public documents, 14.273 ranges of, 14.148 paragraphs abbreviation of term, 10.42 conjunction at beginning of, 5.203 defined, in word processing, 2.11,2.82 indentation of, 2.11,2.12 instead of vertical lists, 6.130 interruptions in, 2.12 manuscript preparation guidelines, 2.12, 2.19,2.20 marking manuscript for, 2.98 marking proofs for, 2.126 notes with multiple, 14.40 symbol for, 2.98,2.126,10.43,14.273 within quotations, 2.19,13.22,13.32-33, 13.35,13.39,13.56 word-processing styles for, 2.82 See also paragraph numbers, as locators paragraph style (indexes). See indexes: run-in style parallel bars. See vertical bars parallel structure, 5.242-45 and auxiliary verbs, 5.245 and correlative conjunctions, 5.244 and en dashes, 6.78 lists and outlines, 6.127 in prepositional phrases, 5.243 subheads, 2.59 paraphrasing, 4.89,11.17,13.4,13.45 parentheses, 6.95-98 and alphabetizing, 16.59,16.60,16.61 back to back, 6.98 font for, 6.5, 6.129 in indexes: authors’ names after titles, 16.50,16.97; continued lines, 16.139; cross-references, 16.18,16.20,16.26, 16.97; endnote locators, 16.111 (see also glosses: in index entries) with in-text citations for quotations, 13.64 marking proofs for, 2.132 other punctuation with: brackets, 6.97, 6.101,14.37,14.38,15.28,15.40; colons, 6.98; commas, 6.18, 6.98; exclamation points, 6.74, 6.98; mul tiple sets of parentheses, 6.97, 6.101; periods, 6.13, 6.98; question marks, 6.70, 6.98 scholarly abbreviations in, 10.42 sentences in, 6.13, 6.98 in source citations: classical Greek and Latin works, 14.250; dissertations and theses, 14.215; with issue num bers, 14.177,15.47; journal articles, 14.171,14.177,14.182,15.9,15.47; legal-style citations, 14.276,14.283- 86,14.288,14.294-95,14.298, 14.300,15.59; newspaper articles, 14.193,14.197; note reference num bers, 14.26; notes, 14.19,14.20,14.39; reprint editions with multiple dates, 15.40; websites and web pages, 14.207 (see also text citations) in table titles and heads, 3.55, 3.56, fig.
3.12 uses: abbreviations, 7.17,10.3; brand names, 8.146; captions, 3.24; clarification of italics for emphasis, 13.62,14.37; enclosing numbers or letters in lists, 6.129; with forthcom ing or in press, 8.188; genus, species, and subspecies, 8.124; glosses or translations, 6.96,11.5,11.10 (see also glosses: in index entries); i.e. and e.g., 6.51; labeling for mathematical expressions, 12.24; letters for parts of books, 8.180; mathematical expressions, 12.45,12.50 (see also delimiters); music dynamics, 7.75; notes to tables, 3.78; parenthetical el- INDEX parentheses (continued) ements in sentences, 6.48, 6.51, 6.85, 6.95; table titles and heads, 3.55, 3.56, fig. 3.12; telephone numbers, 9.57; time zones, 10.41; translations of quotations, 11.12,11.13 See also glosses; parenthetical elements in sentences; punctuation; text citations parenthetical elements in sentences, 6.48, 6.51, 6.85, 6.95 Parliament (UK), 14.298-301 participial phrases, 5.111,5.115,5.231, 6.30, 6.32 participial prepositions, 5.175 participles as adjectives, 5.90,5.111 adverbs modifying, 5.114 te-verbs with, 5.118,5.154 in compound terms, hyphenation of, 7.89 (sec. 2) dangling, 5.115 formation of, 5.110 fused, 5.114 vs. gerunds, 5.113 and negation, 5.231 past, 5.90,5.100, 5.119 and prepositions, 5.175 present, 5.110,5.118,7.41 table titles as, 3.54 word division, 7.41 See also gerunds; participial phrases particles negating, 5.231 in personal names: alphabetizing, 16.71, 16.84; capitalization, 8.4, 8.5,8.7-10 in phrasal verbs, 5.102 parties, political, 8.66 parts and part titles articles, stories, chapters, and such, 8.177 collected works of author, 8.178 division into, 1.48, fig. 1.5 editing of, 2.58 format of terms, 8.179,8.180 generic terms for, 8.179,8.180,14.110 introductions to, 1.38 in letters and diaries, 1.52 manuscript preparation guidelines, 2.17 numerals for, 9.26 pagination, 1.4,1.8,1.46,2.75 parts and part titles (continued) part-title page with text, 1.8 in poetry, 1.51 in running heads, 1.12,1.16 source citations for, 14.147,14.150 submission of, 2.3 in table of contents, 1.38,2.32, fig. 1.5 See also books: parts of parts of books. See books: parts of parts of speech, 5.3,10.42. See also adjec tives; adverbs; articles (definite and indefinite); conjunctions; interjec tions; nouns; prepositions; pronouns; verbs; word usage Party/party, 8.66 passim, 7.55,14.149,16.12 passive voice and dangling participles, 5.115 defined, 5.118 infinitives in, 5.106 linking verbs in, 5.101 long (with by), 5.192 with past participles, 5.119,5.135,5.154 password protection, 2.88 past perfect (pluperfect) tense, 5.126, 5.127, 5.133 past tense, 5.100,5.125,5.126, 5.130, 5.133 patents, 14.258,15.55 pathnames, 6.112. See also file names patronymics, 8.12 pattern matching, 2.79,2.80, p. 984 PCF (process chlorine-free), p. 984 PCN (Preassigned Control Number), 1.34 PDF files annotation of, 1.118,2.71,2.119,2.133 as content proof, 2.140 for continuous publishing model, 1.82, 14.174 defined, p. 984 description, 1.118 for digital artwork, 3.15 edited manuscript sent as, 2.71,2.88 electronic journals as, 1.77 front matter in, 1.122 functional features in, 1.119 hyperlinks in, 1.118,1.121 indexer’s use of, 16.102,16.105 line breaks in, 7.36 for manuscript submission, 2.4,2.8,2.37 navigation, 1.120,1.123 prepress proofs as, 2.107 INDEX PDF files (continued) proofreading in, 1.118,2.2,2.100,2.103, 2.119,2.133 searching, 1.118,2.111,2.133,16.102, 16.105 testing of, 2.139 time and user stamps in, 2.133 pedigree charts, 3.46. See also charts and graphs peer review, 2.5 pencil editing. See manuscript editing: paper-only percent,percentage, 3.82,7.89 (sec. 1), 9.18 percent sign, 3.82, 9.18,10.58 perfect binding, pp. 984-85 performances and rights, 4.14, 4.64 source citations for, 14.201,14.203, 14.261,14.266,14.267,15.57 periodicals copyright registration forms for, 4.48 defined, 14.164 regular departments, columns, and features, 8.177,14.190,14.195 serial rights, 4.64 source citations for: basic citation structure, 14.167; basic information needed, 14.165; initials for authors’ given names, 15.33; journal articles (see journal articles, source citations for); journals vs. magazines, 14.166; legal-style citations, 14.272; maga zines (see under magazines); newspa pers (see under newspapers); reviews, 14.201-4; URLs and DOIs, 14.164 titles: abbreviations of, 10.7,14.170, 15.13,15.46,16.49; articles and features within, 8.177; in building or award names, 8.172; indexing of, 16.48-49; initial the in, 8.170,14.170; italics for, 8.163,8.168; in legal-style citations, 14.272; magazinejournal, review, and such, 8.171; sentence style capitalization, 15.38; treatment in text, 8.168 volume, issue, and page numbers in, 9.27 See also journals; magazines; newspa pers; reviews periods (punctuation), 6.12-15 with abbreviations: academic and pro fessional designations, 10.4,10.21, periods (punctuation) (continued) 10.22; biblical citations, 10.45-47, 14.239; business and commerce, 10.69; at end of sentence, 6.14; eras, 9.34; generally, 10.4; in legal-style ci tations, 14.274; military titles, 10.15; omitted in technical abbreviations, 10.49; organization names, 10.23, 10.26; scholarly, 10.42; slash instead of, 6.109; specific abbreviations, 5.250; time designations, 10.39-41; units of measurement, 10.64 leaders (several spaced periods), 3.64, fig. 3.16 marking manuscript for, 2.93 marking proofs for, 2.132 omitting, 6.14, 6.123,10.4,10.49,14.96 other punctuation with: apostrophes, 6.118; brackets, 6.13; closing quota tion marks, 2.80, 6.9; ellipses, 13.53, 13.56-58,14.97; em dashes, 6.89; ex clamation points, 6.124; parentheses, 6.13, 6.98; question marks, 6.124; 3-em dashes, 6.94 permissible changes to, 11.7,13.7 single space after, 2.9, 6.7, 6.12 slash instead of, 6.109 uses: abbreviations (see with abbrevia tions above); a.m. andp.m., 9.37; cap tions, 3.21,3.23; classical Greek and Latin works, 14.249,14.250; compass points in addresses, 10.34; double or multiple numeration, 1.57,2.28, 2.29,3.11, 3.50,12.25; with#, 14.149; glossaries, 1.61,2.23; imperative sentences, 6.12; indexes, 16.17,16.20, 16.97; initials of personal names, 7.66,10.4,10.12; numerals with sovereigns, 9.41; outlines and lists, 2.21, 6.130, 6.131, 6.132; preceding quotation, 13.17; publishing history on copyright page, 1.25; run-in sub heads, 2.18,2.59; source citations, 14.19,14.21,14.167,14.176,14.254, 15.6; between titles and subtitles in European style, 14.89; transcriptions of discussions or interviews, 13.48; twenty-four-hour system of time, 9.40; words or phrases standing alone, 6.12 See also ellipses; punctuation; URLs INDEX periods of time centuries, 7.8,7.87,7.89,7.89 (sec. 3), 8.71, 9.32, 9.33 cultural, 8.71-74 decades, 9.33 descriptive designations, 8.72 eras, 8.135, 9.34, 9.64,10.38 geological, 8.135 numerical designations, 8.71 prehistoric cultural periods, 8.74 traditional period names, 8.73 See also dates; time designations periphrastic comparatives, 5.85,5.163 periphrastic superlatives, 5.86,5.164 permalinks, 14.9,14.15 permanent compounds, 7.82. See also com pound terms permissible changes to quoted and refer enced materials basic approach, 2.44 capitalization, 8.165,13.7,13.18,13.19, 14.88 cautions about, 2.44,2.61 initial a, an, or the, 8.169 non-English materials, 11.7,11.15-16, 11.19,11.28,13.7 overview, 13.7-8 punctuation, 8.165,8.167,11.7,11.11, 11.19,13.7,13.30-31,14.88 spaces and spacing, 11.7,11.11,11.19 titles of works, 8.165,8.167,11.7,14.88, 14.91 and translations, 11.15,11.16 typography and layout, 13.8 See also editorial interpolations and clarifications permissions, 4.73-83 blanket, 3.32 case studies and field notes, 13.49 checking, 4.79 on copyright page, 1.20,1.30,4.78, fig. 1.4 and Creative Commons licenses, 4.62 criteria for, 13.3 defined, 4.17 and dissertations and theses, 4.60 fees and record keeping, 4.79,4.95, 4.96,4.98,4.101,4.104,4.105 granting, 4.70,4.71 in illustration credit lines, 3.30,3.32, 4.102 permissions {continued) manuscript and editorial concerns, 2.2, 2.3,2.65,4.79 for multiauthor volumes, 2.41 in notes, 14.54 obtaining, 4.75-83,4.95-101; archives’ restrictions, 4.83; for author’s own work, 4.78; automating process of, 4.97; basic principles, 4.75; beyond immediate use, 4.80; communica tions on, 4.95-101; complexity of process, 4.76; independent profes sionals for, 4.76nl; interview and photo releases, 4.77; “orphan works” problem, 4.81,4.82; sample letters, 4.96, fig. 4.3 overview, 4.1 in prefaces, 1.41 role of counsel, 4.74 in separate acknowledgment section, 4.103 special types of material: archival ma terial, 4.83; author’s previously pub lished material, 4.66,4.78; compila tions, 4.105; email addresses, 14.214; illustrations, 2.2,2.3,3.18,3.29, 3.30, 3.32,4.98-101; interviews, 14.211; po etry, 4.95,13.3; previously published materials, 2.2-3,4.66,4.78,4.105, fig. 1.4; tables, 3.77,4.95; unpublished works, 2.3,4.81,4.83,13.3 unnecessary, 4.93 US Supreme Court on, 4.93 See also acknowledgments; copyright; fair-use doctrine; illustration credits and credit lines; source notes persistent URLs (permalinks), 14.9,14.15 person of nouns and pronouns, 5.12,5.30, 5.39, 5.40 of verbs, 5.136,5.138,5.143 personal communications author-date reference system, generally, 15.53 notes and bibliography system, gener ally, 14.214 permissions for, 2.2,2.3 posts on private electronic forums as, 14.210 social media messages as, 14.205, 14.209,14.214,15.52 INDEX personal names. See names, personal personal pronouns. See under pronouns person-first language, 5.260 personification, 5.43,8.37 per vs. slash, 6.109,10.4 PEs (printer’s errors), 2.135, p. 984, p. 985 Philippines, names in, 16.87 philosophical terms and movements, 7.58, 8.79,8.94 phonetics, 6.102,7.64,11.22 photocopying cautions about, 2.6,12.65 classroom and educational use, 4.64,4.66 copyright issues, 4.17,4.35,4.55,4.64, 4.66 for cropping, 3.19 and fair-use doctrine, 1.103 of manuscript, 2.6,2.71,2.91,2.92, 2.116 for musical examples, fig. 3.5 photocopy, use of term, 14.218 for previously published material, 2.43 process and fees for, 1.103,4.97 See also print-on-demand (POD) titles photographs commissioned, 3.33 copyright registration forms for, 4.48 fair use of, 4.90 halftones, 3.3, 3.6, figs. 3.1-2, p. 981 labeled in illustrations, fig. 3.6 original, labeling, 3.16 permissions and credit lines for, 3.29, 3.30, 3.33, 4.98,4.99,4.101 previously published, 2.47 releases for, 4.77 reproduction-quality prints, 2.27 source citations for, 14.235 titles of, 8.198 See also illustrations photomicrographs, 3.27 phrasal (complex) prepositions, 5.174 phrasal adjectives (compound modifiers), 5.79,5.92, 5.93,7.85,7.89 (secs. 2, 3) phrasal adverbs, 5.161 phrasal connectives, misleading, 5.142 phrasal pronouns, 5.27 phrasal verbs, 5.102,5.181 phrases adjectival (compound modifiers), 5.79, 5.92,5.93,7.85,7.89 (secs. 2,3) adverbial, 5.158,5.161,6.31, 6.32,7.89 (sec. 3) phrases (continued) articles as affecting meaning of, 5.76 introductory (see introductory words and phrases) with not, 6.45 with not only ...but also and such, 6.46 parallel structure in, 5.242 parenthetical, 6.48, 6.51, 6.85, 6.95 participial, 5.111,5.115,5.231, 6.30, 6.32 restrictive and nonrestrictive, 5.23, 6.29, 6.30, 6.41, 6.50 standing alone, 6.12 with such as and including, 6.50 with the more, the less, and such, 6.47 used as words, 7.63 verb phrases, 5.104,5.171,5.239,5.245, 7.89 (sec. 2) See also clauses; conjunctions; par enthetical elements in sentences; prepositions physical characteristics, 8.43 physical quantities, 9.13-17 with abbreviations and symbols, 9.16-17 dimensions, 3.27 in general context, 9.13 simple fractions, 9.14 whole numbers plus fractions, 9.15 See also International System of Units; metric system; units of measurement physical science terminology, 8.147-52 abbreviations, 10.63 chemical names and symbols, 7.89 (sec.
1), 8.149,10.63 compound terms, 7.89 (sec. 1) laws and theories, 8.148 mass number, 8.150 metric units, 8.152 radiations, 8.151 resources on, 8.147,10.63 See also scientific and technical termi nology pica, p. 985 Pinyin system, 11.82-85,16.77. See also Chi nese language; Wade-Giles system pitches, musical, 7.71 pixels (resolution), p. 986 place, conjunctions as indicating, 5.201 place-names, 8.44-59 adjectives derived from, 5.69, 8.45 alphabetizing, 16.74,16.90-93 capitalization (see under capitalization) INDEX place-names (continued) compass points and directions, 7.89, 8.46-47,10.34-36 continents, countries, oceans, 8.45 definite article with, 8.59,16.91,16.92 in forewords, 1.40 with Forty Mount, Saint, and such, 10.30, 11.26,16.90,16.93 indexing (see under indexing) institutional names with, 6.39, 6.81 on maps in text, 2.60, 3.20,8.45 mountains, rivers, and such, 8.53-55, 10.30,16.90 in non-English languages, 8.59,11.61, 11.85,11.87,11.99,14.131,16.92 non-English terms in, 8.55,11.26 old vs. present forms, 16.29 plurals of, 7.9, 8.53,8.56,8.57 political divisions, 1.27, 8.51,8.52 popular names and epithets, 8.48 possessives of, 7.17 public places and structures, 8.56-59 punctuation, 5.69, 6.17, 6.39 real vs. metaphorical, 8.50 regions, 8.47,8.54 resources on, 8.44 in source citations: manuscript collec tion depositories, 14.227 with the, 8.45, 8.59,16.91 topographical divisions, 8.53-55 urban areas, 8.49 US states and territories, 10.4,10.8, 10.27,14.130 See also cities and towns; geographical terminology; maps; place of publi cation place of publication, 14.128-32 basic format, 14.127 books published before 1900,14.128 city in, 14.129 country of printing, 1.27,14.130 English names for non-English cities, 14.131 journals, 14.182 in legal-style citations, 14.305 “noplace,” 10.42,14.132 omitted in self-published materials, 14.132,14.137 states, provinces, and countries, 14.130 surmised, 14.132 plagiarism, 1.91,4.75,4.92 plain-text file, p. 985 planets, 8.138-42. See also earth plants, 8.119-30 cultivated varieties (cultivars), 8.130 illustrations, figs. 3.3-4, fig. 3.6 scientific names: authors’ names in, 8.124; divisions higher than genus, 8.126; English derivatives from taxo nomic system, 8.127; genus, species, and subspecies, 8.120-25; hybrids, 8.125; resources on, 8.119 vernacular names, 8.128-30 See also scientific and technical termi nology plates defined, 3.5, p. 985 indexing of, 16.116 lists of, 1.39, fig. 1.7 numbering, 2.28 text references to, 3.9 when to use or omit term, 3.23 See also galleries; illustration credits and credit lines; illustrations Platonic ideas, 8.94 plays divisions of, 8.184 format of, 13.46 quotation of, 13.47,13.57 source citations for: classic English works, 14.253-54; frequently cited works, 13.67; reviews, 14.201,14.203; short forms, 14.254 titles of, 8.183 See also classical Greek and Latin works; dialogue; speech pledges, 8.84 pluperfect (past perfect) tense, 5.126, 5.127, 5.133 plurals, 5.13-16,7.5-15 abbreviations, 7.15, 8.121,10.49,10.52, 10.53,10.65,14.150 alternative forms, 7.6 anomalies of, 5.16 apostrophes in, 6.116,7.13,7.15 centuries, 7.8 collective nouns, 5.7,5.15 compound terms, 7.7 for gender neutrality, 5.255 generally, 5.13 genitives of, 5.20 letters (alphabet), 7.15,7.64,7.65 INDEX plurals (continued) in mathematical expressions, 12.3 Native American group names, 7.10 non-English words, 7.12,11.3 noun coinages, 7.14 numbers, 7.15, 9.53 and phrasal adjectives, 5.92 place-names, 7.9,8.53, 8.56,8.57 political divisions, 8.52 possessive form of, generally, 7.16 (see also possessives) pronouns, 5.52, 5.67 proper names, 5.15,7.9 singular form for, 7.11 singular sense with, 5.14,7.20 standard forms, 7.5 titles of works, 7.12 units of measurement, 9.19,10.65, 10.67,10.68 used as words, 5.14 words in quotation marks, 7.13 words or phrases in italics, 7.12 See also number, of nouns and pronouns plus sign, 7.72,7.78, 9.57,11.130,12.15,12.60 p.m., a.m., 9.37, 9.38,10.41 PNG, p. 985 POD (print-on-demand) titles, 1.28, p. 985 podcasts, 8.189 poetry abbreviations in (e.g., v. for verse), 10.42 book divisions for, 1.51 fair use of, 4.86, 4.87 first lines: indexes of, 16.6,16.144, 16.145; references to, 8.182 interjections in, 5.207 proofreading of, 2.113 quotations from, 13.25-29; centered, 2.20,13.25; and ellipses, 13.57; for mat, 13.25; generic markup of, 2.81; indentation, 2.11,2.20,13.25-27; manuscript preparation guidelines, 2.20; multiple stanzas in, 13.34; in notes, 14.38; and paragraph format, 2.12; permissions for, 4.95,13.3; quo tation marks within poems, 13.28; run in to text, 6.111, 6.113,13.29, 13.34; runover lines, 2.20,13.25, 13.27; text citations for, 2.20,13.71-72 rhyme schemes, 7.69 source citations: classical Greek and Latin works, 14.243; classic English poetry (continued) works, 14.253-54; footnotes keyed to line numbers, 14.53, fig. 14.5; short forms, 14.254; sound recordings of, 14.264; text citations for, 2.20, 13.71-72 terms (e.g., canto, stanza), 8.184 titles of poems, 8.181,14.86 See also classical Greek and Latin works points, p. 985. See also ellipses; periods (punctuation) Polish language, 11.70 political terminology geographical entities in, 8.48 political and economic organizations and movements, 8.66, 8.70 political divisions, 1.27, 8.51, 8.52 unofficial groups and movements, 8.67 See also governmental entities popes, 8.26, 9.41,16.36. See also sovereigns and other rulers pop-up boxes, 15.17 portable document format (PDF) files. See PDF files portrait, p. 985 Portuguese language, 8.8,11.70,16.83 position bars, in e-books, 1.123 positive (absolute) adjectives, 5.84 positive (absolute) adverbs, 5.162 positive questions, 5.235 possessives, 7.16-29 of abbreviations, 6.43, 6.44,7.17,8.189 adjectives after, 5.80 as antecedents of pronouns, 5.29 apostrophes for, generally, 5.50, 6.116 with appositives, 6.28 vs. attributive form, 7.27 basic rule and exceptions, 7.16,7.20-22 of compound terms, 7.24 vs. contractions, 5.50 definition and uses of, 5.20 for... sake expressions, 7.21 and fused participles, 5.114 genitive with, 5.20,7.25 with gerunds, 7.28 in glossed American Sign Language, 11.134 has, have to denote, 5.153 independent, 5.49 for initialisms, 7.17, 8.189 of italicized or quoted terms, 7.29 INDEX possessives (continued) joint (group), 5.22 joint vs. separate possession, 7.23 in names of diseases, 8.144 of nouns plural in form, singular in sense, 7.20 of numbers, 7.17 o/with, 7.20,7.25,7.26,7.29 of plural nouns, basic rule, 7.16 and prepositions, 5.183,7.26 of pronouns: adjectives after, 5.80; generally, 5.35; indefinite pronouns, 5.67; and joint (group) possessives, 5.22; personal, 5.49-50,7.28; relative pronouns, 5.63,5.64 of proper names, 7.17,7.19,7.20,7.27,7.28 text citations following, 15.25 of words ending in unpronounced 5,7.18 See also genitive case post, as prefix, 7.89 (sec. 4) Postal Atlas system, 11.82,11.85 postal codes Canadian provinces and territories, 10.28 in place of publication, 14.130 punctuation with, 6.39,10.27,10.29 vs. standard abbreviations, 10.33 US states and territories, 10.4,10.8, 10.27 zip code with, 10.29 See also addresses, mailing posters, presented at meetings, 14.217 postpositive adjectives, 5.79 PostScript (PS), pp. 979, 985 pound sign (number sign), 2.98,2.125, 3.79, 11.131 pound sterling symbol, 9.20, 9.22 powers of ten (scientific notation), 9.9, 9.11, 9.12 pre, as prefix, 7.89 (sec. 4) Preassigned Control Number (PCN), 1.34 predicate adjectives, 5.79,5.82 predicates compound, 6.23 defined, 5.18 as linking verbs, 5.82, 5.101 nominative, 5.18,5.141 prefaces acknowledgments in, 1.42 author’s and editor’s, order for, 1.41 defined, 1.41 prefaces (continued) editorial method explained in, 1.45 vs. forewords, 5.250 format of term, 8.179,14.110 illustration credits in, 3.30 indexing of, 16.109 material appropriate for, 1.43 and multiauthor volumes, 2.41,2.44 new, noted on title page, 1.26 pagination, 1.7 placement in book, 1.4 quotation issues explained in, 13.7, 13.48,13.58,13.59,13.61 running heads for, 1.11 second or subsequent editions or re printings, 1.41 source citations for, 14.110 submission of, 2.3 prefixes combining forms, 7.82,7.89 (secs. 2, 4) defined, 7.82 division of words with, 7.40 Greek language, 11.121 Hebrew language, 11.93 hyphenation, 6.80,7.40,7.81,7.87-89, 7.89 (secs. 2,4), 8.161 mid- as, 7.87 negating, 5.230 in non-English languages, 11.58,11.93 numerical expressions with, 9.8,9.10, 9.11,10.49,10.55 Russian language, 11.106,11.107 in SI units, 9.10, 9.11,10.49,10.55,10.56 standing alone, 7.89 See also compound terms prehistoric cultural periods, 8.74 preliminary pages (prelims). See front matter prepositional phrases, 5.176-83 active voice to eliminate, 5.192 as adverbs, 5.176, 5.178 adverbs replacing, 5.190 clashing (repeated), 5.181 defined, 5.176 elliptical, 5.182 function, 5.177 with intransitive verbs, 5.98 limiting, 5.187-92 objective (accusative) case in, 5.44, 5.172,5.183 parallel structure in, 5.243 INDEX prepositional phrases (continued) and personal pronouns, 5.44, 5.183 as phrasal connectives, 5.142 placement, 5.178,5.179 pronoun affected by, 5.183 prepositions, 5.172-95 vs. adverbs, 5.184 adverbs modifying, 5.156 appropriate use of, 5.195 avoiding overuse of, 5.187-92 complex (phrasal), 5.174 vs. conjunctions, 5.184 with dates in titles of works, 14.93 defined, 5.172 ending sentences with, 5.180 functional variation, 5.184 in headline-style capitalization, 8.159, 14.284,14.290 idiomatic uses, 5.193-95 in indexing, 16.53,16.68 in Latin expressions, 8.159 like, 5.185,5.250 and nominalizations, 5.189 in non-English languages, 11.58,11.121 with nouns used as adjectives, 5.24 of-genitive, 5.21, 5.191 only, 5.186 participial, 5.175 in phrasal verbs, 5.102 and possessives, 5.183,7.26 simple and compound, examples, 5.173 See also prepositional phrases prepress (preliminary presswork), 2.2, 2.107, p. 985’ prepress services, 3.15 preprints, 1.113,14.173, p. 985. See also elec tronic publications: published before print version present perfect tense, 5.132 present tense, 5.125,5.129 President, president, 8.19, 8.21, 8.22,8.28 presidential documents orders and proclamations, 14.289 programs, 8.75 speeches, 8.76,14.289 treaties, 8.66,8.80,8.81,14.275,14.290 Press, in publishers’ names, 14.134 press releases (news releases), 14.200 press sheets (F&Gs), 2.107, pp. 980, 985 presswork, p. 985 preventive grammar, 5.57 previously published materials of authors, on copyright page, 1.18, figs. 1.3-4 compilations of: and copyright, 2.46, 4.5; illustrations in, 2.47; manuscript preparation guidelines, 2.43-47; notes in, 2.44, 2.45; permissible changes to, 2.44,2.61; permissions for, 4.105; source notes for, 1.50, 2.46,14.54; as work made for hire, 4.10 copyright of, 2.46,4.5 editing of, 2.61,2.91 permissions for, 2.2,2.3,4.66,4.78, 4.105, fig. 1.4 source citations for, 14.54,14.181 See also derivative works; quotations price, 1.75,1.98,2.108,2.109,2.134, 4.63.
See also bar codes; currency prime and double prime symbols, 9.16, 9.17, 10.36,10.61,10.66,11.92 printed sheets. See press sheets printer’s errors (PEs), 2.135, pp. 984,986 print-on-demand (POD) titles, 1.28, p. 986 print technologies digital printing, p. 978 folded-and-gathered sheets (F&Gs), 2.107, pp. 980,985 prepress (preliminary presswork), 2.2, 2.107, p. 985 sheet-fed press, p. 987 web-fed press, p. 990 See also binding privacy, right of, 4.73 prizes and awards, 1.99,8.31, 8.83, 8.115, 8.172 pro, as prefix, 7.89 (sec. 4) probability mathematical copy and expressions, 9.19,10.50,12.57,12.58 in tables, 3.80, fig. 3.22 proceedings of conferences and symposia, 1.18,1.39,4.8,4.59,14.217. See also conferences; multiauthor volumes; special issues, journal process color printing, p. 986 product codes (bar codes), 1.69,1.74,1.75, 1.84 product sign, 12.40 INDEX professional titles. See titles and offices of people programs, names of, 8.66, 8.75. See also conferences; governmental entities; pamphlets, brochures, and reports programs, software. See software progressive (imperfect, continuous) tenses, 5.119, 5.128,5.135 prologues. See introductions promotion and abstracts, 1.76,2.25 artwork for, 4.77,4.100,4.101 and covers and jackets, 1.70 in journals, 1.77,1.84,1.86,2.109 pronouns, 5.27-67 and adjectives, 5.29, 5.49, 5.78,5.80, 5.81 antecedents: absent, 5.30,5.61; defined, 5.28; demonstrative pronoun, 5.52; different genders, 5.57; errors, 5.28; for gender neutrality, 5.255; mul tiple, 5.34; possessives as, 5.29; and pronoun number, 5.32-33,5.42,5.57, 5.62, 5.255; relative pronouns, 5.57, 5.59-62 and appositives, 5.12,5.36,5.51 article as substitute for, 5.78 and assimilation of quoted material into text, 13.12 attribute, 5.45 and bias-free language issues, 5.47-48, 5.252,5.255-56 classes: adjective, 5.49; demonstrative (deictic), 5.52; indefinite, 5.47,5.67, 5.139, 5.256; interrogative, 5.54-55, 5.58; list of, 5.38; reciprocal (each other, one another), 5.53; reflexive and intensive, 5.41,5.51,5.53; relative (see relative pronouns) in contractions, 5.104,5.105 definition and uses of, 5.27-30 distributive, 5.67 em dash between introductory noun and, 6.86 as interjections, 5.208 it, 5.47, 5.115, 5.239,5.240,5.246,8.77, 8.118 for named natural events, 8.77 negating, 5.230, 5.233 personal, 5.39-48; after linking verb, pronouns (continued) 5.45; after than or as... as, 5.46; agreement of noun with, 5.42; case of, 5.44-46; changes in form (nominative, objective, genitive), 5.9, 5.35,5.41 (see also case of nouns and pronouns); compound (-se//forms), 5.51, 5.250; form of, 5.39, 5.41; and gender, 5.43, 5.47,5.48; identification of, 5.40; misuse of, 5.37; possessive, 5.49-50,7.28; and prepositional phrases, 5.44,5.183; singular they, 5.48,5.48n5,5.252,5.256; special uses of, 5.47 (see also he, him, his; she, her, hers) personal, in non-English languages: Danish, 11.70; German, 11.39; glossed American Sign Language, 11.134; Italian, 11.46; Norwegian, 11.70; Polish, 11.70; Swedish, 11.70 person of, 5.30,5.39, 5.40 phrasal, 5.27 possessive: adjectives after, 5.80; gen erally, 5.35; indefinite, 5.67; and joint (group) possessives, 5.22; personal, 5.49-50,7.28; relative pronouns, 5.63,5.64 prepositions with, 5.172 properties, 5.31-37; case (see case of nouns and pronouns); gender, 5.43, 5.47,5.57; number (see number, of nouns and pronouns); overview, 5.31 for religious figures, 8.95 -se//forms of, 5.48, 5.51,5.250 singular they, 5.48, 5.48n5, 5.252, 5.256 for vehicles and vessels, 8.118 proofreader’s marks, 2.119-33 capitalization changes, 2.131 circling comments and instructions, 2.121,2.124,2.127,2.128,2.129, 2.131 conventions in, 2.119 deletions, 2.124,2.130 em spaces, 2.126 examples of, fig. 2.5, fig. 2.7 for extensive changes, 2.122 font changes, 2.131 items to be spelled out, 2.129 legibility, 2.123 list of, fig. 2.6 INDEX proofreader’s marks (continued) paragraph and indentation changes, 2.126 placement of, 2.120 position or alignment (justification) changes, 2.98,2.127 punctuation changes, 2.124,2.132 spacing between letters or words, 2.125 stet, 2.130 transposition of items, 2.128 See also manuscript editing: paper-only proofreading, 2.100-140 assigning responsibility for errors, 2.135-36 defined, 2.100 double-checking in, 2.134 editor’s style sheet for, 2.111,2.112 example, fig. 2.7 extensive changes after, 2.122 global changes, 2.111,2.113,2.122 independent professionals for, 2.101 indexer’s assistance in, 16.125,16.131 of indexes, 2.2,2.105 and metadata, 1.92 of multiauthor volumes, 2.42 in PDF files, 1.118, 2.2,2.100,2.103, 2.119,2.133 process and checks: comparing copy with proofs, 2.110; fonts and type faces, 2.113,2.131,10.8; illustrations and tables, 2.107,2.115,2.127,2.137; master set and routing of corrected version, 2.103; overall appearance, 2.116; page numbers, 2.114,2.134; queries and instructions in, 2.116, 2.118,2.121; quotations, 2.113; running heads, 2.76,2.114; for sense, 2.118; spelling, 2.111; stages of proof, 2.103-9; word division, 2.112 responsibility for, 2.42,2.101 revised proofs, 2.103,2.105 schedule for, 2.2,2.102,16.4 specific types of material: compilations, 2.43; copy in non-Latin alphabets, 11.73; electronic publications, 2.100, 2.103, 2.137-40 standard for, 2.119 See also proofreader’s marks; proofs proofs defined, pp. 984, 986 digital, p. 978 proofs (continued) indexing from, 16.4,16.102,16.105, 16.108,16.117,16.133 marking, for indexing, 16.117-21, fig.
16.1 in production checklist, 2 JI stages, 2.103-9; book cover and jacket, 2.108; F&Gs, 2.107, pp. 980, 985; first pages and galley proofs, 2.104, p. 981; index proofs, 2.105; journal cover, 2.109; master set maintained, 2.103; prepress and press, 2.2,2.107, p. 977; revised, 2.103,2.105, p. 984 See also alterations; proofreader’s marks; proofreading proof symbol, 12.56 proper adjectives defined, 5.69 in non-English languages, 11.26,11.39, 11.70 from place-names, 5.69, 8.45 religious terms, 8.96, 8.97 See also words derived from proper names proper names. See names, proper proper nouns, 5.6. See also names, personal; names, proper; place-names prophets, 8.93 prose extracts. See block quotations protases (conditional clauses), 5.228 proto, as prefix, 7.89 (sec. 4) protocols (network), 14.17, p. 986. See also FTP; HTTP proverbs (maxims), 7.60,7.62,13.5,13.38.
See also figures of speech provinces, Canadian, 10.28,14.130,14.193 pseudo, as prefix, 7.89 (sec. 4) pseudonyms and pseudonymous works blog comments, 14.208 copyright, 4.25 of human research subjects, 13.49 indexing of, 16.34 source citations for, 14.80-81,15.35 public-access policies, 4.68 publication date in book review and book notes sections, 1.98 journal articles, 1.92,1.95,1.104,14.171 journals: in copyright line, 1.103; cur rent, on home page, 1.86; identified by, 1.79; location in, 1.84,1.87; proof-INDEX publication date (continued) reading of, 2.109, 2.134; and volume number, 1.80 location in works: copyright notice, 1.22, 4.41; copyright page or title page, 1.19,1.20,1.22,1.25,14.142; journals, 1.84,1.87 as metadata, 1.75,1.92 “no date” (n.d.), 10.42,14.145,14.263, 15.18,15.44,15.50,15.54 source citations, author-date reference system: basic structure and punc tuation, 15.5,15.6,15.9,15.25; blog posts, 15.51; editions with more than one date, 15.40; letters in published collections, 15.43; manuscript col lections, 15.54; multimedia content, 15.57; multivolume works published over several years, 15.41; newspapers and magazines, 15.49; “no date,” 15.44,15.50; vs. notes and bibliogra phy system, 15.2,15.3; placement of, 15.14; references arranged by, 15.17- 20; social media content, 15.52 source citations, notes and bibliography system, 14.142-46; artworks, 14.235; basic format, 14.127,14.142; blog posts, 14.208; books published before 1900,14.128; copublications, 14.140; journal articles, 14.171; magazines, 14.165,14.188; multimedia content, 14.261,14.263; multivolume works, 14.117-19,14.144; new impressions and renewal of copyright, 14.143; newspapers, 14.165,14.191,14.197; “no date,” 14.145; periodicals, gener ally, 14.165; reference works, 14.232; social media content, 14.209; specific volume of multivolume work, 14.118; websites and web pages, 14.207 when available in previous year, 1.22 See also forthcoming publication details abbreviations in, 1.21 absence of, 10.42,14.132,14.145 basic formats, 14.127,15.6,15.9,15.14, 15.25 in book review and book notes sections, 1.98 for classical Greek and Latin works, 14.246,14.248 publication details (continued) for dissertations and theses, 14.215 form of work (e.g., microfilm), 14.115 in journals, 1.84,1.87 See also authors’ names; copyright; editions; place of publication; publication date; publishers’ names; source citations public display, and copyright, 4.13,4.14, 4.34 public-domain works and copyright, 1.103 derivative works based on, 4.22 government-produced, 4.21,4.44,4.68, 4.75 illustrations, 3.35 non-US publications as, 4.29 quotations from, 4.75 source citations for, 14.54 when works become, 4.19,4.20, 4.27 See also copyright: duration public laws. See legislation public places and structures buildings and monuments, 8.57,8.68, 8.101, 8.172, 9.48, 9.52,11.26,11.61 non-English names for, 8.59 roads and thoroughfares, 8.56, 9.50, 9.51,10.33,11.26 rooms, offices, and such, 8.58 publishers addresses, 1.19,1.20,1.21,1.84,1.95 agreements with (see publishing agree ments) archival practices of, 1.114 as authors, 14.84 copublications, 14.140 distribution arrangements, 14.141 and fair-use “rules of thumb,” 4.85 generic markup style, 2.81 grants for, 1.31 house style: author communications, 2.68, 2.70; editing for, 2.49; gen erally, 1.116; for indexes, 16.103, 16.106,16.118,16.123,16.129; proofreading for, 2.113; for source citations, 1.108-9,2.64,14.2-4; for tables, 3.86 logos, 1.19,1.69,1.70 main editorial offices, 14.129 manuscript preparation guidelines, 2.7 notes on text, 1.45 INDEX publishers (continued) parent companies, imprints, and such, 14.138- 41 permissions from out-of-business, 4.82 websites of, 1.21,1.59,1.68,1.118, 4.94, 4.97 See also copyright; publication details; publishers’ names; publishing agreements publishers’ names Books and Press in, 14.134 location in works: book review and book notes sections, 1.98; copyright notice, 4.42; covers and jackets, 1.69,1.70; edition details, 1.25; journals, 1.84; title page, 1.19,14.129,14.133 as metadata, 1.75 omissible parts of, 14.134 resources on, 14.133,14.138 self-published or privately published works, 14.137 in source citations, 14.133-41; abbrevi ations in, 14.133-34,14.136; and or ampersand in, 14.135; basic format, 14.127; copublications, 14.140; distributed books, 14.141; imprints, parent companies, and such, 1.19, 14.138- 41; non-English names, 14.135,14.136; preferred form, 14.133; when to omit, 14.128 See also publishers publishing agreements, 4.51-63 author’s alterations limited by, 2.136 author’s warranties, 4.52,4.70,4.72, 4.73 basic provisions, 4.52 and copyright owner’s rights, 4.13 examples of, figs. 4.1-2 illustrations noted in, 3.15 journal articles, 4.55-57, fig. 4.1 multiauthor volumes, 2.42,4.58, 4.59, fig-4.2 new books, 4.52-54 “no derivatives” clause, 4.62 and non-English rights, 4.64 open-access publishing models in, 4.52, 4.66 publisher’s responsibilities: copyright tasks, 4.32; rights and permissions requests, 4.70,4.71,4.76nl; subsidi ary rights, 4.17-18 publishing agreements (continued) rights to reproduction and distribution, 4.51 for self-published materials, 4.63 subsidiary rights (see subsidiary rights) symposia proceedings, 4.59 See also copyright; intellectual property rights; licenses for copyrighted works; permissions publishing history, 1.25,2.46. See also editions publishing schedule basic schedule, 2.2, fig. 2.1, fig. 2.2 indexing, 2.2,2.67,2.70,16.4,16.101 journals, 2.102, fig. 2.2 manuscript editing, 2.2,2.52 proofreading, 2.2,2.102,16.4 volume editor’s responsibility for, 2.41 PubMed, 1.111,4.68 punctuation and alphabetizing, 16.59,16.60,16.61 author queries about, 2.69 basic function of, 6.1 in captions, 3.21,3.23, 3.24, 6.14 and electronic file cleanup, 2.80 in electronic publications, 6.2, 6.4, 6.8, 6.84 errors in (see errors in grammar and punctuation) with interjections, 5.206, 6.35 and italics, 6.2,14.89 marking manuscript for, 2.93,2.95,2.96 marking proofs for, 2.124,2.132 in mathematical copy, 12.7,12.18,12.38, 12.54 with mathematical expressions, 12.15, 12.18-20,12.28,12.35,12.42 multiple marks at same place, 6.122-26, table 6.1 non-English materials (see under non English materials) permissible changes to, 8.165, 8.167,11.7, 11.11,11.19,13.7,13.30-31,14.88 spaces following, 2.9, 6.7, 6.12, 6.62, 14.116 tables (see under tables) with trademark symbols, 8.153 typographic considerations, 6.2-6, 6.14 of URLs, DOIs, and such in text, 2.13, 6.8, 6.104, 6.112,7.46,14.17-18 forces and no, 5.250 See also abbreviations, general; abbrevi-INDEX punctuation (continued) ations, specific; braces in mathemat ical expressions; brackets; colons; commas; dashes; ellipses; exclama tion points; guillemets; hyphens and hyphenation; parentheses; periods (punctuation); question marks; quotation marks; quotations; semi colons; slashes; suspension points; typographic considerations quaint characters, p. 983 quantities. See International System of Units; metric system; numbers; physical quantities; scientific and technical terminology; units of mea surement quasi, compound terms with, 7.89 (sec. 3) queens. See sovereigns and other rulers queries. See author queries question marks, 6.68-70 vs. exclamation mark, 6.72 inverted, in Spanish works, 11.19,11.62 marking proofs for, 2.132 in non-English materials, 11.19,11.62, 11.102,11.116 other punctuation with: commas, 6.42, 6.125; ellipses, 11.102,13.54; em dashes, 6.89; exclamation points, 6.126; generally, 6.70; parentheses, 6.70, 6.98; periods, 6.124; quotation marks, 6.10, 6.70,13.69 in relation to surrounding text, 6.70 in source citations: publication details, surmised, 14.132,15.44; with text citations, 13.69 in titles of works, 6.124-26,14.92,14.96, 16.54 uses, other: declarative or imperative sentences, 6.68; editorial doubt, 6.68,14.132,15.34,15.44; indirect questions, 6.69; interpolation for missing or illegible words, 13.59; within sentences, 6.69-70 when to omit, 6.69 See also punctuation; questions questions alternative, 5.212 in American Sign Language, 11.135 as cleft sentences, 5.246 comma with, 6.42 questions (continued) conjunctions for, 5.201,5.202 declarative structure in, 5.213 defined, 5.212 exceptional types of, 5.213 exclamations as, 6.72 exclamatory, 5.213 indirect, 5.201, 6.42, 6.69 negation in, 5.104, 5.235 and pronouns, 5.54-55, 5.58 question marks with, 6.68 rhetorical, 5.213 tag, 5.213 who or whom with, 5.66 wh- questions, 5.212 within sentences, 6.42, 6.63, 6.65, 6.69-70 word order in, 5.224 yes-no, 5.212, 5.213 See also question marks quite, 5.90 quotation marks alternatives to, 6.91,11.11,11.19,13.42 in British style, 6.9,13.30 closing, 6.9-11,13.68,13.70, table 6.1 directional or “smart,” 2.85, 6.115,13.7 double vs. single, 7.58,13.30-31,13.33, 13.63 (see also single below) editing of, 2.61 and electronic file cleanup, 2.80 font for, 6.6 inverted, split-level, 11.41 marking proofs for, 2.132 non-English materials: Chinese, 11.89; French, 11.30; German, 11.41; for guillemets, 11.7,11.11,11.19,13.7; Italian, 11.47; Japanese, 11.89; quotations, 11.3; Spanish, 11.63; translations, 11.5,11.12; transliterated materials, 2.80 omission of, 13.36-38,13.45 other punctuation with, 6.9-11, 6.114; colons, 6.10, 6.65,13.16; com mas, 2.80, 6.9, 6.40; in computer terminology, 7.79; ellipses, 13.41; em dashes, 6.87, 6.90; exclamation points, 6.10, 6.74,13.30,13.69; in non-English languages, 11.29,11.41, 11.47,11.63; parentheses, 6.98; peri ods, 2.80, 6.9; question marks, 6.10, 6.70,13.69; quotation marks (single INDEX quotation marks (continued) and double), 6.11, 6.120, 8.177,13.30- 31; semicolons, 6.10; summary, table 6.1 and paragraphing, 13.32-35 permissible changes to, 13.7 plurals of words in, 7.13 possessives of terms in, 7.29 vs. primes and double primes, 9.16, 10.36 scare quotes, 7.57,11.47,14.94 single: directional or “smart” char acters for, 2.80,2.85, 6.115, 6.117; vs. double, 7.58,13.30-31,13.33; double next to, 6.11, 6.120, 8.177; for horticultural cultivars, 8.130; in non- English-language materials, 11.30, 11.63,11.77; in terms quoted within titles, 14.94 speech, dialogue, and conversation, 13.32,13.39-45; alternatives to quo tation marks, 6.91,11.11,11.19,13.42; direct discourse, 13.39,13.44; indirect discourse, 13.45; single-word speech, 13.40; unspoken discourse, 13.43 uses, other: common expressions and figures of speech, 7.60; computer ter minology, 7.79; in editorial interpola tions, 13.63; episodes and segments in television, radio, and podcasts, 8.189; epithets and nicknames, 8.34; first lines of poems, 8.182; glossed American Sign Language, 11.128, 11.133; lectures,-8.87; non-English materials (see above); notes keyed to text by line or page numbers, 14.53; with so-called, 7.59; songs and shorter musical compositions, 8.194; and text citations for quotations, 13.68- 69; titles of works (see under titles of works); unpublished materials, 14.292; within poems, 13.28; words and phrases used as words, 7.63 See also guillemets; punctuation; quo tations quotations basic principles: accuracy, 13.6; attri bution and fair use, 4.64,4.75, 4.84, 4.92; commonly known facts and ex pressions, 13.5; credit and permission for, 13.3; danger of skewed meaning, quotations (continued) 13.51; in modern scholarship, 13.2; paraphrase vs. quotation, 4.89,11.17, 13.4 beginning in text, 13.23 brackets in: adjustments in capitaliza tion, 13.18,13.21,13.53; adjustments in tenses and pronouns, 13.12; “brackets in the original,” 6.99; editorial interpolations and clarifica tions, 6.99, 6.103 (see also editorial interpolations and clarifications); glosses or translations, 6.96, 6.99; intervening words of text, 13.23; missing or illegible words, 13.59; other punctuation with, 6.103; sic, 13.61; text citations, 13.7,15.28,15.40 capitalization, 8.162,13.18-21,13.53 in context of original, 4.84,4.86 cross-checking of, 2.32,2.33,13.6 editing, 2.61,2.69,11.16 editorial interpolations and clarifica tions (see editorial interpolations and clarifications) errors in original source, 13.7,13.61 generic markup of, 2.74,2.80,2.81 indentation: block quotations, 2.11, 2.19; drama excerpts, 13.8,13.47; interviews and discussions, 13.48; paragraphs within, 2.19,13.22; poetry extracts, 2.11,2.20,13.25-27 interruptions in, 13.14 length of, 13.10,13.25 manuscript preparation guidelines for, 2.11,2.19,2.20,13.9 non-English materials in English context (see under non-English materials) non-English terms in, 6.96 note references or notes within, 13.7, 13.58 original spelling in, 7.3 overview, 13.1 paragraphs within, 2.19,13.22,13.32-33, 13.35,13.39,13.56 vs. paraphrasing, 4.89,11.17,13.4 permissible changes (see permissible changes to quoted and referenced materials) poetry (see under poetry) proofreading of, 2.113 punctuation: brackets (see brackets INDEX quotations (continued) in above); colons, 6.10, 6.63, 6.65, 13.16; commas, 6.40,13.14; ellipses, 13.50-58; non-English materials in English context, 11.3,11.19; periods, 13.17; quotation marks (see quotation marks); slashes, 13.29 quotations within, 13.30-31,13.33 relation to text: assimilation, 3.15, 13.11-12; initial capital or lowercase letter, 13.18-21; introductory phrases and words before, 13.14-16; poetry, 13.25-29; run-in vs. block, 13.9-10; text following, 2.12,2.21,13.24 from secondary sources, 14.260,14.291, 15.56 sic with, 6.73,7.55,13.61 sources for (see source citations; text citations for below) and syntax, 13.11,13.18-20 text citations for, 13.64-72; block quo tations, 13.70-72,15.26; in brackets, 13.7,15.28,15.40; “brackets in the original” in, 6.99; epigraphs, 1.37, 1.49,13.36,14.52; errors in original source, 13.7; frequently cited works, 13.67; full, 13.65; ibid, in, 13.66; non English materials (see non-English materials above); parentheses with, 13.64; placement of, 15.26; poetry, 2.20,13.71-72; run-in quotations, 13.68-69,15.26; shortened, 13.66-67, 13.70 text citations within original, 13.7 in titles of works, 8.162,14.94 within notes, 11.13,14.38 word forms for, 5.250 See also block quotations; dialogue; di rect address; editorial interpolations and clarifications; ellipses; epigraphs; fair-use doctrine; guillemets; max ims; poetry; punctuation; quotation marks; run-in quotations; source citations; speech quoted in, 14.260,15.56 Qur’an (Koran), 8.103,11.77,14.241 q.v. (quod vide), 10.42 race, 8.38, 8.39. See also groups of people radiations, 8.151 radical sign, 12.42-44 radio and radio programs, 8.189 ragged right (flush left) justification, 2.10, 3.73,7.47,16.136, fig. 3.19, p. 986. See also justification ranges (numbers). See inclusive (continu ing) numbers rarely, 5.230 rather, 5.201 ratios, 6.62, 9.58 re, as prefix, 7.89 (sec. 4) readers’ comments on blog posts, 14.208,15.51 in newspapers, 14.196 on online multimedia content, 14.267 on social media posts, 14.209,15.52 reading lists, 1.78 reciprocal pronouns (each other, one an other), 5.53 record group numbers, 14.292 recoverability, principle of, 5.229 recto pages abbreviation, 14.154-55 always odd-numbered, 1.5 beginning of gallery, 3.6 defined, p. 986 first journal page, 1.81 first page of index, 1.65 in folio editions, 14.155 parts of book starting on, 1.4,1.49,1.51, 1.53,1.54,1.65 proofreading for correct use of, 2.114, 2.115 running heads for, 1.11-15 in signed signatures, 14.154 specified on final manuscript for produc tion, 2.75, 2.76, 2.77 See also page numbers; pagination; verso pages redlining, p. 986. See also tracking changes reference lists abbreviations in, generally, 15.15 authors’ names (see authors’ names in source citations) vs. bibliographies, 14.71,15.3 cross-checking of, 2.32,2.63,15.21 cross-references to other titles in, 15.42 editing of, 2.63,2.64 and electronic file cleanup, 2.80 format: examples and variations, 15.9, 15.22, fig. 15.1; hanging (flush- and-hang) indentation, 1.63,2.24; INDEX reference lists (continued) manuscript preparation guidelines, 2.24,14.67,15.17; overview, 15.5-6; page numbers and other locators, 15.8; type size, 1.63 headings for, 15.5,15.10 hyperlinks to and from (see hyperlinks for source citations) importance of, 15.10 manuscript preparation guidelines for authors, 15.17 metadata for, 1.92 multiauthor volumes, 15.10 order of entries: alphabetical, 2.63,15.11, fig. 15.1; chronological, 2.63,14.71, 15.18,15.19; same-author, same- year works, 15.20; single vs. several authors, 15.16; works with multiple publication dates, 15.40 placement in book, 1.4,1.63,15.10 publication date, placement of, 15.14 publication details: editions with mul tiple, 15.40; forthcoming works, 15.18,15.45; letters in published col lections, 15.43; multivolume works published over several years, 15.41; “no date,” 15.18,15.44 scholarly symbols in, 10.43 secondary sources quoted, 15.56 submission of, 2.3 3-em dash in, 15.17-20; caveats, 15.17; chronological order for repeated names, 1.124,15.18,15.19; with ed ited, translated, or compiled works, 15.19; with same author, same year, 15.20 titles as entries, 15.39 See also author-date reference system; bibliographies; source citations; text citations; titles of works in source citations reference works HTML format for, 1.118 hyperlinks in, 1.118,1.120 indexes for, 16.26 for indexing, 16.32 for manuscript editing, 2.54 online versions, 1.118,1.120,1.121,14.233 organizing principle of, 1.123 source citations for, 14.103,14.232-34 titles of, 8.191 reference works (continued) See also catalogs; dictionaries; encyclo pedias; “resources on” under specific topics reflexive and intensive pronouns, 5.41, 5.51,5.53 reflowable formats illustration placement in, 3.8 and indexes, 16.13 line breaks in, 6.82, 6.90, 6.113,7.36, 7.47,14.18 location information, 1.6 navigation in, 1.123,2.35,14.22,14.160, 14.174,15.23 and notes, 14.49 and text breaks, 1.58 regions, national and world, 8.47, 8.54. See also geographical terminology; place- names; political terminology regression testing, 2.138 regular expressions, p. 984. See also pattern matching relationship terms, 7.89 (sec. 3), 8.36 relative adjectives, 5.56 relative adverbial conjunctions, 5.202 relative clauses, 3.54, 5.56, 5.60, 5.225, 5.226, 5.227, 6.27 relative pronouns, 5.56-66 antecedents absent with, 5.30,5.61 antecedents of, 5.57,5.59-62 in cleft sentences, 5.246,5.247 compound, 5.65 defined, 5.56 -ever suffix, 5.65, 5.66 gender, number, and case of, 5.57,5.66, 5.140 for gender neutrality, 5.255 and grammatical ellipses, 5.226, 5.247 one in antecedent, 5.62 positional nuances of, 5.58 possessive forms of, 5.63, 5.64 in remote clauses, 5.60 as subjects, 5.140 See also relative clauses; that; what; which; who, whom, whose religions and religious concerns buildings, 8.101, 9.48 concepts, 8.94,8.99, 8.108, 8.109 events: holidays, 8.89; services and rites, 8.110; theologically important (e.g., the Exodus), 8.108 INDEX religions and religious concerns (continued) groups: Church, church, 8.98, 8.101; councils, synods, and such, 8.102; denominations, sects, orders, and movements, 8.97; jurisdictions and divisions of, 8.100; major religions, 8.96 names and titles: abbreviations, 10.18, 10.22; alternatives to God, 8.92; capitalization, 8.19, 8.26, 8.33, 8.36; deities, 8.91,8.92; honorifics, 8.26, 8.33,10.18,16.39; indexing of, 16.37,16.39; kinship terms, 8.36; in non-English languages, 11.26,11.88; numbered places of worship, 9.48; numerals with, 9.41, 9.48; popes and patriarchs, 8.26, 9.41,16.36,16.37; pronouns for, 8.95; religious figures, 8.93, 8.95,10.20,14.83,16.42 religious objects, 8.111 See also Bible; biblical citations; religious works religious works Qur’an (Koran), 8.103,11.77,14.241 resources on, 14.238,14.241 source citations for, 6.62, 9.26,10.48, 14.238-41 treatment in text: concepts, 8.94, 8.99, 8.108, 8.109; prayers, creeds, and such, 8.107; scriptures, 8.103, 8.104 Upanishads, 8.103,14.241 See also Bible; religions and religious concerns repetition, author queries about, 2.69 reporters (for court cases and decisions), 14.276,14.277,14.278,14.279, 14.294,14.298 reports. See pamphlets, brochures, and reports reprints abbreviation of reprint or reprinted, 10.42 copyright dates of, 1.23, fig. 1.2 defined, p. 986 e-books as, 14.159 vs. editions, 1.23,1.26 editorial interpolations and clarifica tions in, 6.99 introduction for, 1.47 prefaces for, 1.41 rights to, 4.55, 4.64, 4.78, 4.101, fig. 4.1 reprints (continued) source citations for, 14.54,14.114, 14.159,15.40 unnumbered source notes, 1.50,14.54 See also impressions reproduction and copyright, 4.13,4.34, 4.51, 4.64. See also photocopying resolution (pixels), p. 986 resolutions, 14.283. See also legislation; US Congress restrictive and nonrestrictive clauses and phrases, 5.23, 6.27, 6.28, 6.29, 6.30, 6.41, 6.50 retractions, 1.91 Reverend, Rev., 10.18,16.39 reverse italics, 8.173,14.95 reviews basic format, 1.98 of books, 1.87,1.94,1.98,4.10,14.153, 14.202 quoted on paperback covers, 1.70 source citations for, 14.153,14.201-4 revised and expanded, 1.26 revised editions. See editions revised proofs, 2.103,2.105, p. 984 revision dates maps, 14.237 online reference works, 14.233 websites and web pages, 14.13,14.207, 15.50 revisions. See tracking changes revolutions, 8.113 RGB mode, 3.3, p. 986 rhetorical questions, 5.213 rhyme schemes, 7.69 right or left indent, 2.11 rights. See copyright; intellectual property rights; publishing agreements; sub sidiary rights rights to distribution outside the US, 4.34, 4.64,4.76 rivers, 8.53, 8.54, 8.55 roads and thoroughfares abbreviations of, 10.33 names of, 8.56,11.26 numbers of, 9.50, 9.51 See also addresses, mailing roman (type) abbreviations, 10.7,10.42,12.11,12.36, 12.64,16.116 Anon, and pseud, 14.79 INDEX roman (type) (continued) apostrophe and s in possessives of itali cized terms, 7.29 artworks of antiquity, 8.198 case names in citations, 14.272 city or state names with newspaper titles, 14.193,14.194 as default, 7.49, p. 987 defined, p. 987 figure in text, 3.9 for gene products, 8.132 generic terms for parts of books, 8.179, 8.180 index cross-references, 16.22 indexes of authors, first lines, and titles, 16.145 italicized terms within titles (reverse italics), 8.173,14.95 journal and magazine article titles in source citations, 14.169,14.188 key terms after first occurrence, 7.56 Latin terms, 7.55,10.7,10.42,13.66 legal-style citations, 14.272,14.273, 14.276,14.288,14.290 letters as shapes, 7.67 letters standing for names, 7.66 maps, descriptions of, 14.237 marking proofs for, 2.131 mathematical expressions, 12.11,12.36, 12.64,12.65 names of apps, devices, and operating systems, 8.155 names of letters, 7.68 non-English words, 7.53-55 opus numbers, 8.196 performers on recordings, 8.197 periodical departments or columns, 8.177,14.195 phonetic symbols, 7.64 proper names used as words, 7.63 scholastic letter grades, 7.65 in scriptural citations, 14.239,14.241 for s in plural form of italicized terms, 7.12,12.3 titles of series and editions, 14.124, 8.176 titles of websites, 14.206, 8.191 titles with quotation marks (see titles of works: quotation marks for) transliterated terms after first occur rence, 11.75,11.110 roman (type) (continued) UK legal and public documents, 14.298, 14.300 volume numbers, 14.23,14.171,15.9 words for musical pitches, 7.71 See also boldface; italics; quotation marks; special characters; trans literation and transliterated text; typographic considerations Romanian language, 11.70 romanization. See transliteration and trans literated text roman numerals vs. arabic numerals, 14.147, table 9.1 capitals: enzyme names, 8.133; musical chords, 7.73; with personal names, 6.43,7.42, 9.42,10.8,16.37,16.41, 16.66; sequels, 9.43; sovereigns and other rulers, 9.41; vessels and such, 9.44 Chicago’s preference for arabic over, 9.67 defined, p. 987 divisions within legal instruments, 9.28 general principles, 9.65 inclusive ranges, 9.61,16.14 musical chords, 7.73 in pagination, 1.4,1.7,1.9,2.38,2.57, 9.26 subtrahends (back counters), 9.66 See also arabic numerals; numbers rooms, offices, and such, 8.58 royalties failure to pay, 4.37 multiauthor works, 4.58 permissions fees from, 4.104 publishing agreements on, 4.52 and subsidiary rights, 4.64,4.69 royalty. See nobility; sovereigns and other rulers R. R. Bowker Company, 1.32,1.74 rulers. See sovereigns and other rulers rules (lines) above continued footnotes, 14.41, fig. 14.1 avoided in illustrations, 3.19 in tables, 3.53,3.57, 3.58,3.74, 3.86, figs. 3.13-14, fig. 3.16 run in, p. 987. See also indexes: run-in style; run-in lists; run-in quotations run-in lists, 6.128, 6.129, 6.131,7.45 “run in” mark, 2.126 INDEX run-in quotations assimilation into text, 13.11-12 vs. block, 13.9-10 capitalization, 13.19,13.21 converting block quotations to, 13.31 defined, 13.9 multiple paragraphs in, 13.32-33,13.35 poetry, 6.111, 6.113,13.29,13.34 punctuation: colons, 13.16; commas, 13.14; quotation marks, 13.30,13.68; slashes, 13.29 text citations for, 13.68-69,15.26 See also text citations run-in style (indexes). See under indexes run-in subheads, 1.56,2.18,2.59, 6.14 running feet, 1.10,1.83 running heads, 1.10-16 choice of content for, 1.11,2.76,14.47 defined, 1.10, p. 987 manuscript preparation guidelines, 2.76 no punctuation following, 6.14 not always used in electronic publica tions, 1.10,1.83,1.123,14.47 omission of, 1.12,1.16,1.49,2.115, 3.86, 3.87 proofreading of, 2.76, 2.114 publisher’s responsibility for, 2.3 shortening of, 2.76 specific instances: appendixes, 1.14; back matter, 1.14,1.15; electronic publications, 1.12; endnotes {see un der endnotes); front matter, 1.11,1.16; and illustrations, 1.8,1.16, 2.115; in dexes, 1.14,16.6; journals, 1.83,2.76; reference works, 1.13; and tables, 1.16,2.115,3.86,3.87; text, 1.12 title metadata for, 1.12 running sheets. See press sheets runover lines defined,p. 987 drama quotations, 13.47 hanging indentation for, 2.24,16.137 indexes, 16.26,16.28,16.137 mathematical expressions, 12.23 outlines, 6.132 poetry extracts, 2.20,13.25,13.27 tables, 3.61, 3.73, fig. 3.12 vertical lists, 2.21, 6.130 Russian language, 11.98-108 alphabet {see Cyrillic alphabet) capitalization, 8.12,11.99,11.100 Russian language {continued) names, 8.12,11.98 punctuation, 11.101-3 quotations and dialogue, 11.101 resources on, 11.98 special characters, 11.98, table 11.3 titles of works, 11.100 word division, 11.70,11.104-8 saddle stitching (saddle wiring), p. 987 saddle wiring (saddle stitching), p. 987 Saint, St.
alphabetizing of, 16.74,16.93 for names of saints, 8.93,10.20,16.42 personal names with, 16.74 place-names with, 10.30,11.26,16.90, 16.93 when to spell out, 10.20 See also saints saints as authors, 14.83 indexing names of, 16.42 non-English names of, 11.26 treatment in text, 8.93,10.20 Sanskrit language, 11.74 sans serif, p. 987 scalars, 12.52 scale, p. 987 scaling, 3.19 scans and scanning copyright issues, 4.17 defined, p. 987 for electronic versions of older works, 14.162 for illustrations, 2.27,2.47,3.15 OCR for, 2.43, p. 983 scarcely, 5.230 scare quotes, 7.57,11.47,14.94 schedules. See publishing schedule schemas (document type definitions), p.
979 scholarly works, generally abbreviations and symbols used in, 10.42-43 anonymity of research subjects in, 13.49 books and journals as core, 1.1-2 and citation management software, 14.5 conflicts of interest noted in, 1.95 editors’ notes in, 1.45 fair-use tradition in, 4.87 licensing revenue as supporting, 4.66 INDEX scholarly works, generally (continued) retaining original meaning of quota tions, 13.51 role of quotations in, 13.2 role of source citations in, 14.1 scientific and technical terminology abbreviations: astronomical and astro- physical, 9.10,10.61-62; botanical and zoological, 8.121,8.122, 8.123, 8.124; capitalization, 10.6,10.49; chemical elements, 10.63; latitude and longitude, 10.36; list of, 10.49; periods omitted in, 10.4; resources on, 10.1; SI units, 10.49,10.51-59; statistics, 10.50; units of measure ment, 10.64-68 astronomical and astrophysical, 8.137- 42,9.10,10.60-62 botanical and zoological (see botanical terminology; zoological terminology) data as plural, 5.14,5.250 genetic, 8.131-33 geological, 8.134-36,10.42 hyphenation of, 7.89 medical, 8.143-46 physical and chemical, 7.89 (sec. 1, chemical terms), 8.147-52,10.63 resources on, 8.119,8.128-31,8.134, 8.137,8.143, 8.147 See also International System of Units; metric system; scientific and techni cal works; units of measurement scientific and technical works abstracts, 1.93,1.95 • abstracts for, 1.76 author’s warranties for, 4.73 continuous publishing model, 1.82,1.113 errata, 1.68 hyperlinks for source citations in, 14.6 illustrations, 2.28,3.11, 3.25, 3.27, figs.
3.6-9 indexes for, 1.110-11,16.6,16.7,16.11, 16.26 letters to the editor, 1.101 numbers, generally, 9.1,9.3, 9.54 open-access publishing models, 4.61 percent symbol in, 9.18 preprints, 1.113 publication history, 1.104 punctuation, 6.9 sections and subsections in, 1.57 scientific and technical works (continued) source citations for: access dates, 14.12, 14.257; author-date reference system preferred, 14.3,15.5; databases, 14.257; et al. in, 14.76; initials for au thors’ given names, 15.12,15.33; num bered reference system, 1.108,14.28; titles of works, 8.168,15.13,15.38 subhead levels in, 1.56 tables in, 3.48 titles, 8.168,15.13,15.38,15.46 twenty-four-hour system of time in, 9.39 units of measurement in, 9.16,10.64-68 See also computer terminology; mathematical copy; mathematical expressions; scientific and technical terminology scientific notation (powers of ten), 9.9, 9.11, 9.12 screen, p. 987 screen margins, 1.122 screen names, 14.208,14.209,15.52,16.64 screen numbers, in e-books, 1.123 scriptures, 8.103,8.104,14.238-41. See also Bible; biblical citations; religious works sculpture, 8.57,8.198,14.235 search engines feature definition documents for, 2.139 vs. indexes, 1.110,1.111,16.2,16.7 journal-specific, 1.86 and keywords, 1.76,1.93,1.111,1.120, 2.25 and metadata, 1.75,1.76,1.92 and plurals, 7.5 searching in electronic publications, 14.22,16.2 in manuscript files, 2.36,2.79,2.80 in PDF files, 1.118,2.111,2.133,16.102, 16.105 See also search engines seasons, 1.80, 8.88,14.171. See also dates secondary sources, 14.260,14.291,15.56 section mark, 3.79,10.43,14.273 section numbers, as locators double or multiple numeration, 1.57 in electronic publications, 1.123,14.160 in indexes, 16.108 legal and public documents, 14.273, 14.283,14.295 newspapers, 14.191 INDEX sections and subsections abbreviations of section, 10.42,14.150 in bibliographies, 14.63,14.263, fig. 14.9 cross-references to, 2.35 in journals, 1.83,1.87,1.95 newspapers, 14.191 openings, decorative initials for, 13.37 in reference lists, 15.11 in scientific and technical works, 1.57 symbol for, 3.79,10.43,14.273 use of term, 1.48 See also section numbers, as locators; subheads see, 14.42. See also cross-references, in indexes seldom, 5.230 selected bibliographies, 14.64, fig. 14.9 self, as prefix, 7.89 (sec. 3) self, as suffix, 5.48,5.51,5.250 self-published materials checking markup for, 2.73 copyright page, 1.20,1.21 design templates for, 2.82 editing for, 2.48,2.52 illustrations in, 3.15 ISBN and bar codes, 1.32 keywords for, 1.76 metadata for, 1.75,1.76 PCN (Preassigned Control Number), 1.34 place of publication, 14.132,14.137 proofreading for, 2.101,2.137 reliability of materials, 14.14 retail platforms for, 1.75,1.76,2.73,4.63, 14.132,14.137 source citations for, 14.132,14.137 title page, 1.19 semantic markup, 2.82nl, 7.49 semi, as prefix, 5.250,7.89 (sec. 4) semicolons, 6.56-60 marking proofs for, 2.132 other punctuation with: closing quo tation marks, 6.10; ellipses, 13.54; parentheses, 6.98 in source citations: classical Greek and Latin works, 14.245; copublications, 14.140; more than one subtitle, 14.90; notes with multiple citations, 14.57; and or with double titles, 14.91; text citations, 15.7,15.22,15.24,15.28, 15.30 uses: before conjunction, 6.59; before semicolons (continued) conjunctive adverbs, 6.57; before that is, namely, for example, and such, 6.51, 6.58; in complex series, 6.60; independent clauses, 6.56- 59; indexes, 16.17,16.20,16.23, 16.25,16.96; lists, 6.129, 6.131; in mathematical copy, 12.7,12.18,12.38, 12.54; in non-English languages, 11.7, 11.116; for series of elements in sen tence, 6.60, 6.129; tables, 3.80; titles of works, 8.165, 8.167,11.7,14.90-91 See also colons; punctuation sentence adverbs, 5.157 sentence fragments. See ellipses, gram matical sentences adverbs modifying, 5.157 beginnings of: conjunctions, 5.203, 5.204,5.250; lowercased names, 8.4, 8.5; mathematical symbols, 12.7; numbers, 9.5; percentages, 9.18; years, 9.5, 9.29;yes or no, 6.34,13.40 cleft, 5.246-48 deliberately incomplete, 13.55 ellipses after complete, 13.53 glossed American Sign Language, 11.133 and list format, 6.129, 6.130 and mathematical expressions, 12.7, 12.18 one-word, 5.97 parallel structure in, 5.242-45 in parentheses, 6.13, 6.98 parenthetical elements in, 6.48, 6.51, 6.85, 6.95 pauses or breaks, 6.16, 6.87 (see also interjections) prepositions ending, 5.180 questions within, 6.42, 6.63, 6.65, 6.69-70 series of elements in (see series of ele ments in sentence) series of related, 6.61, 6.63 subjects and predicates of, 5.18 traditional structures, 5.217-20 types of, 5.211-16; complex, 5.219, 5.225; compound, 5.218; compound complex, 5.220; declarative (statements), 5.211, 5.246, 6.12, 6.68; elliptical (see ellipses, grammatical); exclamations, 5.216, 6.35; imperative INDEX sentences (continued) (directives) (see imperative sen tences); interrogative (see questions); simple, 5.217 URLs within, 6.8 See also clauses; syntax sentence-style capitalization titles of works: basic principles and examples, 8.158; classical Greek and Latin works, 11.54,14.247; “down style,” 8.1; in journal reference lists, 8.158,8.168,15.13,15.38; non English, 11.6,11.27,14.98-99,14.123, 14.194; translations, 14.99 uses, other: captions, 3.21; chart titles, 3.44; first lines of poems, 8.182; non-English words and phrases, 11.4; subheads, 1.56,2.18,2.59, 8.158; tables, 3.54, 3.55,3.56,3.59 See also headline-style capitalization September 11,2001, 9.35 sequels, numbers for, 8.189, 9.43 Serbian language, 11.70 serial (Oxford) commas, 6.19-21, 6.23, 8.165,14.135. See also commas serial publications, copyright registration forms for, 4.48. See also periodicals serial rights, first or second, 4.64 series, 8.176 series of elements in sentence and ampersands, 6.21 colons to introduce, 6.61, 6.67 commas within, 6.19-21, 6.23,8.165, 14.135 and em dash, 6.86 semicolons within, 6.60, 6.129 series of lectures. See lectures and lecture series series of works ISBN and ISSN for, 1.32 source citations for, 14.86,14.123-26, 14.218 titles of series: in book review and book notes sections, 1.98; capitalized but not italicized, 8.2, 8.176,14.86, 14.123-24; on covers or jackets, 1.70; in front matter, 1.4,1.18; proofread ing of, 2.108 unfinished date ranges for, 6.79 working papers in, 14.218 series of works (continued) See also multivolume works; volumes and volume numbers serif, p. 987 sewing (signatures), 8.14, p. 987 sexual orientation, terms for, 8.41 SGML, p. 987 shading, 3.19,3.43, 3.64 shall, 5.131,5.231,5.250 shan't, 5.231 s/he, 5.250, 5.252, 5.256 she, her, hers, 5.49,5.50 sheet-fed press, p. 987 ships and other craft, 8.2,8.116-18 shortened citations, 14.29-36 vs. abbreviated titles, 14.59 anonymous works in text citations, 15.34 authors’ names, 14.30-32,14.185 basic structure, 14.30 biblical citations, 14.239 and bibliography, 14.61,14.64 blog posts and comments, 14.208 chapters, 14.108 classical Greek and Latin works, 14.244 criteria for, 14.29 cross-reference to full citation, 14.31 databases, 14.276 examples and variations, 14.23 vs. ibid., 1.124,13.66,14.34,14.48 idem, op. cit., loc. cit., discouraged, 14.35-36 journal articles, 14.23,14.33,14.185 journal titles, 15.13,15.46 in legal style, 14.275,14.276,14.279 letters in published collections, 14.111 manuscript collections, 14.229 multiauthor volumes, 14.32,14.108 multivolume works, 14.118 newspapers, 14.196 in notes, 14.19,14.48 poetry extracts, 13.72 quotations, 13.66-67,13.70 same-author, same-year works in text citations, 15.29 secondary sources, 14.291 short forms for DOIs and URLs, 14.10 in table source notes, 3.77 titles as reference list entries, 15.39 very long titles, 14.97 should, 5.149,5.150,5.250 INDEX SI. See International System of Units; metric system sic, 6.73,7.55,13.61 sideheads, pp. 987-88 side sewing, p. 988 signatures (book), 1.41, 3.6,14.154, p. 988 signatures (name), 1.41,10.11 signed languages, 11.125. See also American Sign Language signs, quotation of, 7.61. See also symbols, general signs and notices, 7.61 simple adverbs, 5.160 simple sentences, 5.217 simultaneous print/electronic publishing model books, 1.2,14.159 common use of, 1.2 and formal markup languages, 1.117 and formats, 1.77 and illustrations, 1.107, 3.10, 3.17, 3.26 and ISSNs, 1.79,1.86 and letters to the editor, 1.101 noting differences, 1.78,1.81,1.87, 14.16 page ranges noted, 1.81 proofreading for, 2.140 source citation considerations, 14.16 and tables, 1.106 and version control, 1.112 See also multiple formats, publications in since, 5.201, 5.202, 5.250 singular. See number, of nouns and pro nouns Sir, Dame, 8.32,16.38 site license, 1.86 site map, 2.137 slang, 5.95,7.57 slant. See slashes slashes, 6.105-13 in abbreviations, 6.109,10.4 alternative terms for, 6.105 vs. backslashes, 6.105, 6.112 in dates, 6.107, 6.108, 6.113 dividing poetry lines, 6.111, 6.113,13.29, 13.34 double, dividing poetry stanzas, 13.34 double, in URLs, 14.18 in fractions, 6.110, 6.113,12.45,12.47 in generic markup, 2.81 and line breaks, 6.113 slashes (continued) in mathematical expressions, 6.110, 12.15,12.42,12.45,12.47 in publishers’ names, 14.138 resources on, 6.105 signifying alternatives, 6.106 in source citations for journal articles, 14.171 trailing, 14.17 in two-year spans, 6.107, 6.113 in URLs, DOIs, and such, 6.112,14.17, 14.18 See also punctuation small caps (capitals) for acronyms and initialisms, 10.8 after ornamental text break, 1.58 a.m. and p.m., 9.37,10.41 defined, p. 988 for emphasis, 7.52 for era abbreviations, 9.34 full-size capitals compared to, 7.52 full-size capitals preferred to, 9.34,10.8, 10.38 glosses in ASL, 11.128-35 lowercase preferred to, 10.41 marking proofs for, 2.131 and permissible changes, 13.8 proofreading for, 2.113 See also capital letters “smart” (directional) quotation marks and apostrophes, 2.85, 6.115,13.7 Smyth sewing, p. 988 so, 5.88, 5.202-3, 6.22 so-called, 7.59 social media content defined, 14.205 electronic mailing lists, 14.210 and personal communications, 14.205, 14.209.14.214.15.52 source citations for, generally, 14.209, 15.52 titles for, 8.191,14.206 URLs for, 14.10,14.209 See also blogs and blog posts social titles, 10.16,10.17 socioeconomic terminology. See groups of people soft vs. hard hyphens, 2.96 software conversion errors, 2.80, 2.113,2.137 names of apps, 8.155 INDEX software (continued) small caps in, 10.8 sorting functions, 14.65,14.67,15.17, 16.5,16.57,16.104,16.123 and special characters, 2.16,11.45,11.72 specific types: artwork, 3.4, 3.15; citation management, 14.5,14.12; document comparison, 2.86; indexing, 16.5, 16.57,16.104,16.117,16.123; manu script editing, 2.79,2.80; manuscript preparation, 2.4,2.7,2.77; mathe matical copy, 12.2,12.2nl, 12.9, table 12.1; page layout, 6.121,7.36; PDF proofreading tools, 2.133 See also computer terminology; da tabases; file formats and devices; tracking changes; Unicode standard; word-processing software solid (closed) compounds, 7.82,7.83,7.88. See also compound terms solidus. See slashes some, 5.7 songs, 8.194. See also musical works sort, sorts, 5.52 so that, 5.201 sound recordings contributions to, as work made for hire, 4.10 discographies, 1.63,14.262,14.263, 15.57, fig. 14.12 source citations for, 14.263,14.264, 15.57 titles of, 8.197 See also multimedia content source citations, 14.1-5 author queries about, 2.62,2.63,2.64, 2.69 basic elements, 14.100 basic systems, 1.108,13.64,14.2-3 citation management software, 14.5, 14.12 editing of, 2.62-64 flexibility and consistency, 2.64,14.4 house style for, 1.108-9,2.64,14.4 hyperlinks for (see hyperlinks for source citations) pop-up boxes in, 15.17 purpose of, 14.1 remedies for excessive annotation: abbreviations, 14.59-60, fig. 14.7; source citations (continued) avoiding overlong notes, 14.56; combining notes, 14.57; footnotes vs. endnotes, 14.45,14.59; text citations, 14.34,14.58 and secondary sources, 14.260,15.56 shortened citations (see shortened citations) style manuals for, 2.54,14.3 See also author-date reference system; bibliographies; illustration credits and credit lines; notes; notes and bibliography system; reference lists; source notes; and specific types of sources (e.g., books) and specific elements (e.g., authors’ names) source notes previously published materials, 1.50, 2.46, 4.78,14.54 tables, 2.31,2.61, 3.76-77, 3.81 See also attribution; illustration credits and credit lines; source citations; text citations South, southern, 8.46,8.47 South Asian languages, 11.109,16.79, table 11.2 sovereigns and other rulers Arabic rulers of older times, 8.14 as authors, 14.83 indexing of, 16.37,16.66,16.98 and legal and public documents, 14.298, 14.302 numerals for, 9.41,16.66 titles of, 8.23 See also nobility Soviet Union, former, 10.31 sp., spp., 8.122,8.123 Space bar, 2.12 spacecraft, 8.117,9.44. See also vehicles and vessels spaces and spacing, 6.119-21 and abbreviations, 9.16, 9.55-56,10.5, 10.10,10.12,10.49,10.58,10.66 in biblical citations, 9.26 in captions, 3.23 characters in different fonts, 6.5 decimals, 9.55 and electronic file cleanup, 2.80 in electronic publications, 6.11, 6.120, 9.55,9.56 INDEX spaces and spacing (continued) em spaces, 2.126, 3.23, 3.54, 6.120 en spaces, 6.120,13.29 with ff.y 14.149 hair spaces, 6.5, 6.11, 6.120, p. 988 indexes, 16.28,16.59,16.111,16.131 initials, 6.121,7.66, 8.4,10.12,14.74, 15.33 and justification, 2.125, 6.120, 6.121 marking manuscript for, 2.98 marking proofs for, 2.125 in mathematical expressions, 12.16-17, 12.19,12.21-22,12.42 medium, 12.23 with multiplication sign, for plant hybrids, 8.125 names with particles, 8.5 nonbreaking (see nonbreaking spaces) non-English materials and quotations: French, 11.28-29,11.31-32; Italian, 11.47,11.48; permissible changes, 11.7,11.11,11.19; Spanish, 11.63,11.65 numbers, 6.62, 6.121, 9.55-58 with percent symbol, 9.18 with punctuation: ampersands, 10.10; colons, 2.9, 6.62,14.89,14.116, 14.151,14.152,14.177,15.48; ellipses, 6.121,11.49,11.65,13.50,13.54, 13.55,13.58; en dashes, 6.79,14.144; generally, 2.9, 6.7, 6.12, 6.62,14.116; parentheses, back to back, 6.98; slashes, 6.106, 6.111,13.29,13.34, 14.138; 2-em dashes, 6.93 between sentences, 2.9,2.80 with suspended hyphens, 7.88,7.89 (sec. 1) in table titles, 3.54 thick, 12.23 thin: defined, p. 988; in French mate rials, 11.28-29,11.31-32; in math ematical expressions, 12.17,12.42; for overlapping characters, 6.5; in poetry quotations, 13.29; between single and double quotation marks, 6.11, 6.120 with units of measurement, 6.119, 6.121, 7.44, 9.16 vertical lists, 6.130 width of, 6.120 within paragraphs, 2.12 spaces and spacing (continued) See also leading; lines (text); line spac ing; margins Spanish language, 11.60-69 abbreviation of Spanish, 10.42 capitalization, 8.11,11.61 dialogue, 11.64 names, 8.11,16.83,16.84 punctuation: ellipses, 11.65; em dashes, 11.63,11.64; guillemets and quota tion marks, 11.63,11.64; question marks and exclamation points, inverted, 11.19,11.62 resources on, 11.60,11.67 special characters, 11.69 word division, 11.66-68 spanner heads (decked heads), 3.53, 3.57, fig. 3.13, fig. 3.16 speakers’ names (dialogue and drama), 13.8,13.39,13.46,13.48. See also dialogue; drama; lectures and lecture series; speech spec (specification), p. 988 special characters and alphabetizing, 16.67 in archaic fonts, 13.7 author’s list of, 2.3, 2.16,11.21,11.122, 12.13 bitmapped images for, 2.137 breathing marks, 2.80,11.110,11.112 conversion errors in, 2.80,2.137 described in angle brackets, 2.16 eszett (E) vs. Greek beta, 11.45 generic markup of, 2.81 in indexes, 16.29 listed, tables 11.1-5 manuscript preparation guidelines, 2.3, 2.16.11.21.11.122.12.13 in mathematical expressions, 12.10, 12.12-13,12.52,12.68, tables 12.3-4 non-English languages: African lan guages, 11.24; Arabic, 11.76,11.77, 11.81.11.92, table 11.2; Azeri, 11.70; Czech, 11.70; Finnish, 11.70; French, 11.37; generally, 11.2,11.21,11.70, 11.72-73, table 11.1-5; German, 11.45; Greek, 11.110-15,11.117, table 11.4; Hawaiian, 11.70,11.70nl; Hebrew, 11.92, table 11.2; Hungarian, 11.70; Icelandic, 11.70; Italian, 11.53; Jap-INDEX special characters (continued) anese, 11.87, table 11.2; Latin, 11.59; Moldavian, 11.70; Montenegrin, 11.70; Old and Middle English, 11.70,11.122- 24; Portuguese, 11.70; resources on, 11.45,11.82,16.67; Romanian, 11.70; Russian, 11.98, table 11.3; Serbian, 11.70; South Asian languages, 11.109, table 11.2; Spanish, 11.69; specialized fonts for, 11.21,11.72,11.90,11.110; Turkish, 11.70; Unicode for, generally, 11.2; Uzbek, 11.70 proofreading, 2.132 resources on, 11.76,16.67 style sheet for, 2.55 and transliteration, 2.80,11.2,11.72, 11.74,11.90 See also diphthongs; ligatures; math ematical expressions; symbols, specific; Unicode standard special issues, journal, 1.84,1.87,1.89, 1.102,4.59,14.178 species. See genus and species names speech colloquial (see colloquial speech) direct discourse, 13.39,13.44 discussions and interviews, 13.48 drama, 13.46-47 grammatical ellipses in, 5.229,13.41 indirect discourse, 13.45 interjections in, 5.207 numerals in direct discourse, 13.44 paused, interrupted, or faltering, 6.87, 6.89,13.41,13.50,1358 punctuation: colons, 6.63, 6.65,13.16; commas, 13.14; ellipses, 13.41; em dashes, 13.41; quotation marks, 13.32, 13.39-45 questions in, 5.213 single-word speech, 13.40 unspoken discourse, 13.43 See also dialogue; direct address; quo tations speeches presidential, 8.76,14.289 source citations for, 14.217,14.267, 14.289,15.57 treatment of names and titles, 8.70, 8.76,8.188 See also lectures and lecture series spelling of alphabet letters, 7.68 author queries about, 2.69 British-style, 7.3,16.29 checking, 2.89 common errors of, 2.111 consistency, 2.65,2.89,8.3 contractions, 7.30 discipline-specific, 7.2 editor’s note on, 1.45 in indexes, 16.29,16.33,16.71,16.133 interjections, 7.31 with ligatures, 7.34,7.35 metric units, 8.152 non-US, 7.3 orthographic reforms, 11.38,11.43,11.45, 11.67.11.70 permissible changes to, 13.7 personal names, 6.106,8.3,11.78,14.74, 16.33.16.71 proofreading of, 2.111 resources on, 7.1 retaining original in quotations, 7.3 and sic, 13.61 of spelled-out abbreviations or numbers, 2.129 See also plurals; possessives; word divi sion; and specific languages spell-out mark, 2.129 spine and spine copy, 1.69,1.70,1.71,1.84, 2.108,2.109, p. 988 spiritual leaders. See clerical titles; religions and religious concerns split infinitives, 5.108,5.171. See also infinitives sports, 8.68,8.70,8.78,9.19 spread, p. 988 square brackets. See brackets Sr. (“Senior”), 6.43,7.42,10.19,16.41 stable URLs (permalinks), 14.9,14.15 stacked style (indexes). See indexes: in dented style stage directions, 13.46 stage names, 16.34 stamping, p. 988 standard deviations, notation of, 10.50, 12.59, fig. 3.8, table 12.3 standards, source citations for, 14.259 statements (declarative sentences), 5.211, 5.246, 6.12, 6.68 INDEX states (US) abbreviations and postal codes, 10.4, 10.8,10.27,14.130 in addresses, punctuation, 6.39,10.27, 10.29 capitalization, 8.51 constitutions, 14.280 judicial bodies, 14.279 laws, 14.288 in newspaper titles, 14.193 in place of publication, 14.130 statistics abbreviations in, 10.50, table 12.3 mathematical expressions of, 12.58-60, table 12.3 means and standard deviations, 10.50, 12.59, fig. 3.8, table 12.3 resources on, 12.57 See also mathematical copy; mathemati cal expressions status. See honorifics; titles and offices of people statutes. See legislation step, as prefix, 7.89 (sec. 3) stereotypes. See bias-free language stet, 2.130 sticky notes, 2.92 stochastic theory, 12.59 stock agencies, material from, 3.36,4.99, 4.101 stratigraphy, 8.136. See also geological terminology streaming formats, 4.64, 8.189,14.263 stream of consciousness, 13.43 streets, avenues, and such, 8.56,9.51,10.33-34,11.26,11.61. See also addresses, mailing; roads and thoroughfares strike-through, 2.84 strong (irregular) verbs, 5.100 stub column, 3.59-64 abbreviations in, 3.62 and alignment, 3.70 defined, 3.52, p. 988 heads for, 3.56, 3.59, fig. 3.15 leaders with, 3.64, fig. 3.16 for matrixes, 3.68, figs. 3.17-18 omission of, fig. 3.19 runover lines in, 3.61, fig. 3.12 subheads in, 3.60, fig. 3.13 See also tables style defined, 2.49 mechanical editing for, 1.116,2.48,2.49 style, compound terms with, 7.89 (sec. 3) style manuals, 2.54,5.250,14.3 styles (word processing), 2.18,2.21,2.82, 2.83,2.84, fig. 2.4 style sheets (editorial) defined, p. 988 manuscript editing, 2.55,2.83,2.111, fig. 2.3 proofreader’s use of, 2.111,2.112 style sheets (electronic), pp. 988, 990 sub, as prefix, 7.89 (sec. 4) subentries. See index entries: subentries subheads, 1.55-57 as alternative to page numbers, 14.160, 14.174 capitalization, 1.56,2.18,2.59,8.158 cross-checking of, 2.32,2.59 defined, p. 988 editing of, 2.59 generic markup of, 2.59,2.80,2.81 levels, placement, and typographic style, 1.56,2.59 manuscript preparation guidelines, 2.18 no punctuation following, 6.14 and note reference numbers, 14.27 numbers with or instead of, 1.57 period at end omitted, 2.18 proofreading of, 2.113,2.134 run-in, 1.56,2.18,2.59, 6.14 sideheads, 1.56, p. 987 single, 1.56 specific uses: appendix, 1.59; bibliog raphies, 14.63,14.263; endnotes, 1.62,2.81,14.46, fig. 14.2; journal articles, 1.97; list of illustrations, fig. 1.6; running heads, 1.12,1.13; table of contents, 1.38,1.87,2.32,2.57,2.59; tables, 3.58,3.60,3.61, figs. 3.13-4 text following, 1.55,2.59 See also sections and subsections subject headings, 1.75,1.92,1.93. See also keywords subjective case. See nominative (subjective) case subjective complements, 5.101 subjects and disciplines, academic, 7.2, 8.30, 8.85, 8.86 INDEX subject-verb agreement, 5.138,5.139,5.141, 5.143, 5.205,5.250 subject-verb-object (SVO) pattern, 5.221, 5.222 subjunctive mood, 5.123-27 subordinate clauses. See dependent (subor dinate) clauses subordinating conjunctions, 5.175, 5.200-201,5.225, 6.24, 6.26 subordination, 5.225. See also dependent (subordinate) clauses subscriptions to journals, 1.84,1.86,2.109, 2.134. See also access rights subscripts in chemical symbols, 8.149 defined, p. 988 in glossed American Sign Language, 11.134 marking manuscript for, 2.94 mathematical expressions, 12.36-38; alignment, 12.38,12.67; complex, 12.37; and fractions, 12.47; italics for, 12.63; limits, 12.17; marking, 12.67; simple, 12.36; spacing, 12.16; uncer tainties expressed as, 12.60; vector and tensor notation, 12.53,12.54 musical octaves, 7JI See also superscripts subsidiary rights, 4.64-69 author’s electronic use of own works, 4.66 author’s retention of, 4.18 vs. basic rights, 4.17 categories of, 4.64 ’ distribution outside the US, 4.34,4.64, 4.76 economic considerations, 4.65,4.66,4.69 electronic rights, 4.63,4.64,4.65 granting permissions for, 4.70,4.71 moral, 4.15 non-US publications, 4.29-30 and public-access policies, 4.68 translation, 4.34,4.64 university licenses, 4.67 See also copyright; intellectual property rights; permissions substantive editing, 2.48,2.50,2.51 substantive notes, 14.37-39,14.49,16.110 subtitles capitalization, 8.158, 8.159,8.161, 8.164, 14.87,14.89,14.90,14.91 subtitles (continued) cross-checking of, 2.32,2.58 editing of, 2.58 indexing principle on, 16.54 as metadata, 1.75 omission of, 8.168,16.54 or connecting, 8.167,14.91 permissible changes to, 8.165,11.7 placement of, 1.17,1.19,1.49,1.69,1.70 punctuation: colons, 1.19,8.164-65, 14.89-91,14.96,14.169; semicolons, 8.167,14.90 in running heads, 1.12,2.76 with sequel numbers, 9.43 source citations for, 14.87-91,14.165 two or more, 8.167,14.90 subtrahends (back counters), 9.66 sub verbo (s.v.), 1.123,10.42,14.232 sub voce (s.v.), 10.42 such as, 5.250, 6.50 such that, 5.201 suffixes adjectives with: comparative forms, 5.85,5.87-88; double comparative or superlative error, 5.87; overview, 5.68; phrasal, 7.85; superlative forms, 5.86 and adverbs, 5.158-59,5.163-64 and bias-free language, 5.257 compound terms with, 7.89 division of words with, 7.40 and hyphenation, 7.81 negating, 5.230 pronouns: personal (-self), 5.48,5.51, 5.250; relative (-ever), 5.65,5.66 for words derived from proper names, 8.60 See also compound terms summation sign, 12.39 sun, 8.141 super, as prefix, 7.89 (sec. 4) superlative adjectives, 5.86,5.87,7.89 (sec.
1) superlative adverbs, 5.164,7.89 (sec. 1) superscripts celestial coordinates, 10.61 classical Greek and Latin works, 14.248 defined, p. 988 marking manuscript for, 2.94 mass, in chemical terms, 8.150 mathematical expressions, 12.36-38; INDEX superscripts (continued) alignment, 12.38,12.67; complex, 12.37; and fractions, 12.47,12.49; ital ics for, 12.63; marking, 12.67; powers of ten, 9.9; simple, 12.36; spacing, 12.16; uncertainties expressed as, 12.60; vector and tensor multiplica tion, 12.53 musical octaves, 7.72 in non-English words and phrases, 11.27 note reference numbers, 14.19,14.24 in notes to tables, 3.79, fig. 3.24 not used for ordinal numbers, 9.6 See also subscripts supplementary data, electronic books, 1.59,14.112 hyperlinks to, 1.96 illustrations, 3.9, 3.26 journals, 1.78,1.87,1.96,1.114,14.187 noted in print version, 1.78 source citations for, 14.112,14.187 in table of contents, 1.78,1.87 supplements journals, 14.179 newspapers, 14.197 supporting information. See supplementary data, electronic supra, as prefix, 7.89 (sec. 4) suspension points, 13.50. See also ellipses s.v. (sub verbo, sub voce), 1.123,10.42,14.232 SVO (subject-verb-object) pattern, 5.221, 5.222 Swahili language, 11.23-24 Swedish language, 11.70 syllabary, 11.85 symbols, general a or an with, 7.33 in illustrations or captions, 3.7,3.25, 3.45, fig. 3.4, fig. 3.6, figs. 3.8-9 as labels, 3.7, 3.45 as note references, 3.79,14.24,14.25, 14.49,14.51,16.112, fig. 14.4 for page numbers to be supplied later, 2.35 SI abbreviations referred to as, 10.2 in tables, 3.62, 3.82-84, figs. 3.12-13, fig. 3.22 in vertical lists, 2.11 See also manuscript editing: paper-only; mathematical expressions; proof reader’s marks; special characters symbols, specific alignment, vertical and horizontal, 2.127 celestial coordinates, 10.61 chemical, 7.89 (sec. 1), 8.149,10.63 copyediting (see manuscript editing: paper-only) currency, 9.20-25 for large numbers, 9.10, 9.24 mathematical (see mathematical expres sions: signs and symbols) musical, 7.71-75 ornamental text breaks, 1.58 on proofs (see proofreader’s marks) scholarly, 10.43 SI units, 10.52,10.54-55,10.56-58 technical, 10.49 units of measurement, 3.56, 9.16-17, 10.52,10.54-58, fig. 3.12 symposia. See conferences; lectures and lecture series; proceedings of confer ences and symposia; speeches; un published and informally published materials; working papers syntax, 5.210-48 for avoiding gender bias, 5.255 captions, 3.21 clauses overview, 5.225-28 cleft sentences, 5.246-48 defined, 5.210 expletives, 5.239-41 in indexes, 16.10,16.68,16.129 negation, 5.230-38 parallel structure, 5.242-45 and quotations, 13.11,13.18-20 sentence types, 5.211-16 in tables, 3.54,3.59 table stub entries, 3.59 and text citations, 15.28 traditional sentence structures, 5.217-20 word order, 5.216,5.221-24 See also ellipses, grammatical synthetic comparatives, 5.85,5.163 synthetic superlatives, 5.86,5.164 synthetic vs. analytic languages, 5.221 Systeme international d’unites (SI). See International System of Units table of contents abbreviations list noted in, 1.44,14.60 chapter titles in, 1.38,2.17,2.32,2.58, fig. 1.5 INDEX table of contents (continued) components of, 1.38, fig. 1.5 editing of, 2.57-58 in electronic publications, 1.87,1.122, 1.123, fig. 1.11 electronic supplementary data in, 1.78, 1.87 errata in, 1.68,1.90 example, fig. 1.5 final manuscript, 2.75 galleries in, 1.38,1.39, 3.39, fig. 1.7 hyperlinks in, 1.86,1.87,1.90, fig. 1.11 journals, 1.84,1.86,1.87,1.90,2.109, fig. 1.11 multiauthor volumes, 1.38,1.39,2.32, 2.41 omission of, 1.38 online publications, 1.122 page numbers in, 1.38,2.134, fig. 1.5 pagination, 1.7,1.38 and permissions process, 4.105 placement in book, 1.4 placement of, 1.38,1.84 proofreading of, 2.134 running heads after first page of, 1.11 subheads in, 1.38,1.87,2.32,2.57,2.59 submission of, 2.3 tables, 3.47-88, figs. 3.11-25 abbreviation of term, 16.116 abbreviations in, 3.56,3.62,3.67, 3.81, fig. 3.12 accessibility issues, 3.88 in appendixes, 1.62 appropriate use of, 3.48 basic structure, 3.52, fig. 3.11 body and cells: alignment and format ting, 3.70-74, figs. 3.19-21; defined, 3.65; empty, 3.65, 3.67,3.68, 6.92, fig. 3.15, fig. 3.17; multiple values in single cell, 3.69, fig. 3.13; numbers in, 3.72; rows of, 3.70; words in, 3.73, fig. 3.19 broadside, 2.115, 3.51, 3.86, 3.87, fig. 3.16, fig. 3.25 callouts to (see callouts) column heads: alignment, 3.71; cut-in, 3.53, 3.58, fig. 3.14; format, 3.56; for multiple values in single cell, 3.69, fig. 3.13; no period in, 6.14; paren thetical information in, 3.56, fig. 3.12; repeated, 3.86-87, fig. 3.25; rules to tables (continued) separate, 3.53,3.57,3.58, figs. 3.13-14; signs and symbols in, 3.84; spanner (decked), 3.53,3.57, fig. 3.13, fig. 3.16; stub, 3.56,3.59, fig. 3.15; text in notes, 3.86, fig. 3.24 columns: alignment of, 3.71-73, figs. 3.19-20; data in, 3.66; in matrixes, 3.68, figs. 3.17-18; stub, 3.52,3.56, 3.59-64,3.70, figs. 3.15-18; totals, averages, and means, 3.74, fig. 3.21 (see also column heads above) consistency in and across, 3.49, 3.56, 3.59, 3.74, 3.81, 3.86 continued lines in, 3.86-87, fig. 3.25 defined, 3.2, p. 989 doubled up, 3.53, 3.86, fig. 3.23 editing of (see under manuscript editing) electronic files, 2.26,2.31,2.36,2.80, 3.47 in electronic publications, 1.106, 3.48, 3.76 as electronic supplementary data, 1.78 fair use of, 4.91 generic markup for, 3.86 and hyperlinks, 1.96,1.106,3.51 vs. illustrations, 3.2 inclusive numbers in, 3.83 indentation in, 3.60,3.61,3.63, 3.73, 3.74, fig. 3.13, fig. 3.21 indexing of, 16.116,16.141 in letters to the editor, 1.101 lists of (see lists of illustrations and tables) machine-readable, 1.106 manuscript preparation guidelines, 2.3, 2.22,2.26,2.29-30,3.47 matrixes, 3.68, figs. 3.17-18 N, n, 3.55,3.72, 3.75, 3.85, fig. 3.20 “not applicable” and “no data available” in, 3.67 notes to, 3.76-80; abbreviation defi nitions in, 3.62, 3.67; for broadside tables, 3.87; column head text in, 3.86, fig. 3.24; editing, 2.61,3.81; examples of, figs. 3.14-15; format ting, 2.31,3.77; hyperlinks for, 1.106; vs. identical cell contents, 3.66; numbering, 2.22,2.31,3.79; order and placement, 3.76; vs. parentheti cal elements, 3.55; and rounding for INDEX tables (continued) totals, 3.75; significance (probability) levels, 3.80, fig. 3.22; source notes, 2.31,2.61,3.76-77,3.81; to specific parts of table, 3.79, fig. 3.11, figs. 3.13- 15, fig. 3.24; to whole table, 3.78, figs.
3.11- 12, fig. 3.14, fig. 3.17, fig. 3.20, fig. 3.22, fig. 3.24 numbering of, 1.57,2.29,3.50,3.54, 16.116 and pagination, 1.8 permissions for, 3.77,4.95 placement of, 2.62,3.51 proofreading of, 2.115,2.127,2.137 punctuation: ellipses, 3.67,3.68, fig. 3.15, fig. 3.18; em dashes, 3.67, 3.68, 6.92; notes to whole table, 3.78; parenthe ses, 3.54-56, fig. 3.12; semicolons, 3.80; significance (probability) levels, 3.80, fig. 3.22; stub entries, 3.59, 3.60; in table titles, 3.54,3.55 resources on, 3.47 rules in, 3.53,3.57,3.58,3.74,3.86, figs.
3.13-14, fig. 3.16 and running heads, 1.16,2.115,3.86,3.87 runover lines in, 3.61, 3.73, fig. 3.12 shading in, 3.64 signs and symbols in, 3.62, 3.82-84, figs.
3.12- 13, fig. 3.22 source citations for, 14.158 statistical material in, 10.50 syntax in, 3.54, 3.59 titles of, 3.54,3.55 totals in, 3.53, 3.63,3.74-75, 3.81, figs.
3.15-16, fig. 3.21 units of measurement in, 3.56, figs.
3.12- 13 word-processor function for, 2.26,2.36, 3.47 See also captions; illustrations tabs, 2.11,2.12,2.19,2.21,2.24,2.80, 6.130, 6.132. See also indentation; margins tabular material abbreviations in, 10.3,10.24,10.26, 10.27,10.29,10.32 in HTML format, 1.118 manuscript preparation guidelines, 2.29 placement of, 3.51 proofreading of, 2.127 See also columns; tables tag questions, 5.213 tags defined, p. 989 in formal markup, 2.83 in generic markup, 2.81 See also markup for manuscript editing; markup languages, formal taxonomic indexes, 16.6 TCF (totally chlorine-free), p. 989 technical writing. See computer terminol ogy; scientific and technical termi nology; scientific and technical works technology. See computer terminology; sci entific and technical works; software telephone conversations. See personal communications telephone numbers, 6.77, 9.57 television and television programs broadcast networks, 8.189,10.25 reviews of, 14.201,14.203 source citations for, 14.265 titles of programs and episodes, 8.189 See also multimedia content temperature, 9.16,10.36,10.49,10.52, 10.58,10.61 temporary compounds, 7.82 temporary numbers, 2.28, 3.13 tense. See under verbs tenses. See under verbs tensors, 12.38,12.52,12.53,12.54 ter (Dutch particle), 8.10 territories, 10.4,10.27,10.28. See also geo graphical terminology test-case documents, 2.139 testing, electronic formats, 1.92,2.138-39 tests (examinations), 4.10 text. See books; electronic publications; journals; manuscript preparation guidelines for authors textbooks, 1.10,4.10,4.76nl, 4.79,4.104, 7.56,14.62 text citations abbreviations in, 10.3,10.44,14.59 additional material in, 15.24 authors’ names: and anonymous works, 15.34; authors with same last name, 15.22; editor in place of, 15.36; et al. in citation and/or text, 15.29; more than three, 15.29; organizations as, 15.37; pseudonyms as, 15.35; same author, same year, 15.20,15.29; in INDEX text citations (continued) text, not repeated in citation, 15.25 (see also authors’ names) basic approach: examples and vari ations, 15.9; format, 15.5,15.22, fig. 15.2; function, 15.10; order within citation, 15.30; page, volume numbers, and other locators in, 15.8, 15.23; page numbers in, 15.27,15.30; placement in text, 15.25-28 biblical citations in, 10.44,10.45-48, 14.239 classical Greek and Latin works in, 14.242 cross-checking, 2.32,2.63,15.21 ibid, in, 13.66,14.58 and indexing, 16.114,16.115 multiple references in single citation, 15.30 multiple references to same source, 13.66,14.58,15.27 for newspaper articles, 14.198 original, in quotations, 13.7 publication details in, 15.25,15.41, 15.44-45 punctuation: brackets, 13.7,15.28,15.40; commas, 15.7,15.9,15.23,15.29, 15.30,15.44,15.45; semicolons, 15.7, 15.22.15.24.15.28.15.30 for quotations, 13.64-72; block quota tions, 13.70-72,15.26; in brackets, 13.7,15.28,15.40; “brackets in the original” in, 6.99; epigraphs, 1.37, 1.49,13.36,14.52; errors in original source, 13.7; frequently cited works, 13.67; full, 13.65; ibid, in, 13.66; non English materials (see non-English materials: quotations in English context); parentheses with, 13.64; placement of, 15.26; poetry, 2.20, 13.71-72; run-in quotations, 13.68-69, 15.26; shortened, 13.66-67,13.70 reducing notes by using, 14.34,14.58 in relation to surrounding text, 15.25-26 shortened citations in, 13.66-67,13.70, 15.29,15.34 and syntax, 15.28 year plus a, b, c in, 15.20,15.29 See also author-date reference system; reference lists; source citations; titles of works in source citations text messages, 14.214. See also personal communications text references to illustrations and tables vs. callouts, 2.30, 3.8, 3.51 cross-checking of, 2.32,2.65,2.66 and electronic supplementary data, 3.9 format for, 2.30 to numbered illustrations and tables, 3.9, 3.50,8.180 to parts of illustration, 3.12 and placement of feature, 3.8, 3.51 See also callouts text-to-speech tools, 2.82nl, 3.28,7.49, 12.2nl Thai names, 16.85 than, 5A6, 5.183,5.196,5.201 that antecedents of, 5.59, 5.61 in cleft sentences, 5.246 as demonstrative pronoun, generally, 5.52 instead of the fact that, 5.250 in phrasal conjunction (e.g., provided that), 5.197 possessive form of, 5.63 prepositions with, 5.172 punctuation with, 6.27 quotations following, 6.40 as relative pronoun, generally, 5.56 in remote relative clauses, 5.60 as subordinating conjunction, 5.201 vs. which, 5.250, 6.27 that if, 6.26 that is, 6.51, 6.58, 6.88 the in blog titles, 8.192 and coordinate nouns, 5.75 definition of articles, 5.70, 5.71 disregarded in alphabetizing, 14.71, 14.79,14.80,16.48,16.91 with earth, 8.140 with epithets, 8.34 with ethnic and national group names, 8.38 with honorifics, 10.18 and index entries, 16.48-49,16.88, 16.91,16.144 meaning affected by, 5.75, 5.76, 5.77 with names of celestial bodies, 8.138 with non-English words and phrases, 8.59,11.4 INDEX the {continued) in periodical titles, 8.170,14.170,14.188 in personal names, 8.5 in place-names, 8.45, 8.59,16.91 in publishers’ names, 14.134 in titles in running text, 8.169,8.170 with titles of people, 8.21,8.30,10.18 in website titles, 8.191 See also articles (definite and indefinite) their, theirs, 5.49,5.50,5.250 their, they're, there, 5.50,5.250 Thema, 1.75 themself, 5.48 then, for and then, 6.23, 6.57 theorems, numbering of, 12.25,12.56 theories. See laws and theories there, 5.115, 5.239,5.241,5.246,5.250 there, their, they're, 5.50,5.250 therefore, 5.250, 6.49, 6.57 theses. See dissertations and theses they and antecedents, 5.30 as indefinite, 5.47 personal identification with, 5.256 singular use of, 5.48,5.48n5, 5.252, 5.256 thick spaces, 12.23 thin spaces defined, p. 989 in French materials, 11.28-29,11.31-32 in mathematical expressions, 12.17, 12.42 for overlapping characters, 6.5 in poetry quotations, 13.29 between single and double quotation marks, 6.11, 6.120 though, 5.201 3-em dashes and computer sorting, 14.67,15.17 keyboarding for manuscript, 2.14 length of, 6.75 marking manuscript for, 2.96 not appropriate in manuscript bibliogra phies, 14.67 for repeated names in source citations, 1.124,2.14, 6.94,14.65,14.67-71, 14.79,15.17-20 through, 5.205, 6.78,7.8, 9.60 thumbnails, 1.96,1.107,3.8,3.51,4.90, p. 989 thus, 5.250, 6.57 TIFF, p. 989 time designations abbreviations: a.m. andp.m., 9.37, 9.38,10.41; chronology systems, 9.34,9.64,10.38; days of the week, 10.40; eras, 9.34,9.64,10.38; months, 10.39,14.171,14.224,15.15; numerical designations, 9.30, 9.33, 9.35,9.37, 9.39, 9.40; time of day, 10.41; time zones, 8.90,10.41; units (seconds, minutes, etc.), 10.68; and word division, 7.44 adverbs indicating, 5.166 capitalization, 8.88-90 chronology systems, 9.34,9.64,10.38 hyphenation, 7.89 (sec. 1) international systems, 10.41 periods of, 8.71-74; centuries, 7.8,7.87, 7.89,7.89 (sec. 3), 8.71,9.32,9.33; decades, 9.33; descriptive designa tions, 8.72; eras, 8.135,9.34, 9.64, 10.38; geological, 8.135; numerical designations, 8.71; prehistoric cul tural periods, 8.74; traditional period names, 8.73 spelled out, 7.89 (sec. 1), 9.37,9.38, 9.40 subordinating conjunctions for, 5.201 time of day: abbreviations, 10.41; en dashes with, 6.78; noon and midnight, 9.38; seconds and dates included with, 9.40; twenty-four- hour system, 9.39, 9.40 time zones, 8.90,10.41,14.191 See also dates timeless facts, 5.129 time stamps, 14.191,14.207,14.209,14.233 titled persons, 8.32,16.38 title page, 1.17-19 across two pages, 1.18 author of foreword on, 1.40 authors’, editors’, and translators’ names on, 1.19,14.73,14.82,14.101,15.12 distribution information on, 14.141 editing of, 2.57 edition number on, 1.19,1.26,14.113 journal special issue, 1.89 multiauthor volumes, 2.41 new preface noted on, 1.26 in page sequence, 1.4 pagination omitted from, 1.7 INDEX title page (continued) permissible changes to title when citing, 8.165,14.88,14.91 place of publication on, 14.129 proofreading of, 2.134 publication date on, 1.19,14.142 publishers’ names on, 1.19,14.129,14.133 running heads omitted from, 1.11 submission of, 2.3 titles and offices of people abbreviations: academic and profes sional designations, 7.15,10.4,10.16, 10.21,10.22; civil titles, 10.13,10.14; honorifics, 10.18; military titles and ranks, 8.24,10.13,10.15; Saint, 10.20; social titles, 10.4,10.16,10.17 academic, 8.28-31,16.40 in apposition (before name), 8.21, 8.30, 8.36 Arabic rulers of older times, 8.14 capitalization, 8.19-26, 8.28-33; aca demic titles and offices, 8.28, 8.29, 8.31; appositives, 8.21,8.30,8.36; basic rule and exceptions, 8.19-20; civic and academic honors, 8.31; civil titles, 8.19,8.22; descriptive titles, 8.30; “down style” for, 8.1; honorifics, 8.26, 8.33; kinship terms, 8.36; military and quasi-military titles, 8.19,8.24-25; nobility, 8.32; in non-English languages, 11.46,11.61; organizational and corporate titles, 8.27; popes, 8.26; religious titles, 8.19,8.26,8.33,8.36; sovereigns and other rulers, 8.23 civil, 8.19, 8.22,10.13,10.14,14.284 names replaced by, 8.19,8.20 nobility, 8.32,16.38 President, president, 8.19,8.21,8.22, 8.28 religious: abbreviations, 10.18; cap italization, 8.19, 8.26, 8.33,8.36; indexing, 16.39 saints, 8.93,10.20,11.26,14.83,16.42 social titles, 10.4,10.16,10.17 sovereigns and other rulers: indexing, 16.37,16.66,16.98 tfowith, 8.21, 8.30,10.18 (see also honorifics) titles of works, 8.156-201 abbreviations of: italics for, 10.7,14.60; resources on, 15.46; vs. short titles, titles of works (continued) 14.59; in source citations, 14.59-60, 14.170,15.13,15.46, fig. 14.7 alphabetizing, 16.51-54,16.56 articles at beginning of, 8.169-70, 8.192, 14.98,14.170,14.193-94,16.48-49, 16.51-52,16.56 artworks and exhibitions, 8.57, 8.198- 201,14.86,14.235 in award and building names, 8.172 blogs and blog posts, 8.192,14.206, 14.208 in book review and book notes sections, 1.98 capitalization, 8.157-62; basic prin ciples, 8.157; “down style” for, 8.1; electronic mailing list subjects and threads, 14.210; exhibitions, 8.201; forms, 8.187; headline-style, 8.159- 60 (see also headline-style capital ization); hyphenated compounds in headline-style, 8.161; magazine articles, 14.188; manuscript collec tions, 14.223; musical works, 8.194, 8.195,8.196; non-English works (see capitalization of non-English works below); part and chapter titles, 2.58; poetry, 8.182; quotations in titles, 8.162; sentence-style, 8.158 (see also sentence-style capitalization); series and editions, 8.176; subtitles, 8.158, 8.159,8.161, 8.164,14.87,14.89, 14.90,14.91; unpublished and in formally published materials, 8.188; volume, 8.175 capitalization of non-English works: Chinese and Japanese, 11.89; En glish translations, 11.9,11.10,14.99; French, 11.27; generally, 11.6,14.98, 15.6; Latin, 11.54; newspapers, 14.194; series titles, 14.123 in captions, 3.21,3.22 collected works of author, titles within, 8.178 conferences, meetings, or speeches, 8.70 as copyright management information, 4.15 on covers or jackets, 1.69,1.70 cross-checking of, 2.32 dissertations and theses, 8.188,14.215 INDEX titles of works (continued) double titles connected by or, 8.167, 14.91 in epigraph source, 1.37 folktales, fables, nursery rhymes, and such, 8.185 forms, 8.187 inclusive years in, 9.64 indexing (see under indexing) italicized terms within, 8.173 italics for: abbreviations, 10.7,14.60; in abbreviations lists, 14.60; artworks, 8.57, 8.198-201,14.86,14.235; blogs, 8.192,14.86,14.206,14.208; book length work within book, 14.109; books, generally, 8.163, 8.168, 14.86; folktales, fables, nursery rhymes, and such, 8.185; generally, 8.2, 8.163,15.6; journals, 8.168, 14.86,14.170,15.9; legal cases, 8.82, 14.272,14.276; long poems, 8.181; magazine Journal, review, and such, 8.171; movies, television, radio, and podcasts, 8.189; newspapers, 14.86, 14.193; in non-English languages, 11.8,11.89,11.100,14.99; operas and long musical compositions, 8.194; pamphlets and reports, 8.186; peri odicals, generally, 8.163, 8.168; plays, 8.183; recordings, 8.197; in shortened citations, 14.33; standards, 14.259; vessels and vehicles, 8.116; video games, 14.86; within collected works of author, 8.178 journal articles (see under journal articles) journals (see under journals) journal supplements, 14.179 magazine Journal, review, and such, 8.171 magazine articles, 14.188 magazines, 14.190 maps, 8.199,14.237 as metadata, 1.12,1.75,1.92 movies, television, radio, and podcasts, 8.189,14.86 multivolume works, 8.175 musical works, 8.193-97 newspapers (see under newspapers) non-English materials (see under non English materials) titles of works (continued) pamphlets and reports, 8.186 periodical articles and parts of books, 8.177-80 permissible changes to, in citations, 8.165,8.167,11.7,14.88,14.91 for permission requests, 4.95 plurals of, 7.12 poems and plays, 8.181-84 poetry, 8.181,14.86 possessives of, 7.29 in production checklist, 2.77 proofreading of, 2.108,2.134 punctuation: colons, 1.19,8.164-65, 14.89-91,14.93,14.96,14.169; commas, 6.17,8.165, 8.167,11.7,14.91, 14.93,14.229,16.48,16.49; double titles connected by or, 8.167,14.91; exclamation points, 6.124-26,14.92, 14.96,16.54; question marks, 6.124- 26,14.92,14.96,16.54; quotations in titles, 8.162; in relation to surround ing text, 6.41,8.166; semicolons, 8.165,8.167,11.7,14.90-91 quotation marks for: in bibliogra phies, generally, 14.21; chapters, 8.163,14.23,14.86,14.106-7,15.9; conferences and speeches, 8.70; dissertations and theses, 14.215; electronic mailing list subjects and threads, 14.210; folktales, fables, nursery rhymes, and such, 8.185; generally, 8.2, 8.163,15.6; journal articles, 8.163, 8.177,14.86,14.169, 14.188,15.9; magazine articles, 14.188; in non-English languages, 11.8,11.89; in notes, generally, 14.20; poems, 14.86; in shortened citations, 14.33; titles within titles, 8.173,14.94; unpublished materials, 8.188,14.216; website sections and blog posts, 8.191, 8.192,14.206,14.208 quotations in, 8.162,14.94 as running heads, 1.12,1.83 in running text, 8.166 sequels, 8.189, 9.43 series and editions, 8.2,8.176 shortening of, 8.168,13.58 (see also shortened citations) source citations (see titles of works in source citations) INDEX titles of works (continued) subjects not interchangeable with, 8.174 titles within titles, 8.173 trademark protection of, 4.16 unpublished materials, 8.188,14.216 video games, 8.190 websites and web pages, 8.2,8.191-92, 14.206-7,14.272 See also bibliographies; chapter titles; covers of journals and paperback books; half titles; reference lists; title page; titles of works in source citations titles of works in source citations, 14.85-99 abbreviations, 14.59-60,14.170,15.13, 15.46, fig. 14.7; frequently cited works, 13.67,14.59-60, fig. 14.7 articles in, 8.170 author-date reference system: abbrevia tions, 15.13,15.46; anonymous works, 15.34; capitalization, 15.6,15.13; examples and variations, 15.9; gener ally, 15.13; italics or quotation marks for, 15.6,15.9; resources on, 15.46; sentence-style capitalization, 15.38 capitalization: book titles and subtitles, 14.87,14.89-91,14.97; electronic mailing list subjects and threads, 14.210; journal articles, 14.169, 15.38; journals, 8.168,14.86,14.170, 15.9; magazine articles, 14.188; manuscript collections, 14.223; non- English-language works, 14.98-99, 14.123,14.170,14.194,15.6; religious works, 8.103-7; specific vs. generic titles for manuscript collections, 14.223; website and blog titles, 14.206 italics for: in bibliographies, generally, 14.21; blogs, 14.86,14.206,14.208; book-length work within a book, 14.109; books, generally, 14.86; journals, 14.86,14.170; legal cases, 14.272,14.276; newspapers, 14.86, 14.193; in non-English languages, 14.99; in notes, generally, 14.20; in shortened citations, 14.33; standards, 14.259; video games, 14.86 legal-style citations: articles and chapters, 14.272; hearing testimony, titles of works in source citations (continued) 14.284; international entities, 14.305; legal cases, 14.272,14.276; legislation, 14.295,14.300; legislative debates, 14.286-87; legislative reports and documents, 14.285; treaties, 14.290 notes and bibliography system: abbre viations, 14.59-60,14.170, fig. 14.7; alphabetical order for works of single author, 14.71; and other stories and such with, 14.92; anonymous works, 14.79,14.212; author’s name in title, 14.78; capitalization, 14.20-21,14.87, 14.89-90; chapter titles, 14.23,14.86, 14.106- 7,14.272; classical Greek and Latin works, 14.245,14.247,14.248; dates in titles, 14.93; dissertations and theses, 14.215; double titles with or, 14.91; electronic mailing list subjects and threads, 14.210; exclamation points in, 14.92,14.96; footnotes vs. endnotes, 14.48; itali cized terms within (reverse italics), 14.95; magazine articles, 14.188; magazines, 14.190; manuscript collections, 14.222,14.223,14.226; maps, 14.237; multimedia content, 14.261,14.263; non-English works, 14.33,14.89,14.93,14.98-99,14.123, 14.194; older and very long titles, 14.97; periodicals, generally, 14.165; permissible changes, 14.88,14.91; question marks in, 14.92,14.96; reviews, 14.201,14.202; series and editions, 14.123-24; social media content, 14.206,14.209; standards, 14.259; subtitles, 14.87-91,14.165; titles within titles, 14.94; translated titles, 14.99; unpublished materials, 14.216; websites and web pages, 14.206-8,14.272 (see also italics for above-, quotation marks for below) quotation marks for: in bibliographies, generally, 14.21; chapters, 14.23, 14.86.14.106- 7; dissertations and theses, 14.215; electronic mailing list subjects and threads, 14.210; journal articles, 14.86,14.169,14.188; magazine articles, 14.188; in notes, INDEX titles of works in source citations (continued) generally, 14.20; poems, 14.86; in shortened citations, 14.33; titles within titles, 14.94; unpublished materials, 14.216; website sections and blog posts, 14.206,14.208 series titles, 14.123 to in date ranges, 7.8 en dash for, in inclusive numbers, 6.78, 9.60 in headline-style capitalization, 8.159 in infinitive, 5.106, 5.108 with ought, 5.250 for ratios, 9.58 in source citations for letters, 14.228 together with, 5.142 tone marks, 11.84 too, 6.52 totals, in tables, 3.53, 3.63, 3.74-75,3.81, figs. 3.15-16, fig. 3.21 tracking changes (redlining) ambiguity in, 2.84 author’s review of, 2.71,2.89 in author’s review of copyedited manu script, 2.88 conventions of, 2.15,2.84 defined, p. 986 vs. document comparison software, 2.86 example of, fig. 2.4 explained in cover letter, 2.70 removed before submission of manu script, 2.39 when not to use, 2.79,2.85 See also manuscript editing; word processing software trade books, 4.18 trademarks, 4.16, 8.69, 8.130, 8.153-54, 10.6. See also brand names trade paperbacks, 4.64 trailing slashes, in URLs, 14.17 trains, 8.117. See also vehicles and vessels trans, as prefix, 7.89 (sec. 4) transcriptions, 2.61,2.80,10.11,13.6,13.48-49. See also American Sign Language transgender, 8.41 transitive verbs, 5.19,5.98,5.118 translated by, abbreviated or spelled out, 14.20-21,14.103-4,15.6,15.9,15.15 translations and translated works copyright of, 1.20,1.29,4.14,4.75, fig. 1.3 credit for, 11.14 dedication, 1.36 non-English terms in English context, 6.96, 6.100,11.4-5,11.18-19; capital ization, 11.18; glosses, 6.96, 6.100, 11.5; proper names, 11.4; punctua tion, 11.19 punctuation, 6.96, 6.99, 6.100,11.12, 11.19 quotations: including or omitting trans lation, 11.12; paraphrasing vs. transla tion, 11.17; placement of translations, 11.12; retranslation caution, 11.17; source plus translation, 11.13; trans lation adjusted, 11.15; translation credited, 11.14; translation edited, 2.61,11.16; translation plus source, 11.13; typographic style of, 11.11 retranslation, 11.17 rights for, 4.34,4.64 source citations for, 14.99,14.183 subsidiary rights for, 4.34,4.64 titles of works, 11.9,11.10,14.99, fig. 1.3 translator’s notes on, 1.45,14.51 as works made for hire, 4.10 translators biographical notes on, 1.66 notes by, 1.45,14.51, fig. 14.4 See also editors; manuscript preparation guidelines for authors; translators’ names translators’ names on covers or jackets, 1.70 in source citations: abbreviation of term, 10.42,14.20-21,14.103-4,15.6,15.15, 15.19; vs. authors’ names, 14.104; classical Greek and Latin works, 14.246; examples and variations, 14.23,15.9; for journal articles, 14.183; omission of term in text citations, 15.7,15.22,15.36; in place of author, 15.36; shortened citations, 14.32 on title page, 1.19 transliteration and transliterated text, 11.71-75 citing religious works in, 14.241 italics vs. roman type for terms in, 11.75, 11.88-89,11.110 INDEX transliteration and transliterated text (continued) proper names, 11.4 resources on, 11.71,11.74,11.76,11.82-83,11.86,11.91,11.98 and special characters, 2.80,11.2,11.72, 11.74,11.90 titles of works and institutions, 11.6-10 Unicode for non-Latin alphabets, 11.2, 11.72-73,11.90,11.110,11.115,11.122, tables 11.2-5 See also Arabic language; Chinese language; Cyrillic alphabet; Greek alphabet; Hebrew language; Japanese language; Russian language; South Asian languages transportation. See business and commerce; roads and thoroughfares; vehicles and vessels transpose mark, 2.95, 2.128 transposition, marking proofs for, 2.128 treaties and acts source citations for, 14.275,14.290 treatment in text, 8.66, 8.80,8.81 See also legal and public documents trillion, 5.250 trim size, p. 989 Turkish language, 11.70 Turkmen language, 11.70 turn of the century, 9.32 twenty-four-hour system, 9.39,9.40 Twitter, 8.191. See also social media content 2-em dashes length of, 6.75 with letters standing for names (e.g., A---- ), 7.66 marking manuscript for, 2.96 for missing or illegible words, 6.93, 13.59 typefaces. See fonts and typefaces type page, p. 989 typescripts, abbreviation for, 14.226 typesetters and typesetting adjusting and checking tables (check list), 3.86 definition of typesetter, p. 989 errors assigned to, 2.135 Greek alphabet, 11.111 and indexes, 16.134 instructions to, 2.74,2.81,2.93,2.116, 2.118,2.121 typesetters and typesetting (continued) musical examples, 3.4, fig. 3.5 See also fonts and typefaces; typographic considerations type size appendixes, 1.59 bibliographies and reference lists, 1.63 block quotations, 2.19 endnotes, 1.62 indexes, 1.65,16.135 subheads, 2.18 title and subtitle, 1.19 word-processing styles for, 2.82 See also typographic considerations type styles. See boldface; capital letters; fonts and typefaces; italics; roman (type); typographic considerations typewritten manuscripts, 2.6,2.8,2.37, 2.38, 2.43, 2.45 typographic considerations abbreviations: italic vs. roman type, 10.7; line breaks, 7.44; small caps vs. lowercase, 10.41; small vs. full-size capitals, 10.8,10.38; spacing, 10.5 appendixes, 1.59 captions, 3.9,3.22, 3.23, 3.24, 3.30 charts and graphs, 3.42 computer terminology, 7.79 digital artwork submission, 3.15 epigraphs, 13.36 epilogues and afterwords, 1.54 foreword, 1.40 glossaries, 1.61 illustration numbers, 3.9 indexes (see under indexes) journals, 1.84,1.115 letters as shapes, 7.67 list format, 6.128 for manuscript formatting, 2.7, 2.18, 2.19,2.20 markup for, 2.74 mathematical copy, 12.61-68 mathematical expressions, 12.4 notes keyed to text by page numbers, 14.53, fig. 14.6 ornaments for text breaks, 1.58 permissible changes, 13.8 prefaces, 1.41 and proofreading, 2.113 publisher’s, translator’s, and editor’s notes, 1.45 INDEX typographic considerations (continued) punctuation, 6.2-6, 6.14 quotations from non-English-language materials, 11.11 single-glyph fractions, 6.110 subheads, 1.56,2.59 subtitles, 8.164,14.91 table of contents and list of illustrations, 1.39 tables, 3.49,3.54,3.60, 3.63, 3.86 user-defined typography in electronic publications, 1.123 See also boldface; design; fonts and typefaces; italics; special characters; type size ultra, as prefix, 7.89 (sec. 4) un, as prefix, 7.89 (sec. 4) uncertainties, 12.60 undated works, n.d. (no date) for, 10.42, 14.145,14.263,15.18,15.44,15.50, 15.54. See also access dates; publica tion date under, as prefix, 7.89 (sec. 4) underlining (underscore), 2.15,2.80,2.84, 2.97,7.51,12.63,12.65,13.8 underscore. See underlining unexpressed (blind) folios, 1.6,1.7,1.8,1.39, I. 46,1.49,3.6, p. 980 Unicode standard characters not available in, 2.16,2.137 defined, p. 989 directional or “smart” quotes, 6.115, 6.117 en dash and minus sign distinguished, 6.84 and generic markup, 2.81 and IPA notation, 11.22 mathematical signs and symbols, 12.9, 12.12, table 12.1 non-Latin alphabets, 11.2,11.72-73,11.90, II. 110,11.115,11.122, tables 11.2-5 “smart” apostrophes, 6.115, 6.117 spaces defined in, 6.120, 6.121 special characters with Latin alphabet, 11.2,11.21,11.22, table 11.1 uniform resource locators. See URLs unions, 8.70,9.49 United Kingdom CIP (Cataloging-in-Publication) data, 1.34 United Kingdom (continued) currency, 9.20, 9.22, 9.25 Greenwich mean time, 10.41 legal and public documents, 14.297- 303; command papers, 14.302; legal cases, 14.298; parliamentary debates, 14.301; parliamentary publications, 14.299; resources on, 14.297,14.300; statutes, 14.300; unpublished docu ments, 14.303 units of measurement, 10.67 See also British style United Nations, 8.62,14.290,14.304, 14.305 United States legal and public documents: bills and resolutions, 14.283; congressional reports and documents, 14.285; debates before 1873,14.287; debates since 1873,14.286; hearings, 14.284; legal cases, 14.277-79; legislation, 14.282,14.288; municipal ordinances and state laws, 14.288; presidential documents, 14.289; secondary sources on, 14.291; unpublished, 14.292 as word, 7.20,10.32 See also governmental entities; legal and public documents; states (US); and headings beginning with US United States Code (U.S.C.), 14.282,14.289 units of measurement abbreviations for, 10.51-59; derived units, 10.57,10.60; grams, 10.55; non-SI units accepted by SI, 10.59; numbers with, 7.89 (sec. 1), 9.16-17; periods with, 10.64; plurals of, 10.65, 10.67; prefixes, 9.10, 9.11,10.49, 10.55,10.56; referred to as symbols in SI, 10.2; repeated quantities, 9.17; SI, generally, 10.51,10.52; SI base units, 10.54; in tables, 3.56, fig. 3.12; technical, list of, 10.49; US measure, 10.64-68 length, area, and volume, 10.66 numbers with, 7.44,9.16-17,10.49 plurals of, 9.19,10.65,10.67,10.68 spacing with, 6.119, 6.121,7.44, 9.16 symbols for, 3.56, 9.16-17,10.52,10.54- 58, fig. 3.12 in tables, 3.56, figs. 3.12-13 INDEX units of measurement (continued) United Kingdom, 10.67 weight and capacity, 10.67 See also numbers Universal Product Code. See bar codes universities abbreviation of term, 10.42,14.134, 15.15 names of, 8.68 nonexclusive licenses claimed by, 4.67 in publishers’ names, 14.134 student status terms, 8.29 See also academic concerns unless, 5.201 unpublished and informally published materials copyright, 4.26,4.48 defined, 14.146 fair use of, 4.88 permissions for, 2.2,2.3,4.81,13.3 preprints, 1.113,14.173, p. 985 source citations for: article not yet accepted, 14.146,14.172; books not under contract, 14.146,15.45; disser tations and theses, 14.215; file format noted for, 14.225; generally, 14.216; interviews and personal communi cations, 14.211-12,14.214,15.53; lec tures, papers, and posters presented at meetings, 14.217; legal-style citations, 14.292,14.296,14.303; manuscript collections, 14.221-31, 14.255,15.54; musical works, 14.256; permanent records retained by au thor, 14.15,14.207,14.209; preprints, 14.173; private documents, 14.219; websites and blogs, 14.205-10,15.51 titles of, 8.188,14.216 working papers, 14.218 See also blogs and blog posts; disserta tions and theses; interviews and dis cussions; legal and public documents; letters (correspondence); manuscript collections; personal communica tions; websites and web pages until, 5.202, 9.60 up, in compound terms, 7.89 (sec. 2) Upanishads, 8.103,14.241 uppercase, p. 989. See also capitalization; capital letters “up style.” See headline-style capitalization URLs (uniform resource locators) abbreviation of, 10.49 additional publishing information with, 14.7 on copyright page, fig. 1.1 cross-checking of, 2.3,2.32,2.34 defined, p. 989 DOI-based URLs defined, 1.33,1.79,14.8 embedded in electronic publications, 1.121 and line breaks, 2.13, 6.113,7.46,14.18 manuscript preparation guidelines, 2.3, 2.13 permalinks, 14.9,14.15 recommended use in citations, 14.6, 14.16,14.161,15.4 resources on, 14.6nl short forms for, 14.10 for specific types of material: abstracts, 14.186; artworks, 14.235; blogs, 15.51; book reviews, 14.202; books, 14.100, 14.161; databases, vs. names, 14.9, 14.11,14.161,14.164,14.175; disser tations and theses, 14.215; electronic mailing list subjects and threads, 14.210; electronic supplementary data, 14.112,14.187; individual reference entries, 14.234; interviews, 14.213; journal articles, 1.83,14.23, 14.175,15.9 (see also journal articles, source citations for); journal home pages, 1.84; journal special issues and supplements, 14.178-79; lectures and papers, 14.217; legal and public documents, 14.270,14.276,14.305; magazine articles, 14.189; manu script collections, 14.221; multi media content, 14.261; newspaper articles, 14.191; online reference works, 14.233; periodicals, generally, 14.164-65; publications in multiple formats, 14.16; scanned electronic editions, 14.162; scientific databases, 14.257; social media content, 14.10, 14.209; standards, 14.259 treatment in text: hyphenation, 2.13,7.46, 14.18; line breaks, 2.13,6.113,7.46, 14.18; punctuation, 2.13,6.8,6.104, 6.112,7.46,14.17-18; syntax, 14.17 as website identifier, 8.191 See also access dates; DOIs; electronic publications; hyperlinks INDEX usage. See word usage US Congress abbreviations for, 14.281,14.283 bills and resolutions, 14.283 copyright provisos of, 4.26,4.29,4.40 debates before 1873,14.287 debates since 1873,14.286 documents and reports, 14.285 hearings, 14.284 members of, 8.22 numbered electoral districts, 9.46 numbers for successive terms and sessions, 9.45 presidential vetoes, 14.289 terms for, 8.62 See also acts and treaties; legal and public documents US Constitution, 8.80, 9.28,14.280 US Copyright Office, 4.4,4.19,4.27,4.47-50, 4.82. See also copyright use-case documents, 2.138,2.139 user interface specifications, 2.139 US House of Representatives, 8.22, 8.62,14.281,14.283. See also US Congress US ISSN Center, 1.32 US Postal Service, 10.33,10.34. See also postal codes US Senate, 8.22, 8.62,14.281,14.283. See also US Congress US Supreme Court, 4.93,14.277 US vs. United States, 10.32 UTC (coordinated universal time), 10.41 Uzbek language, 11.70 v., 8.82,10.42,14.276 van, 8.122, 8.123 variables, in tables, 3.52 vector graphic, pp. 989-90 vectors, 12.52,12.53 Vedas, 14.241 vehicles and vessels, 8.2, 8.116-18, 9.44 verbal nouns, 5.107. See also infinitives verb phrases, 5.104, 5.171,5.239,5.245,7.89 (sec. 2) verbs, 5.97-155 adjectives as, 5.95 auxiliary (helping) (see auxiliary (help ing) verbs) conjugation: past participles, 5.100, 5.110, 5.119 (see also participles); verbs (continued) present participles, 5.110,5.118,7.41 (see also participles) definitions: auxiliary verbs, 5.103, 5.144; contractions, 5.105; ergative, 5.99; generally, 5.97; linking (copula or connecting), 5.101; phrasal, 5.102; principal and auxiliary (modal), 5.103,5.144; regular and irregular, 5.100; tenses, 5.128; transitive and intransitive, 5.98 -ed vs. -t for past tense and past partici ple, 5.100 finite, 5.35,5.138 gerunds, 5.112-13,5.116,7.28,7.41,7.89 (sec. 2) and indefinite pronouns, 5.67 infinitives, 5.44, 5.106-9, 5.112, 5.153, 5.171 as interjections, 5.208 linking (copula or connecting), 5.45, 5.82,5.93,5.101,5.154,5.170, 6.30 with mass nouns, 5.7 misleading connectives with, 5.142 mood: defined, 5.120; imperative, 5.122, 5.155, 5.160, 5.214,5.255; indicative, 5.121,5.122,5.124,5.129, 5.130; and modal auxiliary verbs, 5.144; sub junctive, 5.123-27 nouns as, 5.25,5.250 nouns formed from (nominalizations), 5.189 with nouns in objective case, 5.19 objects of, 5.35, 5.37, 5.44,5.225, 5.239, 6.67 participles (see participles) and personal pronouns, 5.44, 5.45 phrasal, 5.102,5.181 phrasal adjectives following, 5.93 properties, 5.117-43; agreement of subject with, 5.138,5.139,5.141,5.143, 5.205, 5.250; mood (seeabove)', num ber, 5.137, 5.138, 5.139,5.140,5.141, 5.142, 5.205; overview, 5.117; person, 5.136, 5.138,5.143; and predicate nominative, 5.141; tenses (see below); voice (see active voice; passive voice) in references to works, 5.129, 8.166 tenses: and assimilation of quoted material into text, 13.12; of be-verbs, 5.155; defined, 5.128; future, 5.131; INDEX verbs (continued) future perfect, 5.134; past, 5.100, 5.125,5.126,5.130, 5.133; past indica tive, 5.130; past perfect (pluperfect), 5.126,5.127, 5.133; present, 5.125, 5.129; present indicative, 5.129; present perfect, 5.132; progressive (imperfect, continuous), 5.119,5.128, 5.135 verb phrases, 5.104, 5.171, 5.239, 5.245, 7.89 (sec. 2) verb stem (present indicative), 5.129 See also adverbs; be-verbs vernacular names, 8.128-30 version control and archival practices, 1.114 document comparison software for, 2.86 journal articles, 1.112 in manuscript editing, 2.5,2.78,2.79 multiauthor volumes, 2.41 and preprints, 1.113 in proofreading, 2.103,2.105,2.110 version numbers, 1.28 verso pages abbreviation, 14.154-55 always even-numbered, 1.5 blank after half-title, 1.4,1.18 defined, p. 990 in folio editions, 14.155 last page of journal, 1.81 page after second half title, 1.46 parts of book starting on, 1.4,1.18 proofreading for correct use of, 2.114-15 running heads, 1.11-15 in signed signatures, 14.154 specified on final manuscript for produc tion, 2.75-76 See also page numbers; pagination; recto pages vertical bars in mathematical expressions, 12.26, 12.28,12.33-34,12.51 in notes to tables, 3.79 vertical lists abbreviations in, 10.29 as alternative to tables, 3.51 as block quotations, 13.10 manuscript preparation guidelines, 2.21 in mathematical expressions, 12.16, 12.18 placement of, 2.62 vertical lists (continued) punctuation and format, 2.11,2.21, 6.92, 6.129-31, 6.132 vs. run-in, 6.128, 6.129, 6.131 very, 7.89 (sec. 2) very, much, 5.90,5.250 vice, as prefix, 7.89 (sec. 3) video blogs, 8.192 video games, 8.190,14.86,14.268. See also multimedia content video recordings clips and files, 1.107,1.114,2.4 numbering of, 3.10 source citations for, 14.265 See also multimedia content Vietnamese names, 16.86 virgule. See slashes viz. (videlicet), 10.42 voice active, 5.19,5.115, 5.118,5.119, 5.135, 5.192 passive: and dangling participles, 5.115; defined, 5.118; infinitives in, 5.106; linking verbs in, 5.101; long (with by), 5.192; with past participle, 5.119, 5.135, 5.154 volume editor. See editors; editors’ names; manuscript preparation guidelines for authors; multiauthor volumes volumes and volume numbers abbreviation of volume: in bibliographies and reference lists, generally, 14.21, 15.6,15.15; generally, 10.42,14.150; multivolume works, 8.175,14.116; in notes, generally, 14.20; omission of, 14.152; plural of, 7.15; series of works, 14.123; in text citations, generally, 15.23 defined, p. 990 front matter in series of works, 1.18 journals: arabic numerals for, 9.27; in copyright line, 1.103; on cover, 1.84, 2.109, 2.134; current, on home page, 1.86; defined, 1.80; issue numbering within, 1.80; issues identified by, 1.79; as metadata, 1.92; omission of, 1.80; proofreading of, 2.134; and retractions, 1.91; in source citations, 14.23,14.171,14.177,15.9; in table of contents, 1.87, fig. 1.11 pagination within, 1.81, fig. 1.11 INDEX volumes and volume numbers (continued) punctuation with, 6.62,14.23 in source citations: arabic numerals for, 14.147,14.171; generally, 14.165, 15.23; journals, 14.23,14.171,14.177, 15.9; legal and public documents, 14.276,14.287,14.294,14.301, 14.303; multivolume works, 14.116, 14.118-19,15.41; page numbers with, 14.23,14.116,14.152,14.177,15.23, 15.48; text citations, 15.23; when to omit vol., 14.152 treatment of term volume, 8.175 (see also abbreviation of volume above) See also multivolume works; series of works von, van, van den, 8.8,8.10 vs., v., 10.42 Wade-Giles system, 11.82,11.84-85,16.77.
See also Chinese language; Pinyin system wars, revolutions, battles, and such, 8.113- 14 we, you, they, 5.47 web (term), 7.80,7.89 (sec. 3) web-based publications. See electronic publications; online publications; websites and web pages web browsers, p. 990 web-fed presses, p. 990 websites and web pages of authors, 1.113,1.121,4.66,4.67 campus, 4.66 copyright of, 4.2,4.13 corporate, as resource for usage, 8.69, 10.24 defined, 14.205, p. 990 identifying sites with no formal title, 8.191 of journals, 1.84,1.86,1.87 owner or sponsor of, 14.14,14.207, 14.267 permissions processing via, 4.97 preprints on, 1.113 of publishers, 1.21,1.59,1.68,1.118, 4.94,4.97 source citations for, 14.205-10; access dates, 14.12,14.207,15.50; author date reference system, 15.4,15.50; basic approach, 14.205; italics or websites and web pages (continued) quotation marks for titles, 14.86, 14.206,14.208; revision dates, 14.13, 14.207,15.50; site discontinued before publication, 14.207; in text vs. bibliography, 14.207,14.208; titles, 14.206-8,14.272; website, 14.207 (see also online publications: source citations for) titles of, 8.2, 8.191-92,14.206-8,14.272 See also blogs and blog posts; electronic publications; HTML; hyperlinks; online publications; URLs weight, 7.89 (sec. 1), 10.67. See also units of measurement well, 5.206, 6.34 Wellcome Trust, 4.68 West, western, 8.46,8.47 what as adverbial conjunction, 5.202 antecedent absent with, 5.30, 5.61 in cleft sentences, 5.246 with -ever suffix, 5.65 as interrogative pronoun, 5.54-55 possessive form of, 5.63 as relative pronoun, 5.56 as subordinating conjunction, 5.201 when, 5.201,5.202, 6.24, 6.69 where, 5.201,5.202 whether, 5.201,5.250, 6.40 which antecedent absent with, 5.30 antecedents of, 5.59,5.61 with -ever suffix, 5.65 as interrogative pronoun, 5.54-55 in nonrestrictive clauses, 6.27 possessive form of, 5.63,5.64 prepositions with, 5.172 punctuation with, 6.27 referent of, 5.55 as relative pronoun, 5.56 in remote relative clauses, 5.60 vs. that, 5.250, 6.27 while, 5.250 whiz-deletions, 5.226,5.247 who, whom, whose antecedent absent with, 5.30, 5.61 declension of, 5.56 distinguishing among, 5.250 with -ever suffix, 5.66 for gender neutrality, 5.255 INDEX who, whom, whose (continued) as interrogative pronouns, 5.54-55, 5.58 in nonrestrictive clauses, 6.27 objective pronouns as antecedents of, 5.57 possessive form of, 5.63,5.64 prepositions with, 5.172 as question within sentence, 6.69 referent of, 5.55 in remote relative clauses, 5.60 who vs. whom, 5.66 whoever, whomever, whichever, whatever, 5.65, 5.66,5.250 who’s, whose, 5.50, 5.250 wh- questions, 5.212 why, 5.201,5.202,5.206, 6.69 wide, compound terms with, 7.89 (sec. 3) widows (lines), 2.116,2.117, p. 990 Wikipedia, 8.191 wikis, 14.13, p. 990 wildcards, 2.79 will, would, 5.131,5.134, 5.151,5.231 wills, source citations for, 14.219 wire frames (design mockups), 2.139 with, in source citations for ghostwritten books, 14.105 without, 5.201 woman vs. lady as adjective, 5.259 word breaks. See word division word-by-word alphabetizing, 14.65,16.57, 16.58,16.60,16.61,16.123 word count, for manuscript submission, 2.4,2.52 word division, 7.36-47 abbreviated units of measure, 7.44 basic rules, 7.36,11.20 compound terms, prefixes, and suffixes, 2.112,7.40,7.42,11.44 diphthongs never divided, 7.39,11.33, 11.42,11.50,11.66,11.118 gerunds and present participles, 7.41 non-English words: Arabic, 11.81; French, 11.33-36; generally, 11.20; German, 11.42-44; Greek, 11.118-21; Hebrew, 11.95,11.96; Italian, 11.50- 52; Latin, 11.55-58; Polish, 11.70; proofreading for, 2.112; Russian, 11.70,11.104-8; Spanish, 11.66-68; two-letter word endings, 7.37 pronunciation as basis for, 7.38 proofreading for bad breaks, 2.112 word division (continued) proper names, 7.42 resources on, 7.36 typographic considerations, 7.47 URLs, DOIs, and such in text, 2.13, 6.113,7.46,14.18 after vowel, 7.39,7.42 words not appropriate for, 7.37,11.36 See also line breaks word order, 5.221-24 SVO (subject-verb-object) pattern, 5.221, 5.222 syntactic patterns listed, 5.223 syntactic variations, 5.224 See also inverted word order; syntax word-processing software for author’s review of copyedited manu script, 2.71,2.88 directional or “smart” quotes and apos trophes, 6.115, 6.117 for electronic file cleanup, 2.79 ellipsis character, 13.50 em dash character, 2.14 formatting codes, 2.63, 2.81, 2.82 functions in: character-level formatting, 2.81; commenting, 2.39,2.84, 2.87, 2.88; document comparison, 2.86; equation editing function, 12.2; global changes, 2.69,2.79,2.80, 2.85, 2.111, 2.113, 2.122; hyphenation, 2.13,2.80,7.36,7.38; indentation, 2.11,2.12, 2.19, 2.20, 2.21, 2.24, 2.80, 16.131; keyboard combinations and shortcuts, 7.78; language settings, 2.80; macros, 2.79, 2.80, p. 983; non breaking spaces, 6.121; note function, 2.22,14.24,14.41,14.43; outlines, 2.21,2.36; pattern matching, 2.79, 2.80, p. 984; search and replace, 2.79; sorting, 14.65,14.67,16.5,16.57, 16.104; spell checking, 2.89,2.111; styles, 2.18,2.21, 2.82,2.83, 2.84, fig. 2.4; tables, 2.26, 2.36, 3.47; tabs, 2.11-12, 2.19, 2.21, 2.24, 2.80 for indexing, 16.104,16.117,16.131 for manuscript editing, 2.79, 2.80 special characters, 2.16,11.45,11.72 third-party add-ins, 2.79 and Unicode standard, 11.2 See also manuscript editing; tracking changes INDEX words coinage of, 7.14 (see also words derived from proper names) emphasizing particular, 7.50-52,11.47, 13.62,14.94 en dash to connect, 6.78,6.80, 6.81 introductory (see introductory words and phrases) key terms, 7.56 missing or illegible in original source, 6.93, 6.104,13.59 non-English, italics for, 7.53,7.54,7.55, 11.3,11.5,11.75,11.88,11.94,14.272 vs. numerals (see numbers: spelled out) parallel structure in, 5.242 phonetic transcription of, 6.102 problematic, list of, 5.250 scare quotes for, 7.57,11.47,14.94 single, for questions within sentences, 6.69 so-called with, 7.59 spacing between, 2.9,2.125 stacked at end of line, 2.116 standing alone, 6.12 2-em dash to indicate omitted, 6.93 words and phrases used as, 7.63 See also compound terms; hyphens and hyphenation; non-English words and phrases; sentences; word division; word usage words as words, 5.14, 6.2,7.63 words derived from proper names abbreviations, 10.6,10.49,10.52 adjectives: from names of ethnic and national groups, 8.38; in non-English languages, 11.26,11.39,11.70; from organization names, 8.62; from personal names, 8.60, 8.61; from place-names, 5.69, 8.45; proper adjectives, defined, 5.69; religious terms, 8.96, 8.97 capitalization, 8.1, 8.60-62, 8.96-97 resources on, 8.60, 8.61 word usage, 5.249-50 consistency in, 5.14 and dialect, 5.25 good vs. common, 5.250 vs. grammar, 5.249 jargon, 5.25,5.95,5.174 myths and misunderstandings, 5.171, 5.180,5.203 word usage (continued) preposition usage, 5.195 problematic words and phrases, list of, 5.250 style guides for, 5.250 See also bias-free language; British style; European style; grammar; and specific parts of speech working papers, 14.218. See also unpub lished and informally published materials works, index entries using, 16.50 works-cited lists, 15.5. See also reference lists works made for hire contributors’ chapters as, 4.58 copyright issues, 4.9-12,4.24,4.38 credit line for, 3.33 employer as author of, 4.9 illustrations as, 3.33,4.10,4.75 world, parts of compass points and directions, 8.46, 8.47 continents, countries, oceans, 8.45 political divisions, 1.27,8.51, 8.52 popular names and epithets for, 8.48 real vs. metaphorical names, 8.50 regions, 8.47,8.54 See also place-names world rights, 4.76 world’s fairs, 8.201 World Wide Web, 7.80, p. 990. See also internet would, will, 5.131,5.134,5.151 x andy axes, 3.43,3.52, fig. 3.8 x-height, p. 990 Xhosa language, 11.23 XHTML, p. 990 XML (extensible markup language) angle brackets in, 6.104 defined, p. 990 and electronic workflow, 1.117 vs. generic markup, 2.81 for metadata, 1.75,1.92 overview, 2.83 in plain-text files, p. 985 Unicode standard required by, 11.2 and word-processing styles, 2.82 See also markup languages, formal x-ray, 8.151 INDEX XSL (extensible style sheet language), p. 990 year-month-day (ISO) date format, 6.108, 9.36 years abbreviations, 9.30, 9.64 in ages, 7.89 (sec. 1) commas in, 9.54 editorial doubt, 6.68 inclusive ranges of, 9.60, 9.64,14.144 incomplete range, 6.79 plurals of, 7.15 sentences beginning with, 9.5, 9.29 slashes with two-year spans, 6.107, 6.113 unspecified date of death, 6.79 words vs. numerals for, 9.29 work period known (/?.), 10.42 See also dates; publication date; time designations yes, 5.250,6.34 yes-no questions, 5.212,5.213 yet, 6.22. See also coordinating conjunctions you, understood, 5.51,5.122, 6.22 you,your,yours, andyou’re, 5.30, 5.41,5.47, 5.48,5.49,5.50,5.250 zero (0), 2.35,3.67,3.72, 9.19, 9.20,12.9 zero (implicit) articles, 5.77 zip codes, 10.29. See also addresses, mail ing; postal codes zombie nouns (nominalizations), 5.189 zoological terminology illustrations, fig. 3.7 resources on, 8.119, 8.128 scientific names: author names in, 8.124; divisions higher than genus, 8.126; English derivatives from taxonomic system, 8.127; genus, species, and subspecies, 8.120-24; resources on, 8.119 vernacular names, 8.128-29 See also scientific and technical termi nology The Chicago Manual of Style Designed by Matt Avery Typeset by Graphic Composition, Inc., Bogart, Georgia Book printed and bound by Edwards Brothers Malloy, Ann Arbor, Michigan Jacket printed by Phoenix Color, Hagerstown, Maryland Composed in Lyon, designed by Kai Bernau, and Atlas Grotesk, designed by Kai Bernau, Susana Carvalho, and Christian Schwartz Printed on 50# Glatfelter Offset Bound in Arrestox Linen DISTRIBUTED BY THE CHICAGO DISTRIBUTION CENTER |
13058 | https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch8/5214814-eng.htm | Skip to "About this site"
Language selection
/ Statistique Canada
Search and menus
Search and menus
Search
4 Data exploration 4.3 Frequency distribution
Table of contents
Text begins
Topic navigation
4 Data exploration
4.1 Data exploration tools
4.2 Types of variables
4.3 Frequency distribution
4.4 Measures of central tendency
4.5 Measures of dispersion
4.6 Exercises
4.7 Answers
The frequency (f) of a particular value is the number of times the value occurs in the data. The distribution of a variable is the pattern of frequencies, meaning the set of all possible values and the frequencies associated with these values. Frequency distributions are portrayed as frequency tables or charts.
Frequency distributions can show either the actual number of observations falling in each range or the percentage of observations. In the latter instance, the distribution is called a relative frequency distribution.
Frequency distribution tables can be used for both categorical and numeric variables. Continuous variables should only be used with class intervals, which will be explained shortly.
Let’s look at some examples of frequency distribution and relative frequency distribution for discrete variables.
Example 1 – Constructing a frequency distribution table
A survey was taken on Maple Avenue. In each of 20 homes, people were asked how many cars were registered to their households. The results were recorded as follows:
1, 2, 1, 0, 3, 4, 0, 1, 1, 1, 2, 2, 3, 2, 3, 2, 1, 4, 0, 0
Use the following steps to present this data in a frequency distribution table.
Divide the results (x) into intervals, and then count the number of results in each interval. In this case, the intervals would be the number of households with no car (0), one car (1), two cars (2) and so forth.
Make a table with separate columns for the interval numbers (the number of cars per household), the tallied results, and the frequency of results in each interval. Label these columns Number of cars, Tally and Frequency.
Read the list of data from left to right and place a tally mark in the appropriate row. For example, the first result is a 1, so place a tally mark in the row beside where 1 appears in the interval column (Number of cars). The next result is a 2, so place a tally mark in the row beside the 2, and so on. When you reach your fifth tally mark, draw a tally line through the preceding four marks to make your final frequency calculations easier to read.
Add up the number of tally marks in each row and record them in the final column entitled Frequency.
Your frequency distribution table for this exercise should look like this:
Table 4.3.1 Frequency table for the number of cars registered in each household Table summary This table displays the results of Frequency table for the number of cars registered in each household. The information is grouped by Number of cars (x) (appearing as row headers), Frequency (f) (appearing as column headers).
| Number of cars (x) | Frequency (f) |
| 0 | 4 |
| 1 | 6 |
| 2 | 5 |
| 3 | 3 |
| 4 | 2 |
| 0 true zero or a value rounded to zero |
By looking at this frequency distribution table quickly, we can see that out of 20 households surveyed, 4 households had no cars, 6 households had 1 car, etc.
Example 2 – Constructing a cumulative frequency distribution table
A cumulative frequency distribution table is a more detailed table. It looks almost the same as a frequency distribution table but it has added columns that give the cumulative frequency and the cumulative percentage of the results, as well.
At a recent chess tournament, all 10 of the participants had to fill out a form that gave their names, address and age. The ages of the participants were recorded as follows:
36, 48, 54, 92, 57, 63, 66, 76, 66, 80
Use the following steps to present these data in a cumulative frequency distribution table.
Divide the results into intervals, and then count the number of results in each interval. In this case, intervals of 10 are appropriate. Since 36 is the lowest age and 92 is the highest age, start the intervals at 35 to 44 and end the intervals with 85 to 94.
Create a table similar to the frequency distribution table but with three extra columns.
In the first column or the Lower value column, list the lower value of the result intervals. For example, in the first row, you would put the number 35.
The next column is the Upper value column. Place the upper value of the result intervals. For example, you would put the number 44 in the first row.
The third column is the Frequency column. Record the number of times a result appears between the lower and upper values. In the first row, place the number 1.
The fourth column is the Cumulative frequency column. Here we add the cumulative frequency of the previous row to the frequency of the current row. Since this is the first row, the cumulative frequency is the same as the frequency. However, in the second row, the frequency for the 35–44 interval (i.e., 1) is added to the frequency for the 45–54 interval (i.e. 2). Thus, the cumulative frequency is 3, meaning we have 3 participants in the 34 to 54 age group.
1 + 2 = 3
The next column is the Percentage column. In this column, list the percentage of the frequency. To do this, divide the frequency by the total number of results and multiply by 100. In this case, the frequency of the first row is 1 and the total number of results is 10. The percentage would then be 10.0.
10.0. (1 ÷ 10) X 100 = 10.0
The final column is Cumulative percentage. In this column, divide the cumulative frequency by the total number of results and then to make a percentage, multiply by 100. Note that the last number in this column should always equal 100.0. In this example, the cumulative frequency is 1 and the total number of results is 10, therefore the cumulative percentage of the first row is 10.0.
10.0. (1 ÷ 10) X 100 = 10.0
The cumulative frequency distribution table should look like this:
Table 4.3.2 Ages of participants at a chess tournament Table summary This table displays the results of Ages of participants at a chess tournament. The information is grouped by Lower Value (appearing as row headers), Upper Value, Frequency (f), Cumulative frequency, Percentage and Cumulative percentage (appearing as column headers).
| Lower Value | Upper Value | Frequency (f) | Cumulative frequency | Percentage | Cumulative percentage |
| 35 | 44 | 1 | 1 | 10.0 | 10.0 |
| 45 | 54 | 2 | 3 | 20.0 | 30.0 |
| 55 | 64 | 2 | 5 | 20.0 | 50.0 |
| 65 | 74 | 2 | 7 | 20.0 | 70.0 |
| 75 | 84 | 2 | 9 | 20.0 | 90.0 |
| 85 | 94 | 1 | 10 | 10.0 | 100.0 |
Class intervals
If a variable takes a large number of values, then it is easier to present and handle the data by grouping the values into class intervals. Continuous variables are more likely to be presented in class intervals, while discrete variables can be grouped into class intervals or not.
To illustrate, suppose we set out age ranges for a study of young people, while allowing for the possibility that some older people may also fall into the scope of our study.
The frequency of a class interval is the number of observations that occur in a particular predefined interval. So, for example, if 20 people aged 5 to 9 appear in our study's data, the frequency for the 5–9 interval is 20.
The endpoints of a class interval are the lowest and highest values that a variable can take. So, the intervals in our study are 0 to 4 years, 5 to 9 years, 10 to 14 years, 15 to 19 years, 20 to 24 years, and 25 years and over. The endpoints of the first interval are 0 and 4 if the variable is discrete, and 0 and 4.999 if the variable is continuous. The endpoints of the other class intervals would be determined in the same way.
Class interval width is the difference between the lower endpoint of an interval and the lower endpoint of the next interval. Thus, if our study's continuous intervals are 0 to 4, 5 to 9, etc., the width of the first five intervals is 5, and the last interval is open, since no higher endpoint is assigned to it. The intervals could also be written as 0 to less than 5, 5 to less than 10, 10 to less than 15, 15 to less than 20, 20 to less than 25, and 25 and over.
Rules for data sets that contain a large number of observations
In summary, follow these basic rules when constructing a frequency distribution table for a data set that contains a large number of observations:
find the lowest and highest values of the variables
decide on the width of the class intervals
include all possible values of the variable.
In deciding on the width of the class intervals, you will have to find a compromise between having intervals short enough so that not all of the observations fall in the same interval, but long enough so that you do not end up with only one observation per interval.
It is also important to make sure that the class intervals are mutually exclusive and collectively exhaustive.
Example 3 – Constructing a frequency distribution table for large numbers of observations
Thirty AA batteries were tested to determine how long they would last. The results, to the nearest minute, were recorded as follows:
423, 369, 387, 411, 393, 394, 371, 377, 389, 409, 392, 408, 431, 401, 363, 391, 405, 382, 400, 381, 399, 415, 428, 422, 396, 372, 410, 419, 386, 390
Use the steps in Example 1 and the above rules to help you construct a frequency distribution table.
Answer
The lowest value is 363 and the highest is 431.
Using the given data and a class interval of 10, the interval for the first class is 360 to 369 and includes 363 (the lowest value). Remember, there should always be enough class intervals so that the highest value is included.
The completed frequency distribution table should look like this:
Table 4.3.3 Life of AA batteries, in minutes Table summary This table displays the results of Life of AA batteries. The information is grouped by Battery life, minutes (x) (appearing as row headers), Frequency (f) (appearing as column headers).
| Battery life, minutes (x) | Frequency (f) |
| 360–369 | 2 |
| 370–379 | 3 |
| 380–389 | 5 |
| 390–399 | 7 |
| 400–409 | 5 |
| 410–419 | 4 |
| 420–429 | 3 |
| 430–439 | 1 |
| Total | 30 |
Example 4 – Constructing relative frequency and percentage frequency tables
An analyst studying the data from example 3 might want to know not only how long batteries last, but also what proportion of the batteries falls into each class interval of battery life.
This relative frequency of a particular observation or class interval is found by dividing the frequency (f) by the number of observations (n): that is, (f ÷ n). Thus:
Relative frequency = frequency ÷ number of observations
The percentage frequency is found by multiplying each relative frequency value by 100. Thus:
Percentage frequency = relative frequency X 100 = f ÷ n X 100
Use the data from Example 3 to make a table giving the relative frequency and percentage frequency of each interval of battery life.
Here is what that table looks like:
Table 4.3.4 Life of AA batteries, in minutes Table summary This table displays the results of Life of AA batteries. The information is grouped by Battery life, minutes (x) (appearing as row headers), Frequency (f), Relative frequency and Percent frequency (appearing as column headers).
| Battery life, minutes (x) | Frequency (f) | Relative frequency | Percent frequency |
| 360–369 | 2 | 0.07 | 7 |
| 370–379 | 3 | 0.1 | 10 |
| 380–389 | 5 | 0.17 | 17 |
| 390–399 | 7 | 0.23 | 23 |
| 400–409 | 5 | 0.17 | 17 |
| 410–419 | 4 | 0.13 | 13 |
| 420–429 | 3 | 0.1 | 10 |
| 430–439 | 1 | 0.03 | 3 |
| Total | 30 | 1 | 100 |
An analyst of these data could now say that:
7% of AA batteries have a life of from 360 minutes up to but less than 370 minutes, and that
the probability of any randomly selected AA battery having a life in this range is approximately 0.07.
Example 5 – Visualization of the cumulative relative frequency distribution
As previously shown for example 2, cumulative frequency is used to determine the number of observations that lie below a particular value in a data set. The cumulative frequency is calculated by adding each frequency from a frequency distribution table to the sum of its predecessors. The last value will always be equal to the total for all observations, since all frequencies will already have been added to the previous total. Let’s look at another example of how to calculate the cumulative frequency.
The daily number of rock climbers in Lake Louise, Alberta was recorded over a 30-day period. The results are as follows:
31, 49, 19, 62, 24, 45, 23, 51, 55, 60, 40, 35 54, 26, 57, 37, 43, 65, 18, 41, 50, 56, 4, 54, 39, 52, 35, 51, 63, 42.
The number of rock climbers ranges from 4 to 65. In order to create a frequency table, the data are best grouped in class intervals of 10. Each interval can be one row in the frequency table. The Frequency column lists the number of observations found within a class interval. For example, there are only two values in the interval from 10 to 20, then its frequency is 2 in the table accordingly.
Use the Frequency column to calculate cumulative frequency.
First, add the number from the Frequency column to its predecessor. For example, in the first row, we have only one observation and no predecessors. The cumulative frequency is one. 1 + 0 = 1
However, in the second row, there are two observations. Add these two to the previous cumulative frequency (one), and the result is three.1 + 2 = 3
Record the results in the Cumulative frequency column.
The other entries in the table can be calculated similarly. Results are presented in the table 4.3.5.
Table 4.3.5 Frequency and cumulative frequency of daily number of rock climbers recorded in Lake Louise, Alberta, 30-day period Table summary This table displays the results of Frequency and cumulative frequency of daily number of rock climbers recorded in Lake Louise. The information is grouped by Number of rock climbers (appearing as row headers), Frequency (f) and Cumulative frequency (appearing as column headers).
| Number of rock climbers | Frequency (f) | Cumulative frequency |
| <10 | 1 | 1 |
| 10 to <20 | 2 | 1 + 2 = 3 |
| 20 to <30 | 3 | 3 + 3 = 6 |
| 30 to <40 | 5 | 6 + 5 = 11 |
| 40 to <50 | 6 | 11 + 6 = 17 |
| 50 to <60 | 9 | 17 + 9 = 26 |
| >= 60 | 4 | 26 + 4 = 30 |
Cumulative relative frequency is another way of expressing frequency distribution. It is obtained by calculating the percentage of the cumulative frequency within each interval.
Cumulative percentage is calculated by dividing the cumulative frequency by the total number of observations (n), then multiplying it by 100 (the last value will always be equal to 100%). Thus,
cumulative relative frequency = (cumulative frequency ÷ n) x 100
The fourth column in the table 4.3.6 shows the calculation of the cumulative relative frequency of the daily number of rock climbers recorded in Lake Louise.
Table 4.3.6 Cumulative relative frequency of daily number of rock climbers recorded in Lake Louise, Alberta, 30-day period Table summary This table displays the results of Cumulative relative frequency of daily number of rock climbers recorded in Lake Louise. The information is grouped by Number of rock climbers (appearing as row headers), Frequency (f), Cumulative frequency and Cumulative relative frequency (%) (appearing as column headers).
| Number of rock climbers | Frequency (f) | Cumulative frequency | Cumulative relative frequency (%) |
| <10 | 1 | 1 | 1 ÷ 30 x 100 = 3 |
| 10 to <20 | 2 | 1 + 2 = 3 | 3 ÷ 30 x 100 = 10 |
| 20 to <30 | 3 | 3 + 3 = 6 | 6 ÷ 30 x 100 = 20 |
| 30 to <40 | 5 | 6 + 5 = 11 | 11 ÷ 30 x 100 = 37 |
| 40 to <50 | 6 | 11 + 6 = 17 | 17 ÷ 30 x 100 = 57 |
| 50 to <60 | 9 | 17 + 9 = 26 | 26 ÷ 30 x 100 = 87 |
| >= 60 | 4 | 26 + 4 = 30 | 30 ÷ 30 x 100 = 100 |
The cumulative relative frequency distribution can be visualized with a bar chart or a line chart, like in chart 4.3.1 below. The value on the horizontal axis is the upper bound of the class interval.
Data table for Chart 4.3.1
Data table for chart 4.3.1 Table summary This table displays the results of Data table for chart 4.3.1. The information is grouped by Upper bound of the class interval of daily number of rock climbers (appearing as row headers), Cumulative relative frequency (%) (appearing as column headers).
| Upper bound of the class interval of daily number of rock climbers | Cumulative relative frequency (%) |
| 9 | 3 |
| 19 | 10 |
| 29 | 20 |
| 39 | 37 |
| 49 | 57 |
| 59 | 87 |
| 69 | 100 |
Chart 4.3.1 shows that for the majority of days (57%) in the period, the number of rock climbers was lower or equal to 49.
Frequency distribution can be visualized using:
a pie chart (nominal variable),
a bar chart (nominal or ordinal variable),
a line chart (ordinal or discrete variable),
or a histogram (continuous variable).
These types of charts will be presented in the section 5 on data visualization. But first, we will look at other methods to summarize data using measures of central tendency and dispersion.
Table of contents
Statistics: Power from Data! - Main page
1 Data, statistical information and statistics
2 Sources of data
3 Data gathering and processing
4 Data exploration
5 Data visualization
Bibliography
Glossary
Date modified: |
13059 | https://www.sciencedirect.com/science/article/pii/S1871404814000276 | Acute suppurative parotitis associated with orbital and vascular complications: A case report - ScienceDirect
Skip to main contentSkip to article
Journals & Books
Access throughyour organization
Purchase PDF
Patient Access
Other access options
Search ScienceDirect
Article preview
Abstract
Introduction
Section snippets
References (10)
International Journal of Pediatric Otorhinolaryngology Extra
Volume 9, Issue 3, September 2014, Pages 108-111
Case Report
Acute suppurative parotitis associated with orbital and vascular complications: A case report☆
Author links open overlay panel Allen C.Lam, Kyle J.Chambers, Gillian R.Diercks, Michael S.Cohen
Show more
Add to Mendeley
Share
Cite
rights and content
Abstract
Complications arising from acute suppurative parotitis (ASP) are rare but can be morbid. We report the case of a young female who presented with facial swelling, trismus and diplopia. She was found to have ASP complicated by orbital abscess, cavernous sinus thrombosis, carotid arteritis, and cerebral infarction. Cultures from parotid abscessneedle aspiration, orbitotomy and blood sampling returned fusobacterium necrophorum. After a lengthy course of antibiotics and anticoagulation, the patient recovered to near-baseline status. To the best of our knowledge, this is the first case of ASP with such extensive orbital and vascular complications, and highlights the importance of awareness for such events.
Introduction
Acute suppurative parotitis (ASP) is a rare entity most commonly occurring in debilitated, elderly patients or children under the age of two . It often presents as sudden onset of induration and erythema over the parotid area in the setting of a high fever and chills. Diagnosis can be made from the expression of purulence from Stenson's duct. Radiographic confirmation of an abscess or abscesses may confirm suspicion when clinical history and examination correlate but no purulence is expressed. Staphylococcus aureus and anerobes are the prevailing microbacteria . Treatment involves drainage of the abscess, abundant intravenous fluid resuscitation, and intravenous antibiotics. Complications arising from ASP are even less common but may include cranial nerve palsies, necrotizing fasciitis, and temporal lobe abscess , , . In the setting of complications, multiple consultants may be involved to provide optimal care for the patient. Herein, we present the case of a young female with ASP and serious, life-threatening complications.
Access through your organization
Check access to the full text by signing in through your organization.
Access through your organization
Section snippets
Case report
An 18-year-old otherwise healthy female was transferred to our institution with acute suppurative parotitis for right orbitotomy and drainage of a lateral orbital abscess.
The patient initially presented to a regional hospital emergency department (ED) with sore throat and fever. Monospot and rapid streptococcal screen were negative, but pharyngeal cultures returned positive for group C streptococcus so was given erythromycin for one week. She returned twice more over the ensuing 4 days with
Discussion
Acute suppurative parotitis is thought to originate secondary to a multitude of conditions. Bacteria may reflux via Stenson's duct into the parotid parenchyma due to salivary stasis. Ductal obstruction can lead to stasis of salivary fluid with bacterial superinfection . Parotitis may also originate from hematogenenous spread in a bacteremic patient, as can be seen in newborns . The stereotypical patient is elderly and debilitated who is dehydrated in the post-operative setting. Risk
Conclusion
Acute suppurative parotitis is a rare process with complications arising even less frequently. Initial management of ASP involves commencement of broad-spectrum empiric antibiotic and aggressive hydration. When complications arise, they can be severe, even life-threatening, and should be considered as potential manifestations in patients with ASP. This case highlights an extreme example of such complications arising in a singular patient. It also underlines the importance of coordinated
Financial disclosures
None.
Conflict of interest
None.
Funding
None.
Recommended articles
References (10)
L. Decembrino et al.
Monolateral suppurative parotitis in a neonate and review of literature
Int. J. Pediatr. Otorhinolaryngol. (2012)
J.F. O’Brien et al.
Dexamethasone as adjuvant therapy for severe acute pharyngitis
Ann. Emerg. Med. (1993)
M.A. Cohen et al.
Acute suppurative parotitis with spread to the deep neck spaces
Am. J. Emerg. Med. (1999)
I. Brook
Acute bacterial suppurative parotitis: microbiology and management
J. Craniofac. Surg. (2003)
R.N. Kristensen et al.
Facial nerve palsy caused by parotid gland abscess
J. Laryngol. Otol. (2012)
There are more references available in the full text version of this article.
Cited by (0)
☆
Poster Presentation: American Society of Pediatric Otolaryngology, Arlington, VA, USA. April 25–28, 2013.
View full text
Copyright © 2014 Elsevier Ireland Ltd. All rights reserved.
Recommended articles
Broad overview of engineering of functional nanosystems for skin delivery
International Journal of Pharmaceutics, Volume 532, Issue 2, 2017, pp. 710-728 Ana Henriques Mota, …, Catarina Pinto Reis
### The CCR6-CCL20 axis in humoral immunity and T-B cell immunobiology
Immunobiology, Volume 224, Issue 3, 2019, pp. 449-454 Adrian Y.S.Lee, Heinrich Körner
### Development and validation of an HPLC-fluorescence method for the quantification of IR780-oleyl dye in lipid nanoparticles
International Journal of Pharmaceutics, Volume 532, Issue 2, 2017, pp. 779-789 Mathieu Varache, …, Anne-Claude Couffin
### Stapes footplate deformity leading to perilymphatic fistula and recurrent meningitis
International Journal of Pediatric Otorhinolaryngology Extra, Volume 17, 2017, pp. 31-35 Joseph R.Gonzalez, …, Stephen P.Cass
### Hepatic mesenchymal hamartoma: The role of radiology in diagnosis and management
Radiology Case Reports, Volume 16, Issue 8, 2021, pp. 2139-2141 Tran Phan Ninh, …, Nguyen Minh Duc
### Antimicrobial treatment of ENT infections
Infectious Diseases Now, Volume 53, Issue 8, Supplement, 2023, Article 104785 R.Cohen, …, V.Couloigner
Show 3 more articles
About ScienceDirect
Remote access
Contact and support
Terms and conditions
Privacy policy
Cookies are used by this site.Cookie settings
All content on this site: Copyright © 2025 Elsevier B.V., its licensors, and contributors. All rights are reserved, including those for text and data mining, AI training, and similar technologies. For all open access content, the relevant licensing terms apply.
We use cookies that are necessary to make our site work. We may also use additional cookies to analyze, improve, and personalize our content and your digital experience. You can manage your cookie preferences using the “Cookie Settings” link. For more information, see ourCookie Policy
Cookie Settings Accept all cookies
Cookie Preference Center
We use cookies which are necessary to make our site work. We may also use additional cookies to analyse, improve and personalise our content and your digital experience. For more information, see our Cookie Policy and the list of Google Ad-Tech Vendors.
You may choose not to allow some types of cookies. However, blocking some types may impact your experience of our site and the services we are able to offer. See the different category headings below to find out more or change your settings.
You may also be able to exercise your privacy choices as described in our Privacy Policy
Allow all
Manage Consent Preferences
Strictly Necessary Cookies
Always active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work.
Cookie Details List
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site.
Cookie Details List
Contextual Advertising Cookies
[x] Contextual Advertising Cookies
These cookies are used for properly showing banner advertisements on our site and associated functions such as limiting the number of times ads are shown to each user.
Cookie Details List
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Confirm my choices |
13060 | https://www.youtube.com/watch?v=gvoPGPMRcr4 | Temporal And Spatial Summation In Neurons Explained (With Passive Membrane Properties) | Clip
Science With Tal
9180 subscribers
92 likes
Description
7325 views
Posted: 19 Apr 2023
Welcome to Science With Tal!
In this video, we will cover how synaptic summation occurs. We will consider temporal and spatial summation and to do so, we will use the passive membrane properties to derive equations to understand these concepts. Both types of summation are very important to understand because they represent a facet of integration that our neurons perform constantly.
This conversation on neurons in the CNS derives from important concepts that are described in two previous full discussions:
1- To watch the first part (Signal propagation in the Neuron), make sure to go to:
2- To watch the second (The Neuromuscular Junction as the model of the chemical synapse) make sure to go to:
To improve the quality of my content, I highly value the feedback from the viewer so do not hesitate to give any feedback in the comment section.
TIMESTAMPS
0:00 Introduction
0:15 Introduction to synaptic summation
2:43 Temporal summation: derivation of necessary equations (RC circuit model)
10:22 Temporal summation: numerical example
14:14 Temporal summation: general intuition on time constant
15:50 A word on spatial summation
16:43 Synaptic summation summary
18:26 Conclusion
RESOURCES
Here is a list of the resources that I’ve used to produce this video. (Author(s): title resource)
Dale Purves: Neuroscience (6th edition)
Eric Kandel: Principles of neural science (6th edition)
Lodish: Molecular Cell Biology (9th edition)
Niswender, Colleen M, and P Jeffrey Conn: Metabotropic glutamate Receptors: physiology, pharmacology, and disease
Bünemann, M et al: Activation and deactivation kinetics of alpha 2A- and alpha 2C-adrenergic receptor-activated G protein-activated inwardly rectifying K+ channel currents
Howlett, Allyn C et al: CB(1) cannabinoid receptors and their associated proteins
Morris, R G: D.O. Hebb: The Organization of Behavior, Wiley: New York; 1949
Stent, G S: A physiological mechanism fo Hebb’s postulate of learning
To have more information on these resources, you can refer to the conclusion section where a more formal citation is provided.
CREDITS
Writing: Tal Klimenko
Voice: Tal Klimenko
Animations: Tal Klimenko
Drawings: Tal Klimenko
Editing: Tal Klimenko
Introductory jingle: Thierry Du Sablond
Conclusion music: lukrembo - sunflower (
8 comments
Transcript:
Introduction foreign Introduction to synaptic summation [Music] now back to the schematic where we've established the differences between the neuromuscular Junction and the neurons in the central nervous system we have two final but very complex differences to discuss which is the fact that the postsynaptic cell received many inputs and these inputs are generally weak such that it takes many excitatory inputs to generate an action potential in the postsynaptic cell these two statements seem rather simple and encompassing but it hides a few key aspects that are fundamental to neurons if we recall the neuromuscular Junction remember that when the motor neuron fired and released acetylcholine on the muscle fiber it always caused an action potential that then caused the contraction of the muscle this event is essentially a one-to-one the motor neuron fires and the muscle contracts which is rather intuitive and follows how we expect muscles to behave hence the communication between the motor neuron and the muscle fiber is as straightforward as it can be in the central nervous system neuronal communication can become highly complex because the weak inputs might not always be sufficient to reach action potential threshold in any case neurons must add the inputs they receive to reach the threshold this aspect is often referred to as summation as you might remember from our discussions on the action potential the action potential is most often initiated at The Icon initial segment the reasoning behind why this place is so important in deciding whether the neurons fires or not is because it has the lowest action potential threshold across the neuron remember that the definition of the threshold as defined in our action potential discussion is the membrane potential at which the inward current which is mostly carried by sodium outweighs the outward current from potassium and or chloride it turns out that the axon hillock is the place which usually has the highest density of voltage-gated sodium channels and thus provides the most inward current to the cell another important point to remember is that before the neuron hits the threshold the signals in the cell are all flowing passively which means that they Decay over time and are not self-sustained like the action potential for that reason when the cell adds the inputs together to reach threshold this summation can be done both in time and space which introduces us to the concept of temporal and spatial summation Temporal summation: derivation of necessary equations (RC circuit model) let's start with temporal summation which is defined as the summation of two or more consecutive signals to explain this principle I want to reconsider the passive membrane properties that I have introduced previously such that I can better explain some new Concepts if I go too quickly on these Concepts in this discussion I recommend watching the videos on the equivalent circuit model and the RC neuron to get additional insights on these Concepts alright we've established that there are three fundamental passive membrane properties which cover in the temporal and spatial time course of passive propagation the three properties are the membrane capacitance the membrane resistance and the axial resistance to give a brief description for each let's start with the membrane capacitance this property arises by the separation of charges between the membrane the separation of charge which can be labeled as Delta Q creates a voltage difference Delta V due to the electric field between the charges this voltage difference is what we know as the membrane potential and it is related to Delta Q by the membrane capacitance constant another consequence of charge separation is the fact that when charges enter the neuron here in this instance positive charges it causes charges on the opposite side to feel a repulsion and move away this charge movement creates a current which we've called the capacitive current IC from Delta Q equals CM Delta V we can derive the equation of the capacitive current if we consider the definition of the current which is simply the derivative of charge over time now the second property is the membrane resistance which arises from the fact that the membrane has ion channels that let ions flow in and out of the cell for our temporal summation purposes there are two distinct types of channels that we need to take into account leak channels and the light indicated synaptic receptors as we've discussed in the equivalent circuit model video ion channels obey Ohm's law and thus the current that flows through them is equal to the voltage difference divided by the membrane resistance another important note about them is that the movement of charges through these channels is powered by the electrochemical gradients of the ions that flow through them accordingly we've derived a modified version of Ohm's law that encompasses all of these different properties note that g here stands for the conductance and it represents the inverse of the resistance now in terms of their roles leak channels are the channels that are open at rest and govern the resting membrane potential there are some resistance or conductance depending on how you see it is known as the leak resistance RL this value is also sometimes referred to as the input resistance when it comes to synaptic channels they are the ones that only open during synaptic events it is very important to keep in mind that their opening changes the total resistance of the cell but this is a detail I'll come back to shortly the third and final property is the isol resistance and it corresponds to the internal resistance that charges fill when they move inside the dendrites or axons the equation that models this property is essentially the same as the equation that models electron flow in electrical wires now to understand temporal summation we only need the two first properties membrane capacitance and membrane resistance another tool that will be relevant for us to understand how temporal summation occur is to have the equivalent circuit model of the synapse this circuit basically models the neuron as an RC circuit which is very convenient for us because from that we will be able to determine equations that will allow us to see how the membrane potential changes as a function of time based on our passive membrane properties now in our system there are two Dynamic states to consider during the synapse when the synaptic channels open and after the synapse when the neuron returns to Baseline let's start with during the setups to find the equation of the membrane potential as a function of time we can establish that the current as a function of time during the synapse will be the sum of the current coming from the capacitor and the resistors next we can substitute the currents for the capacitor and the resistor by their definitions here he gets a bit tricky because as I mentioned a few moments ago synaptic channels will be active during the synapse and for that reason they will affect the resistance of this system based on electric circuit laws we can find the total sum of the resistances by taking the inverse of the sums of the inverse of each resistance I will note this new resistance that occurs during the synapse as RSL with this new resistance we can now continue the derivation of the equation by multiplying each side by the presynaptic leak resistance from here we can Define two new quantities which are the steady state potential and the time constant you will later see the relevance when we go over them in a bit more detail but for the moment they allow us to make the equation less clustered for those of you who have watched the RC neuron video I changed the notation of Vmax to VSS because I feel like VSS which stands for the steady state is more telling on what VSS is rather than VMAX nevertheless by substituting these variables we can rearrange the equation to get this form the next step in our derivation will actually be to integrate the differential equation we came up with I will let you go through the steps on your own but basically we end up getting this exponential equation where the membrane potential as a function of time is equal to the steady state voltage times the exponential Factor this should suffice us for the first state now when the synapse is closed and the membrane potential returns to Baseline the conditions change first the current as a function of time now equals zero because the currents are only mediated by the capacitor and the resting resistors which pass current to return to the resting state but there is no additional source of current now another important distinction is that since the synaptic channels are now closed we only have to take into consideration the resistance of the leaked channels RL from there the derivation follows the same steps as before and yields us this equation you will notice that since the added current in the system is now equal to zero there is no steady state potential to consider instead since this Decay picks up right after the end of the Channel's opening we now have to consider the membrane potential at time 0 but I'll come back to that aspect very shortly at the end you will notice that the two equations are very similar to each other but there are two key differences first the equation that models the membrane potential when the synaptic channels are open describes an exponential growth whereas the equation that models the state when the synaptic channels are closed describes an exponential decay secondly when the synaptic channels open we need to take into account the resistance of the synaptic channels and that is reflected in the value of the steady state voltage and the time constant whereas when channels close it is only the leak resistance that is to be taken into account because of this the growth and Decay will have different time constants and thus different time courses by the way due to the fact that we add the inverse of the resistance the total resistance will always decrease when we add a new resistance hence the time constant when the synaptic channels are open will always be smaller than when the channels are closed with this being said let's now get into Temporal summation: numerical example temporal summation and see how what we just derived helps us understand that aspect to exemplify temporal summation I want to give a numerical example of what a problem might look like because I believe that's more telling than better for our comprehensions so let's imagine that the membrane of this postsynaptic cell receives an excitatory connection that fires twice at 100 Hertz we want to figure out what will be the membrane potential at the end of the second pulse and we will assume that the currents are constant here we make this assumption to keep the steady state potential constant as you will see later we'll also imagine that based on the values of the resistance and capacitance of our postsynaptic neuron the time constant during the synaptic activation is one millisecond and when the channels close the time constant becomes 3 milliseconds furthermore we'll imagine that the current injection from the channels lasts 7 milliseconds and that the steady state potential is about 10 millivolts to better visualize what's going on let's also plot the membrane potential and the synaptic current as a function of time okay so to solve this problem I suggest to First understand when this synapse is active or not based on the frequency and the injection time as a side note the use of frequency to describe temporal Dynamics in neurons happens very often as a reminder frequency which is measured in hertz corresponds to the inverse of the time interval between each pulse in our problem a frequency of 100 Hertz means that the interval between each pulse is therefore 10 milliseconds with this information we can see that the first pulls lasts 7 milliseconds and 3 milliseconds after its end another one is fired for 7 milliseconds again in the plot of the membrane potential we can also plot the steady state potential which follows the same shape as the current now that the time frame is established we can use our exponential equations to find the final membrane potential from 0 to 7 milliseconds plugging our numbers into the growth equation gives us that the membrane potential at 7 milliseconds will be plus 10 millivolts in cases like this where the time elapsed is significantly larger than the time constant about 5 times Tau we can usually say that the membrane potential has reached the steady state potential for that reason the time constant is a good indicator to approximate where the membrane potential actually is in relation to the steady state if for example the time constant was 30 milliseconds well we would have known that just by eyeballing the equation that the depolarization is far from the steady state moving on we now need to compute by how much the signal decays and the three millisecond interval between the end of the first pulse and the start of the second by plugging the numbers we get that the signal decays to 3.7 millivolts in cases like this where T equals Tau there is another trick that can help you save some time indeed when time and Tau are equal the exponential Factor simply becomes e to the negative 1 or 1 minus E to the negative 1 which respectively gives 0.37 and 0.63 hence here I only had to multiply 10 by 0.37 instead of entering the exponential term in my calculator which can be a bit tedious at times finally we can add the membrane potential at the end of the second pulse by adding 10 millivolts to 3.7 to give a final answer of 13.7 millivolts in a case like this where the time constant and steady state potential stays the same for each pulse there is no need to compute the depolarization again we can just take what we found in the first step I hope this problem wasn't too difficult to follow through if you have any questions make sure to leave them in the comments to end our discussion on Temporal summation: general intuition on time constant temporal summation I want to briefly discuss the time constant as we've seen in the problem the time constant gives us a good estimate of where the Decay and growth Parts might end up now if in this problem the open and closed time constants were something small let's say 0.1 millisecond and 0.3 milliseconds you can see that the results are drastically different indeed the time constants for the growth and the Decay are so quick relative to the current injection that no summation can even occur this means that we to reach the threshold in these conditions the presynaptic cell we need to fire at considerably higher frequencies or inject very high depolarizing pulses on the other hand if in the problem the open and close time constant were very large let's say 10 and 30 milliseconds there would be some summation but the depolarizations only reach a fraction of the steady state potential for that reason there is a certain trade-off between low and large time constants if they are too small then the potentials will rise quickly to the steady state but it also means that the potentials will Decay quickly and prevent summation on the other hand if the time constant is too large it takes too much time for the postsynaptic potential to reach the steady state that it only reaches a certain fraction of it but at least the slow Decay increases the likelihood for consecutive signals to Summit hence we can say that as a rule of thumb neurons that have longer time constants are more likely to sum it in time because longer time constant cause longer decade times and thus more time for signals to Summit A word on spatial summation all right now let's move on to spatial summation which occurs when two or more synaptic potentials are added in space in temporal summation we focused a lot on the time constant and as you can imagine here we will focus a lot on the space constant now although there are two different measures they can be interpreted in pretty much the same way by the way if ever you are unfamiliar with the space constant I highly suggest you to watch this video on the cable theory model of the neuron where right thoroughly explain how the constant can be derived and what it means from the video we covered that signals Decay as a function of space and at a high value of the space constant media Decay slower relative to a smaller value hence if we imagine two different pulses that are activated at the same time the likelihood that the two will be summed together will be much greater if the space constant is higher since the signals will have decayed less Synaptic summation summary in summary there are two main types of summation that we can consider temporal and spatial the main elements that can give us the most insight as to the single summation are the time and space constants in general the bigger they are the less the signals Decay and the more probable it is they will sum it one thing to keep in mind here is that the two forms of summation always happen at the same time it is only in our idolized theoretical worlds that the two can be analyzed separately but in real neurons the synaptic connections get integrated in time in space simultaneously also I looked at summation through the lens of excitatory connections but keep in mind that inhibitory connections also get summed and participate in the signals the neuron receives all right back to our schematic on the differences between the neuromuscular Junction and the central nervous system we now have covered metabotropic receptors the different neurotransmitters and synaptic summation which already gives us a pretty good idea on how neurons in the central nervous system are different from neurons in the neuromuscular Junction the final aspect that I want to cover now is a bit hidden under the lines of the differences shown here as I mentioned multiple times neurons in the central nervous system experience very weak inputs but in very large quantities whereas the neuromuscular Junction is essentially one large input on the muscle this difference as we've seen leads to the concept of synaptic integration but it also leads to the idea of plasticity basically due to the fact that the inputs are so small in the central nervous system it leaves room for them to be modulated so to increase or decrease their strength which is known as plasticity Conclusion [Music] thank you for watching this video if there was anything unclear or there was a mistake somewhere in the video make sure to let me know in the comment section if you enjoyed this video and found it useful you can consider leaving a like and subscribing to support the channel on the right you will see the informational resources that I've used to produce this video thank you again for watching and I'll see you in our next discussion [Music] foreign |
13061 | https://math.answers.com/basic-math/Which_triangular_numbers_are_prime_numbers | Which triangular numbers are prime numbers? - Answers
Create
0
Log in
Subjects>Math>Basic Math
Which triangular numbers are prime numbers?
Anonymous ∙ 15 y ago
Updated: 4/28/2022
3 is a triangular number that is also prime.
Here is a list of some of the first few triangular numbers:
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275
Wiki User ∙ 15 y ago
Copy
Add Your Answer
What else can I help you with?
Search
Continue Learning about Basic Math
### Is the number 707 a triangular number? 707 is not a triangular number. The closest triangular numbers to 707 are 703 and 741.
### What are the five triangular numbers that follow 21? The five triangular numbers that follow 21 are 28, 36, 45, 55, and 66.
### Are any two prime numbers relatively prime? Any two prime numbers will be relatively prime. Numbers are relatively prime if they do not have any prime factors in common. Prime numbers have only themselves as prime factors, so all prime numbers are relatively prime to the others.
### What are products of prime numbers? Products of prime numbers are composite numbers.
### What are the opposite of prime numbers? The opposite of prime numbers are composite numbers.
Related Questions
Trending Questions
What two prime numbers add up to 26?How do you factor x to the sixth minus y to the sixth?Cube root for 13824 by prime factorization?What is 6 percent of 12000?How do you make 5.43 as a mixed number?What are all the composite factors of 216?Is 0.77 equialent to 0.077?What does base con- mean?If you wear a size 12 in woman's what size would you wear in misses?When its KE is half of its maximum value how much PE does it have?Prime factors of 256?How much is 18 nickels?What is 849 rounded to the nearest 100 is?What is the name given to 100 percent h20?What is 1.61 rounded to the nearest 10th?What number can go into 720?Can you average percentages?How do you write 5000.00 iN WRITING?How do you write 846 hundredths as decimal?What is 275 x 5?
Resources
LeaderboardAll TagsUnanswered
Top Categories
AlgebraChemistryBiologyWorld HistoryEnglish Language ArtsPsychologyComputer ScienceEconomics
Product
Community GuidelinesHonor CodeFlashcard MakerStudy GuidesMath SolverFAQ
Company
About UsContact UsTerms of ServicePrivacy PolicyDisclaimerCookie PolicyIP Issues
Copyright ©2025 Answers.com. All Rights Reserved. The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Answers. |
13062 | https://www.mckissock.com/blog/land-surveyor/writing-legal-land-descriptions-everything-land-surveyors-need-to-know/ | Skip to content
McKissock Learning Logo
Skip to content
Writing Legal Land Descriptions: Everything Land Surveyors Need to Know
Facebook
X
LinkedIn
LinkCopied
Land surveyors play a crucial role in ensuring the accuracy and legal integrity of property boundaries, making them integral to real estate transactions, construction projects, and land development. One of their key responsibilities is drafting land descriptions. These descriptions, typically used in deeds, title reports, and contracts, are more than just technical jargon—they are legally binding statements that define the exact boundaries of a property.
Crafting these descriptions requires precision, understanding of legal terms, and an in-depth knowledge of surveying principles.
In this blog, we’ll explore the significance of legal land descriptions, key methods for writing them, common challenges faced by land surveyors, and best practices to ensure the creation of legally sound and accurate land descriptions.
The Importance of Legal Land Descriptions
A legal land description is the formal and precise way of identifying a parcel of land. It is critical because it is used in legal documents to ensure that property boundaries are clear and unambiguous. Whether for a deed, title, mortgage, or lease, these descriptions provide a clear understanding of what land is being transferred or encumbered.
Legal land descriptions often appear in several important documents, including:
Deeds and Titles: These are used to prove ownership and to transfer property rights.
Easement Agreements: These outline the right to use a portion of the land for specific purposes.
Land Contracts: When parties enter a sale agreement, a legal land description specifies what is being bought or sold.
Boundary Disputes: A well-written description can help resolve conflicts between property owners regarding boundary lines.
Land descriptions are essential in all legal contexts and contracts involving real estate transactions. Therefore, accuracy in writing and interpreting these descriptions is paramount.
Key Elements of Legal Land Descriptions
Legal land descriptions contain specific elements to ensure clarity and avoid misunderstandings. The most common formats include metes and bounds, lot and block, and rectangular survey descriptions. Each has unique components:
1. Metes and Bounds
The metes and bounds system is the most traditional method for describing land. It uses physical landmarks and measurements to define the boundaries of a property. This description starts at a known point (called the point of beginning) and follows the boundaries of the property by stating distances, directions (often in terms of compass bearings), and landmarks.
Key elements in this system include:
Point of Beginning (POB): The starting point from which measurements are taken.
Course and Distance: The direction and length of each boundary line.
Landmarks or Monuments: Natural or artificial objects used to define the boundary.
For example, a metes and bounds description may state, “Beginning at a stone at the southwest corner of Lot 10, then north 300 feet to a pine tree, then east 200 feet to a fence, and so on.”
2. Lot and Block System
This method is primarily used for subdivisions and developments. It references a parcel’s location within a larger plat or survey map. The description will typically include the lot number, block number, subdivision name, and county or township.
For example: “Lot 3, Block 4 of Green Acres Subdivision, according to the Plat thereof recorded in Plat Book 6, Page 5, of the Public Records of XYZ County.”
This system is often easier for surveyors, as it relies on already recorded plans and maps. However, it’s vital to ensure that the subdivision maps are up-to-date and accurately reflect the land’s current legal boundaries.
3. Rectangular Survey System (Public Land Survey System)
Used primarily in the western United States, the rectangular survey system divides land into a series of rectangles using baseline and meridian lines. This system uses a grid, consisting of townships, sections, and ranges, to describe land. This system is often used in larger rural or undeveloped areas.
For example: “The SE ¼ of Section 15, Township 25 North, Range 3 East of the XYZ Meridian.”
This system relies heavily on established governmental surveys and is especially useful for large tracts of land.
Drafting a Legal Land Description
Writing a legal land description involves more than simply stating boundary lines. It requires a thorough understanding of property surveys and a commitment to legal accuracy. Here are some key steps to follow when drafting a legal land description:
1. Conduct a Thorough Survey
Before drafting a legal land description, it’s essential to conduct an accurate and detailed land survey. This includes measuring the property’s boundaries, identifying any monuments or physical markers, and ensuring that the survey is up to date. If the property has been subdivided, referencing the correct plat is necessary.
2. Use Clear and Consistent Language
Language is key when drafting legal descriptions. It’s important to use consistent terminology and standard symbols. Avoid ambiguous terms like “approximately,” as they can lead to misunderstandings. Legal descriptions should be clear, concise, and free from any room for misinterpretation.
3. Reference Specific Documents
Legal land descriptions often reference recorded documents, such as plats, surveys, and deeds, that provide additional context and history of the property. This is particularly important when dealing with older properties where historical data can help clarify boundaries.
4. Include All Necessary Components
Ensure that the land description includes all the required elements, such as:
The point of beginning (POB),
The boundary lines (including course, distance, and direction),
Monuments or landmarks that mark the boundaries, and
A closing point that confirms the description returns to the point of beginning.
5. Follow Legal Requirements
Specific legal requirements for land descriptions can vary by state. Land surveyors must be familiar with their state’s regulations, which may require additional details or specific formatting.
Common Challenges in Writing Legal Land Descriptions
Writing legal land descriptions can be a complex task with several potential challenges. Some of the most common issues faced by land surveyors include:
Inaccurate or incomplete surveys: If a previous survey was poorly done or if the land has been altered, the boundaries may not be clearly defined, making it difficult to create an accurate legal description.
Misunderstandings in legal terminology: Legal descriptions require precision in language, and small errors or imprecise wording can lead to confusion or even litigation.
Boundary disputes: Disputes over property boundaries are common, especially in areas where markers or monuments may have been removed or altered.
Changes over time: Land can change over time, especially with the construction of roads, fences, or other structures. Surveyors must account for these changes when drafting a legal land description.
Best Practices for Drafting Legal Land Descriptions
To avoid these common challenges and ensure accuracy, land surveyors should adhere to the following best practices:
Double-check all measurements and references before finalizing the description.
Review local regulations to ensure compliance with state or county-specific requirements.
Consult historical records when necessary to resolve potential issues with previous surveys or boundary descriptions.
Work with legal professionals to review land descriptions in situations involving complex or disputed properties.
Use technology like GIS (Geographic Information Systems) or digital mapping tools to assist in drafting accurate and precise legal descriptions.
Conclusion
Legal land descriptions are an essential part of property transactions and land development. For licensed land surveyors, the ability to craft precise, clear, and legally sound descriptions is a requirement. By understanding the different methods for drafting descriptions, adhering to best practices, and staying up to date with regulations, you can ensure that your work remains accurate and valuable in a legal context.
If you’re looking to enhance your skills in drafting legal land descriptions or need a refresher on surveying techniques, consider enrolling in our Land Descriptions continuing education course, which can provide the tools and knowledge to keep your skills sharp and compliant with industry standards.
Facebook
X
LinkedIn
LinkCopied
Explore more on
Land Surveyor
Editor’s note: this post was originally published on July 15, 2025.
Related
Construction Staking: A Critical Role for Land Surveyors
By
Melanie Klag
### What Is a FEMA Elevation Certificate? A Guide for Licensed Land Surveyors
By
Melanie Klag
### Prescriptive Easement vs Adverse Possession: A Land Surveyor’s Guide
By
Melanie Klag
### Understanding Road Surveys and the Role of a Road Surveyor
By
Melanie Klag
### How to Use Drones in Land Surveying
By
Melanie Klag
### ALTA Land Title Survey: What You Need to Know
By
Melanie Klag
Cookie Notice
McKissock (“we” or “our”) uses cookies to offer you a better website browsing experience, personalize content and ads, and analyze our site traffic. We also share information about your use of our site with our social media, advertising, and analytics partners. By clicking “Accept All” on this banner, you are providing your consent to our collection and use of these cookies. Learn more by reading our Privacy Policy. You can manage or opt-out of this sharing by selecting the "Manage Cookies" link. |
13063 | https://www.youtube.com/watch?v=_VAvhI1cqps | LEARN: Inverse Operations Multiplication And Division
The Maths Guy!
35600 subscribers
33 likes
Description
7787 views
Posted: 6 Feb 2024
www.mathshelter.com
In this video we will learn about inverse operations between multiplication and division.
Multiplication and division are special because they are the opposite of each other and therefore can be used to 'cancel each other out'
We can also use this knowledge to help us solve a missing number question such as:
15 @user-zr8pm2nk2w+ ? = 32
or even check our answers to make sure we have done multiplication or division questions correctly !
Watch to find out how!
About me:
I am a teacher from the UK who’s passion is Maths and travel – I have taught in the UK, Australia, Thailand, Moscow, and Bali
The aim of this channel is to help as many people as possible understand the key concepts of Mathematics.
If you have found any of the content on my channel helpful then please consider subscribing to the channel, it really helps me out.
You can also find me on my website:
www.mathshelter.com
Or you can follow me on Facebook:
If you are in Year 6 and preparing for your Year 6 SAT’s then you can also join my Year 6 Maths Preparation course! This course aims to give you all the skills you will need to smash your Year 6 SATs exams!
If you would like to support me in another way then you can ‘buy me a coffee’ using this link here:
Thank you for watching.. see you in another video!
The Maths Guy
6 comments
Transcript:
[Music] what is up everybody and welcome back to the channel today we're going to be looking at inverse operations of multiplication and division let's waste no time let's go okay so for today's lesson we need to remember that every number sentence has a number family but what does that mean well it means that this is a number sentence 5 10 = 50 that is a complete number sentence but it's part of a number family and in a number family we actually have four different number sentences and in these number sentences we're going to still use the same numbers the five the 10 and the 50 but we can rearrange them in different ways to express that number sentence so let's have a look what I mean well 5 10 = 50 but also 10 5 = 50 so this is our second second number sentence in the number family but so far both of my questions have used multiplication well multiplication has a very special link to another of the processes division multiplication and division are best friends because a multiplication and division question can cancel each other out and therefore they can show the inverse operations let's see what I mean so so far I've had my multiplication and I've had the five and the 10 and at the start before the equal sign now because I'm looking at the opposite process division I'm going to be focusing on moving the answer to the other side of the equal sign so for this number sentence I'm going to start by putting the 50 at the beginning and I'm going to say that 50 divided by and I'm going to take the 10 to start with 10 equal 5 and that's correct 50 ID 10 does equal 5 but there's another number sentence that I can still write I could have 50 / 5 = 10 but why is this really important to know well inverse means that we can cancel out the operation or we can check to make sure our question is correct so we originally looked at this question here 5 10 = 50 well this number sentence will have the opposite somewhere in the number family so let's find it so in this we have the 50 after the equal sign so if it's going to be opposite we're looking for the 50 at the beginning so it's going to be one of these two then we have five at the beginning so we're looking for five at the end so I think we've managed to find our inverse operation and our inverse question would be this one so what it means is that number sentence one and number sentence three are best friends in the family they're still part of the same family but they're like best buddies and therefore number two and number four are also best buddies CU they would be the inverse of each other as well let's check we have 10 at the start 10 at the end 50 at the end 50 at the start absolute opposites but again why is this helpful well sometimes you will see a question like this it will say something 5 = 50 and we call a question like this a missing number question because we have a missing number in the question part of the question rather than the answer but now we know about inverse operations we know that we can just bring this 50 to the beginning of our sentence invert my operation which was multiplication we know the opposite is division so now that we have the 50 at the start and we have the division we can put my five in place and now look my question mark is where it should be at the end and I can solve this and say 50 / 5 = 10 so therefore going back to the original question this question mark must be 10 so 10 5 = 50 so understanding number families and understanding inverse operations are very closely linked let's have a look at the red chili challenge 42 / 6 = 7 so let's find all the number number sentences for this so this is our first one our second one would be 42 / 7 = 6 little tip remember when we're dividing our larger number will often be at the start so there are my two division questions what are my multiplication ones well I would have 7 6 = 42 ided some something equals 6 so this question here now I can look at my number families and notice that my other division question starts with 42 / 6 so therefore I can bring my six and swap it with the place of the question mark and that would leave me with 42 / 6 equal question mark which we now know equals 7 so with my division to find a missing number I didn't even need to use the inverse I can just look at my number families to help me so let's summarize we've established that multiplication and division are opposites so if I have a question like 3 5 = 15 the opposite and the inverse would mean putting my 15 to the start of my question and dividing it 15 / 3 = 5 and both of these questions are part of a larger number Family where we would also have 5 3 = 15 and 15 ided by 5 = 3 so if we're ever trying to find the inverse of something and we're struggling draw your number family first and then you will often be able to use that to help you so now it's your turn have a go at finding the remaining number of family sentences for these two questions remember a number family is made up of four number sentences I've given you the first one what are the other three press pause on the video now put your answers in the comment section good luck and there you have it that is inverse operations for multiplication and division quite a challenging lesson but very important if we can understand the number family relationships it will make understanding multiplication and division links much easier to understand hopefully this video has been helpful for you if it has subscribe to the channel and head over to the math shelter.com where you're going to find loads more videos like this covering everything that your group needs to know about maths but for now guys see you in another video peace out |
13064 | https://flexbooks.ck12.org/cbook/ck-12-probability-and-statistics-concepts/section/8.1/primary/lesson/combinations-and-permutations-pst/ | Skip to content
Math
Elementary Math
Grade 1
Grade 2
Grade 3
Grade 4
Grade 5
Interactive
Math 6
Math 7
Math 8
Algebra I
Geometry
Algebra II
Conventional
Math 6
Math 7
Math 8
Algebra I
Geometry
Algebra II
Probability & Statistics
Trigonometry
Math Analysis
Precalculus
Calculus
What's the difference?
Science
Grade K to 5
Earth Science
Life Science
Physical Science
Biology
Chemistry
Physics
Advanced Biology
FlexLets
Math FlexLets
Science FlexLets
English
Writing
Spelling
Social Studies
Economics
Geography
Government
History
World History
Philosophy
Sociology
More
Astronomy
Engineering
Health
Photography
Technology
College
College Algebra
College Precalculus
Linear Algebra
College Human Biology
The Universe
Adult Education
Basic Education
High School Diploma
High School Equivalency
Career Technical Ed
English as 2nd Language
Country
Bhutan
Brasil
Chile
Georgia
India
Translations
Spanish
Korean
Deutsch
Chinese
Greek
Polski
EXPLORE
Flexi
A FREE Digital Tutor for Every Student
FlexBooks 2.0
Customizable, digital textbooks in a new, interactive platform
FlexBooks
Customizable, digital textbooks
Schools
FlexBooks from schools and districts near you
Study Guides
Quick review with key information for each concept
Adaptive Practice
Building knowledge at each student’s skill level
Simulations
Interactive Physics & Chemistry Simulations
PLIX
Play. Learn. Interact. eXplore.
CCSS Math
Concepts and FlexBooks aligned to Common Core
NGSS
Concepts aligned to Next Generation Science Standards
Certified Educator
Stand out as an educator. Become CK-12 Certified.
Webinars
Live and archived sessions to learn about CK-12
Other Resources
CK-12 Resources
Concept Map
Testimonials
CK-12 Mission
Meet the Team
CK-12 Helpdesk
FlexLets
Know the essentials.
Pick a Subject
Donate
Sign Up
8.1
Combinations and Permutations
Written by:CK-12
Fact-checked by:The CK-12 Editorial Team
Last Modified: Sep 01, 2025
Combinations and Permutations
Combinations and Permutations are each different methods of counting the possible number of ways that the members of a set may be selected or arranged. The difference between the two is that permutations consider the order in which objects are placed to be important, and combinations only consider which objects are chosen, ignoring the order.
One way to remember the difference between the two is to consider the meanings of the root words: combine and permute. You may not be familiar with the word ‘permute’, which basically means to mix up the order of something, but you are likely quite comfortable with ‘combine’, which means group things together.
Permutations: If you are trying to find the number of ways that different items can be put in order, meaning that 1, 2, 3 is one permutation, and 3, 2, 1 is another permutation, each counting as separate entries, you are identifying the count of the different ways you could permutate or permute (mix up) the items. It is quite possible to have multiple permutations using exactly the same items.
Combinations: If you are identifying the number of different groups of items, meaning that 1, 2, 3 and 3, 2, 1 are the same group, regardless of order, then you are counting the number of ways you could combine (group up) the items. Combinations can also be thought of as putting items in a bucket, the order in which you drop the items in does not change which items are in the bucket when you finish. In order to have different combinations, you need different items, you can’t just drop them into the bucket in a different order.
Understanding the Difference Between Combinations and Permutations
Suppose you have 12 tabs open in your web browser, and you are curious how many ways they could be reorganized. Is this a combination problem, or a permutation problem, and why?
Since you are counting the number of different ways the same 12 tabs could be organized or put in order, this is a permutation problem. Each different permutation will have the same twelve tabs, they will just be in a different order.
Your sports team is planning a trip to a nearby town, about 2 hours away by bus. The driver will allow each rider to bring 2 electronic devices for the trip. You have access to a smartphone, a tablet, an mp3 player, and a laptop computer, and you are curious how many options that allows you to choose. Is this a combination problem, or a permutation problem, and why?
Regardless of which item you choose first, and which you choose second, the same two items will result in the same devices available to you for the trip, so order does not matter. This is a combinations problem.
In the United States, telephone numbers are numerical series composed of a 3-digit area code, a 3-digit exchange, and a 4-digit subscriber line identifier. For each of the over 250 different area codes, there are 7,920,000 useable phone numbers (some numbers, particularly those starting with 0 or 1, are unavailable for public use). Is the count of useable phone numbers a product of a permutation calculation, or a combination calculation, and why?
Phone numbers are a permutation function, since 555-1234 would ring a different person than 555-4321.
Earlier Problem Revisited
Evan, his girlfriend, and 5 of his other friends are going to the movies. Evan wants to be sure that he sits next to his girlfriend, and is curious how many different ways he and his friends can be arranged to sit in the same row. Is this a combination problem or a permutation problem? How can you tell?
This is a permutation problem, since Evan is interested in learning how many different ways he could order the same 7 friends. If he is curious about how many different groups of 3 friends could be chosen to go get drinks and popcorn after they have seats, then he would be dealing with a combination problem.
Examples
Example 1
The back of the box of jellybeans that I have says that there are hundreds of different flavor combinations possible by eating multiple beans at the same time. If I wanted to calculate how many possibilities there were if eating only two beans at a time, would that be a permutation calculation, or a combination calculation?
Since two jellybeans of given flavors will result in the same combined flavor regardless of which I grab from the box first, this is a combination problem.
Example 2
Tuscany has 500 songs in her MP3 collection, and she wants to make a playlist for her boyfriend. The trick is that she wants to hide a secret message in the playlist by choosing songs so that the first word of each song forms a message. If she wanted to know how many different messages were possible from a playlist 15 songs long, including messages that are gibberish, is this a permutation or combination problem?
Since the message would be different if the same words were read in a different order, this is a permutation problem.
Example 3
Scott has 5 dogs, and he feeds them one at a time so they don’t fight over the food. If he wants to discover how many different ways he could choose to order them for feeding, is he considering permutations or combinations?
The order of feeding is all the Scott is considering, so this is a permutation problem. The combination of 5 dogs is the same regardless.
Example 4
Vicki is making gift bags for a party. She has 8 kinds of gifts and the bags hold 3 gifts each. If she wants to know how many different gift bags she can make, is she dealing with permutations or combinations?
If Vicki chooses a particular group of 3 gifts for a bag, that bag will end up the same regardless of which of the 3 she chooses first, second, or third. This is a combination problem.
Review
For problems 1 – 14, identify each situation as either a permutation or a combination consideration:
An ice cream store has fifteen different flavors of ice cream, you wonder how many different three-scoop bowls can be made.
There is a red, a blue, a green, and a yellow chair around the table. You wonder how many ways can four friends sit around the table.
Two friends decide to stay home on a Saturday and watch a movie marathon. Of the twenty movies the friends have to choose from, each friend chooses a first, second, and third choice. How many different six-movie marathons are possible?
There are seven rides at the carnival, but you can only afford to ride four of them, how many different groups of four rides are there?
How many five-letter groups can you make from the word “grandmother”?
Fourteen friends decide to visit each other’s houses to trick-or-treat, but they only have time to choose seven of the houses. How many ways could the friends put the houses in order for visiting?
How many different pizzas can be made, assuming no double toppings, from seven toppings?
How many 5-card hands are possible with 20 different cards?
You are responsible for selecting a lead and an understudy for a school play.
You are team captain for a tug-of-war, how many different teams could you create from a pool of 30 players?
Scott is a safety-conscious rider, and knows he should wear a leather jacket and helmet when he rides a motorcycle. There are 10 different helmets and 7 different jackets to choose from, he wonders many options are possible.
John is practicing a card trick to show his granddaughter, he asks her to pick three cards from the deck and put them back into the pile. John wants to know many groups of three he needs to guess from to get it right.
Robin is kind of a health food fanatic, and she is making whole wheat pizza with organic sauce, for dinner. She has spinach, kale, fresh tomato, organic anchovy, free-range chicken, and grass-fed beef. Her kids want to know how many different pizzas she could make.
Review (Answers)
Click HERE to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
Student Sign Up
Are you a teacher?
Having issues? Click here
By signing up, I confirm that I have read and agree to the Terms of use and Privacy Policy
Already have an account?
No Results Found
Your search did not match anything in .
This lesson has been added to your library. |
13065 | https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-expressions-and-variables/whole-numbers-integers/a/whole-numbers-integers | Use of cookies
Cookies are small files placed on your device that collect information when you use Khan Academy. Strictly necessary cookies are used to make our site work and are required. Other types of cookies are used to improve your experience, to analyze how Khan Academy is used, and to market our service. You can allow or disallow these other cookies by checking or unchecking the boxes below. You can learn more in our cookie policy
Privacy Preference Center
When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
More information
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
Certain cookies and other technologies are essential in order to enable our Service to provide the features you have requested, such as making it possible for you to access our product and information related to your account.
For example, each time you log into our Service, a Strictly Necessary Cookie authenticates that it is you logging in and allows you to use the Service without having to re-enter your password when you visit a new page or new unit during your browsing session.
Functional Cookies
These cookies provide you with a more tailored experience and allow you to make certain selections on our Service. For example, these cookies store information such as your preferred language and website preferences.
Targeting Cookies
These cookies are used on a limited basis, only on pages directed to adults (teachers, donors, or parents). We use these cookies to inform our own digital marketing and help us connect with people who are interested in our Service and our mission.
We do not use cookies to serve third party ads on our Service.
Performance Cookies
These cookies and other technologies allow us to understand how you interact with our Service (e.g., how often you use our Service, where you are accessing the Service from and the content that you’re interacting with). Analytic cookies enable us to support and improve how our Service operates.
For example, we use Google Analytics cookies to help us measure traffic and usage trends for the Service, and to understand more about the demographics of our users.
We also may use web beacons to gauge the effectiveness of certain communications and the effectiveness of our marketing campaigns via HTML emails.
Cookie List
Consent Leg.Interest
label
label
label |
13066 | https://www.youtube.com/watch?v=YzwIDQi0Qp4 | Parallelogram Proofs
Ms. Smith's Math Tutorials
11200 subscribers
275 likes
Description
18658 views
Posted: 22 Jul 2019
Ms. Smith's Math Tutorials
edit note: There are a couple accidental misspellings in the video (Clearly I moving too fast as I was writing!). The first is at 10:30 with the word "bisects" and the second is at 20:03 with the word "opposite"
12 comments
Transcript:
[Music] welcome to miss miss math tutorials I'm Miss Smith in this video we're gonna be talking about parallelogram proofs we're gonna do three different examples different proofs where we're either trying to prove something is a parallelogram or we're told it's a parallelogram and we're trying to prove something else there are three videos I'm gonna recommend you watching before you watch this one those videos are gonna be and I'll link them I'll link them to this video my video on triangle congruence my video on intro to proofs and my video on parallelogram properties if you get stuck at any point you might want to refer back to one of those videos where I talk about some of these things in much greater detail when we are approaching this problems that we're starting with so it says our given statement is that Kats is a parallelogram okay so we've got this shape here see 80s cats and they're telling us right up front hey this is a parallelogram and they're wanting us to prove that triangle C 80 so this triangle right here so that's kind of like if you thought of it like maybe the top triangle in this picture is congruent to triangle TSC so this bottom triangle so we need to prove that the top triangle is congruent to the bottom triangle knowing that this is a parallelogram and I get it that struggle with proofs is you wanted to say what it is if if C 80 s is a parallelogram then yes those two triangles will be congruent we just know that but that's the tricky thing about proofs is you have to prove it right so step-by-step line by line why is it true so proofs can be tricky for that reason but we're gonna take this line by line and we'll get through it together we always start with what we know okay what have they given us so our first statement is going to be exactly what they told us cats or C ATS is a parallelogram and how do we know this well we know it our reason is because it was given to us so then we have to think okay what's gonna be the best way to go about proving that these two triangles are congruent I always go for side side side so remember in our triangle congruence we talked about the different ways to prove triangles are congruent to me side side side is always the easiest if you can go that direction in this first example that's what I'm gonna do I'm gonna do just the easiest thing which I'm gonna shoot for a set proving that all the sides have a congruent partner let's think about if Katz is a parallelogram then going back to those parallelogram properties I know that this side would have to be congruent to this side right that's one of the basic principles of a parallelogram so I know that line CA is congruent to line TS right and notice I've now marked them I always think it's a good idea that as you do these proofs you mark the picture as you go that's just my personal preference and that's what I'm gonna show like what do I know about this that is true and actually mark it on the picture so you can kind of follow along and how do I know that CA is congruent to TS well it goes back to that rule of parallelogram so I always just note that as opposite sides of a parallelogram are congruent and some teachers may be totally okay with you kind of short handing a bit like I noticed I use the symbol for congruent instead of actually writing out congruent different teachers are gonna have different preferences so just make sure you're paying attention to what your teacher requires but for this video I'm just gonna use some shorthand so let's think of what else we could say so we said ca is congruent to TS we could also say that C CS is congruent to TA right because if this is a parallelogram than our opposite sides have to be congruent so let's write that one down so for our statement number three we know that lines CS is congruent to line TA and how do we know that well for the same reason to this number two we know that opposite sides of a parallelogram are congruent so I'm going to rewrite that again okay so what else do we know so again think about what we're going for I said right at the beginning I wanted to try to prove side-side-side so so far I've got this side is congruent to this side and this side is congruent to this side but let me now look at that interior right this if I were to take those two triangles and separate them this middle line right here this diagonal is actually two separate lines right there just overlaying each other if I were to separate these triangles so we can say that line C T so from here to here is congruent to and you could either say C T again or I like to change it up I like to then go T C right that's just personal preference so that this line is congruent to this line and how do we know that well that's a very special reason that we call the reflexive property so in the reflexive property what we're saying is something is equal to itself and it's a very common property that we use improves so now at this point we can say that triangle CA T is congruent to triangle ts see that these two triangles are congruent to each other based on the triangle congruence of side-side-side right all of the sides are congruent to each other side side side so we can now say our proving statement it's a triangle CA T is congruent to triangle T SC and we know this through the triangle congruence rule of side-side-side and some of your teachers may want you to write out side-side-side I just like to use SSS so in this example they also give us the statement are given is that frog f ro G is a parallelogram so they're telling us right up front hey this shape is a parallelogram which means there's a lot of different things we could prove about it knowing that it is a parallelogram but they specifically want us to prove that triangle FS G so f SG so this like top triangle right here that it is congruent to triangle OSR so Oh s R so that's this bottom triangle so we're trying to prove the top triangle is congruent to the bottom triangle and this can get tricky because there's four different triangles that we're looking at in this shape right there's this one this one this one and this one but you want to focus on what we are trying to prove is congruent so again we always start with our given statement so our first statement is exactly what they told us f R og and our reason how do we know that because it was given because they told us up front it's always gonna be how our proofs are typically gonna start so let's think about what we're trying to prove where we're trying to go and what we probably already already know hopefully if you know your parallelogram properties if we know this is a parallelogram and I'm trying to prove that these two triangles are congruent to each other again I always think the best way if it's possible the best way to go about that is side-side-side I just in my mind that's kind of just what instantly comes to my mind are there other options absolutely and the same thing with the one we just did on on on the first page another teacher could have shown you the same example and gone about it a different way and that is totally fine okay as long as you are is a legitimate reason to prove it then you are good I'm just showing you I like to just show the easiest way at least in my mind so let's think about what we know we know that F G would have to be congruent to O are right and how do we know that well if it's truly a parallelogram then our opposite sides would be congruent so let's go ahead and note that and for our reason so our statement is FGM is congruent to oo art and we know that because opposite signs so we use the same statement on the other page so let's think about the next thing that we could do and again there are a lot of different ways somebody might try to tackle you know that those are vertical angles that's totally acceptable whatever whatever pops into your mind I definitely again I'm getting going first side side side so I noticed I have diagonals here and if this is truly if F ro G is truly a parallelogram then I know that the diagonals bisect each other and remember when you bisect something that means you cut it perfectly in half we know that F o and G are bisect each other so let's state that so o bisects G are and how do we know that well parallelogram diagonals bisect each other if these two lines truly are dissecting each other then let's break up down what that actually means so that means that FS would have to be congruent to OS right notice I used to double tick mark instead of a single because the single was already taken down here so let's state that so FS is congruent to OS and how do we know that well if this angle are sorry if this line is being bisected by that line then that's just a definition of a bisector and some teachers are okay with you shorting like definition to death right with a period so definition of a bicep and then if we know FS is congruent to OS well then we can also say that RS is congruent to GS right for the same reason it's the definition of a bisector so let's note that and at this point hopefully you now see that we've said okay this side is congruent to this side the sides congruent to that side and that's that's congruent to that side so we can prove that triangle FSG is congruent to triangle OS r through side side side right so let's note that you let's look at one more example together so notice this example is a little bit different they've given us some angle congruence statements which we'll look at in detail but they're wanting me to actually prove that this ABCD this this shape is a parallelogram so this one's a little different from the other ones the other ones just told us outright hey it's a parallelogram prove these two triangles are congruent but in this case they're giving us just a little bit of info and saying hey actually work up to proving this is a parallelogram so this one's a little bit of a different challenge let's start by just noting what the Givens are and actually marking them on the picture and then going from there seeing what makes sense our first statement that they tell us is angle BAC is congruent to angle D C a and we know that has to be true because they gave it to us right it's a given so let's go ahead and Mark that on our picture so we've got triangle be a C now be careful you don't mark this whole side because that's not what they're saying right they're just saying be AC so it's just this part right here so that angle is congruent to angle D C a so this angle right here so those two angles are congruent and then they also tell us in our second given statement that angle B C a is congruent to angle D a C so let's and we know that's true because again it's a given so let's actually mark that on our picture and see what that looks like so it's saying angle b c a b c a remember it's always the middle letter that's the actual like turning point BCA and this one is congruent so i'm going to use a little double arc for that one it is congruent to DAC D a/c so it's congruent to this one right there and then at this point that's all they've told us up front and they're saying prove ABCD is in fact a parallelogram so the first thing that I immediately notice is that we've got this middle kind of shared side here so really this is two triangles if I were to separate them out like similar to the one we just did this is actually two different sides they're just overlaying each other so anytime you have that you can go ahead and mark it as congruent so let's go ahead and note that on here that we are saying line AC is congruent to and I like to say AC is congruent to CA just personal preference right they're the same line so AC is congruent to ca and we know that because of the reflexive property and again that reflexive property is something that's really commonly used with us saying you know something is equal to itself notice that we now have enough information to actually prove that these two triangles to the top triangle and the bottom triangle are actually congruent to each other and we can do that because notice we have angle side angle and angle side angle or a si is a way to prove triangle congruence so let's go ahead and state that so we know that triangle a B C so the top triangle is congruent to triangle C D a and the reason we know that's true is because of the angle-side-angle property okay so you might be thinking how does that help us prove that they're a parallelogram right that they create a parallelogram well here's the thing there's another property and again I touch on this in my intro to proofs video but there's a property called we just shorthand it by saying cpctc corresponding parts of congruent triangles are congruent okay so basically what that saying is that if I can prove that these two are triangles you know maybe I haven't said anything about these sides yet right this side BC and this side da but here's the thing if they are corresponding parts of congruent triangles then they are also congruent so that's where that cpctc comes in so corresponding parts of congruent triangles are congruent and it's not just these sides right I could say that same thing about these sides so corresponding parts of congruent triangles are congruent so once we know that two triangles are congruent we can say a lot of things okay pretty much any corresponding part on those triangles is congruent so let's let's pick a side now and and target it so I immediately went for this BC da these two sides so let's go ahead and do that I can now say that side BC is congruent to side da and the reason I can say that is corresponding parts of congruent triangles are congruent cpctc so I immediately after I went to those I immediately went to those so let's let's tackle that can I also say that line a B is congruent to line CD those are corresponding parts right and how can I say that well again we're using that cpctc corresponding parts of congruent triangles are congruent so remember I can prove a parallelogram if I can prove that the two opposite sides are congruent and I've went over that in that parallelogram properties video that if you can prove that these are congruent and these are congruent then you've got a parallelogram okay so we at this point have proved that so we're ready for our proof statement so ABCD is in fact a parallelogram and how do we know that in a parallelogram and you could short you could write this a ton of different ways but I'm just gonna write it that in a parallelogram opposite sides are congruent so that kind of wraps up our parallelogram proofs videos we went through three different examples if you have any questions then definitely leave me one in the comments this has been miss miss math tutorials |
13067 | https://www.aafp.org/pubs/afp/issues/2020/1015/p457.html | AFP 2020: Reflections, New Fellows, Updates, and AFP Statistics | AAFP
This website stores cookies on your computer. These cookies are used to improve the website and provide more personalized services to you, both on this website and through other media. To find out more, please see our Privacy Policy.
ACCEPT
Advertisement
AAFPAAFP
AAFP FoundationFoundation
AFP JournalAFP
FPM JournalFPM
FUTURE (formerly National Conference)FUTURE
FMXFMX
familydoctor.orgfamilydoctor.org
shopping_cart
account_circle Log In
Menu menu Search search
chevron_left Issues chevron_right
chevron_left AFP By Topic chevron_right
chevron_left Collections chevron_right
chevron_left CME Quiz chevron_right
chevron_left Blog chevron_right
chevron_left Multimedia chevron_right
chevron_left Subscribe chevron_right
Search search close
Scheduled maintenance is planned for September 26–29. You may experience brief interruptions during this time.
arrow_back PREV PREV ARTICLEOct 15, 2020NEXT NEXT ARTICLE arrow_forward
Editorials
AFP 2020: Reflections, New Fellows, Updates, and AFP Statistics
picture_as_pdf PDFprint Printcomment Comments
SUMI SEXTON, MD, Georgetown University, Medical Center, Washington, District of Columbia
info
Am Fam Physician. 2020;102(8):457-459
assignment
Author disclosure: No relevant financial affiliations.
Describing the state of American Family Physician (AFP) in 2020, its 70th year of publication, can't be done without first acknowledging the dark shadow of the pandemic and emerging public awareness surrounding racial inequities. In these humbling times, AFP has made some positive changes, including our commitment to increase content related to systemic racism and health inequity. We've stepped outside of our usual editing processes to bring a collection of information related to coronavirus disease 2019 (COVID-19) to our website, and we've published more articles online ahead of print. We are also reviving a beloved department, Diary from a Week in Practice, which is now called Diary of a Family Physician, to foster a sense of camaraderie in primary care. We have two new Jay Siwek medical editing fellows and two patient partners on our Editorial Advisory Board. This editorial will also highlight the top 20 AFP articles of 2019, provide an update on AFP's statistics for print circulation and website visits, and thank our peer reviewers.1,2
Systemic Racism and Health Inequity
AFP is committed to reevaluating our curriculum and processes to better address racism and health inequities.3 Part of this commitment involves collaborating with editors across several other family medicine publications to share a common vision for our specialty and resources to help with this call to action. AFP's entire editing team endorsed an action plan, including this collaboration and other goals, that can be found on our website at Critical components of this plan include increasing content on racism, using more inclusive language, and convening a team focused on diversity, equity, and inclusion that is led by a dedicated AFP medical editor.
Jay Siwek Medical Editing Fellows
The AFP team welcomes two fellows this year, Dr. Joanna Drowos and Dr. Natasha Pyzocha. Each brings a unique background and skill set to the journal. Dr. Drowos is an associate professor of family medicine at the Charles E. Schmidt College of Medicine at Florida Atlantic University. She is also associate dean for faculty affairs and director for the Community and Preventive Medicine Clerkship. Dr. Pyzocha is currently a core physician for 98point6, a text-based virtual primary care service. She previously served in various clinician roles in Fort Carson, Colo. She completed a fellowship in wilderness medicine through the Wilderness Medical Society and serves as a physician volunteer for the U.S. Olympic Committee. You can read more about the new fellows' backgrounds and interests in the AFP Community Blog at
We will soon begin recruiting for the 2021 fellowship. You can read more about the application process at
In addition to medical editing fellows, AFP also has resident and student representatives who help with various projects throughout the year. We thank Dr. Enkhee Tuvshintogs, who completed her residency at the Dignity Health Methodist Hospital of Sacramento Family Medicine Residency Program, where she was chief resident, and Dr. Libby Wetterer, who graduated from Georgetown University School of Medicine, for their valuable input this year. You can find more information about our resident and student representative programs at and
New Content Features
AFP started producing content related to COVID-19 in early February via the community blog. This led to a series of blog posts followed by COVID-19 Daily Research Briefs and online-ahead-of-print editorials, articles, and letters to the editor, which have been compiled on our website at
At the end of 2019, we sunsetted Close-ups, a department that ran in AFP for 12 years and highlighted stories from the patient perspective followed by a commentary from a family physician. Our next venture in sharing the personal side of medicine is to revisit the Diary from a Week in Practice department with a new name and approach. The first Diary of a Family Physician is in this issue of AFP and written by Drs. Annette Chavez and Seiji Hayashi. Dr. Chavez has been in practice for 32 years in the Dayton, Ohio, area with the past 19 years spent as a solo independent family physician at Carillon Family Practice. Dr. Hayashi has extensive experience in primary care leadership, quality improvement, and health policy at the local and national levels. He is currently the chief transformation officer and medical director at Mary's Center, a community health center in Washington, D.C., that provides health care, family literacy, and social services. For information on how to submit as a guest contributor to this department, see
Although Lown Right Care: Reducing Overuse and Underuse is not a new department in AFP, we've added two new contributors who are also AFP's Patient Partners. Including patients in this collaborative way to inform clinical content is not yet common among medical journals. Our objective is to better align medical care with our patients' goals and values.4
AFP's Top 20 Articles of 2019
Don't miss the top 20 most viewed AFP articles on our website last year at The popular topics include treatment of the common cold, type 2 diabetes mellitus therapies, migraine headache prophylaxis, anticoagulation, patellofemoral pain syndrome, concussion, hand-foot-and-mouth disease, and several others, reflecting the broad scope of primary care. Our goal is to highlight all of these topics in AFP Clinical Answers (
Collections of the top 20 research studies for primary care physicians, the top 20 POEMs (Patient-Oriented Evidence that Matters), and the top 20 POEMs consistent with the principles of the Choosing Wisely Campaign, can be found at
Circulation and Website Statistics
The circulation of AFP's print journal is 187,680 (Table 1). To the best of our knowledge, this makes AFP's circulation the second highest among more than 5,000 peer-reviewed medical journals.5 Readership includes all 93,135 active members of the American Academy of Family Physicians (AAFP), approximately 40,000 nonmember family physicians, and approximately 50,000 general internists. AFP has consistently been rated the number one journal among primary care physicians.6
TABLE 1.
American Family Physician Circulation Statistics for 2019
zoom_out_map Enlargeprint Print
| Benchmark | Approximate number |
:---: |
| Print journal | |
| Circulation, twice monthly | 187,680 |
| Online journal | |
| Recipients of electronic table of contents, twice monthly | 108,000 |
| Unique visitors in 2019 | 27,234,422 |
| Unique visitors per month (average) | 2,545,389 |
| Total visits to the website | 39,868,873 |
| Registered unique visitors | 1,255,409 |
| Page views | 49,410,219 |
Note: Unique visitors are visits to the website from a unique browser or device.
In 2019, the AFP website received an average of 2.7 million unique visitors (i.e., from a unique browser or device) per month, making it one of the most popular online medical publications. This means that in addition to visits by almost all AAFP members, the website is viewed by millions of other health care professionals and others who presumably are seeking information about various clinical conditions. Additionally, about 480,000 of the journal's CME quizzes have been taken annually since 2017, and approximately 2.9 million credits were awarded in 2019.
Publication Timelines and Acceptance Rates
On average, review articles are accepted a little under five months after they are submitted, which allows time for peer review and medical editing (Table 2). Articles are typically published eight to nine months after acceptance. A lot goes into the selection of articles for each issue to provide readers with a mix of topics. Online-ahead-of-print articles are typically published online about four months after acceptance. In 2019, we published two articles online ahead of print, whereas we've already had nine in 2020.
TABLE 2.
American Family Physician Publication Timelines and Acceptance Rates for 2019
zoom_out_map Enlargeprint Print
| Benchmark | Number |
:--- |
| Average days to publication | |
| From submission to acceptance | 141 (goal: 150) |
| From acceptance to publication | 257 (goal: 180) |
| From submission to publication | 398 (goal: 330) |
| Manuscript acceptance rates | |
| Accepted solicited manuscript | 97% |
| Approved proposed manuscript† | 87% |
| Accepted unsolicited manuscript (no preapproval process) | 0% |
—About 96% of solicited manuscripts are actually submitted.
†—Only about 10% of proposed manuscripts are approved for submission and, of those, about 92% are actually submitted.
To ensure comprehensive coverage of a curriculum of topics, we solicit most of the articles that are published. We may ask a previous author to update their article, and we also circulate a “call for papers” through multiple family medicine communication platforms. Our preference is for the first author to be an experienced clinician and writer because our readership largely consists of busy physicians who rely on AFP for practical evidence-based content and CME. However, we encourage experienced authors to mentor less experienced coauthors, and we plan to promote a path for diverse coauthorship.
Authors who propose topics go through a formal approval process before submitting a manuscript for consideration, and they are strongly encouraged to read the AFP Authors' Guide ( before contacting us. Roughly 12% of proposed topics are approved. Unapproved proposals are often not selected because the topic has already been covered or does not fit the AFP curriculum. For solicited and approved proposed topics, AFP editors provide authors with detailed guidance on the focus of the article before writing begins. For these reasons, the acceptance rate for solicited manuscripts and approved proposals is high at 95%.
Acknowledging Peer Reviewers
Peer review continues to be an essential and valuable process for AFP, and we are very appreciative of the work reviewers do in helping us develop sound content for our readers.1,7,8 This is the second year that we are highlighting peer reviewers who provided exceptional feedback and received high ratings on the reviews they provided, in addition to recognizing all of our peer reviewers for volunteering their time ( An archive of past AFP peer reviewers is available at
Not only is peer review critical to producing good content, but it is also an important part of learning how to analyze articles and improving one's own authorship skills. We dedicated a portion of our annual editors' meeting this summer to discuss ways to not only find good reviewers, but also ways to cultivate family physician expert reviewers. My hope is to continually improve our processes, enrich the skills of our current and future reviewers, and highlight more exceptional reviewers year after year.
If you are interested in reviewing for AFP, please visit the Reviewers' Guide at The AAFP has authorized CME credit for AAFP members who peer review manuscripts for the journal ( For new reviewers, we highly recommend that you find a mentor for your first review.
In summary, this has been an unparalleled year so far for many reasons, and I am truly thankful for the perseverance of our authors, editors, and reviewers who've kept AFP afloat and even more agile. As always, we welcome reader feedback to help us better meet your needs (afpjournal@aafp.org).
Editor's Note: Dr. Sexton is editor-in-chief of AFP.
expand_more Author Information
Address correspondence to Sumi Sexton, MD, at afpjournal@aafp.org. Reprints are not available from the author.
Author disclosure: No relevant financial affiliations.
expand_more Reference(s)
Sexton S. AFP 2019: new fellow, new features, top 20 Articles of 2018, and AFP statistics. Am Fam Physician. 2019;99(9):538-540. Accessed August 21, 2020.
Siwek J, Sexton SM. AFP 2018: how we're doing. Am Fam Physician. 2018;97(12):772-773. Accessed August 21, 2020.
Sexton SM. From the editor-in-chief. We're listening and taking action on racism and health inequities [published online July 24, 2020]. Am Fam Physician. Accessed August 21, 2020.
Lin KW, Haskell H, James J, et al. Welcoming AFP's patient partners. Am Fam Physician. 2020;101(12):713-714. Accessed August 21, 2020.
BPA Worldwide. American Family Physician brand report and circulation statement. January 2019. Accessed March 18, 2019.
Kantar Media. June 2014 – December 2018 readership summary table 101. Primary care office and hospital.
Siwek J. Acknowledging and thanking our peer reviewers. Am Fam Physician. 2016;93(7):544-545. Accessed August 21, 2020.
Sexton SM. Sexton SM Acknowledging and thanking AFP's peer reviewers for 2017. Am Fam Physician 2018;97(8):502. Accessed August 21, 2020.
Add/View Comments
0 comments
lockLog In to comment
Continue Reading
Oct 15, 2020
Oct 15, 2020
Previous: Formal Training in POCUS Can Positively Impact Family Medicine Practices
Next: Diary of a Family Physician
View the full table of contents chevron_right
Advertisement
More in AFP
More in PubMed
Citation
Related Articles
Most Recent Issue Sep 2025
AFP Email Alerts
Free e-newsletter and email table of contents.
SIGN UP NOW
Copyright © 2020 by the American Academy of Family Physicians.
This content is owned by the AAFP. A person viewing it online may make one printout of the material and may use that printout only for his or her personal, non-commercial reference. This material may not otherwise be downloaded, copied, printed, stored, transmitted or reproduced in any medium, whether now known or later invented, except as authorized in writing by the AAFP.See permissionsfor copyright questions and/or permission requests.
keyboard_tab
Key Sections
AFP Home
Issues
AFP By Topic
Collections
CME Quiz
Blog
Multimedia
Information for
Authors
Reviewers
Subscribers
Advertisers
Services
Subscribe/Renew
Manage Subscription
AFP By Email
Contact AFP
Careers
Using AAFP Content
Back Issues
Other Resources
About AFP
Publication Ethics in AFP
AFP Podcast
Revista Médica AFP Podcast
AFP Digital
EBM Toolkit
FAQs
Copyright© 2025 American Academy of Family Physicians. All Rights Reserved.
About This Site|Privacy Policy|Terms of Use
Connect on Facebook
Connect on X (Twitter)
Connect on Instagram
AAFP Mobile App |
13068 | https://physics-prep.com/index.php/practice-problems-electric-potential-due-to-point-charges-2?view=article&id=1448:practice-problems-electric-potential-due-to-point-charges-solutions&catid=43:physics-2-unit-1 | Unit 1 Physics 2 Workflow
Big Ideas: Unit 1
Presentation: Introduction to Physics 2
Presentation: Basic Electrostatics
Demonstration: Electrostatics and Charging
Practice Problems: Basic Electrostatics
Presentation: The Electric Field
Virtual Activity: The Electric Field
Practice Problems: The Electric Field
Virtual Activity: Motion of a Charged Particle in an E-Field
Presentation: Electric Potential
Practice Problems: Electric Potential
Presentation: Electric Potential Due to Point Charges
Practice Problems: Electric Potential Due to Point Charges
Presentation: Equipotential Surfaces
Practice Problems: Equipotential Surfaces
Video Lab: E-Field Mapping
Virtual Activity: Equipotential Surfaces
Quiz: #1
Presentation: Conductors in Electrostatic Equilibrium
Practice Problems: Conductors in Electrostatic Equilibrium
Presentations: Basic Circuit Analysis
Practice Problems: Basic Circuit Analysis
Lab Activity: Basic Circuit Analysis
Presentation: Kirchhoff's Rules
Practice Problems: Kirchhoff's Rules
Presentation: Short Circuits
Practice Problems: Short Circuits
Presentation: Capacitors
Practice Problems: Capacitors
Presentation: Capacitors and Dielectrics
Practice Problems: Capacitors and Dielectrics
Lab Activity: Capacitor Lab
Presentation: Introduction to RC Circuits
Practice Problems: RC Circuits
Presentation: RC Circuit Charge and Discharge Cycle
Challenge Problem: RC Circuit Analysis
Quiz: #2
Lab Activity: Building Circuits
Review: Unit 1
Test: Unit 1
Physics 2 Click here to see the unit menu
Return to the home page to log out
Do you have questions? Click here to access the class discussion forum.
Practice Problems: Electric Potential Due to Point Charges SolutionsFor all the problems below assume that V = 0 at infinity.
(easy) Determine the electric potential at 0.001 m from a charge of 2pC.
V = kq/r
V = 9x109(2x10-12)/(0.001) = 18 volts
(easy) Refer to the scenario in question #1.
a. If a second charge (-2pC) was the same distance from the point of interest as the first charge, find the total electric potential at that point.
The total potential is the algebraic sum of the potential caused by each charge:
V = Σkq/r
Since the distances are the same for each charge, but the sign is opposite, the total potential is zero in this case.
b. If the second charge was closer to the point of interest would the total potential be positive of negative?
Since the negative charge would be closer than the positive charge, the total potential would be negative.
(easy) Is the magnitude of the electric potential caused by point charges an absolute or a relative value. Explain your answer.
Electric potential is based on electric potential energy. The magnitude of the electric potential energy is relative to an established frame of reference. A convenient F.O.R. is usually utilized wherein a system of two charges separated by infinite distance has a potential energy of zero. Mathematically it can be shown that the electric potential infinitely far from a charge is zero when using the same F.O.R..
(moderate) Two charges are located on corners of a rectangle with a height of 0.05 m and a width of 0.15 m. The first charge (q1= -5x10-6 C) is located at the upper left hand corner, while the second charge (q2 = +2.0 x10-6 C) is at the lower right hand corner. Determine the electric potential at the upper right hand corner of the rectangle.
V = Σ(kq/r)
V = k[(-5x10-6/0.15) + (2x10-6/0.05)]
V = 60000 volts
(moderate) What is the potential difference for a point at the right hand corner (call it point A) of the rectangle in question #2 relative to the lower left hand corner (call it point B)?
VA= 60000 volts
VB = k[(-5x10-6/0.05) + (2x10-6/0.15)] = -780000 volts
ΔV = VA – VB = 60000 - (-780000) = 840000 volts
(moderate) Two charged particles are placed on the x axis of a coordinate system. The first (q1 = 2x10-6 C) is at the origin. The second (q2 = -5x10-6 C) is at x = 1.0 m. Determine a point in between these two charges where the electric potential is zero.
Let x be the distance from q1 to the point of zero potential:
V = Σ(kq/r) = 0
0 = k[(2x10-6/x) – (5x10-6/(1 – x))]
18000/x = 45000/(1 – x)
x = 0.29 m
(moderate) Two charged particles are held in place on the x-axis of a coordinate system. Charge q1 (5 C) is at the origin. Charge q2 (3 C) is at x = 1 m. A relatively small positive test charge (q = 0.01 C, m = 0.001 kg) is released from rest at x = 0.5 m. Will the test charge move to the right or the left? Additionally, use the concepts of electric potential and electric potential energy to determine the speed of the test particle after it moves 0.1 m.
The test particle is repelled from both charges. It begins to move toward the 3C charge because the net force is in that direction.
To find the speed at x = 0.6 m, which is 0.1 m to the right of the initial position, we need to determine the change in K. The change in K is the opposite of the change in U. The change in U is related to the charge and the voltage.
V = k(q1/r1 + q2/r2)
ΔV = k[(5/0.6 + 3/0.4) - (5/0.5 + 3/0.5)]
ΔV = k(5.8) = -1.5x109 volts
ΔU = qΔV
ΔU = (0.01)(-1.5x109) = -1.5x107 J
ΔK = -ΔU = 1.5x107 J
ΔK = K2 – K1 = K2 – 0
1.5x107 = K2 = ½mv2
1.5x107 = ½(0.001)v2
v = 173205 m/s
The "AP" designation is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, products sold on this website.
Powered by Physics Prep LLC. All rights reserved. ©2012-2025 |
13069 | https://www.youtube.com/watch?v=nW0BDJwhqJc | Finding the Missing Angle of a Quadrilateral | Math with Mr. J
Math with Mr. J
1690000 subscribers
2495 likes
Description
224976 views
Posted: 10 Jul 2020
Welcome to Finding the Missing Angle of a Quadrilateral with Mr. J! Need help how to find missing angles of quadrilaterals? You're in the right place!
Whether you're just starting out, or need a quick refresher, this is the video for you if you're looking for help with missing angles in quadrilaterals. Mr. J will go through quadrilateral missing angle examples and explain the steps of how to find the missing angle in a quadrilateral.
About Math with Mr. J: This channel offers instructional videos and mastery checks (practice videos to gauge understanding) that are directly aligned with math standards (4th grade, 5th grade, 6th grade, etc.). Videos can be used to introduce content, reteach content, or as a study tool. Teachers, parents/guardians, and students from around the world have used this channel to help with math content in many different ways. All material is absolutely free.
Click Here to Subscribe to the Greatest Math Channel On Earth:
Follow Mr. J on Twitter: @MrJMath5
Email: math5.mrj@gmail.com
Music:
Hopefully this video is what you're looking for when it comes to finding the missing angle of a quadrilateral.
Have a great rest of your day and thanks again for watching!
262 comments
Transcript:
welcome to math with mr. J in this video I'm going to cover how to find the missing angle within a quadrilateral and we have two examples that we're going to go through together in order to get this down now a very important note at the top of your screen when we're working with quadrilaterals when you add all four angles up it has to equal 360 degrees no matter what type of quadrilateral or what it looks like the magic number is 360 degrees so let's jump right into number one where our missing angle is in the bottom right here represented by that variable X so we need to take a look at our knowns we know we have 120 degrees 180 so the first thing we do is add up those unknowns so 120 degrees plus 100 plus 80 so if we add those up we get 300 degrees so we need to get the 360 so we take what we need to get to 360 degrees and subtract the 300 degrees what we know that will give us what our unknown is so 360 degrees minus 300 degrees equals 60 degrees so x equals 60 degrees that's our missing or unknown angle and it works because if we take a look at it algebraically if we do a hundred degrees plus 120 degrees plus 80 degrees plus X it will have to give us 360 degrees so what we do we isolate our variable we add all of these up and subtract them from 360 and that gives us what x equals so let's take a look at number two and do another example so our unknown is in the top right represented by X so let's take what we do know the 70 132 and 81 degrees add them up and then subtract from 360 so 132 plus 81 degrees plus 70 degrees equals 283 degrees so now we need to do 360 what we need to get to - what we have and see what our missing angle is so a little bit of borrowing action here and we get 77 degrees so x equals 77 degrees so just to recap add up all of your known angle measures and then subtract that from 360 and that will give you your missing angle I hope that helped thanks so much for watching until next time peace |
13070 | https://www.superprof.co.uk/resources/academic/maths/geometry/plane/coplanar.html | Learn Maths from the best
First lesson free!
Chapters
Conditions for Coplanar vectors
Example 1
Example 2
Example 3
Example 4
Example 5
In this article, we will discuss what are coplanar vectors with examples.
The vectors that lie on the same plane in a three-dimensional space are referred to as coplanar vectors. These vectors are parallel to the same plane. We can easily find any two random coplanar vectors in a plane. If two lines are coplanar in a three-dimensional space, then we can represent them in a vector form. The scalar triple product of the coplanar vectors is always equal to zero.
The best Maths tutors available
4.9 (56 reviews)
Sehaj
£60
/h
1st lesson free!
5 (65 reviews)
Intasar
£129
/h
1st lesson free!
5 (47 reviews)
Johann
£50
/h
1st lesson free!
5 (58 reviews)
Poonam
£80
/h
1st lesson free!
4.9 (163 reviews)
Harjinder
£25
/h
1st lesson free!
5 (48 reviews)
Harinder
£25
/h
1st lesson free!
5 (31 reviews)
Hiren
£149
/h
1st lesson free!
4.9 (46 reviews)
Farooq
£50
/h
1st lesson free!
4.9 (56 reviews)
Sehaj
£60
/h
1st lesson free!
5 (65 reviews)
Intasar
£129
/h
1st lesson free!
5 (47 reviews)
Johann
£50
/h
1st lesson free!
5 (58 reviews)
Poonam
£80
/h
1st lesson free!
4.9 (163 reviews)
Harjinder
£25
/h
1st lesson free!
5 (48 reviews)
Harinder
£25
/h
1st lesson free!
5 (31 reviews)
Hiren
£149
/h
1st lesson free!
4.9 (46 reviews)
Farooq
£50
/h
1st lesson free!
Let's go
Conditions for Coplanar vectors
If the scalar triple product of three vectors in 3D space is equal to zero, then we can say that these three vectors are coplanar.
If three vectors in a 3D space are linearly independent, then the vectors are coplanar.
If we have 'n' vectors and a maximum of two vectors linearly independent, then we can say that all the 'n' vectors are coplanar.
To determine whether the three vectors are coplanar or not, we often find the scalar triple product of the three vectors. The scalar triple product, also known as a mixed product, is the scalar product of three vectors. The scalar triple product of three vectors , , and can be mathematically denoted like this:
This product is equal to the dot product of the first vector by the cross product of other two vectors and . It is referred to as a scalar product because just like a dot product, the scalar triple product gives a single number. Mathematically, the scalar triple product is represented as:
=
In the next section, we will solve a couple of examples in which we will determine whether the given vectors are coplanar or not by using the scalar triple product.
Example 1
Determine if point A = (5, 2, 3), B = (1, 6, 7) and C = (4, 2, 5) are coplanar or not.
Solution
The points A, B, and C will be coplanar if the scalar triple product of , , and is equal to zero.
=
First, we will find the cross product of by using a determinant. The elements of the determinant will be the coordinates of these vectors.
We will use the formula for finding a determinant of 3 x 3 matrix to calculate the cross product of and .
Now, we will calculate the dot product of and like this:
Hence, we can conclude that the point A, B, and C are not coplanar because the scalar triple product of the three vectors is not zero.
Example 2
Determine if points A = (0, 1, -1), B = (4, 3, 1) and C = (3, 2, 1) are coplanar or not.
Solution
The points A, B, and C will be coplanar if the scalar triple product of , , and is equal to zero.
=
First, we will find the cross product of by using a determinant. The elements of the determinant will be the coordinates of these vectors.
We will use the formula for finding a determinant of 3 x 3 matrix to calculate the cross product of and .
Now, we will calculate the dot product of and like this:
Hence, we can conclude that the point A, B, and C are coplanar because the scalar triple product of the three vectors is equal to zero.
Example 3
Determine if points A = (5, 1, 1), B = (3, 3, 1) and C = (2, 2, 1) are coplanar or not.
Solution
The points A, B, and C will be coplanar if the scalar triple product of , , and is equal to zero.
=
First, we will find the cross product of by using a determinant. The elements of the determinant will be the coordinates of these vectors.
We will use the formula for finding a determinant of 3 x 3 matrix to calculate the cross product of and .
Now, we will calculate the dot product of and like this:
Hence, we can conclude that the point A, B, and C are not coplanar because the scalar triple product of the three vectors is not equal to zero.
So far, we have solved the problems in which we were given three points. Now, we will start solving the questions in which four points will be given and we will check the coplanarity of those 4 points. So, let us get started.
Example 4
Are four points A = (2, 3, 5), B = (4, 1, 6), C = (1,8, 4), and D = (1, 6, 4) coplanar?
Solution
To find whether these points are coplanar or not, first, we will find , and like this:
= (4 - 2, 1 - 3, 6 - 5) = (2, -2, 1)
= (1 - 2, 8 - 3, 4 - 5) = (-1, 5, -1)
= (1 - 2, 6 - 3, 4 - 5) = (-1, 3, -1)
Now, we will find the determinant of the above vectors like this:
Since, the scalar triple product is not equal to zero, hence the points A, B, C, and D are not coplanar.
Example 5
Are four points A = (1, 5, 7), B = (6, 3, 1), C = (2 ,9, 5), and D = (7, 6, 5) coplanar?
Solution
To find whether these points are coplanar or not, first, we will find , and like this:
= (6 - 1, 3 - 5, 1 - 7) = (5, -2, -6)
= (2 - 1, 9 - 5, 5 - 7) = (-1, 4, -2)
= (7 - 2, 6 - 5, 5 - 7) = (6, 1, -2)
Now, we will find the determinant of the above vectors like this:
Since, the scalar triple product is not equal to zero, hence the points A, B, C, and D are not coplanar.
Did you like this article?
Rate it!
4.38 (8 rating(s))
Loading...
Emma
I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.
Theory
### Irregular Polygons
### Regular Pentagons
### Points, Lines and Planes
### Quadrilaterals and Regular Polygons
### Rhombus
### Triangle
### Diagonals of a Polygon
### Concentric Circles
### Area and Perimeter of a Triangle
### Circular Sectors
### Regular Polygons
### Similar Triangles
### Angles of a Polygon
### Parallelograms
### Orthocenter, Centroid, Circumcenter and Incenter of a Triangle
### Trapezoids
### Circumference
### Equilateral Triangles
### Circles
### Quadrilaterals
### Circular Segments
### Angle Bisectors
### Coplanar
### Angles of the Triangle
### Angles
### Apothems
### Intersection of Three Planes
### Height of a Polygon
### Polygon
### Plane Equation
### Line Segments
### Points
### Medians of a Triangle
### Star Polygons
### Inscribed Polygons
### Sheaf of Planes
### Lune of Hippocrates
### Area and Perimeter of Polygons
### Circumscribed Polygons
### Polygons
### Rectangle
### Right Triangle
### Lines
### Regular Hexagons
### Plane
### Triangles
### Rhomboid
### Sides of a Polygon
### Perpendicular Bisectors
### Intersection of Two Planes
### Squares
Formulas
### Circle Formulas
### Plane Formulas
### Geometry Formulas
### Triangle Formulas
### Area and Perimeter Formulas
### Area Formulas
### Perimeter Formulas
Exercises
### Rectangle Problems
### Trapezoid Problems
### Pythagorean Theorem
### Pythagorean Theorem Worksheet
### Area Word Problems
### Circle Word Problems
### Pythagorean Theorem Word Problems
### Plane Problems
### Square Problems
### Circle Worksheet
### Triangle Problems
### Area Worksheet
Leave us a comment
Cancel reply |
13071 | https://www.researchgate.net/publication/351043667_Comparison_of_the_Sensitivity_and_Specificity_of_Tzanck_Smear_and_Immunofluorescence_Assay_for_the_Diagnosis_of_Cutaneous_Herpes_Simplex_Virus_and_Varicella_Zoster_Virus_Infections_in_a_Real-life_Clin | (PDF) Comparison of the Sensitivity and Specificity of Tzanck Smear and Immunofluorescence Assay for the Diagnosis of Cutaneous Herpes Simplex Virus and Varicella Zoster Virus Infections in a Real-life Clinical Setting
Article PDF Available
Comparison of the Sensitivity and Specificity of Tzanck Smear and Immunofluorescence Assay for the Diagnosis of Cutaneous Herpes Simplex Virus and Varicella Zoster Virus Infections in a Real-life Clinical Setting
April 2021
Siriraj Medical Journal 73(5)
DOI:10.33192/Smj.2021.40
License
CC BY-NC-ND 4.0
Authors:
Chayada Chaiyabutr
Mahidol University
Nuttagarn Jantanapornchai
Nuttagarn Jantanapornchai
This person is not on ResearchGate, or hasn't claimed this research yet.
Chalermkwan Apinuntham
Chalermkwan Apinuntham
This person is not on ResearchGate, or hasn't claimed this research yet.
Charussri Leeyaphan
Mahidol University
Show all 5 authorsHide
Download full-text PDFRead full-text
Download full-text PDF
Read full-text
Download citation
Copy link Link copied
Read full-textDownload citation
Copy link Link copied
Citations (4)References (10)Figures (4)
Abstract and Figures
Objective: This research aim to compare (1) the sensitivities and specificities of Tzanck smears and indirect immunofluorescence assays (IFA) for cutaneous HSV and VZV infections in real-life settings; and (2) the detection rates of the tests for various patient types and lesion morphologies.Materials and Methods: This retrospective study reviewed 440 and 172 samples from patients with clinically suspicious cutaneous HSV and VZV infections, respectively. All patients underwent a Tzanck smear and IFA. The gold standard for the study was agreement of pre- and post-diagnostic coding (determined by a dermatologist) for cutaneous HSV and VZV infections.Results: For HSV infections, the respective sensitivity and specificity of Tzanck smears were 32.8% and 96.6%, whereas those for IFA were 60.7% and 100%. As to VZV infections, the sensitivity and specificity of Tzanck smears were 54.3% and 97.8%, respectively, while the corresponding IFA values were 71.7% and 100%. According to disease characteristics and lesion morphologies, the detection ability of cutaneous HSV by IFA was substantially higher than Tzanck smear especially in immunosuppressed condition. Tzanck smears and IFA demonstrated no statistical difference for early-onset ( 3 days) VZV infections.Conclusion: Tzanck smears and IFA had higher sensitivities for detecting VZV than HSV infections. IFA testing in suspected cutaneous HSV patients with immunosuppressed conditions should be recommended. Despite the overall sensitivity and specificity of IFA being greater than those for Tzanck smears especially in HSV infections, the latter test is a comparable option for early-onset VZV infections.
Demographic data and disease characteristics of the included patients. …
The sensitivities, specificities, positive predictive values (PPV), and negative predictive values (NPV) for the cutaneous HSV and VZV infections. …
Comparison of the sensitivities of detection of the Tzanck smear and IFA for cutaneous HSV infections. …
Comparison of the sensitivities of detection of the Tzanck smear and IFA for cutaneous VZV infections. …
Figures - available via license: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
Content may be subject to copyright.
Discover the world's research
25+ million members
160+ million publication pages
2.3+ billion citations
Join for free
Public Full-text 1
Available via license: CC BY-NC-ND 4.0
Content may be subject to copyright.
Volume 73, No.5: 2021 Siriraj Medical Journal
305
Original Article SMJ
Chayada Chaiyabutr, M.D., Nuttagarn Jantanapornchai, M.D., Chalermkwan Apinuntham, M.D., Charussri
Leeyaphan, M.D., Sukhum Jiamton, M.D., Ph.D.
Department of Dermatology, Faculty of Medicine Siriraj Hospital, Mahidol University, Bangkok 10700, ailand.
Comparison of the Sensitivity and Specicity of
Tzanck Smear and Immunouorescence Assay for
the Diagnosis of Cutaneous Herpes Simplex Virus
and Varicella Zoster Virus Infections in a Real-life
Clinical Setting
ABSTRACT
Objective: This research aims to compare (1) the sensitivity and specificity of Tzanck smear and indirect
immunouorescence assay (IFA) which detect viral antigen for the diagnosis of cutaneous herpes simplex virus
(HSV) and varicella zoster virus (VZV) infections; and (2) the detection rates of the tests among various patient
groups and lesion morphologies.
Materials and Methods: is retrospective study reviewed 440 and 172 samples from patients with clinically
suspicious cutaneous HSV and VZV infections, who underwent both Tzanck smear and IFA, respectively. e gold
standard of the study was dened by showing agreement of diagnostic codes between initial and subsequent visits.
Results: For HSV infections, the respective sensitivity and specicity of Tzanck smear were 32.8% and 96.6%
whereas those of IFA were 60.7% and 100%. As to VZV infections, the sensitivity and specicity of Tzanck smear
were 54.3% and 97.8%, respectively, while the corresponding values of IFA were 71.7% and 100%. According to
disease characteristics and lesion morphologies, IFA provided substantially higher ability to detect HSV than the
Tzanck smear, especially in patients with immunosuppressed conditions. Tzanck smear and IFA demonstrated no
statistically signicant dierence for early-onset VZV infections (≤ 3 days).
Conclusion: e Tzanck smear and IFA had higher sensitivities for detecting VZV than HSV infections. IFA testing
is recommended in patients with immunosuppressed conditions who present with suspected cutaneous HSV
infection. Despite the overall sensitivity and specicity of IFA being greater than those of Tzanck smear especially
in HSV infections, the latter test is comparable option for early-onset VZV infections.
Keywords: Herpes simplex virus; varicella zoster virus; Tzanck smear; immunouorescence (Siriraj Med J 2 021;
73: 305-311)
Corresponding author: Sukhum Jiamton
E-mail: sukhum.jiamton@gmail.com
Received 25 September 2020 Revised 28 December 2020 Accepted 30 December 2020
ORCID ID:
INTRODUCTION
Herpes simplex virus (HSV) and varicella zoster virus
(VZV) are large, enveloped DNA viruses belonging to the
Herpesviridae family.1 Although cutaneous infections of
HSV and VZV are mainly diagnosed by history-taking
and clinical characteristics, laboratory examinations are
sometimes needed for a denite diagnosis.2
Volume 73, No.5: 2021 Siriraj Medical Journal
306
Multiple laboratory options currently available to
diagnose HSV and VZV infections can be categorized into
four groups: (1) morphological tests, such as the Tzanck
smear and tissue histopathology; (2) immunomorphological
tests, like immunouorescence and immunoperoxidase
staining; (3) serological methods, for instance, enzyme-
linked immunosorbent assay and immunoglobulin M/G
titer; and (4) virological testing, for example, viral culture
and viral polymerase chain reaction. A viral culture was
long considered the gold-standard diagnostic test before
the advent of polymerase chain reaction testing.1,3
HSV and VZV are more commonly observed as
cutaneous infections rather than as infections of internal
organs.1,4 Moreover, their cutaneous symptoms are usually
not severe and can be self-limited. Ideally, the chosen
diagnostic test for these infections should be easy to
perform, give a rapid result, and be inexpensive. In
outpatient dermatological settings, the Tzanck smear
and immunouorescence staining are therefore the most
frequently ordered tests at our clinic.
Previous research has found that the sensitivity of
Tzanck smear ranges from 34% to 78% in detecting HSV,
and from 26% to 64% in detecting VZV.5 However, with
a procient technician, the sensitivity of the test may
rise to 80% and its specicity to 90%.6-8 Even though
Tzanck smear is currently considered obsolete in many
countries 1, it still has an important role in developing
countries. ere, the newer testing methodologies are
not only oen deemed to be too expensive, but also
have the drawbacks of slower turnaround times and,
sometimes, a lack of availability.
In terms of immunouorescence testing, previous
studies revealed that the sensitivity and specicity of
immunouorescence staining was greater than those of
Tzanck smear, particularly in the case of VZV infections.
e sensitivity of immunouorescence staining in detecting
cutaneous HSV infections was found to be around 50%
100% compared with the viral culture technique, but
its sensitivity in detecting VZV infections exceeded
that of the viral culture. Moreover, the specicity of
immunouorescence staining was nearly 100%, and it
was able to discriminate between the HSV1/2 and VZV
pathogens.2,8
Earlier studies of the sensitivity and specicity of
the Tzanck smear and immunouorescence testing were
usually performed in a small number of patients, and
compared with those of the viral culture technique as the
gold standard diagnostic modality.9 It is also noteworthy
that few details of the infected patients or the clinical
morphologies of their lesions were reported.2,5,9 us,
the objective of the current research was twofold. e
rst aim was to compare the sensitivity and specicity
of the Tzanck smear and immunouorescence assay for
the diagnosis of cutaneous HSV and VZV infections in a
larger population and in a real-life setting. e secondary
aim of this study was to compare the detection rates of
two tests among various subgroups of patients, durations,
and clinical morphologies of lesions.
MATERIALS AND METHODS
is retrospective research was approved by the
Siriraj Institutional Review Board. (Si 333/2020) e
study reviewed the samples taken from patients with
clinically suspicious cutaneous HSV (ICD-10 B00) and
VZV (ICD-10 B01-B02) infections. During 2012-2019,
the samples had initially been collected from the Infection
Control Clinic of the Department of Dermatology, Faculty
of Medicine Siriraj Hospital, Mahidol University.
All eligible patients had to undergo both a Tzanck
smear and immunofluorescence testing for HSV or
VZV. For each patient, demographic data, onset of
lesions, morphology of the lesions, suspicious diagnosis,
and comorbidities had been collected at their rst visit.
Tzanck-smear and IFA specimens that were reported as
being inadequate for diagnosis were excluded from the
study. e sensitivity and specicity of the tests were
subsequently analyzed only in clinically conrmed cases.
For the purposes of this study, the reference standard
for clinically conrmed diagnosis was an agreement
of diagnostic codes between the rst and subsequent
follow-up visits (determined by a dermatologist).
e Tzanck smear was performed by scraping the
base of lesions with a blunt scalpel blade and spreading
the sample as a thin layer on microscope slides. e slides
were then xed with 100% methyl alcohol for 10 minutes
and stained with eosin solution for 20 seconds. Aer being
rinsed with distilled water, the slides were stained with
3% methylene blue for 60 seconds, rinsed with distilled
water, and allowed to dry. e slides were subsequently
examined under a light microscope. A positive Tzanck
smear was dened as the presence of herpetic cytopathic
eects, such as the presence of multinucleated giant
cells. roughout the 7-year study period, the Tzanck
tests were performed by the same, procient technician,
which obviated inter-rater variability. e typical test
turnaround time was 15 minutes.
As to the immunouorescence staining, our hospital
used the technique of an indirect immunouorescence
assay (IFA) with commercial reagent kit containing HSV
type 1, 2 antibodies (Bio-Rad Laboratories) and VZV
monoclonal antibodies (Merck, Ltd). Specimens scraped
from the base of the lesions were xed in acetone for 15
Chaiyabutr et al.
Volume 73, No.5: 2021 Siriraj Medical Journal
307
Original Article SMJ
minutes before adding a primary antibody. e smear
was then incubated at 37 degrees Celsius for 30 minutes
and rinsed with phosphate-buered saline; pathogen-
specic uorescein-tagged secondary antibodies were
subsequently added, and the mixture was incubated
at 37 degrees Celsius for 30 minutes. e smear was
examined with an epiuorescence microscope by virology
technicians. e test turnaround time was 3 days.
All statistical analyses were undertaken using SPSS
Statistics for Windows, version 18.0 (SPSS Inc., Chicago,
IL, USA). Categorical variables were presented as numbers
or numbers with percentages, while continuous variables
were shown as means with standard deviations. e
sensitivities and specicities of the two tests were calculated.
For the correlation between the tests, Cohen’s Kappa
coecient(
κ
) was reported. A p-value of less than 0.05
was considered to indicate statistical signicance.
RESULTS
A total of 440 and 172 specimens from patients with
clinically suspicious cutaneous HSV and VZV infections,
respectively, were reviewed. e demographic data of the
included patients are detailed in Table 1. e mean age
of the participants was around 50. e majority of them
were tested more than 3 days aer onset of the lesions.
For cutaneous HSV infections, the main characteristic
of the tested lesions was non-vesicle (66.2%) whereas
vesicle was the major type of tested lesions in cutaneous
VZV infections (70.1%).
TABLE 1. Demographic data and disease characteristics of the included patients.
HSV VZV
N = 440 N = 172
N (%) N (%)
Demographic
Sex, Female 239 (54.3) 103 (59.9)
Age (mean) ± SD 51.4 ± 18.4 56.1 ± 18.2
Underlying disease
Hypertension 101 (23.0) 36 (20.9)
Diabetes mellitus 49 (11.1) 20 (11.6)
Autoimmune disease 51 (11.6) 16 (9.3)
Cancer 51 (11.6) 32 (18.6)
HIV (n = 240; n = 62) 45 (18.7) 6 (9.7)
On immunosuppressive drugs 66 (15.0) 26 (15.1)
Disease characteristics
Onset ≤ 3 days (n = 420; n = 167) 165 (39.3) 75 (44.9)
Taken oral acyclovir before testing 29 (6.6) 21 (12.2)
Site
Mucosa 233 (53.0) 8 (4.7)
Skin 207 (47.0) 164 (95.3)
Morphology (n = 420; n = 164)
Vesicle 142 (33.8) 115 (70.1)
Non-vesicle 278 (66.2) 49 (29.9)
Erosion 113 (26.9) 6 (3.7)
Ulcer 105 (25.0) 3 (1.8)
Papule 26 (6.2) 19 (11.6)
Crust 12 (2.9) 12 (7.3)
Erythematous macule 11 (2.6) 8 (4.9)
Verrucous plaque 11 (2.6) 1 (0.6)
Volume 73, No.5: 2021 Siriraj Medical Journal
308
Of 440 specimens, 229 (52%) had clinically conrmed
diagnosis of HSV, while 127 (73.8%) of 172 specimens
had clinically conrmed diagnosis of VZV. Table 2
compares sensitivities, specicities, positive predictive
values, and negative predictive values of the Tzanck smear
and IFA related to the clinically conrmed cases. For
HSV infections, the respective sensitivity and specicity
of Tzanck smear were 32.8% and 96.6% whereas those
of IFA were 60.7% and 100%. As to VZV infections,
the sensitivity and specicity of Tzanck smear were
54.3% and 97.8%, respectively, while the corresponding
values of IFA were 71.7% and 100%. e sensitivity and
specicity of IFA was substantially higher than those
of the Tzanck smear. In addition, the Tzanck smear
and IFA showed higher sensitivities in detecting VZV
infections than HSV infections. e Kappa agreements
between the Tzanck smear and IFA in detecting HSV
and VZV infections were moderate, with the values of
0.4 and 0.5, respectively.
A comparison was made on the sensitivity of the
Tzanck smear and IFA for cutaneous HSV and VZV
infections among various subgroups of patients, durations,
and clinical morphologies of lesions.
In cutaneous HSV infections (Table 3), it was found
that IFA yielded a greater sensitivity in detecting HSV
infections than the Tzanck smear in nearly all subgroups
of patients with statistical signicance. In terms of disease
onset, IFA showed the sensitivity around 60% in both
early-onset (≤ 3 days) and late-onset (> 3 days) HSV
infections whereas the percentage of HSV detection from
the Tzanck smear dropped from 45.2% in early-onset to
24.8% in late-onset HSV infections. Furthermore, the
sensitivity rate of Tzanck smear in patients taken oral
acyclovir before testing was very low (19%) while IFA in
these patients still yielded a sensitivity rate nearly 60%.
Clinical morphologies of the lesions also determined the
sensitivity rates of both tests. IFA also showed a high
sensitivity (around 60%) in detecting HSV in vesicle
and non-vesicle lesions. e detection rate of Tzanck
smear was only 45.7% in vesicle lesions and very low in
non-vesicle lesions (25.4%).
In cutaneous VZV infections (Table 4), even though
IFA yielded a greater sensitivity than Tzanck smear but
the magnitude of dierence was not much as in case of
cutaneous HSV infections. Interestingly, the Tzanck
smear was not statistically dierent from the IFA in some
situations such as early-onset (≤ 3 days) of infection,
non-vesicular lesions and patients who had a history
of taking oral acyclovir before testing.
DISCUSSION
e current research demonstrated that both the
Tzanck smear and IFA had a higher sensitivity in detecting
VZV infections than HSV infections. e comparison of
their sensitivities and specicities revealed that the IFA
was superior overall to the Tzanck smear, corresponding
with earlier ndings.3 e higher sensitivity of IFA was
signicantly shown in nearly all situations of cutaneous
HSV infections.
However, in cutaneous VZV infections, the sensitivity
rate of Tzanck smear was not far dierent from IFA. Our
study showed that the Tzanck smear is still comparable
to the IFA in some VZV-infection situations, such as the
early-onset (≤ 3 days) of infection. is can be explained
by the Tzanck smear having a high ability to detect VZV
infections 5, as well as by a shorter duration of disease
normally resulting in an increase in the sensitivity of
the Tzanck smear.10
e type of lesions is also an important factor in
determining the sensitivity of the two tests. Prior research
TABLE 2. e sensitivities, specicities, positive predictive values (PPV), and negative predictive values (NPV) for
the cutaneous HSV and VZV infections.
HSV infection % Sensitivity % Specicity PPV NPV
Tzanck 32.8 98.6 96 57.5
IFA 60.7 100 100 70.1
VZV infection
Tzanck 54.3 97.8 98.6 43.1
IFA 71.7 100 100 55.6
Abbreviations: PPV: positive predictive value, NPV: negative predictive value
Chaiyabutr et al.
Volume 73, No.5: 2021 Siriraj Medical Journal
309
Original Article SMJ
TABLE 3. Comparison of the sensitivities of detection of the Tzanck smear and IFA for cutaneous HSV infections.
Positive Tzanck smears Positive IFAs P-value
N = 229 N = 229
N (%) N (%)
Demographics
Sex F 54/139 (38.8) 88/139 (63.3) < 0.001
M 21/90 (23.3) 51/90 (56.7) < 0.001
Age ≤ 60 years 46/145 (31.7) 87/145 (60.0) < 0.001
Age > 60 years 29/84 (34.5) 52/84 (61.9) < 0.001
Underlying disease
Hypertension 15/57 (26.3) 33/57 (57.9) 0.001
Diabetes mellitus 9/28 (32.1) 18/28 (64.3) 0.022
Autoimmune disease 6/25 (24.0) 17/25 (68.0) 0.003
Cancer 10/22 (45.5) 14/22 (63.6) 0.388
HIV 8/30 (26.7) 22/30 (73.3) 0.001
On immunosuppressive drugs 9/32 (28.1) 24/32 (75.0) < 0.001
Disease characteristics
Onset ≤ 3 days 42/93 (45.2) 59/93 (63.4) 0.003
Onset > 3 days 31/125 (24.8) 76/125 (60.8) < 0.001
Taken oral acyclovir before testing 4/21 (19.0) 12/21 (57.1) 0.008
Site
Mucosa 26/102 (25.5) 56/102 (54.9) < 0.001
Skin 49/127 (38.6) 83/127 (65.4) <0.001
Morphology
Vesicle 46/102 (45.1) 67/102 (65.7) 0.001
Non-vesicle 28/116 (24.1) 65/116 (56.0) <0.001
Ulcer 12/54 (22.2) 29/54 (53.7) <0.001
Erosion 8/41 (19.5) 24/41 (58.5) <0.001
Hypertrophic 4/9 (44.4) 6/9 (66.7) 0.625
Crust 2/6 (33.3) 2/6 (33.3) 1.000
Papule 2/4 (50.0) 2/4 (50.0) 1.000
Erythematous macule 0/2 (0.00) 2/2 (100) -
Volume 73, No.5: 2021 Siriraj Medical Journal
310
TABLE 4. Comparison of the sensitivities of detection of the Tzanck smear and IFA for cutaneous VZV infections.
Positive Tzanck smears Positive IFAs P-value
N = 127 N = 127
N (%) N (%)
Demographics
Sex F 43/76 (56.6) 56/76 (73.7) 0.011
M 25/51 (49.0) 35/51 (68.0) 0.021
Age ≤ 60 years 40/69 (58.0) 50/69 (72.5) 0.031
Age > 60 years 28/58 (48.3) 41/58 (70.7) 0.007
Underlying disease
Hypertension 16/31 (51.6) 25/31 (80.6) 0.012
Diabetes mellitus 10/16 (62.5) 14/16 (87.5) 0.125
Autoimmune disease 7/12 (58.3) 11/12 (91.7) 0.125
Cancer 10/21 (47.6) 14/21 (66.7) 0.344
HIV 3/5 (60.0) 4/5 (80.0) 1.000
On immunosuppressive drug 15/19 (78.9) 15/19 (78.9) 1.000
Disease characteristics
Onset ≤ 3 days 40/58 (69.0) 47/58 (81.0) 0.092
Onset > 3 days 27/67 (40.3) 43/67 (64.2) 0.002
Taken oral acyclovir before testing 6/20 (30.0) 10/20 (50.0) 0.344
Site
Mucosa 1/3 (33.3) 3/3 (100) –
Skin 67/124 (54.0) 88/124 (71.0) 0.001
Morphology
Vesicle 55/95 (57.9) 75/95 (78.9) <0.001
Non-vesicle 8/26 (30.8) 13/26 (50.0) 0.180
Crust 4/10 (40.0) 7/10 (70.0) 0.250
Papule 2/9 (22.2) 3/9 (33.3) 1.000
Erosion 1/3 (33.3) 0/3 (00.0) -
Erythema 1/2 (50.0) 1/2 (50.0) 1.000
Ulcer 0/2 (00.0) 2/2 (100) -
Chaiyabutr et al.
Volume 73, No.5: 2021 Siriraj Medical Journal
311
Original Article SMJ
has found that vesicles and blisters generally yield higher
sensitivities of detection than other types of lesions
with these two tests 2,10; the present study had a similar
nding. However, our work determined that there was
no statistical dierence in the sensitivities of detection of
the Tzanck smear and IFA for non-vesicular lesions of
VZV. It is possible that the non-vesicular lesions which
were usually in the late stage of infection might have a low
number of virus and was therefore comparable dicult
for both tests to yield the positive result 1, or the number
of specimens enrolled in the non-vesicular-VZV group
might not be enough to provide a statistically signicant
dierence.
Furthermore, in terms of underlying disease of the
patients, the sensitivity of IFA in cutaneous HSV infections
was prominently higher with statistical signicance
compared to Tzanck smear particularly in patients with
immunosuppressive conditions including HIV infection
and taking immunosuppressive agents. e detection
ability of HSV by Tzanck smear in these patients was
around 30% which was substantially lower than IFA (above
70%). IFA testing in suspected cutaneous HSV patients
with immunosuppressed conditions is recommended.
Whether the underlying disease would aect the yield of
Tzanck smear or IFA test in cutaneous VZV infections
was dicult to conclude. As the majority of underlying
diseases or comorbidity subgroups in cutaneous VZV
infections contained a small number of patients.
There are some limitations in this study. The
reference standard for confirmed cases used in this
study was a clinical diagnosis by dermatologists on two
separate occasions, rather than a standard laboratory
investigation like viral culture or polymerase chain reaction
testing. e explanation is that this was a retrospective
study conducted at a dermatology outpatient clinic in
a developing country and in a real-life clinical setting,
where dermatologists need to make prompt diagnosis
without the ready utilization of sophisticated laboratory
testing. For example, the use of viral culture tends to
be avoided because specimens need to be promptly
transported on ice to a laboratory, refrigerated-culture
media are required, and long turnaround times are
involved. Polymerase chain reaction testing, generally
recognized as the platinum standard for VZV and HSV
infections, has a higher sensitivity and specicity than
any other test. Nevertheless, its relatively high cost and
limited accessibility are problematic for developing
countries.
In addition, the number of patients with morphology
of vesicle were substantially higher in VZV (70.1%) than
HSV (33.8%). is might aect the overall sensitivity of both
tests and was another limitation of our study. However,
focusing in subgroup analysis based on morphology of
the lesions, Tzanck smear and IFA still yielded higher
sensitivity in VZV than HSV in either vesicle or non-
vesicle subgroup.
In conclusion, this study revealed the sensitivity and
specicity of the Tzanck smear and IFA which could be
used as a benchmark in a real-life setting. e tests had
a higher sensitivity in detecting VZV infections than
HSV infections. Even though IFA had an overall higher
sensitivity and specicity than the Tzanck smear, the
Tzanck smear is a comparable option to IFA for early-
onset VZV infections. is information is valuable,
especially in an outpatient dermatologic clinic, where
prompt diagnosis of HSV and VZV infections is required.
ACKNOWLEDGMENT
e authors thank Ms. Orawan Supapueng for her
assistance with the statistical analyses.
Conicts of interest: All authors have neither conicts
of interest nor nancial support to declare.
Funding: None
REFERENCES
1.Levin MJ, Weinberg A, Schmid DS. Herpes Simplex Virus and
Varicella-Zoster Virus. Microbiol Spectr, 2016.p.4.
2.Zirn JR, Tompkins SD, Huie C, Shea CR, Rapid detection
and distinction of cutaneous herpesvirus infections by direct
immunouorescence. J Am Acad Dermatol 1995;33: 724-8.
3.Frisch S, Guo AM, Diagnostic methods and management
strategies of herpes simplex and herpes zoster infections. Clin
Geriatr Med 2013;29:501-26.
4.Stuart-Harris C, e epidemiology and clinical presentation
of herpes virus infections. J Antimicrob Chemother 1983;12
Suppl B:1-8.
5.Nahass GT, Goldstein BA, Zhu WY, Sering U, Penneys NS,
Leonardi CL, Comparison of Tzanck smear, viral culture,
and DNA diagnostic methods in detection of herpes simplex
and varicella-zoster infection. JAMA 1992;268:2541-4.
6.Oranje AP, Folkers E, e Tzanck smear: old, but still of
inestimable value. Pediatr Dermatol 1988;5:127-9.
7.Grossman MC, Silvers DN, e Tzanck smear: can dermatologists
accurately interpret it? J Am Acad Dermatol 1992; 27:403-5.
8.
Fan F, Day S, Lu X, Tang YW, Laboratory diagnosis of HSV and
varicella zoster virus infections. Future Virology 2014;9:721-31.
9.Sadick NS, Swenson PD, Kaufman RL, Kaplan MH, Comparison
of detection of varicella-zoster virus by the Tzanck smear,
direct immunouorescence with a monoclonal antibody, and
virus isolation. J Am Acad Dermatol 1987;17:64-69.
10.Ozcan A, Senol M, Saglam H, Seyhan M, Durmaz R, Aktas E,
et al. Comparison of the Tzanck test and polymerase chain
reaction in the diagnosis of cutaneous herpes simplex and
varicella zoster virus infections. Int J Dermatol 2007;46:1177-9.
Citations (4)
References (10)
... Tzanck smear can detect VZV better than other viral infections, showing high sensitivity and specificity. 22 Serological diagnosis is another method. It is used in cases where viral detection is not possible. ...
Varicella Zoster virus: A re-emerging risk in Immunocompromised populations and potential therapies
Article
Jul 2025
Anushree Anand
R. Sai Krupaa
Manisha Khatri
Varicella Zoster Virus (VZV), from the Herpesviridae family, is responsible for causing Varicella, commonly known as chickenpox, during primary infection and Herpes Zoster, known as shingles, upon. Its reactivation can lead to various neurological complications. With more than half of the world population harboring latent VZV, the global burden of herpes zoster and its associated complications significantly contributes to healthcare costs, underscoring the need for effective vaccination strategies and therapeutic interventions. This review explores the recent advancements in VZV epidemiology, pathogenesis, the clinical spectrum of VZV reactivation, vaccine development, including and therapeutics targeting the latent virus and the immune system.
View
Show abstract
... Despite this, the negative varicella IgM and T-Zanck smear results, combined with the histological findings of vasculitis, reduced the likelihood of DHZ or other viral infections, redirecting the diagnosis towards AHS. Nevertheless, intravenous acyclovir was continued for the full 7-day course recommended for DHZ due to the following considerations: reports have documented that immunocompromised patients can develop severe systemic varicella zoster infection, even with negative IgM (7,8), the limited sensitivity of T-Zanck testing for VZV, and the observed clinical improvement (9). The absence of VZV PCR testing at our hospital, a key diagnostic tool for detecting VZV nucleic acids in clinical specimens, limited the ability to definitively confirm or exclude VZV activation in this case (10). ...
Azathioprine Hypersensitivity Syndrome Mimicking Herpes Zoster and Linear IgA Dermatosis Presentation: A Case Report
Article
Feb 2025
ACTA DERM-VENEREOL
Fatimah Budair
Maryam M. Alfaraj
Khaled A Alharthi
Jumana Alratroot
View
... Despite this, the negative varicella IgM and T-Zanck smear results, combined with the histological findings of vasculitis, reduced the likelihood of DHZ or other viral infections, redirecting the diagnosis towards AHS. Nevertheless, intravenous acyclovir was continued for the full 7-day course recommended for DHZ due to the following considerations: reports have documented that immunocompromised patients can develop severe systemic varicella zoster infection, even with negative IgM (7,8), the limited sensitivity of T-Zanck testing for VZV, and the observed clinical improvement (9). The absence of VZV PCR testing at our hospital, a key diagnostic tool for detecting VZV nucleic acids in clinical specimens, limited the ability to definitively confirm or exclude VZV activation in this case (10). ...
Azathioprine Hypersensitivity Syndrome Mimicking Herpes Zoster and Linear IgA Dermatosis Presentation: A Case Report
Article
Full-text available
Feb 2025
ACTA DERM-VENEREOL
Fatimah Budair
Maryam M. Alfaraj
Khaled A Alharthi
Jumana Alratroot
View
Herpes simplex viruses
Chapter
Jan 2024
Alexander S. Maris
Lili Tao
Jonathan E. Schmitz
View
Herpes Simplex Virus and Varicella-Zoster Virus
Article
Full-text available
Jun 2016
Myron J. Levin
Adriana Weinberg
D. Scott Schmid
Herpes Simplex Virus and Varicella-Zoster Virus, Page 1 of 2 Abstract Herpes simplex viruses (HSV) are enveloped large DNA viruses (approximately 152,250 to 154,750 base pairs, depending on HSV type, and 90 transcriptional units). HSVs are α-herpesviruses (family Herpesviridae) and are divided into three major clades. There are two HSV types (HSV 1 and 2) that are genetically distinct but that are colinear and share roughly 83% genomic homology ( 1 ). HSV from mucocutaneous lesions, or from asymptomatic shedding in the oral or genital secretions from an infected contact, enters the skin/mucosa of a new host through minor breaks or abrasions to infect the underlying epithelium. Infection at the dermal–epidermal junction produces characteristic skin lesions. These begin as macules and papules that culminate in vesicles that contain infectious virus ( Fig. 1A ). These form pustules that rupture after 2 days; the resulting ulcers and crusts form within 96 hours. Vesicles and pustules are most likely to contain infectious virus, whereas ulcers and crusts reliably contain HSV DNA. Local vesicle formation is the hallmark of HSV infection, but vesicles may not be appreciated on mucosal surfaces because they rupture shortly after forming ( Fig. 1B ). An essential characteristic of α-herpesviruses during primary infection is entry into a permanent latent relationship with sensory neurons, with the potential for subsequent reactivation to cause recurrent local infections ( 2 ). Diagnostic Microbiology of the Immunocompromised Host, Second Edition
View
Show abstract
Laboratory diagnosis of HSV and varicella zoster virus infections
Article
Aug 2014
Feinan Fan
Stephen Day
Xuedong Lu
Yi-Wei Tang
HSV and varicella zoster virus (VZV) are common pathogens of skin and mucous membranes and the CNS. Their rapid and accurate diagnosis is essential for their treatment as well as infection control. Cytological and morphological examination, specific antibody detection methods and virus isolation all have their own limitations in clinical practice. In recent years, molecular methods have become the primary diagnostic methods for the detection and differentiation of HSV1/2 and VZV due to their high sensitivity, specificity, rapid test turnaround time and potential for high throughput and automation. Although molecular assays detect HSV1/2 and VZV more quickly, the clinical significance of positive results may vary in individual patients.
View
Show abstract
Diagnostic Methods and Management Strategies of Herpes Simplex and Herpes Zoster Infections
Article
May 2013
CLIN GERIATR MED
Stephanie Frisch
Aibing Mary Guo
Herpes infections are extremely prevalent in the adult population. Recognizing early signs and symptoms is essential to provide effective treatment. The immunocompromised population presents treatment challenges requiring prolonged antiviral therapy and more frequent recurrences. Viral culture is often considered the gold standard diagnostic technique; however, polymerase chain reaction (PCR) should be done in tandem with culture especially for varicella zoster virus infections. Antivirals can decrease viral shedding, recurrences of herpes simplex, and hasten healing of herpes zoster. Herpes virus can be a challenging entity to treat with significant morbidity (both physically and psychologically).
View
Show abstract
The Tzanck Smear: Old, but Still of Inestimable Value
Article
May 1988
PEDIATR DERMATOL
Arnold P. Oranje M.D
Elzo Folkers M.D
Arnold Oranje
Elzo Folkers
The Tzanck smear, introduced by the Frenchman Tzanck, has been used for many years in the diagnosis of bullous and vesicular dermatoses especially herpes simplex, herpes zoster and varicella. Also, in the vesicular dermatoses of newborns and immunocompromised patients it is an important diagnostic aid. The Tzanck smear is easy to perform, simple, reliable (if one is experienced) and inexpensive. A quick staining can be done hy Hemacolor or Diff-Quik within one minute. The Tzanck smear is certainly useful and very reliable in the diagnosis of herpes smiplex and varicella-zoster infections if obtained in the vesicular stage. The sensitivity of the Tzanck smear exceeds 80% and the specificity 90% when the investigators are experienced. A positive Tzanck smear is an especially reliable and useful diagnostic aid. However, a disadvantage is that the smear cannot differentiate between herpes simplex virus or varicella-zoster virus infections. The Tzanck smear, although old and simple, remains an important aid in the diagnosis of vesicular, pustular and bullous diseases, in particular herpes. Cytology will, however, never replace culturing or histopathoiogy.
View
Show abstract
Comparison of Tzanck Smear, Viral Culture, and DNA Diagnostic Methods in Detection of Herpes Simplex and Varicella-Zoster Infection
Article
Dec 1992
George T. Nahass
Barbara A. Goldstein
Wenyuan Zhu
Craig Leonardi
To compare Tzanck smears, viral cultures, and DNA diagnostic methods using the polymerase chain reaction (PCR) in detection of herpes simplex virus (HSV) or varicella-zoster virus (VZV) infection in clinically suspected cases. A 12-month trial comparing PCR with viral cultures and Tzanck smears in patients with clinically suspected HSV or VZV infection. Both ambulatory and hospitalized patients were recruited from a tertiary referral center and the Miami (Fla) Veterans Affairs Medical Center. Convenience samples of patients clinically suspected to have HSV (n = 48) or VZV (n = 35). To be included in the final analysis patients needed to have a positive Tzanck smear, viral culture, or PCR result. Patients who were clinically suspected to have HSV but had VZV by viral culture or PCR were analyzed in the VZV group. Similarly, patients who were clinically suspected to have VZV, but had HSV by viral culture or PCR were analyzed in the HSV group. Seventy-seven patients were available for final analysis: HSV (n = 30), VZV (n = 32), and 15 control cases who did not have evidence of viral infection. For HSV, PCR detected HSV DNA sequences in 73% of stained smears and 83% of unstained smears. For VZV infection, VZV DNA sequences were detected in 88% of stained smears and 97% of unstained smears. Viral DNA sequences were not detected in the 15 control cases. Viral cultures were positive in 83% and 44% of HSV and VZV cases, respectively. The Tzanck smear was positive in 60% and 75% of HSV and VZV cases, respectively. PCR is a reliable method for detecting HSV and VZV DNA sequences from single stained and unstained Tzanck smears. It is clearly superior to viral culture in identifying VZV infection and is equivalent to conventional culture techniques in identifying cases of HSV.
View
Show abstract
The Tzanck smear: Can dermatologists accurately interpret it?
Article
Oct 1992
J AM ACAD DERMATOL
Melanie C. Grossman
David N. Silvers
The Tzanck preparation is a standard technique for the rapid diagnosis of herpes simplex and varicella-zoster virus infections. This study was designed to determine the ability of practicing dermatologists to interpret Tzanck preparations accurately. Dermatologists at different levels of training interpreted a series of Tzanck preparations under test conditions. Second- and third-year residents had a pooled average for correct responses of 91%; dermatologists in practice less than 10 years, 84%; dermatologists in practice more than 10 years, 67%. Dermatologists are able to use the Tzanck preparation effectively for diagnosing herpetic infections. Second- and third-year residents who are most likely to be diagnosing blistering eruptions in immunosuppressed or otherwise critically ill patients are especially accurate interpreters.
View
Show abstract
Comparison of detection of Varicella-zoster virus by the Tzanck smear, direct immunofluorescence with a monoclonal antibody, and virus isolation
Article
Aug 1987
J AM ACAD DERMATOL
Neil S Sadick
Paul Swenson
Roderick L. Kaufman
Mark Kaplan
A study comparing direct immunofluorescence assay using a new monoclonal antibody specific for a varicella-zoster virus glycoprotein complex, the Tzanck smear, and virus isolation for detection of varicella-zoster virus in 56 patients with clinically apparent herpes zoster is presented. Of 47 patients with clinical herpes zoster and with cultures negative for herpes simplex virus, 30 (64%) had positive Tzanck smears, direct immunofluorescence assay results were positive in 26 (55%), and cultures were positive in only 12 (26%). Both direct immunofluorescence assay and the Tzanck smear were found to be superior to culture technics; however, direct immunofluorescence assay was found to have greater specificity.
View
Show abstract
The Epidemiology and clinical presentation of herpes virus infections
Article
Oct 1983
C Stuart-Harris
The chief characters of infection by the human herpes viruses are considered with particular reference to herpes simplex viruses, types 1 and 2. Infection with type 1 virus is acquired very early in life though infrequently as a true congenital transmission of virus. Primary infections result from direct contact usually with infected saliva or skin vesicles. Kerato-conjunctivitis, when primary, may be severe yet superficial in extent. Vulvo-vaginitis, often acquired in adults as a result of type 2 infection by sexual transmission, can give extensive but superficial ulceration and discharge. Recurrent infections located on the dermatome with the same nerve supply as that of the organ affected primarily occur throughout life and at relatively short intervals. Sensory nerve ganglia harbour the virus particles as latent infection and when reactivation occurs virus spreads along nerve fibres to the skin. The most serious infections occur as disseminated disease with liver involvement in the neonatal period, in infants suffering from malnutrition or those undergoing immunosuppression for malignancies. Eczematous children are then at particular risk from spreading haemorrhagic skin lesions (Kaposi's eruption). Herpes encephalitis, commoner in adults than children, is an insidious severe disease with mortality related to the depth of coma. Antiviral therapy though successful may lead to chronic neurological sequelae. The success of antiviral therapy in herpes partly turns on the ability to bring the drug into close contact with the infected tissues. Latent virus is relatively unaffected by acyclovir and thus far recurrences have continued to occur.
View
Show abstract
Rapid detection and distinction of cutaneous herpesvirus infections by direct immunofluorescence
Article
Dec 1995
J AM ACAD DERMATOL
Jonathan R Zirn
Stacy D. Tompkins
Chester Huie
Christopher R Shea
Optimal management of cutaneous herpes simplex virus (HSV) and varicellazoster virus (VZV) infections requires rapid, accurate distinction between these pathogens. In a mixed-case series of suspected cutaneous herpesvirus infections, we compared the diagnostic utility of viral culture and direct immunofluorescence (DIF) using a panel of fluoresceinated monoclonal antibodies against HSV and VZV. Epifluorescence microscopy of smears and viral culture were performed in parallel on 58 lesions. DIF and culture were equally sensitive (88%) in HSV infections, whereas DIF was four times as sensitive as culture (100% vs 18%) in VZV. DIF either refuted an incorrect clinical diagnosis or permitted definitive laboratory diagnosis of a clinically indeterminate lesion in 7 (12%) of 58 lesions tested. DIF is a rapid, simple, sensitive, specific, cost-effective, and clinically useful technique for detecting and distinguishing cutaneous HSV and VZV infections.
View
Show abstract
Comparison of the Tzanck test and polymerase chain reaction in the diagnosis of cutaneous herpes simplex and varicella zoster virus infections
Article
Dec 2007
INT J DERMATOL
Atilla Ozcan
Mustafa Senol
Hulya Saglam
Ibrahim Halil Ozerol
Background: Although the diagnosis of herpes simplex virus (HSV) and varicella zoster virus (VZV) infections is usually made clinically, the Tzanck test, electron microscopy, viral culture, polymerase chain reaction (PCR), and serologic tests can be utilized to verify the diagnosis. We conducted a study on a total of 98 patients (77 patients with recurrent herpes simplex and 21 patients with herpes zoster) to evaluate the reliability and reproducibility of the Tzanck test in comparison with PCR. In herpes virus infections, the general positivity rates of the Tzanck test and PCR were 61.2% and 79.6%, respectively. The difference between the positivity rates of the two tests was statistically significant. The positivity rates of the tests differed according to the type and duration of the lesions. Although PCR was superior to the Tzanck test, the Tzanck test has also been proven to be a reliable diagnostic method, with a sensitivity of 76.9% and a specificity of 100%. We recommend the use of this easy, quick, reproducible, and inexpensive diagnostic test more often in dermatologic practice, especially in cutaneous herpes virus infections.
View
Show abstract
Show more
Recommended publications
Discover more
Article
Full-text available
Clinical Validation of the Lyra Direct HSV1+2/VZV Assay for Simultaneous Detection and Differentiati...
September 2014 · Journal of Clinical Microbiology
Feinan Fan
Jeffrey Stiles
Albina Mikhlina
[...]
Yi-Wei Tang
We evaluated the Lyra Direct HSV 1+2/VZV multiplex real-time PCR assay for the detection and differentiation of herpes simplex virus 1 (HSV-1), HSV-2, and varicella-zoster virus (VZV) on 695 consecutive cutaneous and mucocutaneous lesion specimens. The intra-assay and interassay coefficient of variation values for the Lyra assay were 0.29 to 1.30% and 2.33 to 2.61%, respectively. The ... [Show full abstract] sensitivities, specificities, and positive and negative predictive values were 93.4 to 95.0%, 96.1 to 96.8%, 78.0 to 80.3%, and 99.0 to 99.1%, respectively, in comparison to those of viral culture. The values were further improved when a resolution analysis was performed with a laboratory-developed PCR assay.
View full-text
Article
Comparison of tzanck test and polymerase chain reaction in the diagnosis of herpes simplex virus and...
May 2011 · Korean Journal of Dermatology
J.K. Kim
H.J. Lee
Dae-Hyun Kim
Moon Soo Yoon
Background: Herpes simplex virus (HSV) and varicella zoster virus (VZV) are two of the most common causes of mucocutaneous vesicular eruptions. Their diagnoses are usually made clinically, but the clinical distinction between HSV and VZV is sometimes difficult. Objective: We investigated the usefulness of polymerase chain reaction (PCR) and Tzanck test in patients with HSV and VZV infections. ... [Show full abstract] Methods: From June 2008 through June 2010, a total of 396 patients (53 patients with HSV and 343 patients with VZV) were included in this study. Wright-stained smears of scrapings from the base of skin lesions and viral DNA amplification by PCR were examined in all patients. We compared the positivity rates of Tzanck test and PCR according to virus type and duration of skin lesions. Results: The overall positivity rates of Tzanck test and PCR were 56.6% and 86.9%, respectively, and the difference between the positivity rates of the two tests was statistically significant (p<0.0001). The Tzanck test was positive in 41.5% and 58.9% of the HSV and VZV cases, respectively, and the difference was statistically significant (p=0.0259). For PCR, there were no significant differences between the HSV and VZV cases. The positivity rate of the Tzanck test decreased in the old skin lesions over 7 days; however, PCR method showed no significant differences in positivity rates according to duration of the skin lesions. Conclusion: PCR is a more sensitive method for the diagnoses of HSV and VZV infections, and it can be utilized for diagnosis even of old skin lesions.
Read more
Article
Clinical utility of loop-mediated isothermal amplification assay for the diagnosis of common alpha h...
December 2013 · The Journal of Dermatology
Tsukane Kobayashi
Akiko Yagami
Kayoko Suzuki
[...]
Kayoko Matsunaga
Loop-mediated isothermal amplification (LAMP) is a nucleic acid amplification method with a high specificity, efficiency and speed. No reports exist regarding the usefulness of LAMP for clinically suspected skin infections caused by herpes simplex virus (HSV) or varicella zoster virus (VZV). The aim of this study was to evaluate the clinical usefulness of LAMP in the diagnosis of common cutaneous ... [Show full abstract] alpha herpesvirus (HSV type 1 and 2, and VZV) infections. LAMP and real-time polymerase chain reaction (PCR) were performed using swab samples collected from 106 patients with clinically suspected alpha herpesvirus skin infections. The results of LAMP performed with DNA extraction did not differ from those performed without DNA extraction. The sensitivity of LAMP tested against real-time PCR was 96% in herpes simplex, 78% in eczema herpeticum, 93% in herpes zoster and 100% in varicella. No viral DNA was detected by LAMP in all negative real-time PCR samples. Viral DNA load was significantly lower in samples with false-negative LAMP results than in the LAMP-positive samples. LAMP enables confirmation of clinically suspected cutaneous HSV and VZV infections. However, the sensitivity of LAMP is lower than real-time PCR. The accuracy of LAMP may increase if sufficient viral DNA is obtained from lesions. LAMP performed without DNA extraction remains sensitive; thus, LAMP represents a quick and economical method for the diagnosis of common alpha herpesvirus skin infections.
Read more
Article
Herpes zoster as a first manifestation of HIV infection
January 2014 · Dermatología Revista Mexicana
S. Pérez-Cortés
M.G. Olguín-García
Larissa Dorina López-Cepeda
[...]
L. Alonzo-Romero Pareyón
Herpes zoster is a skin disease caused by reactivation of latent varicella zoster virus in the sensory ganglia dorsal root. The incidence increases significantly in HIV positive patients. In these patients, it can be seen more affection, with necrotic lesions and dissemination, tends to be persistent, with a torpid clinical course, and has a higher rate of complications. This paper reports a ... [Show full abstract] brief review of the literature and presents two cases of herpes zoster as the first manifestation of infection with the HIV.
Read more
Last Updated: 18 Aug 2025
Discover the world's research
Join ResearchGate to find the people and research you need to help your work.
Join for free
ResearchGate iOS App Get it from the App Store now.
Install
Keep up with your stats and more
Access scientific knowledge from anywhere
or
Discover by subject area
Recruit researchers
Join for free
LoginEmail Tip: Most researchers use their institutional email address as their ResearchGate login Password Forgot password? - [x] Keep me logged in Log in or Continue with Google Welcome back! Please log in. Email · HintTip: Most researchers use their institutional email address as their ResearchGate login Password Forgot password? - [x] Keep me logged in Log in or Continue with Google No account? Sign up
Company
About us
News
Careers
Support
Help Center
Business solutions
Advertising
Recruiting
© 2008-2025 ResearchGate GmbH. All rights reserved.
Terms
Privacy
Copyright
Imprint
Consent preferences |
13072 | https://www.depauw.edu/learn/econexcel/busanalytics/book/Tex_files/6DElasticity.pdf | Elasticity You probably have heard of the price elasticity of demand, but you may not know what it means or how to use the concept.
Our goal is to truly understand elasticity and be comfortable using it. At its most fundamental level, it is simply a numerical measure of responsiveness.
In the initial version of the Lifeguard Problem, the lifeguard entered the wa-ter after running roughly 56 meters. When maximum running speed doubled to 10 m/sec, ceteris paribus, the optimal solution changed to running almost 80 meters. We can (and did) show this on a graph, but is there a faster way to summarize comparative statics analysis? Yes, in a word, elasticity.
We have been working on a constrained optimization problem where you sipped one brandy and smoked four cigars when the doctor set a limit of 5 total brandies and cigars. When that limit was relaxed and you were al-lowed a total of 6 units, you chose 1.25 brandies and 4.75 cigars. Again, we can (and did) show this on a graph, but elasticity captures the relation-ship between the amount consumed and the total allowed in a single number.
We proceed by reviewing a few general ideas about elasticity and what it is trying to convey. Then we move to actual computations and practice inter-preting elasticity values.
The more examples you see, the more the concept will stick. Be sure to keep an eye out for the repeated pattern in the elasticity. We always have an optimal solution that is responding to a shock and the elasticity measures if the response is weak or strong.
1 Elasticity Basics Elasticity is a pure number (it has no units) that measures the sensitivity or responsiveness of one variable when another changes. Elasticity, respon-siveness, and sensitivity are synonyms. An elasticity number expresses the impact one variable has on another. The closer the elasticity is to zero, the more insensitive or inelastic the relationship is between two variables.
Elasticity is often expressed as “the something elasticity of something,” like the price elasticity of demand. The first something, the price, is always the exogenous variable; the second something, in this case demand (the amount purchased), is the response or optimal value being tracked.
A less common, but perhaps clearer, way to express the cause and effect is to say, “the elasticity of something with respect to something.” The elasticity of demand with respect to price makes it clear that demand depends on and responds to the price.
Unlike the difference between the new and initial values, elasticity is com-puted as the ratio of percentage changes in the values. The endogenous or response variable always goes in the numerator and the exogenous or shock variable is always in the denominator. Thus, the x elasticity of y is %∆y %∆x.
The percentage change, new−initial initial , is the change (or difference), new minus initial, divided by the initial value. This affects the units in the computation.
The units in the numerator and denominator of the percentage change cancel and we are left with a percent as the units. If we compute the percentage change in apples from 2 to 3 apples, we get a 50% in increase. The change (or delta), however, is +1 apple.
If we divide one percentage change by another, as we do with an elasticity computation, %∆y %∆x, the percents cancel and we get a unitless number. Thus, elasticity is a pure number with no units. So if the price elasticity of demand for apples is −1.2, there are no apples, dollars, percents, or any other units.
It’s just −1.2.
The −1.2 can be used to compute the percentage change in apples if the price of apples increases by 10%. We simply multiply −1.2 by 10% to get −12%.
Or, if the price of apples falls by 20%, we know that the quantity demanded of apples will rise by 24% (−1.2 × −20%).
2 We can also use an elasticity to compute the exogenous shock needed to pro-duce a given percentage change in the endogenous variable. If ApplesRUs, Inc knew that the price elasticity of demand for apples was −1.2 and they wanted to increase apples sold by 6%, then they would lower prices by 5% (6% divided by −1.2 ).
Elasticity is a ratio of percentages changes so there are three numbers in-volved: the elasticity, the percentage change in the numerator, and the per-centage change in the denominator. We are given and use two of the three to find the third one: 1. Given %∆x and %∆y, find the elasticity: %∆y %∆x.
2. Given %∆x and elasticity, find the %∆y : elasticity × %∆x.
3. Given %∆y and elasticity, find the %∆x: %∆y elasticity.
The lack of units in an elasticity measure means we can compare wildly dif-ferent things. No matter the underlying units of the variables, we can put the dimensionless elasticity number on a common yardstick and interpret it.
Figure 6.11 shows the possible values that an elasticity can take, along with the names we give particular values.
Figure 6.11: Elasticity on the number line.
Empirically, elasticities are usually low numbers around one (in absolute value). An elasticity of +2 is extremely responsive or elastic because the response is twice the shock. It means that a 1% increase in the exogenous variable generates a 2% increase in the endogenous variable.
3 The sign of the elasticity indicates direction (a qualitative statement about the relationship between the two variables). Zero means that there is no re-lationship—i.e., that the exogenous variable does not influence the response variable at all. Thus, −2 is extremely responsive like +2, but the variables are inversely related so a 1% increase in the exogenous variable leads to a 2% decrease in the endogenous variable.
One (both positive and negative) is an important marker on the elasticity number line because it tells you if the given percentage change in an exoge-nous variable results in a smaller percentage change (when the elasticity is less than one), an equal percentage change (elasticity equal to one), or greater percentage change (elasticity greater than one) in the endogenous variable.
The adjective perfectly is used to identify two extreme cases. If the elasticity is zero, it is perfectly inelastic and this means there is no response at all to a shock. This is rare, usually optimal values of endogenous variables adjusts to changes in the environment.
Perfectly elastic means the elasticity measure is infinity (positive or nega-tive).
This means that the tiniest little change in an exogenous variable triggers a massive response in the endogenous variable. Again, this is a rare, limiting case for elasticity.
Elasticities can be confusing. There is a lot to remember. Below are six com-mon misconceptions and issues surrounding elasticity. Reading these typical mistakes will help you better understand this fundamental, but easily misin-terpreted, concept.
1. Elasticity is about the relationship between two variables, not just the change in one variable. Thus, do not confuse a negative elasticity as meaning that the response variable must decrease. The negative means that the two variables move in opposite directions. So, if the age elas-ticity of time playing sports is negative, that means both that time playing sports falls as age increases and time playing sports rises as age decreases.
2. Elasticity is a local phenomenon. The elasticity will usually change if we analyze a different initial value of the exogenous variable. Thus, any one measure of elasticity is a local or point value that applies only to the change in the exogenous variable under consideration from that 4 starting point. You should not think of a price elasticity of demand of −0.6 as applying to an entire demand curve. Instead, it is a statement about the movement in price from one value to another value close by, say $3.00/unit to $3.01/unit. The price elasticity of demand from $4.00/unit to $4.01/unit may be different. There are constant elasticity functions, where the elasticity is the same all along the function, but they are a special case.
3. Elasticity can be calculated for different size changes. To compute the x elasticity of y, we can go from one point to another, %∆y %∆x, but the size of the change in x can vary. The computed elasticity will be different depending on the size of the shock if the relationship is non-linear.
4. Elasticity always puts the response variable in the numerator. Do not confuse the numerator and denominator in the computation. In the x elasticity of y, x is the exogenous or shock variable and y is the endoge-nous or response variable. Students will often compute the reciprocal of the correct elasticity. Avoid this common mistake by always checking to make sure that the variable in the numerator responds or is driven by the variable in the denominator.
5. Remember that elasticity is unitless. The x elasticity of y of 0.2 is not 20%. It is 0.2. It means that a 1% increase in x leads to a 0.2% increase in y.
6. Perhaps the single most confusing thing about elasticity is its relation-ship to the slope: Do not confuse elasticity with slope. This is easy to forget and deserves careful consideration. Remember that elasticity is a percentage change calculation, %∆y %∆x, while a slope is merely the rise over the run, ∆y ∆x.
Economists, unlike chemists or physicists, often gloss over the units of vari-ables and results. If we carefully consider the units involved, we can ensure that the difference between the slope and elasticity is crystal clear.
The slope is a quantitative measure in the units of the two variables being compared. If Q = P 2 , then the slope, ∆Q ∆P = 1 2. This says that an increase in P of $1/unit will lead to an increase in Q of 1 2 a unit. Thus, the slope would be measured in units squared per dollar (so that when multiplied by the price, we end up with just units of Q).
5 Elasticity, on the other hand, is a quantitative measure based on percentage changes and is, therefore, unitless. The P elasticity of Q = 1 says that a 1% increase in P leads to a 1% increase in Q. It does not say anything about the actual, numerical $/unit increase in P, but speaks of the percent-age increase in P. Elasticity focuses on the percentage change in Q, not the change in terms of number of units.
Thus, elasticity and slope are two different ways to measure the responsive-ness of a variable as another variable changes. Elasticity uses percentage changes, %∆y %∆x, while the slope does not, ∆y ∆x. They are two different ways to measure the effect of a shock and confusing them is a common mistake.
Computing Elasticity When the total allowed B and C went from 5 to 6, you changed B from 1 to 1.25 and C from 4 to 4.75. We can compute two elasticities with these numbers.
The total allowed elasticity of brandies is %∆B %∆T = ∆B B ∆T T = newB−initialB initialB newT−initialT initialT = 1.25−1 1 6−5 5 = 0.25 0.2 = 1.25 The total allowed elasticity of brandies is 1.25 because we had a 20% increase in T (from 5 to 6) and this led to a slightly bigger, 25% increase in brandies (from 1 to 1.25). Thus, we say that the brandies response is elastic, or pretty responsive. Figure 6.11 shows that any elasticity greater than 1 in absolute value is said to be elastic.
The total allowed elasticity of cigars is %∆C %∆T = ∆C C ∆T T = newC−initialC initialC newT−initialT initialT = 4.75−4 4 6−5 5 = 0.1875 0.2 = 0.9375 The total allowed elasticity of cigars is 0.9375 because we had a 20% increase in T (from 5 to 6) and this led to a slightly smaller, 18.75% increase in cigars (from 4 to 4.75). Thus, we say that the cigars response is inelastic, or unre-sponsive. Figure 6.11 shows that any elasticity less than 1 in absolute value is said to be inelastic.
6 Since we know the slope of the optimal brandies as a function of total allowed is 0.25 and the total allowed elasticity of brandies is 1.25, that is conclusive proof that elasticity and slope are different. Another way we can show the difference is with a little algebra.
%∆B %∆T = ∆B B ∆T T = ∆B B T ∆T = ∆B ∆T T B The last term shows we can compute the elasticity by multiplying the slope by T B. This shows that elasticity is slope times the ratio of the exogenous to the endogenous variable values. In this example, 0.25 times 5/1 is 1.25.
We can also show that elasticity changes as you change the point from which it is measured.
STEP In your CS1 sheet, put the label %DB/%DT in cell F8 and then change the two Ds to Symbol font. In cell F10, enter the formula =((C10 -C9)/C9)/((A10 - A9)/A9) and fill it down.
Cell F10 reproduces the 1.25 elasticity we computed earlier, but notice how the elasticities get smaller as T rises. Again, this shows that elasticity is not slope since the slope stays constant while the elasticity changes.
Elasticity Practice Work on these elasticity computations and questions to improve your under-standing. Answers are provided in the appendix (according to step number).
STEP 1. Compute the max run speed elasticity of distance on sand in the lifeguard problem as max run speed is increased from 5 m/sec to 10 m/sec. Interpret your result.
STEP 2. Compute the IR (infection rate) elasticity of group size as IR falls from 5% to 2%. Recall that optimal group size rose from 5 to 8.
Interpret your result.
STEP 3. Compute the slope of C = f(T) and use it to compute the total allowed elasticity of cigars at T = 5. Does your number agree with the 0.9375 value we found earlier?
7 STEP 4.
Compute the slope of C = f(T) and the total allowed elasticity of cigars from T = 9 to 10. Does the slope or elasticity change compared to the elasticity from T = 5 to 6? What does this show?
Figure 6.12: Smoking rates in Japan and the United States.
Source: ourworldindata.org/smoking Cigarettes have been extensively studied. The average number of cigarettes sold per day in the United States and Japan since 1900 is shown in Figure 6.12.
Visit ourworldindata.org/smoking to see an interactive version of this chart and to add other countries. The pattern is the same around the world—rising smoking rates reach a peak, then they decline. Today, a little over 10% of American adults smoke, down from 40% at the peak.
Governments want to reduce cigarette consumption to improve public health.
Banning advertising is common, as is taxing cigarettes. The idea is that in-creasing the total price consumers must pay (the price plus the tax) will reduce consumption. Whether this works depends on the price elasticity of demand (the quantity purchased).
STEP 5. To reduce cigarette consumption in response to a tax, what are governments hoping is true about the price elasticity of demand for cigarettes?
8 STEP 6. What do you think is a good guess for the price elasticity of demand for cigarettes? Explain your answer.
STEP 7. How do you think the price elasticity of demand for cigarettes compares between adult and teenage smokers? Explain your answer.
Takeaways Comparative statics is how economists view the world and elasticity is how they communicate comparative statics results.
You want to be able to interpret and compute it: Interpret: The closer an elasticity is to zero, the less responsive the endoge-nous variable is to a particular shock.
Compute: The exogenous variable elasticity of the endogenous variable is always the percentage change in the endogenous variable divided by the per-centage change in the exogenous variable.
There are other ways to compute elasticities. The ratio of percentage changes is the simplest, most basic approach.
References The economics literature on cigarette smoking is vast.
Frank A. Sloan, V. Kerry Smith, and Donald H. Taylor, “Information, Addiction, and Bad ‘Choices’: Lessons from a Century of Cigarettes,” Economics Letters, Vol.
77 (2002), pp. 147-155, is an accessible, informative starting point.
For a broader, historical review, see Allan M. Brandt, The Cigarette Century: The Rise, Fall, and Deadly Persistence of the Product That Defined America (2007).
9 Appendix 1. The max run speed elasticity of distance on sand is 0.43 and this is quite inelastic, or unresponsive. Max run speed was doubled (so 100% increase) and distance on sand did increase, but only by 43%.
2. The IR elasticity of group size is −1 and this is unit elastic. IR fell by 60% (from 5 to 2) and group size rose by 60% (from 5 to 8). The minus sign means the two variables are inversely related.
3. The slope of C = f(T) is 3/4, so multiplying this by 5/4 is 15/16 which does agree with the 0.9375 value in the text.
4. The slope of C = f(T) stays constant at 0.75, but the elasticity in-creases from 0.9375 to roughly 0.9643. This shows that elasticity is a local phenomenon that changes depending on the value of the exogenous variable at which it is computed.
5. Governments hope that the cigarette demand is elastic, meaning that the price elasticity of demand is high. This way, small increases in taxes will produce big decreases in cigarette consumption.
6. Many studies have produced a variety of results, but the price elasticity of demand for cigarettes is expected to be inelastic, so less than one in absolute value. A good guess would be −0.6.
7. Teenage smokers are more price sensitive since they are not as addicted yet and typically have lower incomes than adults.
If adults are at −0.6, teenagers might be at −1.4.
10 |
13073 | https://www.literarysalon.ca/post/what-makes-a-literary-classic | Error
TemplatesWix.com
Looks like this domain isn't
connected to a website yet
Is this your domain?
Connect it to your Wix website in just a few easy steps:
Go to Wix.com> Subscriptions > Domains
Click Use a Domain You Already Own
Follow the steps to connect your domain to your website
Need more help?
Please contact our Support Team |
13074 | https://en.wikipedia.org/wiki/Oriented_matroid | Jump to content
Oriented matroid
Русский
Українська
Edit links
From Wikipedia, the free encyclopedia
Abstraction of ordered linear algebra
An oriented matroid is a mathematical structure that abstracts the properties of directed graphs, vector arrangements over ordered fields, and hyperplane arrangements over ordered fields. In comparison, an ordinary (i.e., non-oriented) matroid abstracts the dependence properties that are common both to graphs, which are not necessarily directed, and to arrangements of vectors over fields, which are not necessarily ordered.
All oriented matroids have an underlying matroid. Thus, results on ordinary matroids can be applied to oriented matroids. However, the converse is false; some matroids cannot become an oriented matroid by orienting an underlying structure (e.g., circuits or independent sets).
The distinction between matroids and oriented matroids is discussed further below.
Matroids are often useful in areas such as dimension theory and algorithms.
Because of an oriented matroid's inclusion of additional details about the oriented nature of a structure,
its usefulness extends further into several areas including geometry and optimization.
History
[edit]
The first appearance of oriented matroids was in a 1966 article by George J. Minty and was confined to regular matroids.
Subsequently R.T. Rockefellar (1969) suggested the problem of generalizing Minty's concept to real vector spaces. His proposal helped lead to the development of the general theory.
Background
[edit]
Main article: Signed set
In order to abstract the concept of orientation on the edges of a graph to sets, one needs the ability to assign "direction" to the elements of a set. The way this achieved is with the following definition of signed sets.
A signed set, , combines a set of objects, , with an ordered bipartition of that set into two disjoint subsets: and .
: The members of are called the positive elements; members of are the negative elements.
The set is called the support of .
The empty signed set, , is defined as the empty set combined with an (ordered) bipartition of it into two empty sets: and .
The signed set is the opposite of , written , if and
Given an element of the support, we will write for a positive element and for a negative element. In this way, a signed set is just adding negative signs to distinguished elements. This will make sense as a "direction" only when we consider orientations of larger structures. Then the sign of each element will encode its direction relative to this orientation.
Axiomatizations
[edit]
See also: Matroid § Bases and circuits, Axiomatic system, and Cryptomorphism
Like ordinary matroids, several equivalent systems of axioms exist. (Such structures that possess multiple equivalent axiomatizations are called cryptomorphic.)
Circuit axioms
[edit]
Let be any set. We refer to as the ground set. Let be a collection of signed sets, each of which is supported by a subset of .
If the following axioms hold for , then equivalently is the set of signed circuits
for an oriented matroid on .
(C0)
(C1) (symmetric)
(C2) (incomparable)
(C3) (weak elimination)
Vector Axioms
[edit]
The composition of signed sets and is the signed set defined by , , and . The vectors of an oriented matroid are the compositions of circuits. The vectors of an oriented matroid satisfy the following axioms:
for all ,
for all , and , there is a , such that
,
, and
.
The covectors of an oriented matroid are the vectors of its dual oriented matroid.
Chirotope axioms
[edit]
Let be as above. For each non-negative integer , a chirotope of rank is a function that satisfies the following axioms:
(B0) (non-trivial): is not identically zero
(B1) (alternating): For any permutation and , , where is the sign of the permutation.
(B2) (exchange): For any such that for each , then we also have .
The term chirotope is derived from the mathematical notion of chirality, which is a concept abstracted from chemistry, where it is used to distinguish molecules that have the same structure except for a reflection.
Equivalence
[edit]
Every chirotope of rank gives rise to a set of bases of a matroid on consisting of those -element subsets that assigns a nonzero value. The chirotope can then sign the circuits of that matroid. If is a circuit of the described matroid, then where is a basis. Then can be signed with positive elements
and negative elements the complement. Thus a chirotope gives rise to the oriented bases of an oriented matroid. In this sense, (B0) is the nonempty axiom for bases and (B2) is the basis exchange property.
Examples
[edit]
Oriented matroids are often introduced (e.g., Bachem and Kern) as an abstraction for directed graphs or systems of linear inequalities. Below are the explicit constructions.
Directed graphs
[edit]
Main article: Directed graph
See also: Flow network
Given a digraph, we define a signed circuit from the standard circuit of the graph by the following method. The support of the signed circuit is the standard set of edges in a minimal cycle. We go along the cycle in the clockwise or anticlockwise direction assigning those edges whose orientation agrees with the direction to the positive elements and those edges whose orientation disagrees with the direction to the negative elements . If is the set of all such , then is the set of signed circuits of an oriented matroid on the set of edges of the directed graph.
If we consider the directed graph on the right, then we can see that there are only two (graph) circuits, namely and . Then there are only four possible signed circuits corresponding to clockwise and anticlockwise orientations, namely , , , and . These four sets form the set of signed circuits of an oriented matroid on the set .
This is opposed to the ordinary directed graphic matroid for the same graph, whose only circuit is , as the other edges cannot form a directed cycle without flipping the direction of some of them.
Linear algebra
[edit]
See also: Matroid § Matroids from linear algebra
If is any finite subset of , then the set of minimal linearly dependent sets forms the circuit set of a matroid on . To extend this construction to oriented matroids, for each circuit there is a minimal linear dependence
with . Then the signed circuit has positive elements and negative elements . The set of all such forms the set of signed circuits of an oriented matroid on . Oriented matroids that can be realized this way are called representable.
Given the same set of vectors , we can define the same oriented matroid with a chirotope . For any let
where the right hand side of the equation is the sign of the determinant. Then is the chirotope of the same oriented matroid on the set .
Hyperplane arrangements
[edit]
Main article: Arrangement of hyperplanes
A real hyperplane arrangement is a finite set of hyperplanes in , each containing the origin. By picking one side of each hyperplane to be the positive side, we obtain an arrangement of half-spaces. A half-space arrangement breaks down the ambient space into a finite collection of cells, each defined by which side of each hyperplane it lands on. That is, assign each point to the signed set with if is on the positive side of and if is on the negative side of . This collection of signed sets defines the set of covectors of the oriented matroid, which are the vectors of the dual oriented matroid.
Each rank-3 oriented matroid is equivalent to an arrangement of pseudolines, and each oriented matroid which is also uniform is equivalent to a simple pseudoline arrangement (where every 2 lines cross exactly once, and no 3 lines cross at the same point). Tools for dealing with one form can therefore be used to analyze its equivalent form for either study.
Convex polytope
[edit]
Main article: Convex polytope
Günter M. Ziegler introduces oriented matroids via convex polytopes.
Results
[edit]
Orientability
[edit]
A standard matroid is called orientable if its circuits are the supports of signed circuits of some oriented matroid. It is known that all real representable matroids are orientable. It is also known that the class of orientable matroids is closed under taking minors, however the list of forbidden minors for orientable matroids is known to be infinite. In this sense, oriented matroids is a much stricter formalization than regular matroids.
Duality
[edit]
See also: Matroid § Duality
Just as a matroid has a unique duals, an oriented matroid has a unique dual, often called its "orthogonal dual". What this means is that the underlying matroids are dual and that the cocircuits are signed so that they are "orthogonal" to every circuit. Two signed sets are said to be orthogonal if the intersection of their supports is empty or if the restriction of their positive elements to the intersection and negative elements to the intersection form two nonidentical and non-opposite signed sets. The existence and uniqueness of the dual oriented matroid depends on the fact that every signed circuit is orthogonal to every signed cocircuit.
To see why orthogonality is necessary for uniqueness one needs only to look to the digraph example above. We know that for planar graphs the dual of the circuit matroid is the circuit matroid of the graph's planar dual. Thus there are as many different dual pairs of oriented matroids based on the matroid of the graph as there are ways to orient the graph and in a corresponding way its dual.
To see the explicit construction of this unique orthogonal dual oriented matroid, consider an oriented matroid's chirotope . If we consider a list of elements of as a cyclic permutation then we define to be the sign of the associated permutation. If is defined as
then is the chirotope of the unique orthogonal dual oriented matroid.
Topological representation
[edit]
Not all oriented matroids are representable—that is, not all have realizations as point configurations, or, equivalently, hyperplane arrangements. However, in some sense, all oriented matroids come close to having realizations as hyperplane arrangements. In particular, the Folkman–Lawrence topological representation theorem states that any oriented matroid has a realization as an arrangement of pseudospheres. A -dimensional pseudosphere is an embedding of such that there exists a homeomorphism so that embeds as an equator of . In this sense a pseudosphere is just a tame sphere (as opposed to wild spheres). A pseudosphere arrangement in is a collection of pseudospheres that intersect along pseudospheres. Finally, the Folkman–Lawrence topological representation theorem states that every oriented matroid of rank can be obtained from a pseudosphere arrangement in . It is named after Jon Folkman and Jim Lawrence, who published it in 1978.
Geometry
[edit]
Main article: Combinatorial geometry
See also: Convex polytope and Zonotope
The theory of oriented matroids has influenced the development of combinatorial geometry, especially the theory of convex polytopes, zonotopes, and configurations of vectors (equivalently, arrangements of hyperplanes). Many results—Carathéodory's theorem, Helly's theorem, Radon's theorem, the Hahn–Banach theorem, the Krein–Milman theorem, the lemma of Farkas—can be formulated using appropriate oriented matroids.
Optimization
[edit]
See also: Linear programming, Quadratic programming, and Criss-cross algorithm
The development of an axiom system for oriented matroids was initiated by R. Tyrrell Rockafellar to describe the sign patterns of the matrices arising through the pivoting operations of Dantzig's simplex algorithm; Rockafellar was inspired by Albert W. Tucker's studies of such sign patterns in "Tucker tableaux".
The theory of oriented matroids has led to breakthroughs in combinatorial optimization. In linear programming, it was the language in which Robert G. Bland formulated his pivoting rule, by which the simplex algorithm avoids cycles. Similarly, it was used by Terlaky and Zhang to prove that their criss-cross algorithms have finite termination for linear programming problems. Similar results were made in convex quadratic programming by Todd and Terlaky. It has been applied to linear-fractional programming, quadratic-programming problems, and linear complementarity problems.
Outside of combinatorial optimization, oriented matroid theory also appears in convex minimization in Rockafellar's theory of "monotropic programming" and related notions of "fortified descent". Similarly, matroid theory has influenced the development of combinatorial algorithms, particularly the greedy algorithm. More generally, a greedoid is useful for studying the finite termination of algorithms.
References
[edit]
^ R. Tyrrell Rockafellar 1969. Anders Björner et alia, Chapters 1-3. Jürgen Bokowski, Chapter 1. Günter M. Ziegler, Chapter 7.
^ Björner et alia, Chapters 1-3. Bokowski, Chapters 1-4.
^ Because matroids and oriented matroids are abstractions of other mathematical abstractions, nearly all the relevant books are written for mathematical scientists rather than for the general public. For learning about oriented matroids, a good preparation is to study the textbook on linear optimization by Nering and Tucker, which is infused with oriented-matroid ideas, and then to proceed to Ziegler's lectures on polytopes.
^ Björner et alia, Chapter 7.9.
^ G. J. Minty (1966), On the axiomatic foundations of the theories of directed linear graphs, electrical networks, and network programming. J. Math. Mech. 15: 485–520. Reprinted in D. R. Fulkerson, ed., Graph Theory, M.A.A. Study No. 12, Mathematical Association of America.
^ Björner et alia, Chapter 3.5
^ Björner, Anders; Las Vergnas, Michel; Sturmfels, Bernd; White, Neil; Ziegler, Günter (1999). Oriented Matroids. Encyclopedia of Mathematics and Its Applications. Vol. 46 (2nd ed.). Cambridge University Press. ISBN 978-0-521-77750-6. OCLC 776950824. Zbl 0944.52006.
^
Felsner, Stefan; Goodman, Jacob E. (2017). "Pseudoline Arrangements" (PDF). Handbook of Discrete and Computational Geometry (3rd ed.). Chapman and Hall/CRC. ISBN 9781315119601.
^ Björner et alia, Chapter 7.9
^ Björner et alia, Chapter 3.4
^ Björner et alia, Chapter 3.6
^ Björner et alia, Chapter 5.2
^ Bachem and Kern, Chapters 1–2 and 4–9. Björner et alia, Chapters 1–8. Ziegler, Chapter 7–8. Bokowski, Chapters 7–10.
^ Bachem and Wanka, Chapters 1–2, 5, 7–9. Björner et alia, Chapter 8.
^ Rockafellar, R. Tyrrell (1969). "The elementary vectors of a subspace of (1967)" (PDF). In R. C. Bose; Thomas A. Dowling (eds.). Combinatorial Mathematics and its Applications. The University of North Carolina Monograph Series in Probability and Statistics. Chapel Hill, North Carolina: University of North Carolina Press. pp. 104–127. MR 0278972. Archived from the original (PDF) on 2011-09-27. Retrieved 2011-03-23.
^ Björner et alia, Chapters 8–9. Fukuda and Terlaky. Compare Ziegler.
^ Illés, Szirmai & Terlaky (1999)
^ Fukuda & Terlaky (1997)
^ Fukuda & Terlaky (1997, p. 385)
^ Fukuda & Namiki (1994, p. 367)
^ Rockafellar 1984 and 1998.
^ Lawler. Rockafellar 1984 and 1998.
Further reading
[edit]
Books
[edit]
Bachem, Achim; Kern, Walter (1992). Linear Programming Duality: An Introduction to Oriented Matroids. Universitext. Springer-Verlag. doi:10.1007/978-3-642-58152-6. ISBN 978-3-540-55417-2. MR 1230380.
Björner, Anders; Las Vergnas, Michel; Sturmfels, Bernd; White, Neil; Ziegler, Günter (1999). Oriented Matroids. Encyclopedia of Mathematics and Its Applications. Vol. 46 (2nd ed.). Cambridge University Press. ISBN 978-0-521-77750-6. Zbl 0944.52006.
Bokowski, Jürgen (2006). Computational oriented matroids. Equivalence classes of matrices within a natural framework. Cambridge University Press. ISBN 978-0-521-84930-2. Zbl 1120.52011.
Lawler, Eugene (2001). Combinatorial Optimization: Networks and Matroids. Dover. ISBN 978-0-486-41453-9. Zbl 1058.90057.
Evar D. Nering and Albert W. Tucker, 1993, Linear Programs and Related Problems, Academic Press. (elementary)
Rockafellar, R. Tyrrell (1984). Network Flows and Monotropic Optimization. Wiley-Interscience. republished by Athena Scientific of Dimitri Bertsekas, 1998.
Ziegler, Günter M. (1994). Lectures on Polytopes. New York: Springer-Verlag.
Richter-Gebert, Jürgen; Ziegler, Günter M. (1997). "Oriented Matroids". In Goodman, Jacob E.; O'Rourke, Joseph (eds.). Handbook of Discrete and Computational Geometry. Boca Raton: CRC Press. pp. 111–132. ISBN 9780849385247.
Articles
[edit]
A. Bachem, A. Wanka, Separation theorems for oriented matroids, Discrete Math. 70 (1988) 303–310.
Bland, Robert G. (1977). "New finite pivoting rules for the simplex method". Mathematics of Operations Research. 2 (2): 103–107. doi:10.1287/moor.2.2.103.
Folkman, Jon; Lawrence, Jim (October 1978). "Oriented Matroids". Journal of Combinatorial Theory. Series B. 25 (2): 199–236. doi:10.1016/0095-8956(78)90039-4.
Fukuda, Komei; Terlaky, Tamás (1997). Thomas M. Liebling; Dominique de Werra (eds.). "Criss-cross methods: A fresh view on pivot algorithms" (PDF). Mathematical Programming, Series B. 79 (1–3). Amsterdam: North-Holland Publishing Co.: 369–395. doi:10.1007/BF02614325. MR 1464775. S2CID 2794181.
Fukuda, Komei; Namiki, Makoto (March 1994). "On extremal behaviors of Murty's least index method". Mathematical Programming. 64 (1): 365–370. doi:10.1007/BF01582581. MR 1286455. S2CID 21476636.
Illés, Tibor; Szirmai, Ákos; Terlaky, Tamás (1999). "The finite criss-cross method for hyperbolic programming". European Journal of Operational Research. 114 (1): 198–214. CiteSeerX 10.1.1.36.7090. doi:10.1016/S0377-2217(98)00049-6. ISSN 0377-2217. Archived from the original on 2011-07-21. Retrieved 2011-03-23.
R. Tyrrell Rockafellar (1969). The elementary vectors of a subspace of , in Combinatorial Mathematics and its Applications, R. C. Bose and T. A. Dowling (eds.), Univ. of North Carolina Press, pp. 104–127.
Roos, C. (1990). "An exponential example for Terlaky's pivoting rule for the criss-cross simplex method". Mathematical Programming. Series A. 46 (1): 79–84. doi:10.1007/BF01585729. MR 1045573. S2CID 33463483.
Terlaky, T. (1985). "A convergent criss-cross method". Optimization: A Journal of Mathematical Programming and Operations Research. 16 (5): 683–690. doi:10.1080/02331938508843067. ISSN 0233-1934. MR 0798939.
Terlaky, Tamás (1987). "A finite crisscross method for oriented matroids". Journal of Combinatorial Theory. Series B. 42 (3): 319–327. doi:10.1016/0095-8956(87)90049-9. ISSN 0095-8956. MR 0888684.
Terlaky, Tamás; Zhang, Shu Zhong (1993). "Pivot rules for linear programming: A Survey on recent theoretical developments". Annals of Operations Research. 46–47 (1): 203–233. CiteSeerX 10.1.1.36.7658. doi:10.1007/BF02096264. ISSN 0254-5330. MR 1260019. S2CID 6058077.
Todd, Michael J. (1985). "Linear and quadratic programming in oriented matroids". Journal of Combinatorial Theory. Series B. 39 (2): 105–133. doi:10.1016/0095-8956(85)90042-5.
Wang, Zhe Min (1987). "A finite conformal-elimination free algorithm over oriented matroid programming". Chinese Annals of Mathematics (Shuxue Niankan B Ji). Series B. 8 (1): 120–125. ISSN 0252-9599. MR 0886756.
On the web
[edit]
Ziegler, Günter (1998). "Oriented Matroids Today". The Electronic Journal of Combinatorics: DS4: Sep 10–1998. doi:10.37236/25.
Malkevitch, Joseph. "Oriented Matroids: The Power of Unification". Feature Column. American Mathematical Society. Retrieved 2009-09-14.
Retrieved from "
Category:
Oriented matroids
Hidden categories:
Articles with short description
Short description matches Wikidata
CS1: long volume value |
13075 | https://www.youtube.com/watch?v=7zOMnC-BAFY | Solving Exact Differential Equations (Differential Equations 29)
Professor Leonard
1020000 subscribers
2216 likes
Description
115836 views
Posted: 20 Mar 2019
Some practice and further explanation solving Exact Differential Equations.
124 comments
Transcript:
Introduction hi welcome to a video where we're gonna we are gonna practice solving some exact differential equation finding potential functions from these exact differential equations and the last video we recovered was was what they were why why they are the way they are where they come from and how to solve them right now we're just gonna practice and like always I'm gonna give you a lots of examples because I wants you to try them on your own I'm gonna give you some that I think are a little more difficult course and that way we can see how to work with initial values so that's our last problem I'll show you where that is if you want to skip there and just see that one that's fine I will also show you that whether you integrate M with respect to X or and respect to Y first it doesn't matter so I'm gonna show you that on last example as well so let's get right to it well I'm gonna discuss with you and kind of review what an executive resolutely equation is as we go through this but mostly this video is about the technique of solving them so we're just going to get some examples under a belt that way you feel very confident going for the worst thing we can have is for you to go forward and not feel like you know what you're doing that's an awful feeling so I'm gonna make sure that you do feel good about your understanding this stuff because you really can't get it so let's Is that an exact differential take a look at this and the first question that you're gonna be asked is is that an exact differential equation and you're gonna look at it and probably you're gonna go yep and that's wrong that's that's a mistake and the mistake is that's not an exact differential equation of some potential function unless it satisfies a very specific quality and that the mixed partials would be equal why is that because in order for a and exists for a potential function f of X y equals C to have this sort of a form of a total differential or an exact differential the mixed partials would have to be equal same it came from some function where it was continuous and differentiable on some region and so what we're doing is going backwards from that we're saying the this is only an exact differential equation of some potential function if the mixed partials are equal because if that's a case then it is coming from some function that was continuous and differentiable on some sort of region and would have given us this differential I hope you understand the the very specific nuance there that it's really important you're absolutely gonna go yeah because if the mixed partials aren't equal then no that is not an exact differential equation of some sort of potential function so we're going to check that first it's a first step last video I gave you in some nice steps on how you go through we're going to explore those so step number one we're going to typically call this and and this and M is the piece this this partial derivative of F with respect to X that's joined to the DX and the N is this piece this partial derivative of F with respect to Y that's joined to the dy and we're gonna say all right all right well well if M is the partial derivative of F with respect to X so this is M and and it is the partial derivative of F with respect to Y this is bad so that's all well and good how do i do my test guys if M is already the partial derivative of F with respect to X then the mixed partial would take this and just do the partial derivative of M with respect to Y or in this case if you don't remember partial derivatives you really need to go back and check that out in this case we say well that'd be like an X sorry that would be 0 this would be 1 over X because those x's would be held to be constants now they don't go to 0 when you take a derivative son like this they are treated like constant coefficients so you get 1 over that constant so like derivative of Y over 2 wouldn't giving you 1 over 2 1/2 1 so we're gonna give you one half here we're just gonna get 1 over X I hope that makes sense to you so so one more time I'm not going to go through this an awful lot like really slowly because that was the last video we say all right hey this piece that that is and that's what we call them so if I want to find a mixed partial I just take the derivative of M with respect to Y and would give me X then Y likewise the partial derivative of Y with respect to X says well this is already the partial with respect to Y already so if I want to find the partial with respect to X after that the mixed partial I would take the derivative of n with respect to Y well let's I said that wrong find the partial derivative in respect to X this is already a partial with respect to Y so I'm just gonna go I want the partial to a 2x a mixed personal and then that's that's fine I'm just look at them take it with respect to X so all the Y terms that's a constant that's gone but the derivative of Ln X with respect to X will give you 1 or X oh that's nice that right there when that happens what that's telling you is that yes now you can make the determination that this is an exact differential equation for some potential function f of X y equals a constant now we can say that now we can say because these mixed partials are equal they're they're representing a function within that a function that is continuous and differentiable on some sort of a region whose differential form or total differential would have been this exactly this now we can say it's worth more time to go backwards because this is true we can now find that potential function how you do it next I do this every time I always write my mixed partials here I always do my structure here partial derivative F let's wait to x and y respectively and then I look at because I want to see it I really want to look at it I've asked the question which one is easier is it easier to take an integral of this with respect to X or this with respect to Y because the point is if we have a derivative how you undo derivatives if we have a derivative of function how you on your derivatives is with an integral we're about to take an integral and you're gonna have to choose either M take the integral of M with respect to x to undo X getting the function f or and with respect to Y getting rid of the Y I'm doing that derivative and getting F by and taking integral it straight to what either one works it doesn't matter sometimes they look about the same so this would not be hard to take a an integral with respect to X that's not particularly hard to do an integral with respect to Y it really doesn't it doesn't matter so much I would say that one looks easier to me because we get this this natural log it's fine Ln X we already defined that the X would have to be positive even to be up there so we're not gonna have any strange domain issues with that so I'm gonna do that so if they look about the same most of the time I would suggest to you because the most experts do this to just pick the integral of M with respect to X because we're more familiar with that but if this is considerably easier to do with respect to Y that integral then you do that one and I think it showed that in the last example last video I will show it again velocities include this video so let's do it again the idea that going through idea make sure it Semantics represents the exact eventual of a potential function then we're going to undo this derivative with the integral we're going to take the derivative again with respect to the opposite variable and set it equal to this thing and we'll talk about that right now so let's semantics over here we go so if we have this as the partial derivative at respect X then F itself would equal some integral of M with respect to X great let's do that it's not too bad then this would equal let's see X to the fourth over 4 plus y Ln X I don't need the absolute value because we've already defined that that X would have to be positive just by happiness this function the domain is implied so this would add the easy integral this one hey that's a constant that's like two alright so we would have two Ln of X but now the Y is held constant when we're taking a derivative there's my X now here's the issue I spent a long time on this and last video because this is coming from a partial derivative because we're integrating this if we wanted to get this back every function that would have just Y's in it would go to 0 for instance if I took this I took a derivative with respect to X it would have to equal M and let's see what that would do the derivative of this with respect to X is XQ the derivative of this with respect to X is y over X the derivative of this everything would be in terms of Y is 0 with respect to X that's why instead of just like a plus C you've got a function of wise because under partial derivatives with respect to X all of them will disappear we have to represent that it's incredibly important for you to have that don't put a plus C here put a function of Y because this is coming from a partial derivative with respect to X all the Y's would have cancelled so if you're not ok with this right now you need to go through just listen to where this is coming from one more time and then we're gonna move on right now so we know that's a partial of F with respect to X we undo it with an integral we say because it came from a partial all the Y's had we taken that derivative would have gone to 0 so we're representative of the function of Y that we don't know now here's the magic the cool part this is your function f of X Y this this is it it just has a piece that's gonna be all Y's that we don't know now we do know something about a piece of that differential that's all wise you know n we know n is the partial derivative of F with respect to Y so so wait a minute if this is the function of F just a little piece I don't know and this is a function of F under a derivative with respect to Y let's take a derivative of this piece with respect to Y that would be zero it's a derivative of some X's with respect to Y that's zero this would be just Ln X this would be the derivative another function with respect to Y so we've said okay man this is the function just integrate M with respect to X this is the function sum wise I don't know this says some wise I dunno let's take a derivative of this with respect to Y oh yeah that's that's pretty easy but wait a minute is it this itself the derivative of F with respect to Y yeah and they have to be equal this is what's gonna let you solve for that missing function or actually the derivative the missing function of Y is that you don't know so take the derivative respect to Y from what you found upon integrating really quickly I'd say integrate X derivative of Y integrate Y derivative X and then you set them equal to the opposite thing so you take your M integrate set equal take a derivative set equal to n if you integrate n take a derivative set equal to M so here we say this derivative of F with respect to Y that we got from here these are the same exact thing you have to set them equal so this is going to equal and I mentioned also in the last video that a lot of things are going to cancel out and yes you should end with something that just has wise in all of your X things all of your X terms they need to go away because when we take our integral here we say man how do you undo derivatives we just we just do an integral so if we integrate both sides we're going to get that G of Y is third like you we also talked about how the plus C's not really relevant I'll talk about why that is in just a moment but we integrated em with respect to X we take a derivative with respect to Y and now we set it equal to what it has to be equal to and we solve for if we solve them the the derivative of the function we didn't know we integrated it gives us it gives us the missing piece so now right back up here man elicits lion that is your function of a Beth but now we've actually managed to fix all that so we managed to figure out the missing piece here so we know that the function of f with respect to x and y is what it is this and this and this it just took us some time to find it took us the realization that taking a derivative the strength of y is going to say equal to end solving for the derivative of that let's just do a very basic integral to find that missing piece and because the potential functions are always level curves they're always two variables set equal to some constant we're going to show that that you will see so the final answer here the potential function that would give us that exact differential form is this I mentioned to you a minute ago why we don't need a scene if we had a plus C that would be right here is a constant I would just have to subtract that to the other side and would still be an arbitrary constant so it really doesn't matter to include most most people don't show a plus C here if you did you would just subtract it over and you'd get a different arbitrary constant which doesn't really make a difference I hope that that makes sense I hope that my explanation to you it kind of reinforced the process of finding these these potential functions from exact differential equations and and really importantly that you can't just look at it without doing that mixed partial and say that's an executive force equation the exactness of it the exact difference relation part of it comes from testing the mixed partials and then saying okay now it is I can guarantee an exact differential equation from some potential function and we go through and solve it that way so the next one I'm gonna go quite a bit quicker I think ivory explained the entire last lesson right here which is good you should see it again but but really it's check your mixed take either amber and integrate with respect to the variable you have a derivative of then take a derivative of the different variables set it equal to the opposite thing and that should make sense when we get the hang of it so I'm gonna go through one more here I'll give you about three more and then we'll have some great practice under our bill all right so let's do it Mixed Partials the first thing we want to check is some mixed mixed partials let's see if this is actually exact differential equation so we're gonna call the part that's next to the DX m and the apartment that's next to dy and so we know we know based on the form of our differential here that if this is an exact differential equation or exact differential form of some sort of potential function then this right here is our M and that's the parcel Evon strike to X so I would write that down and this would represent the partial of F with respect to Y if this is going to be the exact differential equation we call this M and this end now let's find our mixed partials because M represents the partial derivative of F with respect to X we just have to take that piece with respect to Y so here we say let's see with respect to 2y that would be 0 this would be I'm gonna factor my head for I'll show to you but this would be derivative of the first times the second plus the first times the derivative the second now let's see here would be we have Y we got e to the XY but by the chain rule we get an X so that's e to the X y plus y and XY remember that this derivative with respect to welcome is zero but this is a product rule so derivative the first is why they the second plus the first that's our Y derivative the second is e to the X Y times chain rule chain rule says derivative of X Y gives you X because that would be held like a constant there man okay or the derivative of F with respect to Y then X is a mixed partial so now X this would go to zero and then this with respect to X would give you derivative the first is one that's e to the XY plus leave the first that's X derivative the second let's eat to the XY but the derivative of this would be a I can't a chain rule and you say the derivative of XY with respect to X now actually be the variable Y would be held constant and those are exactly the same thing I'm taking some work to go through those if you get a product rule but this is exactly same as that do the community of videos of multiplication what that says is that hey that is now called an exact difference equation of some potential function let's find it now we're gonna look back here and say which one would be easier for us to find and in your integral would it be easier to define the integral of M with respect to X or and with respect to Y I would say that one with respect to X so let's do that does it really matter it doesn't really matter sometimes it matters a lot because once way easier but it doesn't really matter you're gonna get the same answer so let's do this let's say that the partial of F with respect to X this is where we're starting that's M that's 1 plus y e to the XY okay so if that's the derivative of F with respect to X as far as impartial jury was concerned then if we integrated with respect to X it should give us our function back mostly the only thing it won't do well it won't give us that that thumb in any sort of terms that just had y's and then we call that a function of Y those would have all gone to 0 so we're going to say that yes this function is going to be the integral and let's write X let's see how we do this we take that would be held like a number like 2 so be a wide exponential would be give us that back but yeah if you did it you so you did that you said there you'd get bu d poles with respect to X Y the X what's really happen is you have an integral of 1 DX plus an integral of Y e to the X Y DX this is going to give you a viously an X but in this case when you do your U sub and you say hey D u is why the aect it replaces this whole piece you call it UD u and then we'd say X plus e to the U but U is X Y so we're going to get this X plus e to the X Y it's good to see this integration techniques from time to time anyway you feel you're gonna do the same thing I'm gonna erase this but just to use some here ok so but if we find now we have it wrong first off they were wrong the reason why we have it wrong is because every time you do an integral you should get a constant or you should get something that would be held constant considering where it came from because this is a partial with respect to X any function of Y would have been held is a constant it would have gone to 0 and so we're gonna say in terms of just align them would have gone to 0 here so we've picked em we integrated X let me have a missing function of Y now we're going to take a derivative respect to Y so we picked the derivative of M and integrated we're going to take as derivative of Y and that's not hard to do this is 0 this is going to be X e to the X Y by the chain rule and the derivative of the missing function of Y and that has to be equal to M so so again it's going to start I'm gonna say it a lot here integrate and derivative equal to n or integrate n derivative equal to M and so here we did integrate M derivative has to be equal to end and it's write it down so it's nice to see we just took a partial derivative F restrictive why it must be equal to this then if you are not cancelling out your terms that have X here you're doing something wrong the this missing function can only be in terms of Y and so this this stuff a lot of its going to cancel we're just gonna get to what man that's kind of nice because we know how to undo derivatives since this is the derivative of Y I'm sorry G with right to Y we know that we can integrate both sides and say this is just gonna be whatever the integral of 2y dy is or I squared so that missing piece is just Y squared since we've got our function right here and we just don't know that piece this is the only reason why we did all this stuff we did this another partial derivative to set it equal so we can solve for that piece and now let's put it back so f of X Y it's this stuff it's the X it's e to the X Y and it's the function G which we just solve for as Y squared it's nice and we know that that's going to equal a constant coming from the level curves that's where that's our potential function that we just found so Lord we're gonna write it that way one more thing if you're able to solve for y here a lot of times that happens so if you're able to solve it for Y and that's the BET's final so we can't in this case that's not it's implicitly defined but you you could do that also if you had an initial value that's where we plug in we say like hey y is 0 is 1 we plug in 0 for X we plug in 1 for y let us solve for that arbitrary constant into a particular solution so I hope that's making sense we've got a few more to do here I'll be back in just a second let's get started on a couple more Practice if you can I pause this at least try to do this much of a good set up properly on both of these examples because we're gonna go through them fairly quickly we've kind of got got through the time of explanation like the full explanations and now just sort of really making sense your head now's our time for practice and in a in a class I like and you are my students right now I would have you do it just on your own I'd be walking around seeing how you're doing I can't do that but I would really like you to walk through that so let's do this together now that you've hopefully plus a bit you and try this on your own the number one thing we need to get straight is the fact that this is not an exact difference equation until we satisfy that test so let's call this M and and and let's write down these things so if if M represents the partial derivative of f of X which this is the difference performance what that does then is this piece and I'm always writing this down it's really helpful for us our brains to see what's going on most of us are visual learners and so by writing this down you are cueing your brain and you're seeing it your cueing your brain to see M is this thing the partial of F with X is this thing likewise for n what that means is you're less likely to integrate the right thing with respect to the problem variable okay so now that we have the partial of F expect X partial of F its vector Y we're going to find our mixed partials we're going to see if this is an exact differential equation so we're gonna find a partial of M with respect to Y that's 0 that's 1 over 1 plus y squared and I always set it up like this ok now the partial derivative of n with respect to X this is 1 this is a held constant so we get 1 over 1 plus y squared that right there's what you're looking for actually if you're not want any more work if they're different it's harder because we're gonna find an integrating factor in the next video that's good let us do stuff like that but it's much more work and so you kind of want that to happen so you gotta deal with an integrating factor to change this into the exact differential so that right there says that's an exact differential let's keep on going pick one of these that looks easier to integrate either this with respect to X I'm doing the partial or this with respect to Y I'm doing the partial this with respect to X looks much easier to me than integrating this with respect to Y so I'm going to pick this one I'm going to integrate this so because the partial of F with respect to X is X plus tan inverse Y then the function of F itself would be the integral of that so we're going to integrate x + tan inverse Y with respect to X and that's not that hard well we do need to understand is that we are integrating we're not taking derivatives here so yeah you're going to get this x squared over 2 that's not a problem but this doesn't go to 0 that's held is it constant so that's like 2 which means you would get 2 X or x times tan inverse Y but you're also going to get a constant only that constant would be everything that would just have wise because it's coming from a partial derivative with respect to X so we're gonna say G of Y we use G as a function for y is an age for the function of X mostly okay hey this is the function you know what we have we have all the X terms that we that we need this this is it this is going to be in a final answer what we don't know is are there any terms that would just have wise if there are then taking a partial derivative of F with respect to Y zero this is just x times 1 over let's see you have 1 over y squared plus 1 so this is a multiplication that will be held at constant like 2 you would get 1 over 1 plus y squared that's at 10 inverse Y plus G prime of Y if there are any functions and any terms that just have wise in them taking a partial derivative will let us solve for those because this right here has to be equal to n this is and we took a partial site to why that is and so let's set an equal and you go they said all the next stuff is gonna cancel under it is it just looks really funny so I thought what I actually told you was solve for this and then a lot of the all the next stuff is gonna cancel so if I subtract this and the first derivative of G with respect to Y is X plus y over 1 plus y squared minus this piece x over 1 plus y squared and I hope you're seeing it that has a common governor already if I subtract my X's are gonna cancel G prime of Y is just Y over 1 plus y squared and there's only one more thing we have to do we've got to integrate this to figure out the missing function of G so we're going to integrate looks like we're gonna have a new sub here integrate with respect to why you only have why is shoot it's it as a function of Y let's solve for y dy let's say this is 1 over u D u over 2 and though that's going to give us is this Ln there's a 1/2 Ln u we're not going to do the absolute value this can only be positive anyway and we would get a plus C but that's we determine that's really not necessary because if we had a plus C then right here when we put that G of Y if we had a plus C when we put that right back here that plus C is going to get absorbed into another arbitrary constant anyway so so weren't done we found the function with a missing function what took a derivative found the derivative of G respective Y with Y we solved for the function itself and now we're ready to put it back so the function is we have an x squared over 2 from we've got this X 10 verse 1 that stuff is gonna stay that's that's fine we want that the missing function of y was all we really wanted so if you have X's when you're done you've probably done something wrong and we would say that right there is going to be equal to C and so if you would have had this plus C right here it would have been just absorbed anyway I'm gonna leave it right there but I hope that I'm making sense to you I hope that when you're when you're walking through this on your own you're able to follow the logic down right now it should start being internalized for you saying yeah I understand that if I'm gonna integrate M with respect to X then I don't have a missing function of Y and if I can't agree and with respect to Y another missing function of X I hope you understand that taking the partial derivative of the variable that you that's opposite of what you just integrated with respect to is how you solve for that missing piece so take an integral M you can take a partial of wine set equal to n take an integral of n take a partial expected X it's that equal to n it's just the opposite let's try the bottle that looks nasty let's try this one and where we started we start by thinking if this is an exact different equation then that would have to be the partial of F with respect to X that whole gorgeous piece would have to be the partial of F with respect to Y and man we're gonna write that down we rinse in it and is this and it's all of this and now we're gonna do our test we're gonna find our mixed partials so we're going to take the partial derivative over and with respect to Y and n with respect to X we're going to see what we have so and with respect to y man well that part's pretty easy this part remember that this would be 2 X Y to the negative 1 so I'm showing this you should show this the derivative would be let's see negative 2 X to the X Y to the negative 2 so negative 2 x over Y squared that one's not bad with respect to Y that would be negative six Y over X to the 4 all right next one now a good thing that's with respect to X because that's going to go to 0 I like that but with respect to X so this is this isn't seen to Y X to the negative 3 negative 3 that's negative 6 X to the negative 4 so we have a negative 6 we still have a wine we know over X to the fourth now this next one with four members is respect to X so we have two x over y squared this would be with respect to X so 1 over just a function 1 would give us 0 oh great hey they're the same look at that they're out of order but that doesn't matter right now what we've proved is that that's an exact differential equation for some potential function let's find it now which one is easier to man which one it's easier to integrate either this respect to Y that's an that I don't want to do that or this with respect to X and either don't look good but this one looks a lot better do you see the point I would not want to integrate this with respect to Y it looks really junky don't do that do the easier one not giving me like what four in a row now where this is the easier or it doesn't matter so the next one I'm gonna give you this would be easier this victor y so we'll check that out in just a moment so let's let's go for it we know right now that the first or the partial respect to X is this junk this two x over Y minus three y cubed over X to the fourth because that is the partial of F with respect to X getting the function itself that potential function is sellable involves an integral of that that M and I'm doing the derivative with respect to the same variable so I'm doing that with respect to X okay well we can do that it's not super hard just a partial derivative for goodness sakes so x classy 2x over wife if we integrate that with respect to X Y is held like a constant so you think about this like two thirds X or something let's see we add one to the X exponent so I'll be to divide by the new exponent that would be divided by two we're going to get x squared over Y remember with respect to X with respect to X this is treated as 3y squared X to the negative four where that's just like a number like five or something let's add one to the exponent that's negative three divided by the new exponent that's divided by negative three so this would be a 3y squared X to the negative three over negative three those threes you can cancel we're going to get X to the third on the bottom and our signs being challenged but most importantly where I see a lot of students just kind of make a mistake here what they do is they'll put a plus either forget it and then we'll take a partial derivative they'll set it equal and everything will cancel and they go what about or nothing will cancel and let me get the problem I'm gonna solve great thing you have to understand that there is a constant here but it's any terms that would have been in Y's only those would be held constant so we need to have that now we say if that's our function we're missing a piece let's take an with respect to the variable that I didn't just integrate with respect to you so integrate X take derivative Y this right here remember that's a negative one so let's see negative x squared over Y squared with respect to Y here would give us a plus two Y over X cubed and then we get this derivative of Y so let's see we picked em integrated X derivative Y derivative Y is and we set it equal so this has to be equal to all this stuff hey that's wonderful because if we add this to both sides this is gone everything in for me actually you have to be gone here if we subtract this from both sides and this is gone and we get that the derivative of that function of Y that we really want to know is 1 over Y squared sorry I said the wrong one over two squared Y so the function were missing it would be the integral of that so the integral of Y to the negative 1/2 d1 that's nice that means the function we're missing is pretty basic let's see we add one to be Y to the 1/2 over 1/2 that's 2 square root 5 perfect we just found the missing function we just wrap this up we say hey now that we found this we know that the function I was looking for is this piece and the missing piece that I found by taking the partial inspector Y and then integrating after saying that equal to and it's just 2 square root Y that's really good that's about as far as we go we can't solve for y if we had an initial value we plug it in right there to solve for that that see man am I am I getting through am i am i really making it sink in we've done a lot of examples taking a long time I understand that but it's useful because there's some well I wouldn't say it's just incredibly hard to do there's some nuance there's some really fascinating things that have to be done just right and reverse really follow them and the better I can explain it the more likely it is to stick with you we have one more what I'm going to do in the last example I'm going to show you how we can pick this and integrate Y and then take a derivative with X and set that equal to n so I'm gonna do that one we'll also talk about initial value so hang on I'll be right back alright we're on a roll so speak so let's finish strong we got something to look really nasty don't worry it's not that bad it just looks really bad this example is just one that I want to really reinforce that it doesn't matter what you pick I've been talking a long time about that and where initial values come into play because they'd be remissed pleasure put something in there for you as far as that one so let's go through it I'm not gonna go super slow let's let's make sure that we're all just understanding where this come from we want to try on your own that man the best thing that happened right now is if you do it on your own get right if you don't that's okay you just to learn it but try it so is it an exact difference equation let's find out man let's find out let's call this M and this end and let's see what happens when we find or mixed parcel so if this right here is an exact differential equation then this would be the partial with respect to X and likewise this would be the partial F with respect to Y now it's fine our mixed partials it means if this is already a partial with respect to X take it again with respect to Y and that gives you this right here partial M inspector why would give me to the X plus X the X looked bad not bad for since this is already that partial districtwide I mean you can't respect X now that's the zero but that is a product rule so you're gonna have derivative of first that's one times a second PPS plus the first x times the derivative of the second P DX and that's exactly what we want our mixed partials to be equals saying this is coming from some potential function that's continuous and differentiable on some sort of region that's it that's all it says it says we can basically keep going now now we look at M and n which is easier to integrate it this with respect to X really you want any rate that that's horrible this rate that's not super fun I don't want to integrate this because this is a problem not a problem this is a problem this would make it really nasty now could you do it yes do you have to no don't let's look at this take this and say can I find the integral of this with respect to Y you know but I don't have any wise exactly that means that that's all like a constant that means it's really easy to integrate so let's take n let's say hey the man the partial derivative of F with respect to Y is N and it's really easy it's e XE to the X plus 2 we see if that is the partial of F with respect to Y then the function itself can be found by undoing that derivative with integral with respect to Y so how do we do that there are wise yeah no they're all constants so our function of f with respect to x and y would then be X e to the x times y plus 2 times y we see with the two pretty easily write and say the derivative of 2y give us 2 so the integral of 2 would give us 2y likewise here X e to the X is constant when we're talking about Y as our variable the only issue here is that we have to understand this that because this came from a partial derivative with respect to Y and we're integrating with respect to Y a function of X that would have gone to zero when we took a partial derivative check it out if I take a partial derivative of this with respect to Y this would give us that this would give us this and anything and only X's would go to zero that's why this exists to say hey there's some X's here that you probably don't know let's find them out we just integrated with respect to Y they gave us something that has X as we don't know because we integrated with respect to Y I'm doing the partial with respect to Y we haven't really considered the partial respect to X yet let's do that let's find the partial with respect to the variable you didn't just integrate so the partial of F with respect to X is well now we are going to have this this product rule so with respect to X this is going to give us X plus e to the X the wrong gonna give us e to the X plus X e to the x times y that's a big deal because we have this product rule here as B multiplied by Y that Y is a constant right so we'd have to distribute across that product rule I would encourage you to do it this way first and then distribute it but there are a couple ways you can't do this so the main thing here is to recognize yeah that's a product rule derivative the first one times a second plus the first times through the second but you are multiplying by Y can you see how a lot of people would if they forget those parentheses are going to really mess us up so be careful with that the second thing that I want to do I want to acknowledge that that's a Y with respect to X that goes away this right here says hey you can even this derivative of some function of X with respect to X it's still going to be there if you have any terms involving X and then we understand that that partial derivative with respect to X is what we called em so it's gotta be equal to M this you see a super-nice you know well if I distribute this so it's a tracking stuff whole reason why I got into math and I just love crossing stuff cuz he didn't make sound effects like Star Wars when I was uh it's probably a horrible ending but I could roam in the starboard generation and that's probably too much information for you guys anyway but when I was 3 years old my parents started let me watch Star Wars bad decision by the time I was 4 I'm not even joking you I had memorized every line to the first sorcerer who ever stood for so dorky but like as a four-year-old I just walk around like it doesn't matter anyway I love crosses to fell and everything crosses out have you done something wrong no you haven't done anything wrong all this means is that you have no function of X that's just X's where's that what do you do then when you take an integral it's just gonna give you a constant so if the first derivative of a function of X is giving you 0 then that function of X that looks really awkward yeah it really is awkward but that means that all you had was at most a constant here no X's you'd have at most a constant so and let's call that C sub 1 because I want to really show you one last time where that comes from so we had our function with the missing X there actually aren't any nice admission X's this is this is it you can kind of confirm that by taking a partial with respect to X and maybe this partial inspected wine getting exactly that and go oh yeah really aren't any missing X's but we go through the process we confirm that that that right there has no X's it would just be at most a constant so let's right so our missing piece is a constant now if we solve this for just our X's and Y's if we subtract that constant DC how a constant minus a constant is still a constant I'd really encourage you to do that first before you start trying to solve for your initial value otherwise it gets a little bit awkward so now right now now that we've solved for the the potential function now we get to use our initial value so now we plug in 0 for X and negative 1 for y and we can solve for that constant doing here don't wait don't do it later do it here because it's solved for that constant you say oh that's nice I don't spend the time to solve for y which we are going to do until I solve for the constant that'd be silly so if I solve for the constant now all the variables are here that is the constant it's much easier than solving for y and then trying to find the C basically better to undo everything you just did so plug in your initial value here x is 0 Y is negative 1 0 times all that stuff is 0 sorry I meant to see you so C is negative 2 ok let's plug that in let's solve for y how the world and solve for y well on the left hand side we could factor good week so if we factor this factor 2y and divide we actually have an explicitly defined function that's a particular solution and what it's saying is that we've now found the exact curve the exact function that has this as its exact potential equation or derivative in differential form if you want man I hope it's making sense to you I hope that you really seeing that the the finer points of the technique can you trudged through it yeah it's not it's fine it's it's much better if you understand it my job my goal is to always provide that level of intimacy with the sounds of hawkers with this problem with this sort of technique I hope I've done that I hope you understand that how these things relate the the very important point that you don't just assume it's exact until it meets the whole mixed partials being equal otherwise you do not just start this without checking that because you can go a long way and waste a lot of time when they're not exact I will teach you how to make them exact in the next video when we talk about an integrating factor and how to use that so I hope you've enjoyed it I hope that it's making sense to you hopefully you're encouraged that you're getting this and I'll see you for the next video have a wonderful day you |
13076 | https://artofproblemsolving.com/wiki/index.php/Modular_arithmetic/Introduction?srsltid=AfmBOopkgIrlg7P_YaLR7tJ-aChRB9QXZWLknEpCO6VpB4SkrCA1MYap | Art of Problem Solving
Modular arithmetic/Introduction - AoPS Wiki
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical CampusVisit the Virtual Campus
Sign In
Register
online school
Class ScheduleRecommendationsOlympiad CoursesFree Sessions
books tore
AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates
community
ForumsContestsSearchHelp
resources
math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten
contests on aopsPractice Math ContestsUSABO
newsAoPS BlogWebinars
view all 0
Sign In
Register
AoPS Wiki
ResourcesAops Wiki Modular arithmetic/Introduction
Page
ArticleDiscussionView sourceHistory
Toolbox
Recent changesRandom pageHelpWhat links hereSpecial pages
Search
Modular arithmetic/Introduction
Modular arithmetic is a special type of arithmetic that involves only integers. This goal of this article is to explain the basics of modular arithmetic while presenting a progression of more difficult and more interesting problems that are easily solved using modular arithmetic.
Contents
1 Introductory Video
2 Understand Modular Arithmetic
3 Residue
4 Congruence
4.1 Examples
4.2 Sample Problem
4.2.1 Solution:
4.2.2 Another Solution:
4.2.3 Another Solution:
5 Making Computation Easier
5.1 Addition
5.1.1 Problem
5.1.2 Solution
5.1.3 Why we only need to use remainders
5.1.4 Solution using modular arithmetic
5.1.5 Addition rule
5.1.6 Proof of the addition rule
5.2 Subtraction
5.2.1 Problem
5.2.2 Solution
5.2.3 Subtraction rule
5.3 Multiplication
5.3.1 Problem
5.3.2 Solution
5.3.3 Solution using modular arithmetic
5.3.4 Multiplication rule
5.4 Exponentiation
5.4.1 Problem #1
5.4.2 Problem #2
5.4.3 Problem #3
6 Summary of Useful Facts
7 Problem Applications
8 Applications of Modular Arithmetic
9 Resources
10 See also
Introductory Video
Understand Modular Arithmetic
Let's use a clock as an example, except let's replace the at the top of the clock with a .
This is the way in which we count in modulo 12. When we add to , we arrive back at . The same is true in any other modulus (modular arithmetic system). In modulo , we count
We can also count backwards in modulo 5. Any time we subtract 1 from 0, we get 4. So, the integers from to , when written in modulo 5, are
where is the same as in modulo 5. Because all integers can be expressed as , , , , or in modulo 5, we give these integers their own name: the residue classes modulo 5. In general, for a natural number that is greater than 1, the modulo residues are the integers that are whole numbers less than :
This just relates each integer to its remainder from the Division Theorem. While this may not seem all that useful at first, counting in this way can help us solve an enormous array of number theory problems much more easily!
Residue
We say that is the modulo-residue of when , and .
Congruence
There is a mathematical way of saying that all of the integers are the same as one of the modulo 5 residues. For instance, we say that 7 and 2 are congruent modulo 5. We write this using the symbol : In other words, this means in base 5, these integers have the same residue modulo 5:
The (mod 5) part just tells us that we are working with the integers modulo 5. In modulo 5, two integers are congruent when their difference is a multiple of 5. In general, two integers and are congruent modulo when is a multiple of . In other words, when is an integer. Otherwise, , which means that and are not congruent modulo .
Examples
because is a multiple of .
because , which is an integer.
because , which is not a multiple of .
because , which is not an integer.
Sample Problem
Find the modulo residue of .
Solution:
Since R , we know that
and is the modulo residue of .
Another Solution:
Since , we know that
We can now solve it easily
and is the modulo residue of
Another Solution:
We know is a multiple of since is a multiple of . Thus, and is the modulo residue of .
Making Computation Easier
We don't always need to perform tedious computations to discover solutions to interesting problems. If all we need to know about are remainders when integers are divided by , then we can work directly with those remainders in modulo . This can be more easily understood with a few examples.
Addition
Problem
Suppose we want to find the units digit of the following sum:
We could find their sum, which is , and note that the units digit is . However, we could find the units digit with far less calculation.
Solution
We can simply add the units digits of the addends:
The units digit of this sum is , which must be the same as the units digit of the four-digit sum we computed earlier.
Why we only need to use remainders
We can rewrite each of the integers in terms of multiples of and remainders:
.
When we add all four integers, we get
At this point, we already see the units digits grouped apart and added to a multiple of (which will not affect the units digit of the sum):
.
Solution using modular arithmetic
Now let's look back at this solution, using modular arithmetic from the start. Note that
Because we only need the modulo residue of the sum, we add just the residues of the summands:
so the units digit of the sum is just .
Addition rule
In general, when , and are integers and is a positive integer such that
the following is always true:
.
And as we did in the problem above, we can apply more pairs of equivalent integers to both sides, just repeating this simple principle.
Proof of the addition rule
Let , and where and are integers. Adding the two equations we get:
Which is equivalent to saying
Subtraction
The same shortcut that works with addition of remainders works also with subtraction.
Problem
Find the remainder when the difference between and is divided by .
Solution
Note that and . So,
Thus,
so 1 is the remainder when the difference is divided by . (Perform the subtraction yourself, divide by , and see!)
Subtraction rule
When , and are integers and is a positive integer such that
the following is always true:
Multiplication
Modular arithmetic provides an even larger advantage when multiplying than when adding or subtracting. Let's take a look at a problem that demonstrates the point.
Problem
Jerry has boxes of soda in his truck. The cans of soda in each box are packed oddly so that there are cans of soda in each box. Jerry plans to pack the sodas into cases of cans to sell. After making as many complete cases as possible, how many sodas will Jerry have leftover?
Solution
First, we note that this word problem is asking us to find the remainder when the product is divided by .
Now, we can write each and in terms of multiples of and remainders:
This gives us a nice way to view their product:
Using FOIL, we get that this equals
We can already see that each part of the product is a multiple of , except the product of the remainders when each and are divided by 12. That part of the product is , which leaves a remainder of when divided by . So, Jerry has sodas leftover after making as many cases of as possible.
Solution using modular arithmetic
First, we note that
Thus,
meaning there are sodas leftover. Yeah, that was much easier.
Multiplication rule
When , and are integers and is a positive integer such that
The following is always true:
.
Exponentiation
Since exponentiation is just repeated multiplication, it makes sense that modular arithmetic would make many problems involving exponents easier. In fact, the advantage in computation is even larger and we explore it a great deal more in the intermediate modular arithmetic article.
Note to everybody: Exponentiation is very useful as in the following problem:
Problem #1
What is the last digit of if there are 1000 7s as exponents and only one 7 in the middle?
We can solve this problem using mods. This can also be stated as . After that, we see that 7 is congruent to -1 in mod 4, so we can use this fact to replace the 7s with -1s, because 7 has a pattern of repetitive period 4 for the units digit. is simply 1, so therefore , which really is the last digit.
Problem #2
What are the tens and units digits of ?
We could (in theory) solve this problem by trying to compute , but this would be extremely time-consuming. Moreover, it would give us much more information than we need. Since we want only the tens and units digits of the number in question, it suffices to find the remainder when the number is divided by . In other words, all of the information we need can be found using arithmetic mod .
We begin by writing down the first few powers of mod :
A pattern emerges! We see that So for any positive integer , we have (mod ). In particular, we can write
.
By the "multiplication" property above, then, it follows that
(mod ).
Therefore, by the definition of congruence, differs from by a multiple of . Since both integers are positive, this means that they share the same tens and units digits. Those digits are and , respectively.
Problem #3
Can you find a number that is both a multiple of but not a multiple of and a perfect square?
No, you cannot. Rewriting the question, we see that it asks us to find an integer that satisfies .
Taking mod on both sides, we find that . Now, all we are missing is proof that no matter what is, will never be a multiple of plus , so we work with cases:
This assures us that it is impossible to find such a number.
Summary of Useful Facts
Consider four integers and a positive integer such that and . In modular arithmetic, the following identities hold:
Addition: .
Subtraction: .
Multiplication: .
Division: , where is a positive integer that divides and .
Exponentiation: where is a positive integer.
Problem Applications
Applications of Modular Arithmetic
Modular arithmetic is an extremely flexible problem solving tool. The following topics are just a few applications and extensions of its use:
Divisibility rules
Linear congruences
Resources
The AoPS Introduction to Number Theory by Mathew Crawford.
The AoPS Introduction to Number Theory Course. Thousands of students have learned more about modular arithmetic and problem solving from this 12 week class.
See also
Intermediate modular arithmetic
Olympiad modular arithmetic
Retrieved from "
Category:
Introductory Mathematics Topics
Art of Problem Solving is an
ACS WASC Accredited School
aops programs
AoPS Online
Beast Academy
AoPS Academy
About
About AoPS
Our Team
Our History
Jobs
AoPS Blog
Site Info
Terms
Privacy
Contact Us
follow us
Subscribe for news and updates
© 2025 AoPS Incorporated
© 2025 Art of Problem Solving
About Us•Contact Us•Terms•Privacy
Copyright © 2025 Art of Problem Solving
Something appears to not have loaded correctly.
Click to refresh. |
13077 | https://www.youtube.com/watch?v=JL8CymzB4CU | Physics Unit 2 Uniformly Accelerated Motion Average velocity and instantaneous velocity
Waliin Academy
7170 subscribers
114 likes
Description
2877 views
Posted: 20 Nov 2023
Physics Unit 2 Uniformly Accelerated Motion
Average velocity and instantaneous velocity
Speed is a quantity that describes how fast a body moves.
Speed is the rate at which an object changes its location. Like distance, speed is a scalar quantity because it has a magnitude but no direction.
Since speed is a rate, it depends on the time interval of motion. Its symbol is v.
In other words, speed is the distance covered by a moving body per unit time.
The SI unit of speed is meter per second (m/s).
Other units of speed include kilometer per hour (km/h) and miles per hour (mi/h).
Speed and average speed are not the same although they are derived from the same formula.
The average speed is defined as the total distance travelled divided by the total time it takes to travel that distance:
During a typical trip to school by car, the car undergoes a series of changes in its speed.
If you were to look at the speedometer readings at regular intervals, you would notice that it changes.
22 comments
Transcript:
[Music] section 2.2 average velocity and instaneous velocity average velocity the rate of change of distance with time is called speed but the rate of change of displacement is known as velocity unlike speed velocity is a vector quantity for when an object travels a certain distance with different velocities it motion is specified by it average velocity for the average velocity of a body is defined as the body displacement the change of displacement divided by the time interval the change of time during which that displacement occurs velocity the change of let s not and S be its positions at instances T not and the T respectively for mathematically you can express average velocity as average velocity is equal to change in displacement over change in time change in displacement means final displacement minus initial displacement change in time is final time minus initial time forus initial position final time minus initial time where minus t not is change in time T minus t and is the starting time which is obviously Z the SI unit for average velocity is met per second met this but there are also many other unit such as kilomet per hour mile per hour or also written as MP H mile per hour and cm per second in common use how kilometer mile per hour average speed of an object is obtained by dividing the total distance traveled by the total time taken average speed is equal to total distance traveled over total time taken s total time taken if the motion is in the same direction along straight line the average speed is the same as the magnitude of the average velocity however this is always not the case example 2.2 it takes you 10 minute to work with an average velocity of 1.2 m/ second to the north from the first stop to the museum entrance what is your displacement solution 600 second average velocity is equal to 1.2 m/s nor required change in dis change in displacement is equal to average velocity times change in time change in 1.2 in displacement this means the displacement has a magnitude of 720 m in a direction to the north a passenger on the bus took 5 Seconds to move for M to a seat on provided Place forward what is his average velocity make solution format change in displacement is equal to 4 M change in time is equal to 5 Second requ aage velocity average velocity average velocity is equal to change in displacement over change in time the average velocity gives information based on initial point and final point of the passenger but you can't get information from average velocity whether he stops goes slowly or goes fast in the middle of his to know this you can use another physical quantity which is called instantaneous velocity instaneous example 2 a car travels at a constant 50 km per hour 400 kilm it then speeds up to 100 kilm per hour and is driving another 100 kilomet what is the car's average speed for the 200 kilm trip solution format distance one is equal to 100 kilom F V1 speed 1 is equal to 50 km per hour second distance K V2 equal 100 km hour total distance traveled speed required average speed average speed is equal to total distance traveled over total time taken fore foree the change of time is equal to distance one over speed one 100 kilm ided V1 50 km per hour shant change of time t2al speed is equal to total distance traveled 200 kilom kilid total time taken for average speeding 66.7 kilom per hour [Music] k |
13078 | https://www.khanacademy.org/standards/MO.Math/6.RP | Standards Mapping - Missouri Math | Khan Academy
Skip to main content
If you're seeing this message, it means we're having trouble loading external resources on our website.
If you're behind a web filter, please make sure that the domains .kastatic.org and .kasandbox.org are unblocked.
Explore
Browse By Standards
Explore Khanmigo
Math: Pre-K - 8th grade
Math: High school & college
Math: Multiple grades
Math: Illustrative Math-aligned
Math: Eureka Math-aligned
Math: Get ready courses
Test prep
Science
Economics
Reading & language arts
Computing
Life skills
Social studies
Partner courses
Khan for educators
Select a category to view its courses
Search
AI for Teachers FreeDonateLog inSign up
Search for courses, skills, and videos
Free to Use. Not Free to Make.
“Let’s get straight to it: Less than 1% of users donate to Khan Academy.
If you believe that a free, world-class education can change lives, then we need your help. With your support, Khan Academy can reach even more learners and transform even more lives. Monthly gifts go further—enabling us to support learners every single day. Will you join us?”
Sal Khan, Founder and CEO
Select gift frequency
One time
Recurring
Monthly
Yearly
Select amount
$12
$20
$30
$40
Other
Give now
By donating, you agree to our terms of service and privacy policy.
#### STANDARDS
>
US-MO
Math
Grade 1
Number Sense
Number Sense and Operations in Base Ten
Relationships and Algebraic Thinking
Geometry and Measurement
Data and Statistics
Grade 2
Number Sense and Operations in Base Ten
Relationships and Algebraic Thinking
Geometry and Measurement
Data and Statistics
Grade 3
Number Sense and Operations in Base Ten
Number Sense and Operations in Fractions
Relationships and Algebraic Thinking
Geometry and Measurement
Data and Statistics
Grade 4
Number Sense and Operations in Base Ten
Number Sense and Operations in Fractions
Relationships and Algebraic Thinking
Geometry and Measurement
Data and Statistics
Grade 5
Number Sense and Operations in Base Ten
Number Sense and Operations in Fractions
Relationships and Algebraic Thinking
Geometry and Measurement
Data and Statistics
Grade 6
Ratios and Proportional Relationships
Number Sense and Operations
Expressions, Equations and Inequalities
Geometry and Measurement
Data Analysis, Statistics and Probability
Grade 7
Ratios and Proportional Relationships
Number Sense and Operations
Expressions, Equations and Inequalities
Geometry and Measurement
Data Analysis, Statistics and Probability
Grade 8
Number Sense and Operations
Expressions, Equations and Inequalities
Geometry and Measurement
Data Analysis, Statistics and Probability
Functions
Algebra 1
Number and Quantity
Seeing Structure in Expressions
Creating Equations
Reasoning with Equations and Inequalities
Arithmetic with Polynomials and Rational Expressions
Interpreting Functions
Building Functions
Linear, Quadratic and Exponential Models
Data and Statistical Analysis
Algebra 2
Number and Quantity
Seeing Structure in Expressions
Reasoning with Equations and Inequalities
Arithmetic with Polynomials and Rational Expressions
Interpreting Functions
Building Functions
Modeling
Data and Statistical Analysis
Geometry
Congruence
Similarity, Right Triangles, and Trigonometry
Circles
Exploring Geometric Properties with Equations
Geometric Measurement and Dimension
Modeling with Geometry
Conditional Probability and Rules of Probability
Missouri Math
Grade 6: Ratios and Proportional Relationships
6.RP.A ------ Understand and use ratios to solve problems. --------------------------------------------
6.RP.A.1
Fully covered
Understand a ratio as a comparison of two quantities and represent these comparisons.
Basic ratios
Basic ratios
Equivalent ratios
Equivalent ratios
Equivalent ratios in the real world
Equivalent ratios with equal groups
Equivalent ratios: recipe
Intro to ratios
Part-part-whole ratios
Part:whole ratios
Ratio review
Ratios and double number lines
Ratios with tape diagrams
Ratios with tape diagrams
Ratios with tape diagrams (part:whole)
Simplify a ratio from a tape diagram
Understanding equivalent ratios
6.RP.A.2
Fully covered
Understand the concept of a unit rate associated with a ratio, and describe the meaning of unit rate.
Comparing rates
Comparing rates example
Equivalent ratio word problems
Interpreting unequal ratios
Intro to rates
Part to whole ratio word problem using tables
Proportion word problem: cookies
Proportion word problem: hot dogs
Proportion word problems
Rate problems
Rate problems
Rate review
Simplify a ratio from a tape diagram
Solving unit price problem
Solving unit rate problem
Understanding equivalent ratios
Unit rates
### 6.RP.A.3 ### Solve problems involving ratios and rates.
6.RP.A.3a
Fully covered
Create tables of equivalent ratios, find missing values in the tables and plot the pairs of values on the Cartesian coordinate plane.
Comparing rates example
Equivalent ratio word problems
Equivalent ratios
Equivalent ratios
Equivalent ratios: recipe
Part-part-whole ratios
Ratio tables
Ratio tables
Ratios and double number lines
Ratios on coordinate plane
Ratios on coordinate plane
Ratios with tape diagrams
Ratios with tape diagrams
Solving ratio problems with tables
6.RP.A.3b
Fully covered
Solve unit rate problems.
Comparing rates
Comparing rates example
Equivalent ratio word problems
Interpreting unequal ratios
Intro to rates
Part to whole ratio word problem using tables
Proportion word problem: cookies
Proportion word problem: hot dogs
Proportion word problems
Rate problems
Rate problems
Rate review
Simplify a ratio from a tape diagram
Solving unit price problem
Solving unit rate problem
Understanding equivalent ratios
Unit rates
6.RP.A.3c
Fully covered
Solve percent problems.
Benchmark percents
Converting between percents, fractions, & decimals
Converting decimals and percents review
Converting percents and fractions review
Converting percents to decimals & fractions example
Find percents visually
Finding a percent
Finding common percentages
Finding percentages with a double number line
Finding percents
Finding the whole with a tape diagram
Fraction, decimal, and percent from visual model
Intro to percents
Meaning of 109%
Percent of a whole number
Percent word problem: recycling cans
Percent word problems
Percents from fraction models
Percents from fraction models
Solving percent problems
The meaning of percent
Ways to rewrite a percentage
6.RP.A.3d
Fully covered
Convert measurement units within and between two systems of measurement.
Ratios and measurement
Ratios and units of measurement
Our mission is to provide a free, world-class education to anyone, anywhere.
Khan Academy is a 501(c)(3) nonprofit organization. Donate or volunteer today!
Site Navigation
About
News
Impact
Our team
Our interns
Our content specialists
Our leadership
Our supporters
Our contributors
Our finances
Careers
Internships
Cookie Preferences
Contact
Help center
Support community
Share your story
Press
Download our apps
Courses
Math: Pre-K - 8th grade
Math: High school & college
Math: Multiple grades
Math: Illustrative Math-aligned
Math: Eureka Math-aligned
Math: Get ready courses
Test prep
Science
Economics
Reading & language arts
Computing
Life skills
Social studies
Partner courses
Khan for educators
Language
English
CountryU.S.IndiaMexicoBrazil
© 2025 Khan Academy
Terms of use
Privacy Policy
Cookie Notice
Accessibility Statement
Use of cookies
Cookies are small files placed on your device that collect information when you use Khan Academy. Strictly necessary cookies are used to make our site work and are required. Other types of cookies are used to improve your experience, to analyze how Khan Academy is used, and to market our service. You can allow or disallow these other cookies by checking or unchecking the boxes below. You can learn more in our cookie policy
Accept All Cookies Strictly Necessary Only
Cookies Settings
Privacy Preference Center
When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
More information
Allow All
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
Certain cookies and other technologies are essential in order to enable our Service to provide the features you have requested, such as making it possible for you to access our product and information related to your account. For example, each time you log into our Service, a Strictly Necessary Cookie authenticates that it is you logging in and allows you to use the Service without having to re-enter your password when you visit a new page or new unit during your browsing session.
Functional Cookies
[x] Functional Cookies
These cookies provide you with a more tailored experience and allow you to make certain selections on our Service. For example, these cookies store information such as your preferred language and website preferences.
Targeting Cookies
[x] Targeting Cookies
These cookies are used on a limited basis, only on pages directed to adults (teachers, donors, or parents). We use these cookies to inform our own digital marketing and help us connect with people who are interested in our Service and our mission. We do not use cookies to serve third party ads on our Service.
Performance Cookies
[x] Performance Cookies
These cookies and other technologies allow us to understand how you interact with our Service (e.g., how often you use our Service, where you are accessing the Service from and the content that you’re interacting with). Analytic cookies enable us to support and improve how our Service operates. For example, we use Google Analytics cookies to help us measure traffic and usage trends for the Service, and to understand more about the demographics of our users. We also may use web beacons to gauge the effectiveness of certain communications and the effectiveness of our marketing campaigns via HTML emails.
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Reject All Confirm My Choices |
13079 | https://www.sciencedirect.com/topics/engineering/dihedral-angle | Skip to Main content
Sign in
Chapters and Articles
You might find these chapters and articles relevant to this topic.
Aluminium powder metallurgy
21.3.4 Dihedral angle
The dihedral angle is similar to the contact angle in that it involves the balance of interface energies. The dihedral angle, ϕ, forms where a grain boundary intersects the liquid and is a measure of grain boundary penetration by a liquid, Fig. 21.9.
The relationship between the dihedral angle, the solid–solid interface energy, σSS, and the solid–liquid interface energy σSL, is given by:
[21.4]
When the energy associated with two solid–liquid interfaces, 2σSL, is less than the solid–solid interface energy, σSS, liquid will penetrate along the grain boundary. As the dihedral angle increases, the penetration of the liquid phase along grain boundaries is reduced and the number of solid/solid contacts increase. Excessive penetration of grain boundaries by liquid can lead to macroscopic swelling (Kaysser et al., 1980; Tabeshfar and Chadwick, 1984; Wanibe et al., 1990; Danninger, 1992), as illustrated in Fig. 21.10. This type of swelling can be alleviated by reducing the σSS/σSL ratio through alloying additions (Jamil and Chadwick, 1985).
View chapterExplore book
Read full chapter
URL:
Book2011, Fundamentals of Aluminium MetallurgyS.H. Huo, ... E. Crossin
Chapter
Basic Concepts and Definitions
2013, Aerodynamics for Engineering Students (Sixth Edition)E.L. Houghton, ... Daniel T. Valentine
Dihedral Angle
If an airplane is viewed from directly ahead, it is seen that the wings are generally not in a single geometric plane but instead inclined to each other at a small angle. Imagine lines drawn on the wings along the locus of the intersections between the chord lines and the section noses, as in Fig. 1.6. Then the angle 2Γ is the dihedral angle of the wings. If the wings are inclined upward, they are said to have dihedral; if inclined downward, they have anhedral.
View chapterExplore book
Read full chapter
URL:
Book2013, Aerodynamics for Engineering Students (Sixth Edition)E.L. Houghton, ... Daniel T. Valentine
Chapter
Organic Chemical Systems, Theory
2003, Encyclopedia of Physical Science and Technology (Third Edition)Josef Michl
I.B.3 Dihedral Angles
A dihedral angle is defined as the angle between two planes, both of which pass through the same bond. One of the planes also contains one of the additional bonds formed by one of the bond termini, and the other plane contains one of the additional bonds formed by the other terminus.
The preferred dihedral angles around a double bond are 0° and 180°, once again counting a lone pair on a terminus as another nearest neighbor. Thus, the usual geometries around CC and NN double bonds are planar (6–9):
6
7
8
9
However, small twisting distortions from planarity are relatively easy at room temperature.
There is a much weaker preference for particular values of the dihedral angle around single bonds, and rotation around such bonds is nearly free. Usually, the value of 0° (“eclipsed”) is avoided, and values of around 60° (“staggered”) to 90° are somewhat preferred, depending on the number of lone pairs on the termini.
The general rules just stated for bond lengths and angles permit the construction of mechanical molecular models either from balls and sticks or on a computer screen. The size of the balls that represent the volume of individual atoms is given by their van der Waals radii. The sum of the van der Waals radii of two atoms represents the distance of most favorable approach of these two atoms if they are not mutually bonded (e.g., atoms on neighboring molecules in a crystal). Values for these quantities are compiled inTable IV. Molecules in which two or more atoms that are not bonded to one another and are located at distances shorter than the sum of the van der Waals radii are strained by steric crowding and are less stable than otherwise expected. Often, it is possible to avoid some of this unfavourable interaction by a distortion of the valence angles.
TABLE IV. Atomic van der Waals Radiia
| Atom | Radius (Å) | Atom | Radius (Å) | Atom | Radius (Å) |
--- --- --- |
| H | 1.2 | O | 1.4 | F | 1.4 |
| N | 1.5 | S | 1.9 | Cl | 1.8 |
| P | 1.9 | Se | 2.0 | Br | 2.0 |
| As | 2.0 | Te | 2.2 | I | 2.2 |
| Sb | 2.2 | | | | |
a
: 1 Å = 100 pm.
View chapterExplore book
Read full chapter
URL:
Reference work2003, Encyclopedia of Physical Science and Technology (Third Edition)Josef Michl
Chapter
POLYCRYSTALLINE MICROSTRUCTURES
2005, SinteringSuk-Joong L. Kang
3.1.2 Dihedral Angle
The angle at the junction between two grains (Figure 3.3) is referred to as the dihedral angle. If the interfacial tensions are in equilibrium at this junction,
(3.3)
is satisfied, where γb is the grain boundary energy of α, γαβ the interfacial energy between α and β, and ϕ the dihedral angle. The dihedral angle is determined only by the interfacial energies and is independent of the pressures in the phases.9 This means that the dihedral angle is constant, irrespective of the pressure of phase β. If β is a vapour phase, ϕ is larger than 120° because γs is usually higher than γb. In general, γs is 2–3 times higher than γb and ϕ is around 150°. If the dihedral angle is constant and the junction edges are randomly distributed in three dimensions, the measured angle with the maximum frequency is the true dihedral angle.10,11
View chapterExplore book
Read full chapter
URL:
Book2005, SinteringSuk-Joong L. Kang
Chapter
Aircraft Conceptual Layout
2014, General Aviation Aircraft DesignSnorri Gudmundsson BScAE, MScAE, FAA DER(ret.)
4.2.3 Wing Dihedral
The dihedral angle is the angle the wing plane makes with the horizontal. It allows the aircraft designer to provide the airplane with roll stability and a way to affect the severity of dynamic modes such as Dutch roll. Its primary effect is on the stability derivative Clβ (dihedral effect). In addition to the dihedral angle, the magnitude of the dihedral effect depends on the vertical location of the wing and sweep angle. Ultimately, the designer must predict the dynamic stability characteristics of the airplane design and evaluate the appropriate dihedral angle.
Figure 4-9 shows an airplane at airspeed U, banking through an angle ϕ. The banking results in a side-slip, which effectively results in a yaw angle β. The yaw angle results in side flow component amounting to U·tan β, which when combined with the change in vertical flow due to the roll causes a net change in angle-of-attack, Δα, on each wing. The subsequent change in lift (ΔL) on each wing is shown in the figure, and it generates a restoring rolling moment (one that tends to rotate the aircraft back to level flight), here denoted by the letter M.
Common dihedral configurations are shown in Figure 4-10. Of these the three on the left are most commonly used. The cranked dihedral is used extensively on the French Jodel and selected Robin aircraft, as well as on some sailplanes. However, it is also featured on the Argentinian FMA IA-58 Pucará twin turboprop ground attack aircraft and of course the McDonnell-Douglas F-4 Phantom.
The gull-wing configuration is fairly rare, being most famously used on the Vought F4U Corsair, where its purpose was to increase the propeller clearance for carrier operations. It was also used for various reasons on the Blohm & Voss BV-137, Caproni Ca-331b Raffica, Dewoitine HD-780, Fairey AS-1 Gannet, Heinkel He-112 B, and Junkers Ju-87 Stuka.
The inverted gull-wing configuration is often used for twin engine seaplanes, where it helps take wing-mounted engines and propellers away from the spray of water. It is featured on the Beriev Be-6, Be-12, Chyetverikov MDR-6, and Moskalev 16 amphibians and seaplanes. It is also used on the Göppingen Gö-3 Minimoa sailplane, and the PZL P-1, PZL P-11, Piaggio P-166, and Supermarine 224 landplanes. The Stinson SR-10 is an example of an aircraft that could fit into this class, featuring a wing whose upper surface has a distinct gull-wing break. However, the lower surface forms a straight line and the spar does not have a break, rendering it more of a transitional form.
View chapterExplore book
Read full chapter
URL:
Book2014, General Aviation Aircraft DesignSnorri Gudmundsson BScAE, MScAE, FAA DER(ret.)
Chapter
Descriptive Geometry 1
2014, Geometry of DesignNam-Ho Kim, ... Harold F. Snider
A5.5a DIHEDRAL ANGLE example
To find the DIHEDRAL ANGLE between planes ABC and ABD.
1.
: Find the TL of the common line AB in AUX view 1.
2.
: Find the PV of line AB in AUX view 2.
3.
: The angle between the EV of ABC and EV of ABD = dihedral angle.
View chapterExplore book
Read full chapter
URL:
Book2014, Geometry of DesignNam-Ho Kim, ... Harold F. Snider
Chapter
Solid-state Sintering
2001, Encyclopedia of Materials: Science and TechnologyJ.E. Blendell
7 Dihedral Angle Effect
In the discussion of the initial and intermediate stages it was assumed that the dihedral angle of the grain boundary with the pore surface is 180°, implying that the energy of the grain boundary is very much lower than the energy of the surface. As shown in Fig. 3, many pores have dihedral angles of less than 180°. For a pore on a grain boundary, the dihedral angle, ψ, is defined by the ratio of the grain boundary energy, γb, to the surface energy, γs, such that
Figure 7 shows the shapes of pores for different values of the dihedral angle. The curvature of the pore surface increases as the dihedral angle increases, so the driving force for pore shrinkage also increases as the dihedral angle increases. When a pore is located on the junction of three or more grains, the curvature of the pore surface is determined by both the dihedral angle and the number of grains the pore contacts. For a three-sided pore when ψ=60° the pore will have straight sides and there will not be a driving force for shrinkage. Hence, this angle is a critical dihedral angle, ψcrit, for pore shrinkage. For a four-sided pore, ψcrit=70.5°. The critical dihedral angle as a function of the number of surrounding grains is shown in Fig. 8 (Kingery and Francois 1967). Thus, when the dihedral angle is too low, pores will be stable in the structure and there will be a limiting density even though removal of the pores would lower the energy of the system. Ultimately, as grain growth occurs the number of grains intersecting the pore will decrease so that pores will become unstable and shrink. However, as the grain size increases, the diffusion distances increase and the shrinkage rate slows down, making pore removal a relatively slow process.
View chapterExplore book
Read full chapter
URL:
Reference work2001, Encyclopedia of Materials: Science and TechnologyJ.E. Blendell
Review article
Blade-end treatment to improve the performance of axial compressors: An overview
2017, Progress in Aerospace SciencesXinqian Zheng, Zhihui Li
3.1 Definition
The detailed definition of the term end-dihedral is shown in Fig. 12. Movement normal to the airfoil section chord line will be termed as dihedral in this paper. Then, the dihedral angle is taken to be positive if the suction surface makes an obtuse angle with the end-wall and the angle is regarded as a negative one with an acute angle..
View article
Read full article
URL:
Journal2017, Progress in Aerospace SciencesXinqian Zheng, Zhihui Li
Chapter
The Anatomy of the Wing
2014, General Aviation Aircraft DesignSnorri Gudmundsson BScAE, MScAE, FAA DER(ret.)
9.3.4 Dihedral or Anhedral, Γ
The dihedral (or anhedral) is the angle the wing makes with respect to the ground plane or the x-y plane (as it is normally called) when viewing the airplane from the front (or back). A dihedral refers to the wingtip being higher (with respect to the ground) than the wing root. The opposite holds true for anhedral. The dihedral affects the lift of the configurations in two ways: due to the tilting of the lift force and how the dihedral changes the AOA of the wing. Consider a wing that has dihedral angle Γ subjected to an AOA given by α. The geometry of the configuration reveals that the AOA seen by the airfoil, and denoted by αN, is reduced by the factor cos Γ (for instance, if Γ = 0°, then αN = α and if Γ = 90°, then αN = 0°, no matter the magnitude of α). Therefore, we can write:
: Impact on AOA:
(9-33)
Similarly, the lift generated in the plane of symmetry is the product of the lift normal to the wing surface, denoted by LN, reduced by the same factor, cos Γ.
: Tilting of lift force:
(9-34)
Since LN can be written as , the lift in the plane of symmetry can be presented as follows:
: Lift of a wing with dihedral:
(9-35)
where
: CLα = lift curve slope of the wing assuming Γ = 0°
: q = dynamic pressure
: S = reference wing area
This result has been experimentally confirmed, for instance see Ref. . It allows wing lift to be determined in terms of variables that refer to a wing with no dihedral. Furthermore, see discussion of V-tails in Section 11.3.4, V-tail or butterfly tail. Typical wings feature a dihedral of 4°–7°. The term cos2 Γ amounts to 0.995 to 0.985 and, therefore, is usually ignored in the estimation of stability derivatives.
Dihedral plays an essential role in the generation of dihedral effect. It is discussed further in Section 4.2.3, Wing dihedral. Values for several classes of aircraft are given in Table 9-6.
TABLE 9-6. Various Wing Layout Properties for Selected Aircraft
| Aircraft | Aspect Ratio | Taper Ratio | Dihedral | Washout | Incidence (Root) | Leading Edge Sweep |
--- --- ---
| AR | λ | Γ | ϕG | iroot | ΛLE |
| COMMERCIAL JETLINERS | | | | | | |
| Airbus A300-600 | 7.7 - - | 28° (0.250C) |
| Airbus A310 | 8.8 | 0.260 | 11.13° (IB) 4.05° (OB) 5.05° | 28° (0.250C) |
| Airbus A320-200 | 9.4 5.2° | 25° (0.250C) |
| Boeing 707 | 7.056 | 0.275 | 7° 2° | 35° (0.250C) |
| Boeing 727 | 7.2 | 0.304 | 3° 2° | 32° (0.250C) |
| Boeing 737-100, -200 | 8.83 | 0.340 | 6° 1° | 25° (0.250C) |
| Boeing 747-100, -200 | 6.96 | 0.245 | 7° 2° | 37.5° (0.250C) |
| Boeing 757 | 7.77 | 0.211 | 5° 3.2° | 25° (0.250C) |
| Boeing 767 | 7.9 | 0.267 | 6° 4.25° | 31.5° (0.250C) |
| Fokker F28 Fellowship | 8.4 | 0.239 | 2.5° - | 17.4° (0.250C) |
| McDonnell-Douglas DC-9/MD87 | 9.62 | 0.156 | 3° 1.25° | 24.5° (0.250C) |
| McDonnell-Douglas DC-10-30/40 | 7.5 | 0.252 | 5.24° (IB) 3.03° (OB) - | 35° (0.250C) |
| MILITARY JET AIRCRAFT | | | | | | |
| Aermacchi MB-339 | 6.1 - - | 11.3° |
| Aero L-39 Albatros | 4.4 2.5° 2° | 6.43° |
| BAC 167 Strikemaster | 5.84 | 0.545 | 6° | 3° | 3°
| Cessna A-37 Dragonfly | 6.2 | 0.682 | 3° | 2.63° | 3.63° | 0° (0.225C) |
| DBD Alpha Jet | 4.8 −6° - | 28° (0.250C) |
| Hawker Siddeley Harrier | 3.175 | 0.354 | −12° | −8° | 1.75° | 34° (0.250C) |
| Microjet 200B | 9.3 5.03° 3° | 0° (0.300C) |
| TURBOPROPS, COMMUTER AND MILITARY | | | | | | |
| ATR 42 | 11.08 | 0.549 | 2.5° 2° | 3.1° (0.250C) |
| ATR 72 | 12.0 | 0.549 | 2.5° 2° | 2.8° (0.250C) |
| De Havilland DHC-6 Twin Otter | 10.1 | 1 - - |
| De Havilland DHC-7 Dash 7 | 10.1 | 0.441 | 4.5° 3° | 3.2° (0.250C) |
| De Havilland DHC-8 Dash 8 | 12.4 2.5° 3° | 3.03° (0.250C) |
| Dornier Do 28 Skyservant | 8.3 | 1 | 1.5° 4° | 0° |
| Fokker F27 Friendship | 12.0 | 0.404 | 2.5° | 2° | 3° | 0° (0.250C) |
| IAI-101, 201 Arava | 10 | 1 | 1.5° 0.45° | 0° |
| LET L-410 | 10.79 1.75° | 2.5° | 2° | 0° (0.250C) |
| Lockheed C-130 Hercules | 10.09 | 0.852 | 2.5° | 3° | 3° | 0° (0.250C) |
| Mitsubishi MU-2 | 7.71 0° | 3° | 2° | 0.35° (0.250C) |
| SAAB 340 | 11.0 | 0.375 | 7° 2° | 3.6° (0.250C) |
| Shorts SD3-30 | 12.3 | 1 | 3° - | 0° |
| Shorts Skyvan Series 3M | 11 | 1 | 2.03° 2.5° | 0° |
| GA SINGLE-ENGINE, PISTON AND TURBOPROP | | | | | | |
| Beechcraft Sierra/Sundowner | 7.5 | 1 | 6.5° | 2° | 3° | 0° |
| Beechcraft T-34C Mentor | 6.22 | 0.412 | 7° | 3° | 4° | 0° (0.250C) |
| Beechcraft V35 Bonanza | 6.2 | 0.5 | 6° | 3° | 4° | 0° (0.250C) |
| Cessna 150/152 | 6.7 | 0.687 | 1° | 1° | 1°
| Cessna 172 Skyhawk | 7.52 | 0.687 | 2.73° | 3° | 1.5°
| Cessna 177 Cardinal | 7.31 | 0.726 | 1.5° | 3° | 3.5° | 0° |
| Cessna 182 Skylane | 7.37 | 0.669 | 1.73° | 3.62° | 0.78° | 0° |
| Cessna 208 Caravan | 9.61 | 0.616 | 3° | 3.22° | 2.62°
| Cessna 210 Centurion | 7.66 | 0.726 | 1.5° | 3° | 1.5°
| Cirrus SR20/22 | 10 | 0.5 | 4.5° | 0° | 0.25° | 0° (0.250C) |
| Pilatus PC-6 Turbo-Porter | 8.4 | 1 | 1° 2° | 0° |
| Piper PA-28 Cherokee Arrow II | 6.11 | 1 | 7° 2° | 0° |
| Piper PA-28 Cherokee Warrior | 7.24 | 0.669 | 7° | 3° | 2°
| Piper PA-46 Malibu | 10.57 4° -
| SIAI-Marchetti SF260 | 6.3 | 0.49 | 6.33° | 2.75° | 2.75° | 0° (0.250C) |
| Valmet L-70 Miltrainer | 6.62 - -
| Valmet L-90 Redigo | 7.25 | 0.6 | 6° | 3° | 3°
| Zlin 142 | 6.4 | 1 | 6° - | −4.33° |
| GA TWIN-ENGINE | | | | | | |
| Beechcraft B58 Baron | 7.16 | 0.42 | 6° | 4° | 4°
| Beechcraft B60 Duke | 7.243 | 0.32 | 6° | 4° | 4°
| Beechcraft B99 Airliner | 7.51 | 0.5 | 7° | 4.8° | 4.8°
| Britten-Norman BN-2A (standard) | 7.4 | 1 | 0° 2° | 0° |
| Cessna 310 | 7.3 | 0.674 | 5° | 3° | 2.5°
| Cessna 337 Skymaster | 7.18 | 0.667 | 3° | 2° | 4.5°
| Partenavia P.68 Victor | 7.7 | 1 | 1° 1.5° | 0° |
| Piaggio P.180 Avanti | 11.8 | 0.352 | 2° 0° | 0° (0.150C) |
| Piper PA-31 Cheyenne | 7.22 5° | 2.5° | 1.5° | 0° (0.300C) |
| Piper PA-31-310 Navajo | 7.22 - | 1.0° (aero) 2.5° (geo) - |
| BUSINESS JETS | | | | | | |
| Cessna M550 Citation II | 7.8 4° | 3° | 2.5°
| Cessna M650 Citation III | 8.94 3° - | 25° (0.250C) |
| Dassault Falcon 20 | 6.4 2° 1.5° | 30° (0.250C) |
| Dassault Falcon 100 | 6.5 2° 1.5° | 30° (0.250C) |
| Gates Learjet 55 | 6.72 | 0.391 - 13° (0.250C) |
| EXPERIMENTAL AND HOMEBUILT | | | | | | |
| Colomban MC 15 Cri Cri | 7.75 | 1 | 4° | 1.5° | 1° | 0° |
| Corby Starlet | 5.00 | 1 | 6° | 3.5° | 2.5° | 0° |
| Rutan Varieze | 9.20 - | −3° - |
| Verilite Model 100 Sunbird | 7.56 | 1 | 2° | 2° | 2° | 0° |
| SAILPLANES | | | | | | |
| Rolladen-Schneider LS4 | 21.4 4° - | 0° |
| Schempp-Hirth Nimbus-3D | 35.9 3° 1.5° | −2° |
| Schleicher ASK 21 | 16.1 4° - | −1.5° (0.250C) |
| Schweizer SGM 2-37 | 18.1 3.5° | 1° | 1° | 0° |
(Based on Ref. )
View chapterExplore book
Read full chapter
URL:
Book2014, General Aviation Aircraft DesignSnorri Gudmundsson BScAE, MScAE, FAA DER(ret.)
Chapter
LAT-DIR Stability and Control
2022, General Aviation Aircraft Design (Second Edition)Snorri Gudmundsson BScAE, MScAE, Ph.D., FAA DER (ret.)
25.4.1 Important Contributions to Rolling Moment
Wing dihedral is one of the most important contributors to rolling moment (see Figure 25-13). The contribution is called dihedral effect. The dihedral angle, denoted by Γ, is positive when the vertical position of the wingtip is above the root. Otherwise, it is called anhedral angle.
Figure 25-14 shows an airplane banking with associated yaw (sideslip). As stated in Section 25.2.4, the leading wing generates higher lift than the trailing one. The difference in loading creates a rolling moment that tends to level the wing. Simultaneously, directional stability tends to neutralize the yaw angle, as illustrated in segments ① through ④ in Figure 25-14. These tendencies comprise our design goal.
Wing sweep is another important contributor to dihedral effect. Figure 25-15 shows an aircraft with an aft swept wing at β = 0 (left) and β ≠ 0 (right). It shows that the yaw (here with nose pointing left) increases the effective span of the right wing by amount ΔbR and reduces its sweep angle by amount β. Simultaneously, the span of the left wing is reduced by amount ΔbL, while its sweep angle increases. This improves the lift generation of the right wing and reduces that of the left wing. The asymmetric lift generates the rolling moment. The greater the sweep, the greater the moment. These and other contributions to the rolling moment will now be presented.
View chapterExplore book
Read full chapter
URL:
Book2022, General Aviation Aircraft Design (Second Edition)Snorri Gudmundsson BScAE, MScAE, Ph.D., FAA DER (ret.)
Related terms:
Energy Engineering
Grain Boundary
Densification
Sintering
Potential Energy
Crystal Structure
Microstructure
Conformers
Side Chain
Substituent
View all Topics |
13080 | https://cds.cern.ch/record/2281430/files/nature23446.pdf | 6 6 | N A T U R E | V O L 5 4 8 | 3 a u g u s t 2 0 1 7 Letter doi:10.1038/nature23446 Observation of the hyperfine spectrum of antihydrogen M. Ahmadi1, B. X. R. Alves2, C. J. Baker3, W . Bertsche4,5, E. Butler6, A. Capra7, C. Carruth8, C. L. Cesar9, M. Charlton3, S. Cohen10, R. Collister7, S. Eriksson3, A. Evans11, N. Evetts12, J. Fajans8, T. Friesen2, M. C. Fujiwara7, D. R. Gill7, A. Gutierrez12,13, J. S. Hangst2, W . N. Hardy12, M. E. Hayden14, C. A. Isaac3, A. Ishida15, M. A. Johnson4,5, S. A. Jones3, S. Jonsell16, L. Kurchaninov7, N. Madsen3, M. Mathers17, D. Maxwell3, J. T. K. McKenna7, S. Menary17, J. M. Michan7,18, T. Momose12, J. J. Munich14, P . Nolan1, K. Olchanski7, A. Olin7,19, P . Pusa1, C. Ø. Rasmussen2, F. Robicheaux20, R. L. Sacramento9, M. Sameed3, E. Sarid21, D. M. Silveira9, S. Stracka7,22, G. Stutter2, C. So11, T. D. Tharp23, J. E. Thompson17, R. I. Thompson11, D. P . van der Werf3,24 & J. S. Wurtele8 The observation of hyperfine structure in atomic hydrogen by Rabi and co-workers1–3 and the measurement4 of the zero-field ground-state splitting at the level of seven parts in 1013 are important achievements of mid-twentieth-century physics. The work that led to these achievements also provided the first evidence for the anomalous magnetic moment of the electron5–8, inspired Schwinger’s relativistic theory of quantum electrodynamics9,10 and gave rise to the hydrogen maser11, which is a critical component of modern navigation, geo-positioning and very-long-baseline interferometry systems. Research at the Antiproton Decelerator at CERN by the ALPHA collaboration extends these enquiries into the antimatter sector. Recently, tools have been developed that enable studies of the hyperfine structure of antihydrogen12—the antimatter counterpart of hydrogen. The goal of such studies is to search for any differences that might exist between this archetypal pair of atoms, and thereby to test the fundamental principles on which quantum field theory is constructed. Magnetic trapping of antihydrogen atoms13,14 provides a means of studying them by combining electromagnetic interaction with detection techniques that are unique to antimatter12,15. Here we report the results of a microwave spectroscopy experiment in which we probe the response of antihydrogen over a controlled range of frequencies. The data reveal clear and distinct signatures of two allowed transitions, from which we obtain a direct, magnetic-field-independent measurement of the hyperfine splitting. From a set of trials involving 194 detected atoms, we determine a splitting of 1,420.4 ± 0.5 megahertz, consistent with expectations for atomic hydrogen at the level of four parts in 104. This observation of the detailed behaviour of a quantum transition in an atom of antihydrogen exemplifies tests of fundamental symmetries such as charge–parity–time in antimatter, and the techniques developed here will enable more-precise such tests.
In an earlier experiment12 using the original ALPHA apparatus16, we demonstrated microwave-induced spin flips in trapped antihydrogen. The current work was carried out using the second- generation ALPHA-2 device (Fig. 1), operating at the CERN Antiproton Decelerator17. Unlike their matter counterparts, antihydrogen atoms must be synthesized18 by merging cold plasmas of antiprotons and positrons in specially configured Penning–Malmberg traps. In the ALPHA-2 device, we typically mix 90,000 antiprotons, slowed and captured from the Antiproton Decelerator, with 1.6 million positrons from a Surko-type accumulator19 to produce about 25,000 antihydrogen atoms. These numbers are monitored in daily baseline measurements that involve ejecting particles onto a multichannel plate detector.
Of the produced antihydrogen atoms, only a few will have low enough kinetic energies (0.54 K in temperature units) to be trapped in our superconducting, multipolar, magnetic-minimum trap. The current state-of-the-art is that about 20 atoms can be trapped from a single mixing sequence, and we have accumulated up to 74 atoms by repetitive mixing (M.A. et al., submitted). A single mixing and capture sequence takes approximately 4 min, the bulk of which is used for preparation of plasmas of appropriate temperature, size and density (M.A. et al., submitted). The actual mixing process takes less than 1 s. The trapped antimatter atoms can survive for at least 1,000 s in the cryo-pumped ultrahigh vacuum of ALPHA-2.
Referring to Fig. 1, the antiproton and positron plasmas are merged in the central Penning trap (yellow electrodes) to produce antihydrogen. An external solenoid magnet provides a uniform 1-T field for the Penning trap. The production region is near the centre of the magnetic-minimum trap, which comprises an octupole coil for transverse confinement of neutral anti-atoms and five short solenoids (‘mirror coils’) that can shape the axial trapping well. The trapping volume is cylindrical, with a diameter of 44.35 mm and length of 280 mm. For the current experiment, only the outer two mirror coils are used to create the axial well.
Antihydrogen atoms that leave the trap and annihilate on the electrodes of the Penning trap are registered by the ALPHA-2 annihilation detector20. This three-layer silicon vertex detector that surrounds the trapping volume (Fig. 1) determines the vertex position of the antiproton annihilation. The amount of trapped antihydrogen can be determined destructively at any time by intentionally ramping down the trapping magnets to release anti-atoms, while monitoring their annihilations. The dominant background in our experiment comes from cosmic rays, which trigger the detector at an average rate of 10.02 ± 0.02 s−1 (all errors herein are one standard deviation).
To distinguish antiproton annihilations from cosmic rays, we use extended versions of our previously developed methods of 1Department of Physics, University of Liverpool, Liverpool L69 7ZE, UK. 2Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark. 3Department of Physics, College of Science, Swansea University, Swansea SA2 8PP, UK. 4School of Physics and Astronomy, University of Manchester, Manchester M12 9PL, UK. 5Cockcroft Institute, Sci-Tech Daresbury, Warrington WA4 4AD, UK. 6Physics Department, CERN, CH-1211 Geneve 23, Switzerland. 7TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia V6T 2A3, Canada. 8Department of Physics, University of California at Berkeley, Berkeley, California 94720-7300, USA. 9Instituto de Fisica, Universidade Federal do Rio de Janeiro, Rio de Janeiro 21941-972, Brazil. 10Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel. 11Department of Physics and Astronomy, University of Calgary, Calgary, Alberta T2N 1N4, Canada. 12Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada. 13Department of Medical Physics and Biomedical Engineering, University College London, London WC1E 6BT, UK. 14Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada. 15Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan. 16Department of Physics, Stockholm University, SE-10691 Stockholm, Sweden. 17Department of Physics and Astronomy, York University, Toronto, Ontario M3J 1P3, Canada. 18École Polytechnique Fédérale de Lausanne (EPFL), Swiss Plasma Center (SPC), Lausanne CH-1015, Switzerland. 19Department of Physics and Astronomy, University of Victoria, Victoria, British Columbia V8P 5C2, Canada. 20Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA. 21Soreq NRC, Yavne 81800, Israel. 22Universita di Pisa and Sezione INFN di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy. 23Physics Department, Marquette University, PO Box 1881, Milwaukee, Wisconsin 53201-1881, USA. 24IRFU, CEA/Saclay, F-91191, Gif-sur-Yvette Cedex, France.
OPEN © 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
3 a u g u s t 2 0 1 7 | V O L 5 4 8 | N A T U R E | 6 7 Letter RESEARCH multivariate analysis12 (Methods). The multivariate analysis used for the current experiment yields a cosmic ray background rate of 0.00559 ± 0.00051 s−1 and an overall detection efficiency for annihilations of 0.596 ± 0.002.
Figure 2 depicts the expected energy levels of ground-state anti hydrogen in a magnetic field B. Atoms in either of the two levels labelled | c〉 and | d〉 have energies that increase with field strength B and can thus be trapped in a minimum-B configuration. The other two states, labelled | a〉 and | b〉 , are expelled from the trap. The essential idea of the experiment is to use microwaves at about 29 GHz to resonantly drive transitions from trapped to un-trapped states as anti- atoms traverse the bottom of the magnetic potential well. At 1 T, the | c〉 → | b〉 and | d〉 → | a〉 transitions correspond to positron spin flips. Calculations for hydrogen (Fig. 3, inset) show that the inhomogeneous magnetic field produces asymmetric line shapes, with sharply defined onsets corresponding to the resonant frequency of either transition at the field minimum. The frequency difference between the two onsets represents the ground-state hyperfine splitting and is independent of the field strength and the number of anti-atoms that are trapped. It is this splitting that we seek to measure in antihydrogen and compare to that in hydrogen9,10.
In Fig. 3 we show the magnetic field strength in ALPHA-2 as a func-tion of position in the trapping volume. The volume of interest is at the centre of the trap, where fields, in the vicinity of the minimum, are about 1 T. Microwaves enter the apparatus through a purpose-built vacuum feed-through and are directed to the trapping volume using a waveguide (Fig. 1). We use an Agilent 8257D PSG frequency synthesizer and a Miteq AMF-4B amplifier to generate the microwave power. We injected 160 mW and 320 mW at the lower and upper transitions, respectively (see the discussion below); both are measured at the feed-through.
The experimental procedure involves producing and trapping anti-hydrogen atoms, removing any residual charged particles from the trap using pulsed electric fields, and then introducing microwaves into the trapping volume. The microwave frequency is stepped up in 300-kHz increments, starting from below the expected onset frequency of the | c〉 → | b〉 transition. The trapped antihydrogen atoms are exposed to microwave fields at each frequency for 4 s. After the first 16 steps, the starting frequency is incremented by + 1,420.4 MHz and the process is repeated to scan through the onset of the | d〉 → | a〉 transition. The silicon vertex detector continuously monitors for the annihilation of antihydrogen atoms that are ejected following a resonant spin flip. The total illumination cycle is 2 × 64 s (16 points spanning the onset of each transition), after which the trap is de-energized in 1.5 s, releasing any remaining antihydrogen.
For the dataset presented here, we repeated this measurement sequence 22 times over a three-day period. A combination of single and double mixing sequences was used, achieving an average trapping rate of about 14 atoms per trial. Each day, before data acquisition, the external solenoid field was reset and the minimum field strength at the centre of the magnetic trap was determined by measuring the electron cyclotron resonance frequency of an electron plasma21. The precision of this measurement is estimated to be ± 0.3 mT (equivalent to 8.4 MHz in electron cyclotron resonance frequency).
The results of the 22 measurement trials are plotted in Fig. 4. Cosmic background contributes 0.492 ± 0.045 events to each 4-s measurement bin. The sums of each day’s trials are combined by aligning the maxima of the lower (| c〉 → | b〉 ) transition, to account for day-to-day variations in magnetic field. The responses observed should not be directly compared to traditional spectral lines or to the calculated distributions in the inset to Fig. 3, because the detailed shape is strongly influenced Magnetic feld, B (T) 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Relative energy (GHz) –20 –15 –10 –5 0 5 10 15 20 |a〉 |b〉 |c〉 |d〉 Trappable low-feld-seeking states |d〉 = |↓ ⇓〉 |c〉 = |↓ ⇑〉 Untrappable high-feld-seeking states |b〉 = |↑ ⇑〉 |a〉 = |↑ ⇓〉 Figure 2 | Ground-state hyperfine energy levels. The energy levels are calculated assuming they are identical to those of hydrogen. The ket notation indicates the positron spin (left; ↓ or ↑ ) and antiproton spin (right; ⇓ or ⇑ ) states in the high-field limit. The shaded region illustrates part of the range of fields in the ALPHA-2 antihydrogen trap, with the minimum at 1.03 T. The full field map is shown in Fig. 3.
0 5 10 15 20 Radial position (mm) 200 0 –200 Axial position (mm) 1.0 1.5 2.0 Magnetic feld, B (T) |c〉 → |b〉 |d〉 → |a〉 Probability Frequency 1,420.4 MHz Figure 3 | Magnetic field strength. The scalar magnitude of the total magnetic field in the atom trap is plotted versus position. The radial position is measured from the symmetry axis of the trap, and the axial origin is the centre of the atom trap, defined by the outermost mirror coils. Inset, the probability of an atom being resonant at a given frequency, calculated for hydrogen; the two low-frequency onsets are separated by the ground-state splitting.
Microwave waveguide Annihilation detector (three layers) Octupole Mirror coils Trap electrodes for antihydrogen production Trap electrodes for positron preparation Trap electrodes for antiproton preparation Positron injection Antiproton injection Figure 1 | The ALPHA-2 central apparatus. A cut-away schematic of the antihydrogen production and trapping region of ALPHA-2 is shown. For clarity, the vacuum wall and the cryostat for the superconducting magnets are not shown. Antiproton and positron plasmas are prepared on either side of the production region before being mixed to form antihydrogen at the centre of the minimum-B trap. All of the components shown are immersed in a uniform, 1-T, axial magnetic field, which is provided by an external solenoid (not illustrated).
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
6 8 | N A T U R E | V O L 5 4 8 | 3 a u g u s t 2 0 1 7 Letter RESEARCH by the rate at which the trap is depopulated as atoms undergo spin-flip transitions.
Figure 4 reveals a qualitative difference between the shapes of the responses for the two transitions. The lower transition features a low-frequency onset and a narrow width; most of the anti-atoms are removed in the first two resonant bins. The more gradual onset and broader width of the upper transition are attributed to a lower ampli-tude of the in situ microwave magnetic field (see simulation results below). As illustrated in Fig. 1, the Penning-trap electrode stack in ALPHA-2 represents a complicated boundary surface for the injected electromagnetic radiation. It is unsurprising that disparate mixtures of standing and travelling waves are established at different frequencies. We can measure the strengths of the microwave electric fields in the vicinity of the two transitions by studying electron cyclotron resonance heating of electron plasmas stored in the trap centre21. These meas-urements lead to the conclusion that, for a given injected power, the microwave electric field strength at the centre of the trap is about seven times stronger at the lower transition frequency than at the higher one. However, we do not know the precise in situ relationship between the microwave electric and magnetic fields; and it is the latter that drives transitions between hyperfine levels. To partially compensate for this imbalance, we injected twice as much power at the upper transition, relative to the lower transition. This reduces the ratio of microwave-field amplitudes at the two transitions from seven to five. Our ability to further balance these amplitudes is currently limited by adverse thermal effects in the cryogenic, ultrahigh-vacuum environment.
Also shown in Fig. 4 are the results of a simulation (Methods) of the expected behaviour of hydrogen atoms in the magnetic environment of our trap. The inputs to this simulation include the trapping magnetic fields that are calculated from measured currents, the amplitudes of the microwave magnetic field that are inferred via electron cyclotron reso nance and assuming plane-wave propagation in vacuum, the starting frequency for the microwave scan, the energy distribution of the trapped atoms, and the expected temporal magnetic field fluctuations, which are based on current-transformer monitoring of the currents in the trapping magnets. The simulation result is scaled to give the same total number of events as the experiment, integrated over both transi-tions (194 detected events). The simulation explicitly accounts for the removal of atoms from the trap as spin-flip events occur.
As described above, our experimental protocol was designed to determine the difference between the onset frequencies for the two transitions, profiting from the expected sharp increase in signal asso-ciated with resonance at the magnetic-field minimum. The slower increase that was observed in the | d〉 → | a〉 transition complicates the determination of the frequency splitting. It is tempting to extract a ‘best’ value for the hyperfine splitting by fitting the hydrogen simulation to the experimental data, particularly because the simulation reproduces the form of the data rather well. Given the fundamental nature of the quantity that is to be extracted, however, we defer any such interpreta-tion until more detailed systematic studies can be performed.
We determine the splitting from Fig. 4 to be 1,420.4 ± 0.5 MHz, which reflects the difference between the low-frequency onsets of the two lines. The uncertainty includes contributions from drifts in the magnetic field that were observed during the scan (0.3 MHz), the procedure used for combining the data from separate days (0.3 MHz), and the determination of the onset frequencies of the two lines (0.3 MHz). This is the only available direct measurement of this fundamental quantity. The precision of our technique can be improved by reducing the step size in the frequency scan, by balancing the micro-wave power at the two transitions and by more precise characterization and stabilization of the magnetic fields in the atom trap. There are also plans to measure the same quantity in zero magnetic field using a beam of antihydrogen22.
The release of the antihydrogen atoms that remain in the trap after the two transitions have been illuminated yielded 9 detected events for the 22 trials. A different multivariate analysis is used for this determi-nation (Methods); the overall efficiency is 0.726 and we expect 1.3 total background events for the 22 trials. We conclude that about 96% of the trapped anti-atoms were removed as a result of a spin flip. Independent measurements using only the lower transition indicate that a micro-wave power sufficient to remove all of the trapped atoms with a 1-s time constant was injected without adverse thermal effects in the cryogenic, ultrahigh-vacuum system. In addition to being a useful diagnostic for optimizing antihydrogen trapping, the ability to selectively control the populations of the trapped quantum states will be useful for future microwave and optical spectroscopy of trapped antihydrogen.
The work described exemplifies a new approach in antimatter physics: the observation of spectral line shapes in antihydrogen. The ability to make a controlled frequency scan over an expected quantum mechanical transition in an atom of antimatter points the way to more precise tests of fundamental symmetries with antihydrogen. Charge–parity–time invariance implies that the detailed shapes—not just the resonance frequencies—of spectral lines for hydrogen and antihydrogen in the same environment must be identical. We will soon be able to use such precise measurements to subject antihydrogen to previously unobtainable scrutiny.
Also of interest is the nuclear magnetic resonance (NMR)-type tran-sition between the | c〉 and | d〉 states, which corresponds to an anti-proton spin flip. Recent advances in trapping efficiency (M.A. et al., submitted) bode well for the feasibility of observing this transition in trapped antihydrogen. The absolute energy scales for the positron and antiproton spin-flip transitions in ALPHA-2 are respectively five and eight orders of magnitude smaller than that of the laser transition that was recently observed15. In addition to probing different interactions in the anti hydrogen Hamiltonian, these energy scales offer very high sensitivity to potential new physics23.
Relative frequency (MHz) 0 10 20 30 40 50 60 70 Counts –1.2 0 1.2 2.4 3.6 1,419.2 1,420.4 1,421.6 1,422.8 1,424.0 Data Simulation Figure 4 | Data and simulation. The number of detected antihydrogen annihilation events (filled blue squares), summed (see the text for details) over 22 trials, is plotted as a function of frequency. Note the discontinuous abscissa. The error bars represent counting statistics only. The expectations from the simulation for hydrogen in the trap are also shown (open red squares). The simulation results are scaled to match the total number (194) of observed events and are aligned to match the onset of the lower transition. The simulation includes the effect of measured fluctuations in the currents in the trapping magnets.
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
3 a u g u s t 2 0 1 7 | V O L 5 4 8 | N A T U R E | 6 9 Letter RESEARCH Online Content Methods, along with any additional Extended Data display items and Source Data, are available in the online version of the paper; references unique to these sections appear only in the online paper.
received 27 May; accepted 30 June 2017. 1. Rabi, I. I., Kellogg, J. M. B. & Zacharias, J. R. The magnetic moment of the proton. Phys. Rev. 46, 157–163 (1934).
2. Rabi, I. I., Kellogg, J. M. B. & Zacharias, J. R. The magnetic moment of the deuteron. Phys. Rev. 46, 163–165 (1934).
3. Kellogg, J. M. B., Rabi, I. I. & Zacharias, J. R. The gyromagnetic properties of the hydrogens. Phys. Rev. 50, 472–481 (1936).
4. Petit, P., Desaintfuscien, M. & Audoin, C. Temperature dependence of the hydrogen maser wall shift in the temperature range 295–395 K. Metrologia 16, 7–14 (1980).
5. Nafe, J. E., Nelson, E. B. & Rabi, I. I. The hyperfine structure of atomic hydrogen and deuterium. Phys. Rev. 71, 914–915 (1947).
6. Nafe, J. E. & Nelson, E. B. The hyperfine structure of hydrogen and deuterium. Phys. Rev. 73, 718–728 (1948).
7. Nelson, E. B. & Nafe, J. E. The hyperfine structure of tritium. Phys. Rev. 75, 1194–1198 (1949).
8. Nelson, E. B. & Nafe, J. E. A comparison of the g value of the electron in hydrogen with that in deuterium. Phys. Rev. 76, 1858–1860 (1949).
9. Schwinger, J. On quantum-electrodynamics and the magnetic moment of the electron. Phys. Rev. 73, 416–417 (1948).
10. Schwinger, J. Quantum electrodynamics. III. The electromagnetic properties of the electron—radiative corrections to scattering. Phys. Rev. 76, 790–817 (1949).
11. Goldenberg, H. M., Kleppner, D. & Ramsey, N. F. Atomic hydrogen maser. Phys. Rev. Lett. 5, 361–362 (1960).
12. Amole, C. et al. Resonant quantum transitions in trapped antihydrogen atoms. Nature 483, 439–443 (2012).
13. Andresen, G. B. et al. Trapped antihydrogen. Nature 468, 673–676 (2010).
14. Andresen, G. B. et al. Confinement of antihydrogen for 1,000 seconds. Nat. Phys. 7, 558–564 (2011).
15. Ahmadi, M. et al. Observation of the 1S–2S transition in trapped antihydrogen. Nature 541, 506–510 (2017).
16. Amole, C. et al. The ALPHA antihydrogen trapping apparatus. Nucl. Instrum. Methods A 735, 319–340 (2014).
17. Maury, S. The antiproton decelerator: AD. Hyperfine Interact. 109, 43–52 (1997).
18. Amoretti, M. et al. Production and detection of cold antihydrogen atoms. Nature 419, 456–459 (2002).
19. Murphy, T. J. & Surko, C. M. Positron trapping in an electrostatic well by inelastic collisions with nitrogen molecules. Phys. Rev. A 46, 5696–5705 (1992).
20. Amole, C. et al. Silicon vertex detector upgrade in the ALPHA experiment. Nucl. Instrum. Methods A 732, 134–136 (2013).
21. Amole, C. et al. In situ electromagnetic field diagnostics with an electron plasma in a Penning–Malmberg trap. New J. Phys. 16, 013037 (2014).
Acknowledgements All authors are members of the ALPHA Collaboration. This work was supported by: the European Research Council through its Advanced Grant programme (J.S.H.); CNPq, FAPERJ and RENAFAE (Brazil); NSERC, NRC/TRIUMF, EHPDS/EHDRS and FQRNT (Canada); FNU (NICE Centre) and Carlsberg Foundation (Denmark); JSPS Postdoctoral Fellowships for Research Abroad (Japan); ISF (Israel); STFC, EPSRC, the Royal Society and the Leverhulme Trust (UK); DOE and NSF (USA); and VR (Sweden). We are grateful for the efforts of the CERN Antiproton Decelerator team. We thank J. Tonoli (CERN) and his staff for help with machining work. We thank the staff of the Superconducting Magnet Division at Brookhaven National Laboratory for collaboration and fabrication of the trapping magnets. We thank C. Marshall (TRIUMF) for his work on the ALPHA-2 cryostat. We thank F. Besenbacher (Aarhus) for support in procuring the ALPHA-2 external solenoid.
Author Contributions This experiment was based on data collected using the ALPHA-2 antihydrogen trapping apparatus, which was designed and constructed by the ALPHA Collaboration using methods developed by the entire collaboration. The entire collaboration participated in the operation of the apparatus and the data-taking activities. The experimental protocol was conceived by M.E.H. and T.F. The microwave hardware was implemented by T.F., W.N.H., A.I., J.J.M. and M.E.H. Detailed analysis of the antiproton annihilation detector data was done by J.T.K.M. and A.O., using methods introduced to ALPHA by S.S. The manuscript was written by J.S.H., T.F., M.E.H., N.M., J.T.K.M. and F.R., and was then edited and improved by the entire collaboration.
Author Information Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests. Readers are welcome to comment on the online version of the paper. Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Correspondence and requests for materials should be addressed to J.S.H. (jeffrey.hangst@cern.ch), T.F. (Tim.Friesen@cern.ch) or M.E.H. (mhayden@sfu.ca).
Reviewer Information Nature thanks R. Pohl and the other anonymous reviewer(s) for their contribution to the peer review of this work.
This work is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) licence. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons licence, users will need to obtain permission from the licence holder to reproduce the material. To view a copy of this licence, visit
22. Kuroda, N. et al. A source of antihydrogen for in-flight hyperfine spectroscopy. Nat. Commun. 5, 3089 (2014).
23. Kostelecký, V. A. & Vargas, A. J. Lorentz and CPT tests with hydrogen, antihydrogen, and related systems. Phys. Rev. D 92, 056002 (2015).
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
Letter RESEARCH 24. Narsky, I. StatPatternRecognition: a C+ + package for statistical analysis of high energy physics data. Preprint at (2005).
25. Narsky, I. Optimization of signal significance by bagging decision trees. Preprint at (2005).
26. Amole, C. et al. Alternative method for reconstruction of antihydrogen annihilation vertices. Hyperfine Interact. 212, 101–107 (2012).
Methods Simulation of the microwave–atom interaction. Our simulation of the micro-wave flip process uses quantum and classical ideas. The motion of antihydrogen through the trap is determined by solving the classical equations of motion using a fourth-order symplectic integrator. The force on the atom arises from the spatially dependent magnetic field. Because the magnetic fields are always large, the state of the antihydrogen is conserved unless the resonance condition is satisfied. While trapped, the atoms are in either the | c〉 or | d〉 states (the low-field-seeking states). These states have a magnetic moment that is approximately equal to that of the positron, which leads to a potential energy of PE ≈ μB/2, where μ = gμB. Here, μB is the Bohr magneton and g is the spin g-factor. The force is obtained by numerically computing the gradient of the magnitude of the magnetic field.
The energies of the hyperfine states are solved using an effective Hamiltonian with the hyperfine splitting and the magnetic moments of hydrogen as inputs. A quadratic interpolation of the energies at three consecutive time steps is used to determine whether the microwave resonance condition is met. Where in space this condition is met depends on the currents in the various magnets and on the microwave frequency. If the resonance condition occurs during the step, then the Landau–Zener approximation is used to obtain a spin-flip probability. The time derivative of the energy separation is calculated from the quadratic interpolation. The matrix element V that couples the | c〉 and | b〉 or | d〉 and | a〉 states depends on the microwave magnetic field BMW perpendicular to the static magnetic field at the position at which the antihydrogen is in resonance; it is well approximated by V = BMWμ/4. To estimate BMW we use the electron-cyclotron- resonance plasma-heating diagnostic discussed in the main text, which measures the microwave electric field EMW perpendicular to the static magnetic field. We then assume BMW is uniform and given by BMW = EMW/c, where c is the speed of light in vacuum. (The precise relationship between EMW and BMW is not known because the boundary conditions imposed by the electrode stack support a complex mixture of standing and travelling wave modes. Errors from making this assump-tion are reduced by averaging.) Resonance conditions are encountered in pairs as atoms pass through the centre of the trap, and we account for the possibility that spins will flip more than once. Simulations do not predict a simple exponential decay of trapped populations when microwaves are present; the rate at which atoms encounter resonance conditions and the probability that they undergo a spin flip as they pass through resonance vary with trajectory. However, as an indication of scale, a microwave intensity of 4 mW cm−2 (corresponding to BMW = 0.6 μ T and V = h × (4 kHz), where h is Planck’s constant) applied just above the onset of either transition will clear atoms in the corresponding state from the trap with a time constant of order 1 s.
Multivariate analysis of detector events. Differentiation of antihydrogen annihi-lations and background events (primarily cosmic rays) is achieved by discerning their distinctive topologies. A multivariate analysis package is used to distinguish between these two populations24,25.
Two independent multivariate analyses were performed for this experiment: a low-background analysis for identifying annihilations during the 128-s microwave window and a high-signal-acceptance analysis to identify annihilations during the 1.5-s trap shutdown. The latter analysis has the same design as used in previous experiments12,16. The former analysis is modified to achieve a much lower back-ground, through the addition of more variables that enhance the signal-to-back-ground discrimination. The additional variables include: the asymmetry in hit count between the two hemispheres defined by the plane perpendicular to the event axis12 and passing through the centre of the trap; the minimum distance of closest approach of any cosmic track candidate to the reconstructed vertex; the polar angle of the vector describing the vertex position26 relative to the centre of the trap; and the average of the ratios between the axial and radial projections of the tracks that originate from the reconstructed vertex.
The signal and background data used for multivariate-analysis training, validation and testing (split equally) comprises a set of 305,706 annihilation events and 236,969 background events.
Data availability. The datasets generated and analysed during this study are available from corresponding author J.S.H. on reasonable request.
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved. |
13081 | https://pubmed.ncbi.nlm.nih.gov/31875103/ | High Prevalence of Asthenopia among a Population of University Students - PubMed
Loading [MathJax]/jax/output/PreviewHTML/jax.js
Clipboard, Search History, and several other advanced features are temporarily unavailable.
Skip to main page content
An official website of the United States government
Here's how you know
The .gov means it’s official.
Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.
The site is secure.
The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.
Log inShow account info
Close
Account
Logged in as:
username
Dashboard
Publications
Account settings
Log out
Access keysNCBI HomepageMyNCBI HomepageMain ContentMain Navigation
Search: Search
AdvancedClipboard
User Guide
Save Email
Send to
Clipboard
My Bibliography
Collections
Citation manager
Display options
Display options
Format
Save citation to file
Format:
Create file Cancel
Email citation
Email address has not been verified. Go to My NCBI account settings to confirm your email and then refresh this page.
To:
Subject:
Body:
Format:
[x] MeSH and other data
Send email Cancel
Add to Collections
Create a new collection
Add to an existing collection
Name your collection:
Name must be less than 100 characters
Choose a collection:
Unable to load your collection due to an error
Please try again
Add Cancel
Add to My Bibliography
My Bibliography
Unable to load your delegates due to an error
Please try again
Add Cancel
Your saved search
Name of saved search:
Search terms:
Test search terms
Would you like email updates of new search results? Saved Search Alert Radio Buttons
Yes
No
Email: (change)
Frequency:
Which day?
Which day?
Report format:
Send at most:
[x] Send even when there aren't any new results
Optional text in email:
Save Cancel
Create a file for external citation management software
Create file Cancel
Your RSS Feed
Name of RSS Feed:
Number of items displayed:
Create RSS Cancel
RSS Link Copy
Full text links
Free PMC article
Full text links
Actions
Cite
Collections
Add to Collections
Create a new collection
Add to an existing collection
Name your collection:
Name must be less than 100 characters
Choose a collection:
Unable to load your collection due to an error
Please try again
Add Cancel
Permalink
Permalink
Copy
Display options
Display options
Format
Page navigation
Title & authors
Abstract
Conflict of interest statement
Figures
Similar articles
Cited by
References
Publication types
Related information
LinkOut - more resources
Review
J Ophthalmic Vis Res
Actions
Search in PubMed
Search in NLM Catalog
Add to Search
. 2019 Oct 24;14(4):474-482.
doi: 10.18502/jovr.v14i4.5455. eCollection 2019 Oct-Dec.
High Prevalence of Asthenopia among a Population of University Students
Hassan Hashemi12,Mohammad Saatchi1,Abbasali Yekta3,Babak Ali3,Hadi Ostadimoghaddam4,Payam Nabovati5,Mohamadreza Aghamirsalim6,Mehdi Khabazkhoob7
Affiliations Expand
Affiliations
1 Noor Research Center for Ophthalmic Epidemiology, Noor Eye Hospital, Tehran, Iran.
2 Noor Ophthalmology Research Center, Noor Eye Hospital, Tehran, Iran.
3 Department of Optometry, School of Paramedical Sciences, Mashhad University of Medical Sciences, Mashhad, Iran.
4 Refractive Errors Research Center, Mashhad University of Medical Sciences, Mashhad, Iran.
5 Rehabilitation Research Center, Department of Optometry, School of Rehabilitation Sciences, Iran University of Medical Sciences, Tehran, Iran.
6 Eye Research Center, Tehran University of Medical Sciences, Tehran, Iran.
7 Department of Psychiatric Nursing and Management, School of Nursing and Midwifery, Shahid Beheshti University of Medical Sciences, Tehran, Iran.
PMID: 31875103
PMCID: PMC6825687
DOI: 10.18502/jovr.v14i4.5455
Item in Clipboard
Review
High Prevalence of Asthenopia among a Population of University Students
Hassan Hashemi et al. J Ophthalmic Vis Res.2019.
Show details
Display options
Display options
Format
J Ophthalmic Vis Res
Actions
Search in PubMed
Search in NLM Catalog
Add to Search
. 2019 Oct 24;14(4):474-482.
doi: 10.18502/jovr.v14i4.5455. eCollection 2019 Oct-Dec.
Authors
Hassan Hashemi12,Mohammad Saatchi1,Abbasali Yekta3,Babak Ali3,Hadi Ostadimoghaddam4,Payam Nabovati5,Mohamadreza Aghamirsalim6,Mehdi Khabazkhoob7
Affiliations
1 Noor Research Center for Ophthalmic Epidemiology, Noor Eye Hospital, Tehran, Iran.
2 Noor Ophthalmology Research Center, Noor Eye Hospital, Tehran, Iran.
3 Department of Optometry, School of Paramedical Sciences, Mashhad University of Medical Sciences, Mashhad, Iran.
4 Refractive Errors Research Center, Mashhad University of Medical Sciences, Mashhad, Iran.
5 Rehabilitation Research Center, Department of Optometry, School of Rehabilitation Sciences, Iran University of Medical Sciences, Tehran, Iran.
6 Eye Research Center, Tehran University of Medical Sciences, Tehran, Iran.
7 Department of Psychiatric Nursing and Management, School of Nursing and Midwifery, Shahid Beheshti University of Medical Sciences, Tehran, Iran.
PMID: 31875103
PMCID: PMC6825687
DOI: 10.18502/jovr.v14i4.5455
Item in Clipboard
Full text links Cite
Display options
Display options
Format
Abstract
Purpose: To determine the prevalence of asthenopia and its associated factors in a sample of university students in Iran.
Methods: In this cross-sectional study, participants were selected using multistage cluster sampling. Presence of at least one of the 10 symptoms-foreign body sensation, diplopia, blurred vision, eye swelling, dry eye, eye pain, difficulty in sustaining visual operations, decreased visual acuity, tearing, and photophobia-was considered as asthenopia. Ocular examinations, including uncorrected/corrected visual acuity measurement, objective/subjective refraction, cover test, amplitude of accommodation (AA), and near point of convergence (NPC) were performed.
Results: Of the 1,462 students (mean age: 22.8 3.1 years), 73% were women. The age- and gender-standardized prevalence was 70.9% (95% confidence interval [CI]: 68.3-73.5), 39.8% (95% CI: 36.4-43.1), and 19.7% (95% CI: 16.0-23.3) based on the presence of at least one, two, and three symptoms, respectively. The prevalence was significantly higher in females (P = 0.048), hyperopic students (P< 0.001), and astigmatic participants (P< 0.001). The mean AA and NPC were 9.7 ± 2.6 D and 10.2 ± 4.2 D (P = 0.008) and 7.0 ± 2.1 cm and 7.7 ± 3.9 cm (P< 0.001) in participants with and without asthenopia, respectively. Multiple regression model revealed age (28-29 years), astigmatism, and NPC as independent associated factors (odds ratios: 3.51, 1.61, and 0.91, respectively).
Conclusion: This study shows relatively high prevalence of asthenopia in university students. Demographic factors and visual system disorders are important risk factors and timely correction of conditions may lead to decreased asthenopia.
Keywords: Astigmatism; Photophobia; Prevalence; Asthenopia.
Copyright © 2019 Hashemi et al.
PubMed Disclaimer
Conflict of interest statement
There is no conflict of interest.
Figures
Figure 1
The prevalence of asthenopic symptoms…
Figure 1
The prevalence of asthenopic symptoms in 1,465 students above 18 years in the…
Figure 1
The prevalence of asthenopic symptoms in 1,465 students above 18 years in the city of Kazerun.
See this image and copyright information in PMC
Similar articles
Near Point of Convergence in Iranian Schoolchildren: Normative Values and Associated Factors.Hashemi H, Nabovati P, Khabazkhoob M, Yekta A, Emamian MH, Fotouhi A.Hashemi H, et al.Strabismus. 2018 Sep;26(3):126-132. doi: 10.1080/09273972.2018.1475493. Epub 2018 May 18.Strabismus. 2018.PMID: 29775110
Near Points of Convergence and Accommodation in a Population of University Students in Iran.Hashemi H, Pakbin M, Ali B, Yekta A, Ostadimoghaddam H, Asharlous A, Aghamirsalim M, Khabazkhoob M.Hashemi H, et al.J Ophthalmic Vis Res. 2019 Jul 18;14(3):306-314. doi: 10.18502/jovr.v14i3.4787. eCollection 2019 Jul-Sep.J Ophthalmic Vis Res. 2019.PMID: 31660110 Free PMC article.Review.
Orthoptic findings and asthenopia in a population of Swedish schoolchildren aged 6 to 16 years.Abdi S, Lennerstrand G, Pansell T, Rydberg A.Abdi S, et al.Strabismus. 2008 Apr-Jun;16(2):47-55. doi: 10.1080/09273970802020243.Strabismus. 2008.PMID: 18484506
Asthenopia in schoolchildren.Vilela MA, Castagno VD, Meucci RD, Fassa AG.Vilela MA, et al.Clin Ophthalmol. 2015 Aug 28;9:1595-603. doi: 10.2147/OPTH.S84976. eCollection 2015.Clin Ophthalmol. 2015.PMID: 26357460 Free PMC article.
[Asthenopia and monitor characteristics].Rechichi C, Scullica L.Rechichi C, et al.J Fr Ophtalmol. 1990;13(8-9):456-60.J Fr Ophtalmol. 1990.PMID: 2081858 Review.French.
See all similar articles
Cited by
Can Nutrition Play a Role in Ameliorating Digital Eye Strain?Lem DW, Gierhart DL, Davey PG.Lem DW, et al.Nutrients. 2022 Sep 27;14(19):4005. doi: 10.3390/nu14194005.Nutrients. 2022.PMID: 36235656 Free PMC article.Review.
Characterizing Refractive Errors, Near Accommodative and Vergence Anomalies and Symptoms in an Optometry Clinic.Wajuihian SO.Wajuihian SO.Br Ir Orthopt J. 2022 Jul 14;18(1):76-92. doi: 10.22599/bioj.267. eCollection 2022.Br Ir Orthopt J. 2022.PMID: 35903147 Free PMC article.
Investigation of the Relationship Between Subjective Symptoms of Visual Fatigue and Visual Functions.Zheng F, Hou F, Chen R, Mei J, Huang P, Chen B, Wang Y.Zheng F, et al.Front Neurosci. 2021 Jul 15;15:686740. doi: 10.3389/fnins.2021.686740. eCollection 2021.Front Neurosci. 2021.PMID: 34335163 Free PMC article.
Asthenopia Among University Students: The Eye of the Digital Generation.Touma Sawaya RI, El Meski N, Saba JB, Lahoud C, Saab L, Haouili M, Shatila M, Aidibe Z, Musharrafieh U.Touma Sawaya RI, et al.J Family Med Prim Care. 2020 Aug 25;9(8):3921-3932. doi: 10.4103/jfmpc.jfmpc_340_20. eCollection 2020 Aug.J Family Med Prim Care. 2020.PMID: 33110788 Free PMC article.
Nonlinear effects of humidex on risk for asthenopia among adults: a national cross-sectional survey in China.Wu X, Chen X, Tian F, Yang M, Lu F, Deng R, Lin N.Wu X, et al.Front Public Health. 2025 Mar 13;13:1515672. doi: 10.3389/fpubh.2025.1515672. eCollection 2025.Front Public Health. 2025.PMID: 40182528 Free PMC article.
See all "Cited by" articles
References
Bhanderi D., Choudhary S., Doshi V. A community-based study of asthenopia in computer operators. Indian Journal of Ophthalmology. 2008;56(1):51. doi: 10.4103/0301-4738.37596. - DOI - PMC - PubMed
García-Muñoz Á., Carbonell-Bonete S., Cacho-Martínez P. Symptomatology associated with accommodative and binocular vision anomalies. Journal of Optometry. 2014;7(4):178–192. doi: 10.1016/j.optom.2014.06.005. - DOI - PMC - PubMed
Ayanniyi A. A., Folorunso F. N., Adepoju F. G. Refractive ocular conditions and reasons for spectacles renewal in a resource-limited economy. BMC Ophthalmology. 2010;10(1) doi: 10.1186/1471-2415-10-12. - DOI - PMC - PubMed
Vilela M. A., Pellanda L. C., Fassa A. G., Castagno V. D. Prevalence of asthenopia in children: a systematic review with meta-analysis. Jornal de Pediatria. 2015;91(4):320–325. doi: 10.1016/j.jped.2014.10.008. - DOI - PubMed
Han CC., Liu R., Liu RR., Zhu ZH., RB Yu., Ma L. Prevalence of asthenopia and its risk factors in Chinese college students. Int J Ophthalmol. 2013:718–722. - PMC - PubMed
Show all 27 references
Publication types
Review
Actions
Search in PubMed
Search in MeSH
Add to Search
Related information
Cited in Books
MedGen
LinkOut - more resources
Full Text Sources
PubMed Central
Full text links[x]
Free PMC article
[x]
Cite
Copy Download .nbib.nbib
Format:
Send To
Clipboard
Email
Save
My Bibliography
Collections
Citation Manager
[x]
NCBI Literature Resources
MeSHPMCBookshelfDisclaimer
The PubMed wordmark and PubMed logo are registered trademarks of the U.S. Department of Health and Human Services (HHS). Unauthorized use of these marks is strictly prohibited.
Follow NCBI
Connect with NLM
National Library of Medicine
8600 Rockville Pike Bethesda, MD 20894
Web Policies
FOIA
HHS Vulnerability Disclosure
Help
Accessibility
Careers
NLM
NIH
HHS
USA.gov |
13082 | https://www.mechref.org/sol/axial_loading/?origin=sidebar | Axial Loading
Reference Pages
Mechanical Engineering
Menu
Home
Statics
Statics
1. Introduction
2. Cartesian coordinates
3. Vectors and scalars
4. Free body diagrams
5. Moments
6. Force systems
7. Reaction forces
8. Trusses
9. Frames and machines
10. Internal forces
11. Friction
12. Geometric properties
13. Hydrostatic fluid pressure
14. Virtual work
Applications
1. Track and field starting blocks
Dynamics
Dynamics
1. Introduction
2. Vectors
3. Vector calculus
4. Coordinate systems
5. Tangential and normal basis
6. Particle kinematics
7. Particle kinetics
8. Rigid body kinematics
9. Geometric properties
10. Rigid body kinetics
11. Multi-body systems
12. Contact And rolling
13. Work and energy
14. Momentum
Applications
1. Shortest flight paths
2. Track transition curves
3. Projectiles with air resistance
4. Celestial velocities
5. Accelerating and braking
6. Banked turns
7. Steering geometry
8. Four-bar linkages
9. Capsizing ships
Solid mechanics
Solid mechanics
1. Stress
2. Strain
3. Material properties
4. Axial loading
5. Torsion
6. Shear moment diagrams
7. Bending
8. Transverse shear
9. Pressure vessels
10. Combined loading
11. Stress transformation
12. Failure theories
13. Beam deflection
14. Buckling
Thermodynamics
Thermodynamics
1. Thermodynamic systems
2. Substances
3. Heat and work
4. Mass and energy balances
5. Complex problems
6. Thermodynamic devices
7. Entropy
8. Cycles
Mechanism design
Mechanism Design
1. Planar Mechanisms
2. Gears
3. Graphical linkage synthesis
4. Instant centers
5. Cams
6. Position velocity acceleration
7. Dynamic force analysis
8. Virtual work
9. Balancing
Machine failure
Machine Failure
1. Stress concentrations
2. Failure theories
3. Threaded components
About
Toggle theme
Scroll back to top
Current page:
Axial loading
Notation and Convention
Saint-Venant's Principle: slender beam case
Force-Deformation Relation
Axially Varying Properties
Principle of Superposition
General Solving Procedure
Statically Indeterminate Problems
General Solving Procedure
Thermal Effects: Temperature Changes
Misfit Problems
Axial Loading
Notation and Convention
Notation
Displacement of points in space: the movement of a point relative to its initial position in space (ex; δ B )
Change in length of a segment: the relative displacement of a point with respect to another point (ex; δ B/A=δ B−δ A=δ 1 ). This can be written multiple ways: Notation. δ A D≡δ D A≡δ A/D≡δ D/A
Sign Conventions for internal forces in a bar.
F>0 : tension
F<0 : compression
δ>0 : elongation
δ<0 : contraction
Saint-Venant's Principle: slender beam case
Notation
Stress analysis very near to the point of application of load P. Saint-Venant's principle: the stress and strain produced at points in a body sufficiently removed from the region of external load application will be the same as the stress and strain produced by any other applied external loading that has the same statically equivalent resultant and is applied to the body within the same region.
farther than the widest dimension of the cross section
Force-Deformation Relation
Relation. #axl-fdr
δ=P L E A
Derivation +
P=σ A
P=E ε A
P=E δ L A
Axial flexibility.
δ=f P=>f=L E A
Axial stiffness.
P=k δ=>k=E A L
Axially Varying Properties
For non-uniform load, material property and cross-section area:
Variable properties. #sts-vpr
δ=∫0 L F(x)E(x)A(x)d x
Derivation +
σ=E ε
F(x)A(x)=E(x)ε(x)
F(x)A(x)=E(x)d δ(x)d x
d δ=F(x)E(x)A(x)d x
Assume variations with x are "mild" (on length scale longer than cross-sectional length scales)
Principle of Superposition
Superposition: If the displacements are (1) small and (2) linearly related to the force components acting, the displacements caused by the components can be added up:
Superposition.
δ=∑i δ i=∑i F i L i E i A i
General Solving Procedure
Draw a FBD
Equilibrium equations: force balance and moment balance
Constitutive equations: stress-strain or force-displacement relations
Compatibility equations: geometric constraints
Statically Determinate Problems
Statically determinate
All internal forces can be obtained from equilibrium analysis only
Statically Indeterminate Problems
Statically indeterminate
Equilibrium does not determine all internal forces.
Thermal Effects: Temperature Changes
Notation
Δ T>0,σ<0 : Compression
Δ T<0,σ>0 : Tension
δ T , ε T present in addition to elastic δ E , ε E (from internal forces). Superposition (small strains):
Total strain.
ε t o t=ε E+ε T
Total displacement.
δ t o t=δ E+δ T
Temperature changes with no applied loads
A rod rests freely on a smooth horizontal surface. Temperature of the rod is raised by Δ T. Rod elongates by an amount.
Displacement from temperature changes.
δ T=α Δ T L
Linear coefficient of thermal expansion α, [α]=1 K,1°C,.... This deformation is associated with an average thermal strain:
Strain from temperature changes.
ε T=δ T L=α T
Temperature changes with statically indeterminate beam
Initially, rod of length L is placed between two supports at a distance L from each other. With no internal forces, there is no stress or strain.
Force balance.
R A=R B=0
Reaction force.
R A=F
After raising the temperature, total elongation of the rod is still zero. The total elongation is given by:
Total elongation.
δ=F L E A+α L Δ T=0
The stress in the rod due to change in temperature is given by:
Stress from temperature changes.
σ=−α E Δ T
Misfit Problems
A misfit problem is one in which there is difference between a design distance and the manufactured length of a material. Some misfits are created intentionally to pre-strain a member. (e.g. spokes in a bicycle wheel or strings in a tennis racket). This type of problem neither modifies the equilibrium equations (1) nor the force-extension relations, (2) but the compatibility equations, (3) need to be modified.
Heads up!
Stress concentration factors build on this content in engineering materials and machine failure.
Stress concentraions
The stress concentration factor is the highest at lowest cross-sectional area.
Stress concentration factor.
K=σ m a x σ a v g
Found experimentally
Solely based on geometry
Hello
Learn More
About Us
Academy for Excellence in Engineering Education (AE3)
Using this page |
13083 | https://courses.lumenlearning.com/wm-prealgebra/chapter/finding-the-least-common-multiple-of-two-numbers/ | Prealgebra
Module 2:The Language of Algebra
Finding the Least Common Multiple of Two Numbers
Learning Outcomes
One of the reasons we find multiples and primes is to use them to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators.
Listing Multiples Method
A common multiple of two numbers is a number that is a multiple of both numbers. Suppose we want to find common multiples of [latex]10[/latex] and [latex]25[/latex]. We can list the first several multiples of each number. Then we look for multiples that are common to both lists—these are the common multiples.
[latex]\begin{array}{c}10\text{ : }10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110\ldots \hfill \ 25\text{ : }25, 50,75, 100, 125\ldots \hfill \end{array}[/latex]
We see that [latex]50[/latex] and [latex]100[/latex] appear in both lists. They are common multiples of [latex]10[/latex] and [latex]25[/latex]. We would find more common multiples if we continued the list of multiples for each.
The smallest number that is a multiple of two numbers is called the least common multiple (LCM). So the least LCM of [latex]10[/latex] and [latex]25[/latex] is [latex]50[/latex].
Find the least common multiple (LCM) of two numbers by listing multiples
example
Find the LCM of [latex]15[/latex] and [latex]20[/latex] by listing multiples.
Solution:
List the first several multiples of [latex]15[/latex] and of [latex]20[/latex]. Identify the first common multiple.
[latex]\begin{array}{l}\text{15: }15,30,45,60,75,90,105,120\hfill \ \text{20: }20,40,60,80,100,120,140,160\hfill \end{array}[/latex]
The smallest number to appear on both lists is [latex]60[/latex], so [latex]60[/latex] is the least common multiple of [latex]15[/latex] and [latex]20[/latex].
Notice that [latex]120[/latex] is on both lists, too. It is a common multiple, but it is not the least common multiple.
try it
In the next video we show an example of how to find the Least Common Multiple by listing multiples of each number.
Prime Factors Method
Another way to find the least common multiple of two numbers is to use their prime factors. We’ll use this method to find the LCM of [latex]12[/latex] and [latex]18[/latex].
We start by finding the prime factorization of each number.
[latex]12=2\cdot 2\cdot 318=2\cdot 3\cdot 3[/latex]
Then we write each number as a product of primes, matching primes vertically when possible.
[latex]\begin{array}{l}12=2\cdot 2\cdot 3\hfill \ 18=2\cdot 3\cdot 3\end{array}[/latex]
Now we bring down the primes in each column. The LCM is the product of these factors.
Notice that the prime factors of [latex]12[/latex] and the prime factors of [latex]18[/latex] are included in the LCM. By matching up the common primes, each common prime factor is used only once. This ensures that [latex]36[/latex] is the least common multiple.
Find the LCM using the prime factors method
example
Find the LCM of [latex]15[/latex] and [latex]18[/latex] using the prime factors method.
Solution:
Write each number as a product of primes. |
Write each number as a product of primes, matching primes vertically when possible. |
Bring down the primes in each column. |
Multiply the factors to get the LCM. | [latex]\text{LCM}=2\cdot 3\cdot 3\cdot 5[/latex]The LCM of [latex]15\text{ and }18\text{ is } 90[/latex].
The LCM of [latex]15\text{ and }18\text{ is } 90[/latex].
try it
example
Find the LCM of [latex]50[/latex] and [latex]100[/latex] using the prime factors method.
Solution:
Write the prime factorization of each number. | [latex]50=2\cdot{5}\cdot{5}\quad\quad\quad{100=2\cdot{2}\cdot{5}\cdot{5}}[/latex]
Write each number as a product of primes, matching primes vertically when possible. | [latex]50=\quad{2\cdot{5}\cdot{5}}[/latex][latex]100=2\cdot{2}\cdot{5}\cdot{5}[/latex]
Bring down the primes in each column. |
Multiply the factors to get the LCM. | [latex]\text{LCM}=2\cdot 2\cdot 5\cdot 5[/latex]The LCM of [latex]50\text{ and } 100\text{ is } 100[/latex].
[latex]100=2\cdot{2}\cdot{5}\cdot{5}[/latex]
The LCM of [latex]50\text{ and } 100\text{ is } 100[/latex].
try it
In the next video we show how to find the Least Common Multiple by using prime factorization.
Candela Citations |
13084 | https://www.allenoverseas.com/blog/tips-to-avoid-negative-marking-in-mcq-based-competitive-exams/ | Tips to avoid negative marking in MCQ-based Competitive exams - ALLEN Overseas
Overeseas Address
Inquiry Now !
[x] I agree with the Terms and Conditions of ALLEN Career Institute Overseas.
X
Where do you want to Login?
Session 2024-25Session 2025-26
New Start early with Qatar Offline Classes with double the advantage of Hybrid learning at Online fees.
New NTA released the NEET 2024 Exam date.
New NTA released the JEE Main 2024 Exam dates.
New ALLEN TALLENTEX Overseas Exam 2024 was conducted successfully on 27th October.
‹›
Apply Now
Exam Past Year Papers
DASA
Olympiads
ICSE
NEET UG
JEE Advanced
JEE Main
CBSE
NTSE
Past Year Papers
JEE Main Paper and Solutions 2025
JEE Advanced Past Papers 2025
JEE Main Paper and Solutions 2024
DASA
Direct Admission of Student Abroad (DASA)
Application Process
DASA 2025 Seat Allotment Procedure
DASA UG Counselling Schedule
DASA UG Seat Matrix 2025
Eligibility Criteria for DASA
Fee & Refund Policy
Reservation Criteria & CIWG
Olympiads
International Biology Olympiad (IBO)
International Chemistry Olympiad (IChO)
International Physics Olympiad IPhO
International Mathematical Olympiad (IMO)
ICSE
Class 6
Class 7
Class 8
Class 9
Class 10
Class 11
Class 12
NEET UG
NEET-UG 2025 Exam Syllabus
NEET 2025 Preparation Tips
NEET 2025 Exam Pattern
NEET 2025 Eligibility Criteria
NEET-UG Application Process
NEET-UG Past Papers
JEE Advanced
JEE Advanced 2025 Application
JEE Advanced 2025 Eligibility
JEE Advanced 2025 Exam Centres
JEE Advanced 2025 Preparation Tips
JEE Advanced 2025 Syllabus
JEE Advanced Admit Card 2025
JEE Advanced Exam Pattern
JEE Advanced Past Papers
JEE Main
JEE Main 2025 Syllabus
JEE Main Eligibility Criteria
JEE Main 2025 Exam Pattern
JEE Main Exam Centres Outside India
JEE Main Application
JEE Main 2025 Preparation Tips
JEE Main 2025 Admit Card
CBSE
Class 6
Class 7
Class 8
Class 9
Class 10
Class 11
Class 12
NTSE
NTSE Admit Card
NTSE Selection Procedure 2024
NTSE Application Procedure 2024
NTSE Exam Date
NTSE Stage 2
NTSE Stage 1
NTSE Exam Pattern
NTSE Eligibility 2024
NTSE Syllabus
TALLENTEX
Discover ALLEN
About ALLEN
Directors’ Message
Leadership’s Vision
Academic Corner
Academic Desk
ALLEN System
ALLEN World Records
Lead With ALLEN
Results
Gallery
Newsroom
Careers
Contact Us
Blog
Exam
DASA
Direct Admission of Student Abroad (DASA)
Application Process
DASA 2025 Seat Allotment Procedure
DASA UG Counselling Schedule
DASA UG Seat Matrix 2025
Eligibility Criteria for DASA
Fee & Refund Policy
Reservation Criteria & CIWG
Olympiads
International Biology Olympiad (IBO)
International Chemistry Olympiad (IChO)
International Physics Olympiad IPhO
International Mathematical Olympiad (IMO)
ICSE
Class 10
Class 11
Class 12
Class 6
Class 9
Class 8
Class 7
NEET UG
NEET 2025 Eligibility Criteria
NEET 2025 Exam Pattern
NEET 2025 Preparation Tips
NEET-UG 2025 Exam Syllabus
NEET-UG Application Process
JEE Advanced
JEE Advanced 2025 Application
JEE Advanced 2025 Eligibility
JEE Advanced 2025 Exam Centres
JEE Advanced 2025 Preparation Tips
JEE Advanced 2025 Syllabus
JEE Advanced Admit Card 2025
JEE Advanced Exam Pattern
JEE Advanced Past Papers
JEE Main
JEE Main 2025 Exam Pattern
JEE Main 2025 Admit Card
JEE Main 2025 Preparation Tips
JEE Main Application
JEE Main Eligibility Criteria
JEE Main Exam Centres Outside India
JEE Main 2025 Syllabus
JEE Main Paper and Solutions 2024
NTSE
NTSE Admit Card
NTSE Selection Procedure 2024
NTSE Application Procedure 2024
NTSE Exam Date
NTSE Stage 2
NTSE Stage 1
NTSE Exam Pattern
NTSE Eligibility 2024
NTSE Syllabus
SELECT COUNTRY
Home
Bahrain
Kuwait
Oman
Qatar
Saudi Arabia
UAE
Apply Now
Home
Blog
Tips to avoid negative marking in MCQ-based Competitive exams
Allen Overseas
Tips to avoid negative marking in MCQ-based Competitive exams
ACIO
November 24, 2023
All the majorcompetitive examinationsin India, includingJEE and NEET, follow the MCQ-based examination pattern. MCQs (for Multiple Choice Questions) are a popular assessment format used in competitive exams to test a candidate’s knowledge, reasoning ability, accuracy and speed in answering questions.
However, one of the major challenges that students face while attempting multiple-choice questions in competitive exams is the risk of negative marking.Negative markingis a scoring scheme in which marks are deducted from the total score for incorrect answers. Here we’ll discuss some practicaltips to avoid negative markingin MCQ-based competitive exams likeJEEandNEET.
Negative Marking in Competitive Exam:
Negative markingis a scoring scheme implemented in all the major competitive exams in which marks are deducted for incorrect answers. That means, for every wrong answer, the prescribed marks will be deducted from the total marks that candidates scored for all the correct answers.
Since negative marking can substantially decrease your score, candidates hesitate to answer even when they have an excellent idea about the correct ones.
Negative Marking in NEET-UG:
The NEET-UG marking scheme is as follows:
(+4) Four marks will be given for every correct answer
(-1) One mark will be deducted for every incorrect answer
(0) No marks will be given for unanswered/un-attempted questions
Negative Marking in JEE Main:
The JEE Main marking scheme for both MCQs and numerical-value questions is as follows:
(+4) Four marks will be given for every correct answer
(-1) One mark will be deducted for every incorrect answer
(0) No marks will be given for unanswered/un-attempted questions
What is the Reason Behind Negative Marking?
Apart from JEE and NEET, other competitive exams also follow the negative marking scheme. The primary goal ofnegative markingis to eliminate applicants. In competitive exams, lakhs of candidates appear and compete for limited seats. Negative marking has been used by authorities to filter out candidates who lack adequate knowledge, skills and confidence for the exam.
Thousands of students lose their selection and dream spot yearly because of negative marking. Hence, it is the primary concern for JEE and NEET aspirants. Here we’ll present some practicaltips to avoid negative markingand maximise your chances of success in JEE or NEET 2023.
Tips to Avoid Negative Marking in JEE/ NEET Examinations:
1. Read the Questions Carefully:
During competitive exams, candidates often feel overwhelmed by the pressure and anxiety and end up making silly mistakes. One of the mostcommon mistakescandidates make is rushing through the questions without correctly understanding them, and in that rush, they tend to mark the wrong answers, which leads to negative marking. While it’s essential to complete the exam within the given time, it’s equally crucial to take the time to carefully read and comprehend the questions before attempting to answer them. Candidates are advised to read the question carefully and underline the keywords like not, incorrect, wrong etc.
2. Avoid Guesswork:
Another common mistake students make while attempting competitive exams isguessing the answers. Guessing can be compelling, but it can also lead to negative markings. If you are not sure about the answer, it is better to leave the question and move on to the next one. Guessing should be your last resort only.
3. Skip Doubtful Questions:
While attempting highly competitive exams like JEE or NEET, you come across questions you are not confident about. In these cases,avoid guessing answers randomlybecause if your guess is incorrect, you will receive negative marking, which is not a risk you should take. However, leaving too many questions unattempted is also disadvantageous. The ideal practice in such circumstances is to take calculated risks and answer questions about which you are at least 80%–90% confident.
4. Don’t Answer at the Last Minute:
An effective exam approach necessitates students to spare the last 10-15 minutes for revision. Hence try not to solve questions in the last moments as you’ll tend to make more mistakes in a rush. Instead, properly utilise every last second for revision and re-checking your answers.
5.Eliminate Wrong Options:
If you are unsure about the answer, tryeliminating the wrong options. Sometimes, the options give you a clue to the correct answer. You can eliminate the obviously wrong options and then try to choose the correct answer from the remaining options. It can increase your chances of getting the answer right and help you avoid negative marking.
6. Keep Calm and Stay Focused:
During the exam, it is essential to stay calm and focused. Panic and anxiety can lead to mistakes, which can result in negative marking. If you feel stressed, take a deep breath and try to relax. Staying focused can help you to answer the questions accurately and avoid negative marking.
7. Practice Mock Tests:
Practice is the key to avoiding negative marking. First, take as many mock tests as possible and carefully analyse your mistakes while attempting them. Then, make conscious efforts to rectify your mistakes. Also, it would be best to work through an issue until you reach the solution rather than bypass it because they believe it to be simple. It can help you to avoid negative marking and increase your chances of success.
8. Manage Your Time:
Time managementis crucial while attempting MCQ-based exams like JEE and NEET. It is essential to allocate time to each section and question. You should answer the easy questions first and then move on to the difficult ones. If you are stuck on a question, it is better to move on to the next one and return to it later. Time management can help you avoid negative marking by ensuring that you have enough time to answer all the questions.
Competitive exams can be challenging, but you can avoid negative marking with the right preparation. Understanding the exam pattern, managing your time, avoiding guesswork, eliminating wrong options, staying calm and focused, and practising mock tests can help you to achieve success in JEE or NEET.
Related Blogs:
Effective Time Management: How To Manage Time During Exams?
Study Smart vs Study Hard: Which is more Productive for Success?
How to Avoid Distractions during Study Hours?
Test Anxiety in Students: Symptoms, Causes, and Treatment
How to Plan your Study Hours Smartly?
Popular Tags
avoid guessing answers randomly
Avoid Negative Marking in JEE
Avoid Negative Marking in NEET
common mistakes in exam
Tips to avoid negative marking
Leave a comment Cancel reply
Your email address will not be published.Required fields are marked
Comment
Name
Email
Website
[x] Save my name, email in this browser for the next time I comment.
ISD CODE
Mobile Number
ISD CODE is a required field
Mobile Number is a required field
Search Here
Popular Tags
avoid guessing answers randomly
Avoid Negative Marking in JEE
Avoid Negative Marking in NEET
common mistakes in exam
Tips to avoid negative marking
Popular Posts
###### CBSE Class 10th 2024: Complete Syllabus, Chapter-wise Weightage, Exam Pattern, Marking Scheme
###### CBSE Class 10 2025 Mathematics: Paper Design | Weightage | Marks | Important Topics | Preparation Tips
###### Math in Daily Life: Uses, Importance and Facts
###### CBSE Class 10th 2025: Syllabus | Exam Pattern | Marking Scheme
Subscribe and Newsletter
Subscribe
Our Support
Support Corner
Call Us :
(+971) 567461832
Mail Us :
enquiry@allenoverseas.com
Request a Call
Click Here
About Us
About ALLEN
Directors’ Message
Leadership’s Vision
Academic Desk
ALLEN System
Lead With ALLEN
Important Links
Results
Gallery
Newsroom
Careers
Contact Us
Countries
Bahrain
Kuwait
Oman
Qatar
Saudi Arabia
UAE
Contact Us
Corporate Office
Unit 408-409, The Business Centre, Khalid Bin Al Waleed Rd - Bur Dubai, (Near Exit Gate-4 of Burjuman Metro Station), Dubai
Academics Office
Unit 601, The Business Centre, Khalid Bin Al Waleed Rd - Bur Dubai, (Near Exit Gate-4 of Burjuman Metro Station), Dubai
enquiry@allenoverseas.com
(+971) 56 746 1832, (+971) 45461696
Follow Us
Payment Options
Contact Us
Corporate Office
Unit 408-409, The Business Centre, Khalid Bin Al Waleed Rd - Bur Dubai, (Near Exit Gate-4 of Burjuman Metro Station), Dubai
Academics Office
Unit 601, The Business Centre, Khalid Bin Al Waleed Rd - Bur Dubai, (Near Exit Gate-4 of Burjuman Metro Station), Dubai
enquiry@allenoverseas.com
(+971) 56 746 1832, (+971) 45461696">(+971) 56 746 1832, (+971) 45461696
About Us
About ALLEN
Directors’ Message
Leadership’s Vision
Academic Desk
ALLEN System
Lead With ALLEN
Important Links
Results
Gallery
Newsroom
Careers
Contact Us
Countries
Bahrain
Kuwait
Oman
Qatar
Saudi Arabia
UAE
Follow Us
Payment Options
Copyright © 2025 Allen Overseas. All Rights Reserved.
Privacy Policy| Terms of Use| Payment Terms| Refund & Cancellation |
13085 | https://www.youtube.com/watch?v=OcCRflNM-Xk | Simple Explanation of the Birthday Paradox
Wrath of Math
264000 subscribers
21210 likes
Description
1319458 views
Posted: 20 Oct 2024
How many people need to be in a room for it to be more likely than not that at least two of them have the same birthday? The surprisingly small answer to this question is what's often dubbed "The Birthday Paradox". With only 23 people being necessary, we'll discuss why this is, look at the math, and a simple reframe of the numbers. #maths #mathematics #paradox
I just launched a math fashion brand at
Check it out and pick up some of the coolest math merch of all time!
Join Wrath of Math to get exclusive videos, lecture notes, and more:
Outro music by Me, with beat by
Follow Wrath of Math on...
● Instagram:
● TikTok:
● X:
Transcript:
here I have a room and it's a circle cuz circular rooms are just so interesting and inside this room we've got two randomly selected people How likely is it that these two people share a birthday your response would probably be not very likely if we just have two random people not very likely that they share a birthday that's certainly true in fact we could calculate exactly How likely it is making a few assumptions we'll just assume that there are 365 possible birthdays we're not going to consider leap years and we'll assume that all days are equally likely to be a randomly selected person's birthday let's say that this person's birthday is Day Day month month year year it's just someday what's the likelihood that this guy has the same exact birthday well of the 365 possible birthdays he could have he'd have to have exactly this one so the probability that he has the same birthday would be 1 over 365 or after Consulting a calculator this is about 0.3% it is very unlikely now of course if there were three random people in this room it would be a little bit more likely that at least two of them share birthday cuz there's just more people that could have a shared birthday if we have four people in the room it it's a little bit more likely that at least two people share a birthday still it's probably unlikely to happen but it's certainly more likely than the first case with only two people and that leads to a very famous Paradox called the birthday Paradox the birthday Paradox is the surprising answer to this question how many people do we need to consider in the problem that we were just discussing so that it is more likely than not that at least two people share a birthday so again the question is how many people need to be in our room or said in more General language how many people do we need to consider so that it's more likely than not at least 50% or we should say greater than 50% chance that at least two of them share a birthday now I'm going to put the answer to this question you can take a minute to think about what you would suspect the answer is it turns out the number of people you need so that it switches from less likely to more likely that at least two people in the room share a birthday the answer is just 23 people this is actually what's called a veritical paradox it's a type of paradox that seems like it can't be true but is in fact demonstrably true so there is a perfectly valid mathematical solution an explanation that justifies the correctness of 23 to this question the MTH necessary to show that 23 people actually makes it more likely than not that at least two of them share a birthday is pretty straightforward but first let's just do a quick reframing of the problem so that instead of the mathematical Precision that we will look at you can just slightly change your gut feeling because most people have the gut feeling that this seems very small because 365 days in the year that's quite a lot of possible birthdays it doesn't seem like having a mere 23 people would make it more likely than not that at least two of them have the exact same birthday it just seems surprising but it's a little less surprising with a simple reframe of the problem and I guess really this isn't a reframe of the problem it's just a reframe of your perspective when we think about the people in this room we could represent each of them with a DOT and why don't I just go ahead and draw all 23 dots just to really drive the point home here 1 2 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 so here are 23 dots I've just scattered them about representing 23 people in a room you have to realize that when we're asking about at least two of these people having the same birthday we're not really asking about individual people of which there's only 23 more so we're considering pairs of people for example this pair of two people do they share a birthday maybe yes maybe no or this pair of people do they share a birthday maybe yes maybe no you can start to see while there's only 23 people there are a ton of different pairs of people that need to be considered here and every pair of people could potentially have the same birthday and so once you start to see the sheer number of comparisons that we have to make even with only 23 people it starts to become a little more intuitive that it would be more likely than not that for at least one pair of people here the answer will be yes they do in fact have the same birthday in fact while the number of people here is only 23 if I were to go forward and draw every possible connection the total number of pairs is something that we uh mathematicians call 23 choose 2 it turns out this is equal to 53 so there's only 23 people but there are in fact 253 pairs of people which starts to make this result just seem a little bit less surprising the fact of the matter is that us humans just don't have a great gut instinct for these probability things our guts will often lead us astray so let's actually see the computation that verifies 23 as a correct answer to this problem now if we just have a bunch of people in a room I've just sketched out some of them here without arms let's consider the probability of them all having a different birthday that's actually a little easier than considering all the possible ways that at least two of them could share a birthday just easier to think about none of them sharing a birthday which is the opposite so the first person has whatever birthday they have whatever we don't really care but the second person what's the probability that the second person has a different birthday from the first person well of all 365 possible days the second person could have 364 of them and we'd still be fine he just can't have that same birthday that the first person has so there's a 364 out of 365 chance that the second person does not share a birthday with the first now what's the probability that the third person doesn't share a birthday with the first two people well each of the first two people have claimed a day and so there's only 363 days remaining so there's a 363 out of 365 chance then that this third person doesn't share birthday with either of the first two people and of course this pattern continues for the fourth person there's a 362 out of 365 chance he doesn't share a birthday with the first three people and for this last person that I've drawn the fifth person there is a 361 out of 365 chance that he doesn't share a birthday with the first four people and again this pattern would continue up through all 23 people that we have and for that 23rd person there would be a 343 out of 365 probability that he does not share birthday with any of the first 22 people because he can't have any of the 22 birthdays they do but the other 343 days are all perfectly fine now remember the original question is about at least two of these people having the same birthday but what we're calculating here is that none of them have the same birthday so we're saying that this should actually be less likely than 50% and perhaps once you actually see this computation that's not as surprising as it could be initially because all of these numbers are less than one so as we multiply them together the product is getting smaller and smaller and in total we have 23 factors all smaller than one they're all making our product smaller in the end and so it's maybe not so shocking that in the end after all of this multiplication that the answer should be less than .5 in the end once you run the numbers this is actually about 0.49 so there's about a 49% chance that in a room full of 23 random people that none of them share a birthday now that means of course that all of the rest of the probability 1 minus 49 that's the chance 0.51 that at least two people in the room do share a birthday and of course that's more likely than not that's greater than the flip of a coin so that's the computation that shows in a room of 23 people it's actually less likely that none of them share a birthday than it is that at least two of them share a birthday but only just barely it's still not super likely that there's a pair of people sharing a birthday I mean it's more likely than not but it's only just barely over 50% another question you might ask is how many people do there need to be in the room so that the probability of at least two of them having a shared birthday is at least 99.9% likely so it's almost a sure thing that at least two people in the room share a birthday how many people do you need for that turns out if you run the numbers it's about 75 people so if you've got a room of 75 people very very likely almost certain that at least two of them share a birthday and that too feels quite surprising now what if we take it just one step further what if we want a 100% chance that at least two people in the room share a birthday how many people do you need for that well going from just 23 people to 75 people basically took us from 51% to nearly 100% to get that last little bit to 100% you're actually going to need it maybe isn't surprising 366 people that's of course because even if you have 365 people it's definitely possible that all of them could have a different birthday if we ignore leap years there's only 365 days in the year and so if we have 366 people it's now impossible that they all have different birthdays just not enough days to go around at least two of them would have to share a birthday that's actually a basic principle in mathematics called the pigeon hole principle if we've got more pigeons than we have holes then at least one of the holes has to have at least two pigeons it's actually a really really useful result but that's a very famous veritical Paradox I think it's a great one to be familiar with particularly if you like to talk to people about math because even to someone who doesn't really know or like math it can be explained pretty simply and is a pretty easy problem to be kind of curious about especially once you hit somebody with that answer of 23 people it's very surprising and most people are going to want an explanation let me know in the comments if you have any questions and be sure to subscribe for more of the swankiest math videos on the internet out up my if you can break it breaking in my past I'm making it up fast so slow down give me the time so I can fake itce T of words and just how I say let me speak my poetry to your fa not in the if you L not if you really |
13086 | http://www.algebra.com/algebra/homework/word/numbers/Numbers_Word_Problems.faq.question.924493.html | SOLUTION: [A number has a ones digit that is half the tens digit. The tens digit is a 8. The hundreds digit is twice the ones digit. The thousands digit is 1 lees then the tens digit. What i
SOLUTION: [A number has a ones digit that is half the tens digit. The tens digit is a 8. The hundreds digit is twice the ones digit. The thousands digit is 1 lees then the tens digit. What i
Algebra->Customizable Word Problem Solvers->Numbers -> SOLUTION: [A number has a ones digit that is half the tens digit. The tens digit is a 8. The hundreds digit is twice the ones digit. The thousands digit is 1 lees then the tens digit. What iLog On
Ad: Over 600 Algebra Word Problems at edhelper.com
Discover more
Word Problems: Numbers, consecutive odd/even, digits WordSolvers SolversLessons LessonsAnswers archive Answers Discover more Mathematics Math Algebra Inc math Click here to see ALL problems on Numbers Word Problems Question 924493: [A number has a ones digit that is half the tens digit. The tens digit is a 8. The hundreds digit is twice the ones digit. The thousands digit is 1 lees then the tens digit. What is the number?] Found 2 solutions by TimothyLamb, MathTherapy: Answer byTimothyLamb(4379) (Show Source): You can put this solution on YOUR website! x = ones y = tens z = hundreds u = thousands --- x = y/2 y = 8 z = 2x u = y - 1 --- put the system of linear equations into standard form --- 2x - y = 0 y = 8 2x - z = 0 y - u = 1 --- copy and paste the above standard form linear equations in to this solver: --- solution: x = 4 y = 8 z = 8 u = 7 the number = 7884 --- by substitution: --- y = 8 x = y/2 x = 4 z = 2x z = 8 u = y - 1 u = 7 --- solution: x = 4 y = 8 z = 8 u = 7 the number = 7884 --- Solve and graph linear equations: --- Solve quadratic equations, quadratic formula: --- Solve systems of linear equations up to 6-equations 6-variables: Answer byMathTherapy(10552) (Show Source): You can put this solution on YOUR website! [A number has a ones digit that is half the tens digit. The tens digit is a 8. The hundreds digit is twice the ones digit. The thousands digit is 1 lees then the tens digit. What is the number?] Tens digit: 8 Units digit: � of tens digit____�(8), or 4 Hundreds digit: 2(4), or 8 Thousands digit: 8 � 1, or 7 Number: 7,884
Discover more
Math
Algebra Inc
math
Mathematics |
13087 | https://math.stackexchange.com/questions/1734852/real-analysis-proving-inequality-without-using-jensens-inequality | calculus - Real analysis - proving inequality without using Jensen's inequality - Mathematics Stack Exchange
Join Mathematics
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Mathematics helpchat
Mathematics Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Real analysis - proving inequality without using Jensen's inequality
Ask Question
Asked 9 years, 5 months ago
Modified9 years, 5 months ago
Viewed 178 times
This question shows research effort; it is useful and clear
1
Save this question.
Show activity on this post.
I wonder how to prove rigorously the following conjecture without the use of Jensen's inequality:
By the use of the convexity/concavity of appropriate functions of 1 real variable, prove that:
∀x,y,z∈[0,π],s i n(x)+sin(y)+sin(z)3≤sin(x+y+z 3)∀x,y,z∈[0,π],s i n(x)+sin(y)+sin(z)3≤sin(x+y+z 3) ()
While inequality in () becomes an inequality if and only if x=y=z.
I know that proof of this conjecture is straightforward if one uses Jensen's inequality for the function f(x):=sin(x)f(x):=sin(x) for x∈[0,π]x∈[0,π] and knowing that function f(x) is at that interval concave.
Any help very appreciated!
calculus
real-analysis
analysis
Share
Share a link to this question
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this question to receive notifications
asked Apr 9, 2016 at 16:50
MITMIT
13 2 2 bronze badges
Add a comment|
1 Answer 1
Sorted by: Reset to default
This answer is useful
1
Save this answer.
Show activity on this post.
Hint: Write (x+y+z)/3(x+y+z)/3 as 1 3 x+2 3 y+z 2 1 3 x+2 3 y+z 2, and then apply the definition of concavity satisfied by the sine function on [0,π][0,π] (namely sin(λ u+(1−λ)v)≥λ sin(u)+(1−λ)sin(u)sin(λ u+(1−λ)v)≥λ sin(u)+(1−λ)sin(u) for 0<λ<1 0<λ<1 and u,v∈[0,π]u,v∈[0,π]) twice.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
answered Apr 9, 2016 at 17:06
John DawkinsJohn Dawkins
30k 1 1 gold badge 23 23 silver badges 40 40 bronze badges
Add a comment|
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
calculus
real-analysis
analysis
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Report this ad
Related
8Young's inequality without using convexity
12Easy but hard question regarding concave functions!
2Proving an inequality using calculus
2Real analysis - proving inequality using concavity of function
10Using Jensen's inequality to prove another inequality?
0Utilizing Jensen's to prove an inequality
3Prove an inequality without using Jensen's inequality
Hot Network Questions
The rule of necessitation seems utterly unreasonable
The geologic realities of a massive well out at Sea
RTC battery and VCC switching circuit
Is existence always locational?
Numbers Interpreted in Smallest Valid Base
Why include unadjusted estimates in a study when reporting adjusted estimates?
For every second-order formula, is there a first-order formula equivalent to it by reification?
Interpret G-code
how do I remove a item from the applications menu
Non-degeneracy of wedge product in cohomology
What is this chess h4 sac known as?
Clinical-tone story about Earth making people violent
Discussing strategy reduces winning chances of everyone!
Who is the target audience of Netanyahu's speech at the United Nations?
Calculating the node voltage
Identifying a movie where a man relives the same day
What’s the usual way to apply for a Saudi business visa from the UAE?
Do we need the author's permission for reference
Suggestions for plotting function of two variables and a parameter with a constraint in the form of an equation
Is it ok to place components "inside" the PCB
What NBA rule caused officials to reset the game clock to 0.3 seconds when a spectator caught the ball with 0.1 seconds left?
What is the feature between the Attendant Call and Ground Call push buttons on a B737 overhead panel?
Origin of Australian slang exclamation "struth" meaning greatly surprised
Does the curvature engine's wake really last forever?
more hot questions
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Mathematics
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
13088 | https://www.youtube.com/watch?v=y2xS0sabRjY | If t is the parameter for one end of a focal chord of the parabola y^2 =4ax, then its length is ...
Doubtnut
3940000 subscribers
7 likes
Description
342 views
Posted: 2 Feb 2022
If t is the parameter for one end of a focal chord of the parabola y^2 =4ax, then its length is :
Class: 11
Subject: MATHS
Chapter: CONIC SECTION
Board:CBSE
You can ask any doubt from class 6-12, JEE, NEET, Teaching, SSC, Defense and Banking exam on Doubtnut App or You can Whatsapp us at - 8400400400
Link -
Join our courses to improve your performance and Clear your concepts from basic for Class 6-12 School and Competitive exams (JEE/NEET) -
Contact Us:
👉 Have Any Query? Ask Us.
🤙 Call: 01247158250
💬 WhatsApp: 8400400400
📧 Email: info@doubtnut.com
🌐 Website:
Welcome to Doubtnut.
Doubtnut is World’s Biggest Platform for Video Solutions of Physics, Chemistry, Maths and Biology Doubts with over 5 million+ Video Solutions. Doubtnut is a Q&A App for Maths, Physics, Chemistry and Biology (up to JEE Advanced and NEET Level), Where You Can Ask Unlimited Questions by Clicking a Picture of Doubt on the Doubtnut App and Get Instant Video Solution.
Subscribe Our YouTube Channels:
✿ Doubtnut:
✿ Class 11-12, JEE & NEET (Hindi):
✿ Class 11-12, JEE & NEET (English)::
✿ Class 6-10 (Hindi):
✿ Class 6-10 (English):
✿ Doubtnut Govt. Exams:
Follow Us:
🔔 Facebook:
🔔 Instagram:
🔔 Telegram:
🔔 Twitter:
🔔 LinkedIn:
Our Telegram Pages:
🔔 Doubtnut Official:
🔔 Doubtnut IIT JEE:
🔔 Doubtnut NEET:
🔔 Doubtnut CBSE Boards:
🔔 Doubtnut UP Boards:
🔔 Doubtnut Bihar Boards:
🔔 Doubtnut Government Exams:
class 11
class 11 organic chemistry
class 11 physics
class 11 chemistry
class 11 maths
class 11 english
class 11 economics
class 11 biology
class 11 statistics
class 11 syllabus physics
class 11 commerce
class 11 syllabus chemistry
class 11 ncert economics
class 11 ncert
class 11 syllabus cbse
class 11 syllabus maths
cbse class 11
class 11 english grammar
class 11 syllabus
class 11 science
class 11 syllabus english
class 11 syllabus 2020-21
cbse class 11 maths
cbse class 11 physics
cbse class 11 books
cbse class 11 english
cbse class 11 chemistry
cbse class 11 biology
cbse class 11 accountancy
cbse class 11 commerce
cbse class 11 science
class 11 syllabus biology
cbse class 11 statistics
class 11 syllabus ncert
class 11 syllabus of english
class 11 syllabus science
class 11 syllabus commerce
Transcript:
without net get instant video solutions to all your maths physics chemistry and biology doubts just click the image of the question crop the question and get instant video solution download doubt and app today hello students let's discuss the question if t is a parameter for one end of the focal point of the parabola y square equal to 4ax then its length is so let's begin with solution of this question so if our parabola is y square equal to 4ax and if first end of the focal point this is given by a and its parameter is t so its end will be 80 square comma 280 basically this is the parametric equation of any point on the parabola y square equal to 4x and we know the concept that if other end if one end of a focal point is t1 let's represent this is basically the parameter t1 so coordinate will be a t1 square and 281 and other end and will be t2 so basically this t2 represent 82 square and 282 so we can say that t1 into t2 this will be equal to minus 1 if this is a focal point so here we will use this concept only that our t1 our t1 is given as t so from here we can say that if t1 is equal to t so we will just substitute t into t2 this will be equal to minus 1 so our t2 will come out to be minus 1 by t so from here the other end of the focal point that will let's name this other end will be b so our b coordinate of b this will be minus 1 by t which is the parameter for other end of focal chord and coordinates are given by a by t square comma minus 2 a by t what we are doing we are just substituting this t y minus 1 by t into this parametric equation so we will get other end of focal code now we have to find the length of the focal cord so here we will use distance formula so a b this is length of focal cord this will be equal to root of x 2 minus x 1 x 2 minus x 1 whole square plus here will be y 2 minus y 1 whole square so from here we'll just substitute x 1 and x 2 as this coordinate that is this a by t square and a t square and this y 1 and y 2 will be minus 2 a by t and this 280 so from here we get our a b and this will be equal to root of here this will be x 2 minus x 1 that is a t square or we can say a by t square minus a t square a by t square minus a t square whole square and here will be this that is y 2 minus y 1 whole square so this will be minus 2 a by t minus 2 8 4 square so minus 2 a by t and minus 2 a t and here will be whole square so we will further solve this so what we will do here this a square will be common from both the brackets so we will just take this a square outside the bracket so we get here a and inside the bracket inside the bracket we get here is 1 by t square minus t square whole square and here will be this will be so taking this minus common and squaring we get here is 2 by t and here will be this 2t whole square so from here we'll just take this two also common and on squaring this two will become four so from here this will be a b and a b will be equal to root of here would be root of and here will be a also and this will be opening this bracket so we get one by t power 4 here here this will be plus t power 4 and here will be this minus 2 and here this will be 2 by t square or we can say this 4 and here will be this 1 by t plus t whole square so what we will do this bracket is actually will provide us the common answer that is what we will do we will just replace this this term by this bracket only so we get here is this will be 1 by t square minus t square so we can open this equation by a plus b into a minus b so we get here is 1 by t plus t whole square and here this will be 1 by t minus t whole square and this will this another term will be plus and here will be 4 multiplied by 1 by t plus t whole square this is our another step so from here we can just simplify this so we get here is a and this 1 by t plus t whole square this will be taken common so from here we can take this outside the root so we get 1 by t plus t here so this will be 1 by t plus t and here in the under root this will be 1 by t minus t 1 by t minus t whole square and here will be this 4 only so from here when we will just open this square so we get 1 by t square plus t square minus 2 and that minus 2 and this plus 4 this will result in a plus 2 sign so our whole equation this will become a multiplied by this 1 by t plus t and here this will be root of so when we will just open this bracket we get here is this 1 by t square here will be plus t square and here will be minus 2 and this will be 4 so this will become t plus 1 by t whole square and on taking this t plus 1 by t 4 square outside the bracket we get our a b this a b will become this is a b basically and this a b will become a multiplied by t plus 1 by t and another t plus 1 by t will result will resulted by this term so we get here also a t plus 1 by t so our answer that is final answer to this question this will be a multiplied by t plus 1 by t whole square so answer to this question that is length of vocal cord whose one parameter is t and another parameter is given by this point is this length a t plus 1 by t whole square thank you for class 6-12 itj and neet level trusted by more than 5 crore students download doubt and after today |
13089 | https://pubs.acs.org/doi/10.1021/acs.jchemed.2c01252 | Opens in a new window Opens an external website Opens an external website in a new window
This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy
Recently Viewedclose modal
You have not visited any articles yet, Please visit some articles to see contents here.
Epistemological Lessons from Inconsistencies in Teachers’ Errors Related to Use of the Mole Ratio in Stoichiometry Calculations: A Cue for Professional Development
Skip to articleSkip to sidebar
Download Hi-Res ImageDownload to MS-PowerPointCite This:J. Chem. Educ. 2023, 100, 7, 2548-2557
Get e-Alerts
This publication is Open Access under the license indicated.
Learn More
Citation
Citation and abstract
Citation and references
More citation options
Abstract
Introduction
Epistemology and Pedagogy
Empirical Discrimination between Coherent and Fragmented Knowledge
Aims
Research Questions
Method
Results and Discussion
Suggestions for Practice
Limitations and Suggestions for Further Investigation
Author Information
Appendix A
References
Cited By
Chemical Education Research
Epistemological Lessons from Inconsistencies in Teachers’ Errors Related to Use of the Mole Ratio in Stoichiometry Calculations: A Cue for Professional Development Click to copy article linkArticle link copied!
Angela Elisabeth Stott
Angela Elisabeth Stott
Department of Mathematics, Natural Sciences and Technology Education, Faculty of Education, University of the Free State, Bloemfontein 9300, South Africa
E-mail: stottae@ufs.ac.za
More by Angela Elisabeth Stott
Open PDF
Journal of Chemical Education
Cite this: J. Chem. Educ. 2023, 100, 7, 2548–2557
Click to copy citationCitation copied!
Published June 15, 2023
Publication History
online
Published
in issue
Copyright © 2023 The Author. Published by American Chemical Society and Division of Chemical Education, Inc. This publication is licensed under
CC-BY 4.0.
Article Views
2730
Altmetric
-
Citations
2
Learn about these metrics
Article Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.
Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.
The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated.
Abstract
Click to copy section linkSection link copied!
Presenters’ pedagogical choices are strongly affected by whether they view knowledge as coherent or fragmented. This mixed-methods survey study seeks to contribute to epistemological discussions and associated pedagogical suggestions in the context of mole ratio related mistakes evident in stoichiometry calculations performed by in-service teachers in a developing world country. Physical sciences teachers in South Africa were asked to solve four stoichiometry calculation questions, and their written solutions (361 sets) were analyzed for consistency of various types of mole ratio related mistakes. The study found high levels of inconsistencies in each observed mistake, supporting the view that most participants’ knowledge lay closer to the fragmented end of the coherent–fragmented epistemological continuum. Evidence is presented for differential cognitive load, overgeneralization, and intuitive heuristics as possible explanations for these mistakes. Pedagogical suggestions are made for addressing such errors in in-service chemistry teacher workshops about stoichiometry calculations, although these may only be applicable where teacher subject matter knowledge tends to be low, such as within the developing world.
This publication is licensed under
CC-BY 4.0
.
License Summary
You are free to share(copy and redistribute) this
article in any medium or format and to adapt(remix,
transform, and build upon) the material for any purpose, even commercially
within the parameters below:
Creative Commons (CC): This is a Creative Commons license.
Attribution (BY): Credit must be given to the creator.
View full license
Disclaimer
This summary highlights only some of the key features and terms of the
actual license. It is not a license and has no legal value. Carefully
review the actual license before using these materials.
License Summary
You are free to share(copy and redistribute)
this article in any medium or format and to
adapt(remix, transform, and build upon) the material for any
purpose, even commercially within the parameters below:
Creative Commons (CC): This is a Creative Commons
license.
Attribution (BY): Credit must be given to the creator.View
full license
Disclaimer
This summary highlights only some of the key features and terms of
the
actual license. It is not a license and has no legal value.
Carefully
review the actual license before using these materials.
### License Summary
You are free to share(copy and redistribute)
this article in any medium or format and to
adapt(remix, transform, and build upon) the material for any
purpose, even commercially within the parameters below:
Creative Commons (CC): This is a Creative Commons
license.
Attribution (BY): Credit must be given to the creator.View
full license
Disclaimer
This summary highlights only some of the key features and terms of
the
actual license. It is not a license and has no legal value.
Carefully
review the actual license before using these materials.
ACS Publications
Copyright © 2023 The Author. Published by American Chemical Society and Division of Chemical Education, Inc.
Subjects
what are subjects
Article subjects are automatically applied from the ACS Subject Taxonomy and describe the scientific concepts and themes of the article.
Chemical calculations
Chemistry education
Hydrogen
Mathematical methods
Reagents
Keywords
what are keywords
Article keywords are supplied by the authors and highlight key terms and topics
of the paper.
High School
Chemical Education Research
Misconceptions
Stoichiometry
Introduction
Click to copy section linkSection link copied!
Using the coefficients of a balanced equation to apply the mole ratio in proportional calculations is central to stoichiometry but is poorly understood and fraught with errors. (1,2) Mole ratio related errors include failing to identify limiting reagents, (3) misusing coefficients, (4) failing to recognize the need to use proportion, (5) conflating ratio and amount, and applying the mole ratio as a mass ratio. (1) Such errors are prevalent among students throughout the world and even among teachers in developing world contexts, such as South Africa. (5) Error analysis can provide valuable insight into the thinking that causes the error, (6) pedagogy which may effectively address the error, (7,8) and, more fundamentally, inferences that can be made about the nature of knowledge and its acquisition, i.e., epistemology.
Epistemology and Pedagogy
Click to copy section linkSection link copied!
The central premise of constructivism is that learners come to formal education with prior knowledge which influences learning, and that learning is enhanced by teachers making pedagogical choices which are informed by this prior knowledge. (9) This premise, applied to teacher professional development workshops, implies that presenters need to understand the nature of the attending teachers’ knowledge (i.e., epistemology) to optimally use the limited time available in such workshops. While there are multiple perspectives regarding epistemology, the two extremes of a longstanding debate regarding the coherence of knowledge are still held by the Classic (10) and Knowledge in Pieces (KiP) (11) theories, as summarized in Table 1.
Table 1. Summary of the Epistemological and Pedagogical Debate between the Classic and the Knowledge in Pieces Theories
| | Classic theory | Knowledge in Pieces theory |
| Seminal references | Posner et al. (1982) | Smith et al. (1994) |
| Novices’ epistemology | Knowledge is coherent. Novices’ knowledge includes coherent misconceptions. | Novice conceptions are fragmented, variable, and may be faulty in specific contexts. |
| Prior learning causes misconceptions. | Misconceptions are overgeneralizations of useful prior knowledge, including implicit knowledge. |
| Misconceptions are resistant to instruction. | Misconceptions are not always resistant to instruction. |
| Pedagogical consequences of this view of epistemology | Misconceptions are barriers to learning. | Misconceptions are resources for learning, being refined and reused during learning. |
| Misconceptions must be confronted and replaced, or, where this is not possible, their status should be decreased relative to scientific conceptions. | Both the value and limitations of misconceptions should be acknowledged and discussed. |
| Instruction should confront misconceptions. | Instruction that confronts misconceptions is misguided and unlikely to succeed. |
This table is derived from a seminal article in which Smith et al. (11) opposed the popular view of epistemology that had emerged during the rise of misconception research in the previous decade, according to which novices come to school with coherent misconceptions, formed from prior experiences, which need to be confronted and displaced since they obstruct learning. (10) In contrast, Smith et al. (11) proposed that novices’ knowledge is fragmented and context-dependent, and, although faulty in some contexts, it is productive in others and so should be seen as a resource for, rather than a barrier to, learning. In their view, scientific conceptions are developed through forming scientifically appropriate contextual boundaries to the generalization of intuitive and learned fragments of knowledge, and through increasing interconnection and context-related enhancement or suppression of these fragments. (12)
Despite the long passage of time since the emergence of this debate, a consensus has not entirely been reached. (13) This is despite the alteration of the Classic theory to include conceptual ecology which allows for changes in status, rather than replacement, of naïve conceptions. (14) Also, proposals of theory merging (15,16) are gaining prominence in chemistry education philosophy, empirically supported by findings that sometimes naïve conceptions are more coherent and sometimes more fragmented. (17) Taber (16) asserted that it is no longer appropriate to frame this debate as a dichotomy since conceptions manifest on a continuum from coherent to fragmented. Consequently, the appropriate question to ask is not whether naïve knowledge is coherent or fragmented, but, for various types of people and contexts of learning: (1) where, along the coherent–fragmented epistemological continuum, each type of knowledge lies and, consequently, (2) which of the general pedagogical orientations suggested by epistemological theories, such as those summarized in Table 1, is more likely to be optimal.
Within chemistry education research, few topics and contexts have been examined empirically in this manner. (9) Those that have include developed-world school-learners’ learning of the particulate structure of matter (18) and atomic structure. (19) However, no empirical studies could be found regarding the epistemology of knowledge about stoichiometry calculations or of the knowledge of in-service chemistry teachers in developing world contexts. Such teachers abound, globally, and since they are known to have limited subject matter knowledge, (20) it is of value to improve understandings of how to help them develop this, (21) particularly for a topic, such as stoichiometry calculations, which is foundational to much of chemistry and tends to be emphasized in chemistry teaching and assessment. (22)
The paucity of inclusion of the fragmented end of the coherent–fragmented epistemological continuum, in empirical studies on chemistry learning, (9) and of its mention in texts aimed at practitioners, is an additional motivation to publish the results of this study. Personally, this is because these results have impacted my epistemology and pedagogy by making me more aware of the fragmentation of knowledge. In empirical studies and practitioner texts, the Classic theory has continued to dominate, sometimes with enhanced nuance through including conceptual ecology and replacing the term misconceptions with alternative or naïve conceptions, but generally still with its pedagogical aims to “avoid”, (23) “replace”, (24) “eliminate”, (25) “correct”/“address”, (26) “overcome”, (27) and “remediate”, (28) students’ misconceptions.
Given the enormous influence that epistemology has on pedagogy, (16,29) it is important to (a) advance the field of empirical research into the epistemological nature of under-researched contexts, and (b) promote the pedagogical implications of such research in ways accessible to practitioners. These are the gaps in the literature that this study contributes to addressing in the context of the knowledge of in-service teachers from developing world contexts regarding stoichiometry calculations.
Empirical Discrimination between Coherent and Fragmented Knowledge
Click to copy section linkSection link copied!
To interpret empirical findings in terms of the coherent–fragmented epistemological continuum, a method is needed to discriminate between the two theories at its poles. Such a method is illustrated in Figure 1.
Central to this method is distinguishing between incorrect answers which are reasonable to the person who made them, and therefore pedagogically significant, and accidental slips, which clearly need less focus and comprehension by the presenter. In this article, the term mistakes refers to all types of incorrect answers, with errors being reasonable, and slips being accidental, subcategories. This is consistent with error-related research in Mathematics education, (7) although it has reversed meanings for mistakes and errors relative to some other authors (see, for example, Wong and Lim (8)). Whether knowledge is coherent or fragmented, slips are to be expected to manifest inconsistently. However, for reasonable mistakes (errors), coherent knowledge should manifest consistently, whereas fragmented knowledge might manifest inconsistently. Therefore, if inconsistencies can be found and they can be explained to be reasonable, we can deduce that the nature of the knowledge under observation lies further on the fragmented end of the coherent–fragmented epistemological continuum, and therefore may be better described, and associated pedagogy better informed, by the KiP, than the Classic, theory.
This raises the question of what may be considered reasonable to the person making the mistake, a question made more complicated by the fact that this is largely determined implicitly. (9) Central to the KiP theory is the notion of phenomenological primitives (pPrims), which are universally held, instinctive implicit assumptions about how the world behaves. (12) Taber and García-Franco (18) found diSessa’s (30) list of pPrims, which had emerged from empirical work in physics education, useful but incomplete, in the context of chemistry education. Talanquer’s (31,32) list of implicit heuristics for “Common Sense Chemistry” and continuing work on implicit assumptions and short-cut reasoning (33−36) expand such literature on implicit knowledge into the realm of chemistry education.
Aims
Click to copy section linkSection link copied!
Given the reasoning, above, this study uses the term mole ratio related mistake to refer to any incorrect written work related to the use of the mole ratio given by the coefficients of a chemical equation. Consistent with the identified gap in the literature, the aims of this study are (1) to determine the extent to which these mistakes are applied consistently, and, where they are applied inconsistently, whether these mistakes are explainable, and therefore likely to be reasonable and significant, i.e., classifiable as errors; (2) to make epistemological deductions regarding the nature, and development, of mole ratio related knowledge; (3) to derive pedagogical suggestions related to the development of mole ratio related knowledge. The common difficulties experienced in developing world contexts (20) improve the likelihood that findings will apply to other developing world contexts too.
Research Questions
Click to copy section linkSection link copied!
To achieve these aims, the following research questions are posed, in reference to 361 sets of individually written answers to four stoichiometry calculation questions, given by South African physical sciences teachers, but assumed to be of relevance more broadly regarding the stoichiometry calculation knowledge of in-service teachers, particularly in the developing world: (1) To what extent did these answers exhibit consistent mole ratio related mistakes? (2) How might inconsistently displayed mole ratio related mistakes be explained?
Method
Click to copy section linkSection link copied!
This pragmatically conducted survey study is guided by the Framework for Integrated Methodologies (FraIM). (37) Consistent with this, both quantitative and qualitative data are drawn on, and findings are presented as transparently as space allows to enable readers to judge the warrantability of the claims made.
Sample and Data Collection
The data consists of 361 sets of written answers by South African physical sciences teachers, to four calculation questions of two versions (see Table 2).
Table 2. Questions Used
| | | | See the Appendix: | |
| Question number and topic | Version | Question | Difficulty | Cognitive complexity |
| 1. Yield with limiting reagent | A | A reaction mixture contains 17 g N2 and 0.5 mol H2. What is the maximum number of moles of NH3 that can be produced? (3H2 + N2 → 2NH3) | Medium | 3 |
| B | A reaction mixture contains 3.5 mol N2 and 18 g H2. What is the maximum number of moles of NH3 that can be produced? (3H2 + N2 → 2NH3) |
| 2. Number of atoms | A | How many atoms of hydrogen are needed to fully react with 17 dm3 of nitrogen gas at STP? | Difficult | 7 |
| B | How many atoms of hydrogen are needed to fully react with 13 dm3 of NH3 gas at STP? |
| 3. Percent yield | A | 7 g H2 react completely with N2 to form 28 g NH3. What is the percentage yield? | Difficult | 5 |
| B | 7 g N2 react completely with H2 to form 8 g NH3. What is the percentage yield? |
| 4. Yield without limiting reagent | A | Calculate the maximum mass of SO3 that could be produced from 1.9 mol of oxygen and excess sulfur. | Medium | 3 |
| B | Calculate the maximum mass of SO3 that could be produced from 6 mol of oxygen and excess sulfur. |
Version A was answered by 171 teachers and B by 190. In fact, considerable overlap exists between these groups of teachers, since tests were administered as pre- (A) and post- (B) tests at an in-service workshop, although this is irrelevant to this article since its focus is the mole ratio consistency within each of the 361 sets of answers, rather than any influence of the intervention. The teachers answered these four questions individually in writing without access to help from external sources.
Novices find stoichiometry difficult because of the cognitive load it offers. (1) Horvat et al. (38) provided guidelines for quantifying this cognitive load by considering a question’s difficulty and interactivity. A question’s difficulty is affected by whether (a) a unit conversion is needed between given and required values, and (b) conversion through molar quantities needs to be performed for both the given and required units. Knaus et al. (39) designed a rubric to quantify the complexity of a task based on its difficulty and the number of concepts it includes. Horvat et al., (38) modified this system by incorporating an interactivity rating. This is related to the number of concepts included, thus inflating the resulting cognitive complexity rating for tasks which include more concepts, since interactivity between concepts is known to result in additional intrinsic cognitive load. (40) The two versions of the questions (A and B in Table 2) differed slightly from one another but retained the same essential elements. Table 6 in Appendix A shows how the questions were analyzed according to Horvat et al.’s (38) framework, used in conjunction with Knaus et al.’s (39) cognitive complexity rating rubric. The outcomes of this analysis are included in the last column of Table 2.
As shown in Table 3, a range of teachers are represented in the sample, improving the generalizability of the findings to the population of South African physical sciences teachers. Furthermore, transparent reporting of these characteristics improves the reader’s ability to judge the likelihood that the findings may be applicable to their context, given the role that context may play on epistemology. (16)
Table 3. Some Characteristics of the Teachers Comprising the Data Set (n = 361)
| Socioeconomic status of the learners taught | Possess a BSc degree | Stoichiometry teaching experience category | n |
| High (teach at a quintile 5 school) n = 40 (11%) | No (n = 19) | Inexperienceda | 12 |
| Moderately experiencedb | 7 |
| Experiencedc | 0 |
| Yes (n = 21) | Inexperienced | 8 |
| Moderately experienced | 3 |
| Experienced | 10 |
| Low (teach at quintile 1–4 schools) n = 201 (89%) | No (n = 201) | Inexperienced | 104 |
| Moderately experienced | 44 |
| Experienced | 53 |
| Yes (n = 118) | Inexperienced | 58 |
| Moderately experienced | 36 |
| Experienced | 24 |
| Incomplete biographical survey | | | 2 |
a
3 years or less.
b
3–10 years.
c
10 years or more.
Before data collection commenced, the relevant institution’s ethics committee for educational research evaluated the research proposal for compliance with relevant laws and institutional guidelines for ethical research and awarded ethical clearance for the study (UFS-HSD2017/1520). All teachers included in the sample gave written informed consent to anonymous inclusion of their data in this study.
Data Analysis
Each written answer was scrutinized to explain the thinking it evidenced. Simultaneously, each answer was coded for the presence of mole ratio mistakes. This was done deductively, i.e., from a list generated from literature. Descriptive statistics were used to determine (a) the number of data sets which displayed each type of mistake, according to the number of questions (/4) in which this mistake was displayed, in partial answer to the first research question; (b) the number of data sets (/361) in which each type of mistake was displayed, for each of the four questions, interpreted in relation to the question characteristics, in partial answer to the second research question. These statistics were supplemented with a qualitative reexamination, performed in greater depth than during the coding phase, of data sets of interest. These were identified through performing electronic queries to expose extents and patterns of inconsistencies in the teachers’ mistakes. These queries were sufficiently sensitive to identify data sets which appeared to indicate consistency for a particular mistake type, despite the mistake type not being evidenced in all four of the questions. Such a situation could exist if the presence of another type of mistake precluded the mistake of interest in a particular question. These data sets were of particular interest since (1) the quantitative analysis had revealed them to be rare, and (2) they may serve as test cases to evaluate knowledge coherence. Also of interest were illustrative examples of each type of mole ratio related mistake, which were also extracted through the data querying process. This qualitative reexamination of these extracted answers involved writing descriptions of the mole ratio related mistakes observed, and observations of their relative consistency across the four questions, as well as motivated interpretations on the reasons why these mistakes were made in each case, drawing on literature. One example is included in this article since it illustrates the insights developed through the analysis.
Results and Discussion
Click to copy section linkSection link copied!
Extent of Consistency of Mole Ratio Related Mistakes
Quantitative Analysis Shows Predominance of Mistake Inconsistency
Figure 2 shows the number of data sets (/361) in which the various coded mole ratio related mistakes were detected, distinguished according to the number of questions (/4) in which these were detected.
Out of the 190 data sets which displayed the failure to recognize the need to use proportion mistake, more than half displayed this mistake in only one of the four questions, and only 16 showed this mistake in all four questions. For all the other mistakes, none of the respondents consistently displayed the mistake in all four questions. In fact, the fraction showing each of these mistakes in only one question was considerably greater than 50% of those who showed the mistake. Additionally, except for the misuse of coefficients mistake, which four respondents exhibited in three out of the four questions, the other mistakes were only detected in one or two of the four questions.
This high degree of mistake inconsistency corresponds to findings by Stamovlasis et al., (41) who argued for a fragmented view of knowledge for the context they studied. However, with the current methodology, measurement of mistake inconsistency may not be incommensurate with the Classic view since slips or the preclusion of one mistake by another may falsely suggest fragmentation. The four data sets which displayed the misuse of coefficient mistake in three of the four questions were analyzed qualitatively in greater detail to examine these possibilities. This analysis suggested even greater fragmentation of knowledge than did the quantitative analysis, as illustrated using one of these sets.
Finer Grained Qualitative Analysis Reveals Even Greater Mistake Inconsistency
The answers analyzed in Table 4 show three types of misuse of coefficients: (1) multiplying hydrogen’s available amount by its coefficient, presumably in an attempt to apply proportion to the mole ratio, in questions 1 and 3; (2) calculating the molar mass (M) of ammonia as M = M × coefficient in questions 1 and 3; (3) giving the mole ratio incorrectly using coefficient × subscript for oxygen in question 4. Although mistakes (1) and (2) are applied consistently to hydrogen and nitrogen respectively, these mistakes are not applied to other chemicals. The M = M × coefficient mistake, for example, is not applied to hydrogen (for which M is calculated correctly) or to SO3 (for which M is calculated correctly in the crossed out working and taken as S’s incorrectly calculated n value within a series of inconsistent substitutions, in the working which has not been crossed out).
Table 4. Sample Answersa
a
Answers, with annotations, for a data set flagged in the electronic query as consistent for the misuse of coefficients mistake, but which qualitative scrutiny reveals as illustrating inconsistency for this and other mole ratio related mistakes.
Regarding use of the mole ratio in proportion and recognition of the need to do this, Table 4 suggests that the mistake classified as both misuse of coefficients and conflation of ratio and amount (questions 1 and 3) seems to have been the respondent’s attempt to apply proportion to the mole ratio. This would mean that the respondent did not fail to recognize the need to use proportion in these questions, as was coded, although the obscure method of executing proportion here was different to the cross-product method used for the other two questions (2 and 4). This method was also applied inconsistently, since in question 2 the mole ratio was not included.
How can the findings given in Table 4 be explained from each of the fragmented and coherent epistemological perspectives? The greater mistake inconsistency on analysis at a finer grain size is entirely consistent with a fragmented view of knowledge. (11) Does this finding pose a difficulty for the coherent view of knowledge? If all or some of the mistakes evident in Table 4 could be classified as accidental slips, rather than reasonable errors, then a case could be made for this person holding a misconception regarding the meanings of coefficients and that accidental slips detracted from the coherence of the errors. However, this seems unlikely since, except for one calculation mistake which appears to be a slip, each of the mistakes is explainable as resulting from a reasonable intuitive heuristic, particularly through overgeneralizing a useful procedure. This is discussed in the table and below. This argument is made tentatively, though, since the respondent did not engage in a think-aloud session during, nor an interview after, the question answering session.
Potential Explanations for Mole Ratio Related Errors
Question Difficulty and Complexity Increase Failure to Recognize the Need to Use Proportion
The frequency of mole ratio related mistakes observed were question-dependent as indicated in Figure 3.
The failure to recognize the need to use proportion mistake occurred approximately double as frequently for questions 2 and 3 (number of atoms and percentage yield), which have higher difficulty (difficult), and complexity (7 and 5, see Table 2) ratings, than for questions 1 and 4 (medium difficulty and complexity rating of 3). This may be understood in terms of the resultant greater cognitive load enhancing the likelihood of mistake-making. (38,40) Although this argument applies to either errors or slips, the pedagogical lessons deducible from this finding seem more consistent with a KiP than a Classic theory, as will be presented in the Suggestions for Practice section.
Errors May Result from Overgeneralizations of Useful Procedures and Priming of Intuitive Heuristics
Failure to Recognize the Need to Use Proportion
The differential recognition of the need to apply proportion to the mole ratio across the questions (Figure 3) suggests that question priming was generally a more appropriate explanation for this mistake, than was a misunderstanding of proportion. Related to the greater difficulty and complexity of questions 2 and 3, discussed above, these questions required information other than what is clearly related to chemical reactions (2: number of atoms of hydrogen, 3: percentage yield). This could have detracted the participants’ attention from the reaction, priming the reduction heuristic, (31) resulting in use of procedures appropriate for single substance unit conversions, where proportion does not need to be applied to the mole ratio. Possibly also contributing to the participants’ attention being drawn away from the reaction, these two questions did not include the term produced. However, they did contain other terms (needed and react) which should have drawn attention to the reaction.
Disregard of Limiting Reagent through Adding Reactant Quantities
The answers given to questions 1 and 4 in Table 4 both include adding the reactant quantities to obtain the product quantity. Although additional mistakes obscure this procedure, it could have yielded the correct answer in question 4, but it is flagged as the disregard of limiting reagent mistake in question 1. This is because this is a useful procedure that is consistent with the Law of Conservation of Mass. However, this procedure is overgeneralized in question 1 since it does not consider the presence of an excess quantity of one of the reagents, and therefore disregarded the Law of Constant Proportions. This may indicate a lack of knowledge regarding limiting reagents, (3) or it may indicate deployment of the fixation heuristic in which a strategy is applied without considering the nature of the problem. (31) The data does not allow interrogation of the likelihood of each of these explanations since interviews or think-aloud sessions were not conducted, and only one question included a limiting reagent, so the consistency of this error could not be determined.
Misuse of Coefficients
Multiplying the molar masses of two reacting substances by each substance’s coefficient in the balanced equation (M × coefficient) is a necessary procedure to convert mole to mass reacting ratios. The respondent whose answers are shown in Table 4 applied this useful procedure in multiple inappropriate and inconsistent ways, as has been discussed. The finding that this person applied one form of this mistake consistently to H2 and another form to NH3 suggests that something about these formulas primed different heuristics. The most obvious suggestion is NH3’s relative complexity. However, SO3’s similar complexity was associated with a different kind of mistake, confusing diagnosis. Despite this confusion, the inconsistency of these mistakes, coupled with each one’s reasonability in terms of overgeneralizing a useful procedure, appears more indicative of knowledge fragmentation than coherence.
Conflation of Ratio and Amount
It is useful to draw submicroscopic diagrams in a manner that suggests that the coefficients in the balanced equation refer to actual numbers of molecules, rather than ratios of these. Although the influence of overgeneralizing this useful practice was evident in discussions the author held with the participants (not included in the data set for this article), it is difficult to see how the calculation mistakes shown in Table 4 could be explained in this way.
Suggestions for Practice
Click to copy section linkSection link copied!
Table 5 provides some pedagogical choices a presenter may make when guided by each of the epistemological positions embodied in the Classic and KiP theories. The main differences between these choices are (1) whether examining the role of question features is a legitimate use of time (yes to a greater extent for a KiP than a Classic perspective) in addition to a more generic, conceptual focus; (2) whether the observed error is viewed as a resource to be harnessed (KiP) or a barrier to be confronted (Classic).
Table 5. Teaching Focus Which May Result from Each Epistemological Position Examined
| | Example of the focus that may be taken in a pedagogical response which is guided by the: | |
| | Classic theory | Knowledge in Pieces theory |
| Failure to recognize the need to use proportion | The presenter is likely to assume that the participants possess a misunderstanding of what proportion is and how to apply it. They may focus on developing this understanding, e.g., through providing examples of cooldrink with concentrate and water mixed in various ways and asking participants to rank the sweetness of the resulting mixture. | The presenter should be open to the possibility that proportion may be understood and so instead of needing to spend time on developing the concepts of proportion, it may be more fruitful to spend this time on examining question features which should alert the answerer to the requirement to use proportion, as well as those which may detract the answerer from this. They should also be aware that more difficult and complex questions may detract the participants’ attention from the realization that proportion should be applied to the mole ratio, and so devote some time to discussing how to decompose question complexity. The presenter should also be open to the possibility that some participants may also need a more conceptual introduction, such as use of the cooldrink activity suggested alongside. |
| Disregard of limiting reagent through adding reactant quantities | The presenter is likely to assume that the participants hold the misconception that all reactants are consumed. They may focus on exposing and correcting this misconception, e.g., through drawing parallels between chemicals reacting and girls putting on shoes, or similar analogies, such as described in le Maire et al. (43) where there may or may not be equivalent numbers of items. | The presenter should acknowledge the correctness of the procedure of adding reactant quantities to find the quantity of product produced, for some contexts, and lead a discussion about examples of questions where such a procedure would not be appropriate, and reasons for this. The girls and shoes, and other, analogies suggested alongside may also be appropriate. |
| Misuse of coefficients | The presenter is likely to assume that the participants hold misconceptions related to the meanings of coefficients and subscripts, particularly at the submicroscopic level. They may focus on exposing these misconceptions through asking questions using examples which involve linking of submicroscopic representations to corresponding symbolic representations. | The presenter should acknowledge the possibility that the participant may be able to convert between symbolic and submicroscopic representations but may have (a) had an intuitive heuristic primed; or (b) been applying a useful procedure in an inappropriate context. To cater for these possibilities, the presenter could give examples of common errors resulting from primed intuitive heuristics/overgeneralized procedures and ask the participants to explain why the error may be reasonable to a student/for which contexts such a procedure would be appropriate, why the procedure is not appropriate in the current context and how to correct this. Use of submicroscopic representations could be useful in this discussion. |
| Conflation of ratio and amount | The presenter is likely to assume that participants hold the misconception that the ratio given in the balanced equation refers to actual amounts of particles. They may try to expose this misconception through asking questions which explicitly refer to actual amounts of substances for a particular equation, and then engage in dialogue about this in a manner designed to create cognitive dissonance. For example, “For the reaction C + O2 → CO2, the equation means that only 1 atom or 1 mol of carbon burns. Explain why you agree or disagree with this statement.” | The presenter should acknowledge that when one represents a balanced equation using submicroscopic diagrams, it is useful to draw these in a manner that suggests that the coefficients refer to actual numbers of molecules. The presenter should acknowledge that the participants may need help with the language “for every” to express ratio. The presenter should acknowledge that it is possible that even if a person may be able to explain a reacting ratio appropriately when referring to the submicroscopic diagrams representing a reaction, they may still conflate ratio and amount in their solution, perhaps due to activation of an intuitive heuristic. Therefore, in addition to the discussions above and alongside, the presenter could present an example in which this kind of error is made and lead a discussion about why this may be reasonable to a learner, why it is not appropriate, and how to correct this. |
| Application of a mole ratio as a mass ratio | The presenter is likely to assume that the participants hold misconceptions about what a mole is, how it is related to atomic and molar mass, and to conflate the various ways there are of measuring “how much”, e.g., “how many”, “how big” and “how heavy”. The presenter may try to expose these misconceptions through asking questions and chairing discussions which promote cognitive dissonance. | In addition to a nonconfrontational version the discussions alongside (see Stott (44)), the presenter could present an example in which this kind of error is made and lead a discussion about why this may be reasonable to a learner, why it is not appropriate, and how to correct this. |
Both the quantitative and qualitative analyses showed the mistakes to be almost only inconsistently applied, and the qualitative analysis revealed that almost all the observed mistakes appeared reasonable. This seems to suggest that KiP─rather than Classic─theory informed pedagogy is more likely to be appropriate. In fact, I had conducted the intervention of which this study forms a part informed by a Classic perspective, as partly described in the first column of Table 5. However, based on the discussion and analysis performed in producing this article, I now think that the approach listed in the second column of Table 5 would have been more appropriate. Additionally, examples of mistakes, such as those included in this article, could be used as discussion points to help participants develop both understanding of stoichiometry and critical thinking skills. (42)
Limitations and Suggestions for Further Investigation
Click to copy section linkSection link copied!
Crucial to the argument given in this article is determining whether a mistake is reasonable or not, a distinction which is generally difficult to make, (9) but may have been facilitated by including think-aloud protocols or interviews with participants. Since this was not possible in this study, interpretation has been used to make these judgements instead. These interpretations have been supported by literature and have been reported transparently for the reader to judge their warrantability since this serves as a proxy for validity within the pragmatic paradigm. (37) As has been discussed, in a few cases, it is possible that misconception coherence may have been masked by accidental slips, and in the absence of interview or think-aloud data, it is not possible to judge the warrantability of this possibility. An additional limitation, and opportunity for future research, is the fact that the efficacy of the pedagogical suggestions has not been tested empirically.
Author Information
Click to copy section linkSection link copied!
Corresponding Author
Angela Elisabeth Stott - Department of Mathematics,
Natural Sciences and Technology Education, Faculty of Education, University of the Free State, Bloemfontein 9300, South Africa;
Email: stottae@ufs.ac.za
Notes
The author declares no competing financial interest.
Appendix A
Click to copy section linkSection link copied!
Table 6 shows how the four questions provided to chemistry teaches were analyzed according to Horvat et al.’s (38) framework and used in conjunction with Knaus et al.’s (39) cognitive complexity rating rubric.
Table 6. Analysis of the Calculation Questions Used
| No. | Topic | Given unit(s) | Required unit(s) | Difficulty level (according to Horvat et al., 2016) | Concepts | Interactivity value (according to Horvat et al., 2016) | Cognitive complexity rating (applying Knaus et al.’s (2011) rubric) |
| 1 | Yield with limiting reagent | mol, mass | mass (version a), moles (version b) | Medium, since given and required units differ, with one being moles | Limiting reagent, Stoichiometric calculation | 0, since only 2 concepts are applied | 3 |
| 2 | Number of atoms | volume | number of particles | Difficult, since given and required units differ, with neither being moles | Avogadro’s Law, Stoichiometric calculation, Submicroscopic particles | 1, since 3 concepts are applied | 6 + 1 = 7 |
| 3 | Percent yield | mass | percent yield | Difficult, since given and required units differ, with neither being moles | Stoichiometric calculation, Percent yield | 0, since only 2 concepts are applied | 5 |
| 4 | Yield without limiting reagent | mol | mass | Medium, since given and required units differ, with one being moles | Chemical equation, Stoichiometric calculation | 0, since only 2 concepts are applied | 3 |
References
Click to copy section linkSection link copied!
This article references
44 other publications.
1
Ramful, A.; Narod, F. B. Proportional Reasoning in the Learning of Chemistry: Levels of Complexity. Math. Educ. Res. J. 2014, 26 (1), 25– 46, DOI:
10.1007/s13394-013-0110-7
Google Scholar
There is no corresponding record for this reference.
2. 2
Ralph, V. R.; Lewis, S. E. Chemistry Topics Posing Incommensurate Difficulty to Students with Low Math Aptitude Scores. Chem. Educ. Res. Pract. 2018, 19 (3), 867– 884, DOI:
10.1039/C8RP00115D
Google Scholar
There is no corresponding record for this reference.
3. 3
Miheso, J. M.; Mavhunga, E. The Retention of Topic Specific Pck: A Longitudinal Study with Beginning Chemistry Teachers. Chem. Educ. Res. Pract. 2020, 21 (3), 789– 805, DOI:
10.1039/D0RP00008F
Google Scholar
There is no corresponding record for this reference.
4. 4
BouJaoude, S.; Barakat, H. Students’ Problem Solving Strategies in Stoichiometry and Their Relationships to Conceptual Understanding and Learning Approaches. Electron. J. Sci. Educ. 2003, 7 (3), 1– 42
Google Scholar
There is no corresponding record for this reference.
5. 5
Stott, A. E. South African Physical Sciences Teachers’ Use of Formulae and Proportion When Answering Reaction-Based Stoichiometry Calculation Questions. Chem. Educ. Res. Pract. 2021, 22 (2), 443– 456, DOI:
10.1039/D0RP00291G
Google Scholar
There is no corresponding record for this reference.
6. 6
Brodie, K. The Power of Professional Learning Communities. Educ. as Chang. 2013, 17 (1), 5– 18, DOI:
10.1080/16823206.2013.773929
Google Scholar
There is no corresponding record for this reference.
7. 7
Gardee, A.; Brodie, K. Relationships Between Teachers’ Interactions with Learner Errors and Learners’ Mathematical Identities. Int. J. Sci. Math. Educ. 2022, 20, 193– 214, DOI:
10.1007/s10763-020-10142-1
Google Scholar
There is no corresponding record for this reference.
8. 8
Wong, S. S. H.; Lim, S. W. H. Prevention–Permission–Promotion: A Review of Approaches to Errors in Learning. Educ. Psychol. 2019, 54 (1), 1– 19, DOI:
10.1080/00461520.2018.1501693
Google Scholar
There is no corresponding record for this reference.
9. 9
Taber, K. S. The Significance of Implicit Knowledge for Learning and Teaching Chemistry. Chem. Educ. Res. Pract. 2014, 15 (4), 447– 461, DOI:
10.1039/C4RP00124A
Google Scholar
9
The significance of implicit knowledge for learning and teaching chemistry
Taber, Keith S.
Chemistry Education Research and Practice
(2014),
15
(4),
447-461CODEN:
CERPCE;
ISSN:1756-1108.
(Royal Society of Chemistry)
This article discusses the nature of implicit knowledge, something which is considered to be highly influential in learning. The notion of implicit knowledge is important in conceptualising studies exploring student thinking and learning in chem., and in considering how the results of such studies should be interpreted to inform teaching. Research on cognition suggests that a good deal of the knowledge that people call upon in interpreting their world and making decisions is not accessible to conscious introspection. This has consequences in chem. education research as individuals are not able to directly report implicit knowledge - so it can only be elicited indirectly. A corollary is that the results of many research studies reporting student conceptions in chem. need to be understood as reflecting - at least in part - cognition drawing upon implicit knowledge. The distinction between explicit and implicit knowledge is an important one in understanding chem. learning given that implicit knowledge operates automatically in cognition without deliberation. This suggests that strategies designed to counter students' alternative conceptions may need to be quite different when such ideas derive from the operation of implicit knowledge rather than students' explicit knowledge. The importance of implicit knowledge elements sometimes labeled as p-prims has been widely recognized in physics education research, and it is argued here that research into student thinking and learning in chem. needs to take more account of the distinction between explicit and implicit knowledge elements if it is to better inform teaching. Research is needed to understand the repertoire and action of implicit knowledge elements active in chem. learning. This will then facilitate the design of studies to test out teaching approaches that can recruit the most suitable implicit knowledge elements to support learning of canonical chem. ideas.
10. 10
Posner, G. J.; Strike, K. A.; Hewson, P. W.; Gertzog, W. A. Accommodation of a Scientific Conception: Towards a Theory of Conceptual Change. Sci. Educ. 1982, 66 (2), 211– 227, DOI:
10.1002/sce.3730660207
Google Scholar
There is no corresponding record for this reference.
11
Smith, J. P.; Disessa, A. A.; Roschelle, J. Misconceptions Reconceived: A Constructivist Analysis of Knowledge in Transition. J. Learn. Sci. 1994, 3 (2), 115– 163, DOI:
10.1207/s15327809jls0302_1
Google Scholar
There is no corresponding record for this reference.
12. 12
diSessa, A. A. A Friendly Introduction to “Knowledge in Pieces”: Modeling Types of Knowledge and Their Roles in Learning. In Invited lectures from the 13th international congress on mathematical education; Springer, 2018; pp 65– 84. DOI:
10.1007/978-3-319-72170-5_5 .
Google Scholar
There is no corresponding record for this reference.
13. 13
Cooper, M. M.; Stowe, R. L. Chemistry Education Research─From Personal Empiricism to Evidence, Theory, and Informed Practice. Chem. Rev. 2018, 118 (12), 6053– 6087, DOI:
10.1021/acs.chemrev.8b00020
Google Scholar
13
Chemistry Education Research-From Personal Empiricism to Evidence, Theory, and Informed Practice
Cooper, Melanie M.; Stowe, Ryan L.
Chemical Reviews (Washington, DC, United States)
(2018),
118
(12),
6053-6087CODEN:
CHREAY;
ISSN:0009-2665.
(American Chemical Society)
This Review of Chem. Education Research (CER) provides an overview of the development of research in chem. education from the early days, when ideas about how to teach chem. and help students learn were guided by practitioner wisdom, to current research that is based on theories of learning and provides evidence from which to make arguments about improving teaching and learning. We introduce the dominant learning theories that have guided CER over the years and attempt to show how they have been integrated into modern research in chem. education. We also provide examples of how this research can be used to inform the development and use of educational materials. Because CER literature is vast, we chose to limit the research we reviewed to those studies that help us answer three driving questions: (1) What should students know and be able to do with that knowledge (2) How will we know that students have developed a coherent and useful understanding of chem. (3) What evidence do we have about how to help students develop a deep and robust understanding of chem.
14. 14
Hewson, P. W.; Lemberger, J. Status as the Hallmark of Conceptual Learning. In Improving science education: The contribution of research; Millar, R., Leach, J., Osborne, J., Eds.; Open University Press: Buckingham, 2001; pp 110– 125.
Google Scholar
There is no corresponding record for this reference.
15. 15
Brown, D. E. Students’ Conceptions as Dynamically Emergent Structures. Sci. Educ. 2014, 23 (7), 1463– 1483, DOI:
10.1007/s11191-013-9655-9
Google Scholar
There is no corresponding record for this reference.
16. 16
Taber, K. S. Alternative Conceptions and the Learning of Chemistry. Isr. J. Chem. 2019, 59 (6–7), 450– 469, DOI:
10.1002/ijch.201800046
Google Scholar
16
Alternative Conceptions and the Learning of Chemistry
Taber, Keith S.
Israel Journal of Chemistry
(2019),
59
(6-7),
450-469CODEN:
ISJCAT;
ISSN:0021-2148.
(Wiley-VCH Verlag GmbH & Co. KGaA)
A review. A great deal of research has indicated that teaching is rarely a matter of introducing learners to material that simply replaces previous ignorance, but is more often a matter of presenting ideas that are somewhat at odds with existing understanding. In subjects such as chem., learners at school and university come to their studies already holding misconceptions or 'alternative conceptions' of subject matter. This has implications for subsequent learning, and so for teaching. This article reviews a no. of key issues: (i), the origins of these alternative conceptions; (ii), the nature of these ideas; and, (iii), how they influence learning of the chem. curriculum. These issues are in turn significant for guidance on (a) how curriculum should be selected and sequenced, and (b) on the pedagogy likely to be most effective in teaching chem. A specific concern reported in chem. education is that one source of alternative conceptions seems to be instruction itself.
17. 17
Nakhleh, M. B. Theories or Fragments? The Debate over Learners’ Naive Ideas about Science. J. Chem. Educ. 2001, 78 (8), 1107– 1113, DOI:
10.1021/ed078p1107.5
Google Scholar
17
Theories or fragments? The debate over learners' naive ideas about science
Nakhleh, Mary B.
Journal of Chemical Education
(2001),
78
(8),
1107CODEN:
JCEDA8;
ISSN:0021-9584.
(Division of Chemical Education of the American Chemical Society)
There is no expanded citation for this reference.
18. 18
Taber, K. S.; García-Franco, A. Learning Processes in Chemistry: Drawing upon Cognitive Resources to Learn about the Particulate Structure of Matter. J. Learn. Sci. 2010, 19 (1), 99– 142, DOI:
10.1080/10508400903452868
Google Scholar
There is no corresponding record for this reference.
19. 19
Zarkadis, N.; Papageorgiou, G.; Stamovlasis, D. Studying the Consistency between and within the Student Mental Models for Atomic Structure. Chem. Educ. Res. Pract. 2017, 18 (4), 893– 902, DOI:
10.1039/C7RP00135E
Google Scholar
19
Studying the consistency between and within the student mental models for atomic structure
Zarkadis, Nikolaos; Papageorgiou, George; Stamovlasis, Dimitrios
Chemistry Education Research and Practice
(2017),
18
(4),
893-902CODEN:
CERPCE;
ISSN:1756-1108.
(Royal Society of Chemistry)
Science education research has revealed a no. of student mental models for at. structure, among which, the one based on Bohr's model seems to be the most dominant. The aim of the current study is to investigate the coherence of these models when students apply them for the explanation of a variety of situations. For this purpose, a set of six tasks describing different everyday situations was given to 225 students of the 10th and 11th grades of secondary schools from Northern Greece. Quant. anal. of the students' responses using Latent Class Anal. (LCA) showed that there is no consistency between models across the tasks and that the context of the task affects the distribution of students' responses across models. Qual. anal. showed a variety of pieces of knowledge from different models that students combine when manipulating the tasks, which possibly causes a lack of consistency within each one of the models. The findings are discussed in terms of between and within model consistency, and the conclusions contribute to the debate concerning the coherent vs. fragmented knowledge hypotheses. The empirical evidence provided by the anal. clearly demonstrates that student mental models for at. structure were not coherent when applied in different everyday situations. Implications for theory and practice are discussed.
20. 20
Guthrie, G. Classroom Change in Developing Countries: From Progressive Cage to Formalistic Frame; Routledge:: London, 2018. DOI:
10.4324/9781351130479 .
Google Scholar
There is no corresponding record for this reference.
21. 21
Rollnick, M.; Mavhunga, E. The Place of Subject Matter Knowledge in Teacher Education. In International handbook of teacher education; Loughran, J., Hamilton, M., Eds.; Springer: Singapore, 2016; pp 423– 452. DOI:
10.1007/978-981-10-0366-0_11 .
Google Scholar
There is no corresponding record for this reference.
22. 22
Makhechane, M.; Qhobela, M. Understanding How Chemistry Teachers Transform Stoichiometry Concepts at Secondary Level in Lesotho. South African J. Chem. 2019, 72 (1), 59– 66, DOI:
10.17159/0379-4350/2019/v72a9
Google Scholar
22
Understanding how chemistry teachers transform stoichiometry concepts at secondary level in Lesotho
Makhechane, Mamohato; Qhobela, Makomosela
South African Journal of Chemistry
(2019),
72
(1),
59-66CODEN:
SAJCDG;
ISSN:0379-4350.
(South African Chemical Institute)
A review. This qual. study sought to investigate how secondary level chem. teachers transform and represent subject matter knowledge (SMK). The study drew its theor. framework from the notion of Pedagogical Content Knowledge (PCK) that foregrounds the teachers' ability to transform and represent SMK for students to understand the content presented. Seven qualified chem. teachers participated in this study which used questionnaires, follow-up interviews, and lesson observations to collect data. The teachers were obsd. teaching the topic of stoichiometry to grade 11 students. Data anal. started with a coding system that resulted in themes of meanings suggested by teachers. The results show that chem. teachers participating in the study were aware of different teaching strategies but their choice of those strategies was not informed by an attempt to transform the concepts. It is argued that teachers' failure to transform concepts will have a neg. impact on students' understanding of these concepts.
23. 23
Reina, M.; This, H.; Reina, A. Improving the Understanding of Chemistry by Using the Right Words: A Clear-Cut Strategy to Avoid Misconceptions When Talking about Elements, Atoms, and Molecules. J. Chem. Educ. 2022, 99 (8), 2999– 3006, DOI:
10.1021/acs.jchemed.2c00411
Google Scholar
23
Improving the understanding of chemistry by using the right words: A clear-cut strategy to avoid misconceptions when talking about elements, atoms and molecules
Reina, Miguel; This, Herve; Reina, Antonio
Journal of Chemical Education
(2022),
99
(8),
2999-3006CODEN:
JCEDA8;
ISSN:0021-9584.
(American Chemical Society and Division of Chemical Education, Inc.)
A language is a system of communication, consisting of a set of sounds or written symbols that enable people to communicate. In chem., a particular language is required in order to represent the phenomenol. world by means of symbols. Choosing the right words and knowing the precise definitions for chem. concepts is needed for avoiding misconceptions. Despite the creation of International Union of Pure and Applied Chem. (IUPAC), fundamental notions are still undergoing const. debate, in particular concerning the conceptual distinction between element, atom, mol., compd., chem. species and substance. In this article, we propose to better distinguish categories of chem. objects and their element, and to define a mol. as a stable and elec. neutral chem. entity that exists with a finite no. of chem. bonds (0 to n, with n finite no.). This new definition implies that noble gases, i.e., reduced to one isolated atom, are made of mols., that all bonded atoms are atoms, and enables distinguishing both of these from compds. We insist that chem. education makes a clear difference between objects and categories, emphasizing the importance of understanding that elements and compds. represent categories of nuclei, whereas chem. entities and substances are a sample of matter. We also point out a no. of nuances in chem. language, primarily in order to avoid misconceptions regarding the macroscopic phys. properties relating to chem. substances and chem. objects.
24. 24
Bierenstiel, M.; Snow, K. Periodic Universe: A Teaching Model for Understanding the Periodic Table of the Elements. J. Chem. Educ. 2019, 96 (7), 1367– 1376, DOI:
10.1021/acs.jchemed.8b00740
Google Scholar
24
Periodic Universe: A Teaching Model for Understanding the Periodic Table of the Elements
Bierenstiel, Matthias; Snow, Kathy
Journal of Chemical Education
(2019),
96
(7),
1367-1376CODEN:
JCEDA8;
ISSN:0021-9584.
(American Chemical Society and Division of Chemical Education, Inc.)
The Periodic Table of the Elements (PTE) is arguably one of the most central topics in chem. This article provides a crit. review of current teaching practices of the PTE to high-school students and undergraduate university students. It also provides a new teaching model, the "periodic universe", for understanding of the PTE through building on its foundational periodicity. Students learn through a guided re-creation of the PTE from simple periodic simulations and interdisciplinary processes. The periodic-universe teaching model directs students to conceptualize, rather than memorize, relationships between the elements by predicting patterns in progressively challenging simulated worlds. This mixed-method research project consists of both quant. (n = 58) and qual. (n = 15) responses from university undergraduate students, demonstrating the utility of inquiry-based learning of the PTE for novice chem. students. Herein, we position the periodic-universe teaching model in the current educational literature of inquiry and interdisciplinary studies by outlining the function of the model and illustrating its effectiveness on the basis of empirical anal. with undergraduate students in a postsecondary-education setting. The results indicate that knowledge of the PTE increases after instruction with the model, and that nonchem.-major students have increased engagement, understanding, and appreciation of the importance of periodicity to chem. when presented with the periodic-universe teaching model.
25. 25
Jusniar, J.; Effendy, E.; Budiasih, E.; Sutrisno, S. Eliminating Misconceptions on Reaction Rate to Enhance Conceptual Understanding of Chemical Equilibrium Using EMBE-R Strategy. Int. J. Instr. 2021, 14 (1), 85– 104, DOI:
10.29333/iji.2021.1416a
Google Scholar
There is no corresponding record for this reference.
26. 26
Cai, S. Harry Potter Themed Digital Escape Room for Addressing Misconceptions in Stoichiometry. J. Chem. Educ. 2022, 99 (7), 2747– 2753, DOI:
10.1021/acs.jchemed.2c00178
Google Scholar
26
Harry Potter Themed Digital Escape Room for Addressing Misconceptions in Stoichiometry
Cai, Shiying
Journal of Chemical Education
(2022),
99
(7),
2747-2753CODEN:
JCEDA8;
ISSN:0021-9584.
(American Chemical Society and Division of Chemical Education, Inc.)
Chem. is often seen as an abstr. and content-heavy subject. Many students struggle with chem. because they attempt to memorize the content without understanding the concepts. As a result, students often have misconceptions. COVID-19 has driven teaching and learning online, and an escape room teaching method, which is a way to enhance student engagement, has gained popularity among educators in higher education. This study examines the effectiveness of teaching through a digital escape room as compared to a typical online lesson with a collaborative learning method to address misconceptions in stoichiometry. A Harry Potter themed digital escape room is created to spark the students' interest in chem. and address misconceptions. Thirty-eight students from the Nanyang Polytechnic Foundation Program participated in this study. The students completed a pretest, a post-test, and a survey, in addn. to participating in the digital escape room and a typical online lesson. Four topics were covered in this study: balancing chem. equations, calcg. empirical formulas, identifying the type of chem. bonding, and interpreting element symbols. Out of these four topics, it was discovered that students tended to have difficulty calcg. empirical formulas. It was found that, on av., students showed a 10% improvement in test scores after being taught through the digital escape room. This result is similar to results obtained from a typical online lesson with a collaborative learning method (9% improvement). This implies that a digital escape room is equally as effective as a typical online lesson with a collaborative learning method at addressing misconceptions. Teaching through a digital escape room has shown potential addnl. benefits of enhancing soft skills, promoting teamwork, the ability to work under time pressure, communication skills, innovation competency, and increasing student motivation. The researcher recommends the use of a digital escape room to complement typical lessons for these addnl. benefits.
27. 27
Tsaparlis, G. Teaching and Learning Electrochemistry. Isr. J. Chem. 2019, 59 (6–7), 478– 492, DOI:
10.1002/ijch.201800071
Google Scholar
27
Teaching and Learning Electrochemistry
Tsaparlis, Georgios
Israel Journal of Chemistry
(2019),
59
(6-7),
478-492CODEN:
ISJCAT;
ISSN:0021-2148.
(Wiley-VCH Verlag GmbH & Co. KGaA)
A review. Numerous conceptual difficulties and misconceptions have been reported in the science and chem. education literature about electrochem. concepts such as electrolytes, redox equations, and about electrochem. (galvanic and electrolytic) cells. Other studies have considered teaching approaches aimed at improving learning and at overcoming misconceptions. This paper reviews these studies and considers strategies and approaches for the effective teaching of electrochem. Then, the review focuses on problem solving in equil. electrochem., esp. problems involving the Nernst equation. The main findings of four studies with Greek university chem. students, dealing with: (i) algorithmic problem-solving ability; (ii) practice on problem solving; (iii) the effect of the format of a problem; and (iv) the construction and evaluation of a website devoted to electrochem. problem solving, are reported.
28. 28
Islamiyah, K. K.; Rahayu, S.; Dasna, I. W. The Effectiveness of Remediation Learning Strategy in Reducing Misconceptions on Chemistry: A Systematic Review. Tadris J. Kegur. dan Ilmu Tarb. 2022, 7 (1), 63– 77, DOI:
10.24042/tadris.v7i1.11140
Google Scholar
There is no corresponding record for this reference.
29. 29
Vosniadou, S. The Development of Students’ Understanding of Science. In Frontiers in Education; Frontiers Media SA, 2019; Vol. 4, p 32. DOI:
10.3389/feduc.2019.00032 .
Google Scholar
There is no corresponding record for this reference.
30. 30
DiSessa, A. A. Toward an Epistemology of Physics. Cogn. Instr. 1993, 10 (2–3), 105– 225, DOI:
10.1080/07370008.1985.9649008
Google Scholar
There is no corresponding record for this reference.
31. 31
Talanquer, V. Commonsense Chemistry: A Model for Understanding Students’ Alternative Conceptions. J. Chem. Educ. 2006, 83 (5), 811, DOI:
10.1021/ed083p811
Google Scholar
31
Commonsense chemistry: a model for understanding students' alternative conceptions
Talanquer, Vicente
Journal of Chemical Education
(2006),
83
(5),
811-816CODEN:
JCEDA8;
ISSN:0021-9584.
(Journal of Chemical Education, Dept. of Chemistry)
The recognition and characterization of student beliefs and prior knowledge seems to be crucial to helping them build scientific understandings. The paper discusses the results of a research project guided by the assumption that a common explanatory framework does exist and it can be described by analyzing the research literature on alternative conceptions in chem.
32. 32
Talanquer, V. Chemistry Education: Ten Heuristics to Tame. J. Chem. Educ. 2014, 91 (8), 1091– 1097, DOI:
10.1021/ed4008765
Google Scholar
32
Chemistry Education: Ten Heuristics To Tame
Talanquer, Vicente
Journal of Chemical Education
(2014),
91
(8),
1091-1097CODEN:
JCEDA8;
ISSN:0021-9584.
(American Chemical Society and Division of Chemical Education, Inc.)
Students in our chem. classes often generate shallow responses to our questions and problems. They fail to recognize relevant cues in making judgments and decisions about the properties of chem. substances and processes, and make hasty generalizations that frequently lead them astray. Results from research in the psychol. of decision making can help us better understand how students approach chem. tasks under conditions of limited knowledge, time, or motivation. In this contribution, I describe 10 cognitive heuristics that are often responsible for biases in student thinking. Helping students tame these heuristics may allow us to foster more meaningful learning in chem. classrooms.
33. 33
Macrie-Shuck, M.; Talanquer, V. Exploring Students’ Explanations of Energy Transfer and Transformation. J. Chem. Educ. 2020, 97 (12), 4225– 4234, DOI:
10.1021/acs.jchemed.0c00984
Google Scholar
33
Exploring students' explanations of energy transfer and transformation
Macrie-Shuck, Michael; Talanquer, Vicente
Journal of Chemical Education
(2020),
97
(12),
4225-4234CODEN:
JCEDA8;
ISSN:0021-9584.
(American Chemical Society and Division of Chemical Education, Inc.)
The ability to build mechanistic explanations of natural phenomena is highly valued in all scientific disciplines, including chem. In this paper, we summarize the results of a qual. study designed to characterize the nature of the mechanistic explanations built by college General Chem. students when analyzing a phys. process involving energy transfer between parts of a system at different temps., and a chem. process involving energy transformation and transfer due to chem. bonding. Prior studies have mostly focused on eliciting "what" students think happens in terms of energy transfer when chem. bonds break or form (e.g., energy is absorbed or released) and in this investigation we provide insights into their reasoning about "how" and "why" it happens. Our findings reveal that many students struggle to build sound mechanistic explanations for energy transfer between objects at different temps. and for energy transformation and transfer during the formation of chem. bonds.
34. 34
Talanquer, V. How Do Students Reason about Chemical Substances and Reactions? In Concepts of matter in science education; Tsaparlis, G., Sevian, H., Eds.; Springer: Dordrecht, 2013; pp 331– 346. DOI:
10.1007/978-94-007-5914-5_16 .
Google Scholar
There is no corresponding record for this reference.
35. 35
Talanquer, V. Progressions in Reasoning about Structure–Property Relationships. Chem. Educ. Res. Pract. 2018, 19 (4), 998– 1009, DOI:
10.1039/C7RP00187H
Google Scholar
There is no corresponding record for this reference.
36. 36
Talanquer, V. The Complexity of Reasoning about and with Chemical Representations. JACS Au 2022, 2, 2658, DOI:
10.1021/jacsau.2c00498
Google Scholar
36
The complexity of reasoning about and with chemical representations
Talanquer, Vicente
JACS Au
(2022),
2
(12),
2658-2669CODEN:
JAAUCR;
ISSN:2691-3704.
(American Chemical Society)
External visual representations of chem. entities and processes (chem. representations) play a crit. role in chem. thinking and practice. They support reasoning by serving as bridges between the macroscopic world and the chem. models that help us make sense of the properties and behaviors of substances in our surroundings. Consequently, many chem. education research studies have been carried out to explore and foster students representational competency in our discipline. Nevertheless, in this perspective I argue that investigations in this area would benefit from a more in-depth anal. of how the distinctive characteristics of chem. representations affect student reasoning. I identify four dimensions of variation in these representations (iconicity, quantitativeness, granularity, dimensionality) that affect students ability to interpret, connect, generate, and use chem. representations. I discuss how these features influence the unpacking or packing of information during different types of tasks, affecting sense-making and perceptual competency. Implications for chem. education research and practice are considered.
37. 37
Plowright, D. Using Mixed Methods: Frameworks for an Integrated Methodology; SAGE Publications: London, 2011.
Google Scholar
There is no corresponding record for this reference.
38. 38
Horvat, S.; Segedinac, M. D.; Milenković, D. D.; Hrin, T. N. Development of Procedure for the Assessment of Cognitive Complexity of Stoichiometric Tasks. Maced. J. Chem. Chem. Eng. 2016, 35 (2), 275– 284, DOI:
10.20450/mjcce.2016.893
Google Scholar
38
Development of procedure for the assessment of cognitive complexity of stoichiometric tasks
Horvat, Sasa; Segedinac, Mirjana D.; Milenkovic, Dusica D.; Hrin, Tamara N.
Macedonian Journal of Chemistry and Chemical Engineering
(2016),
35
(2),
275-284CODEN:
MJCCA9;
ISSN:1857-5552.
(Society of Chemists and Technologists of Macedonia)
The aim of this study was the creation of a procedure for detg. the cognitive complexity of stoichiometric tasks and its validation. The created procedure included an assessment of the difficulty of concepts and skills, and an assessment of the concepts' interactivity. There were 82 students who participated in the study, with an educational profile of a pharmaceutical technician. As a research instrument for assessing performance, test of knowledge was used. Each task in the test was followed by a seven-point Likert scale for the evaluation of invested mental effort. The validity of this instrument for the assessment of cognitive complexity was confirmed by a series of linear regression anal. where extremely high values of correlation coeffs. are obtained among the examd. variables: student's performance and invested mental effort (dependent variables) and cognitive complexity (independent variable).
39. 39
Knaus, K.; Murphy, K.; Blecking, A.; Holme, T. A Valid and Reliable Instrument for Cognitive Complexity Rating Assignment of Chemistry Exam Items. J. Chem. Educ. 2011, 88 (5), 554– 560, DOI:
10.1021/ed900070y
Google Scholar
39
A Valid and Reliable Instrument for Cognitive Complexity Rating Assignment of Chemistry Exam Items
Knaus, Karen; Murphy, Kristen; Blecking, Anja; Holme, Thomas
Journal of Chemical Education
(2011),
88
(5),
554-560CODEN:
JCEDA8;
ISSN:0021-9584.
(American Chemical Society and Division of Chemical Education, Inc.)
The design and use of a valid and reliable instrument for the assignment of cognitive complexity ratings to chem. exam items is described in this paper. Use of such an instrument provides a simple method to quantify the cognitive demands of chem. exam items. Instrument validity was established in two different ways: statistically significant correlations between expert-based cognitive complexity ratings and student performance (as measured through statistical difficulty of items), and statistically significant correlations between expert-based cognitive complexity ratings and student mental effort ratings. Key benefits assocd. with instrument use include an enhanced understanding of the cognitive complexity of chem. assessment tasks and as a means for characterizing exam content for the measurement of cognitive development.
40. 40
Klepsch, M.; Seufert, T. Understanding Instructional Design Effects by Differentiated Measurement of Intrinsic, Extraneous, and Germane Cognitive Load. Instr. Sci. 2020, 48 (1), 45– 77, DOI:
10.1007/s11251-020-09502-9
Google Scholar
There is no corresponding record for this reference.
41. 41
Stamovlasis, D.; Papageorgiou, G.; Tsitsipis, G. The Coherent versus Fragmented Knowledge Hypotheses for the Structure of Matter: An Investigation with a Robust Statistical Methodology. Chem. Educ. Res. Pract. 2013, 14 (4), 485– 495, DOI:
10.1039/C3RP00042G
Google Scholar
41
The coherent versus fragmented knowledge hypotheses for the structure of matter: an investigation with a robust statistical methodology
Stamovlasis, Dimitrios; Papageorgiou, George; Tsitsipis, Georgios
Chemistry Education Research and Practice
(2013),
14
(4),
485-495CODEN:
CERPCE;
ISSN:1756-1108.
(Royal Society of Chemistry)
Research in cognitive psychol. and science education has acknowledged two antagonistic theor. perspectives concerning students' knowledge. One perspective considers students' knowledge as coherent or theory-like, explaining the learning process in terms of framework theories and stable mental models, whereas the other considers it fragmented and based on the existence of knowledge elements that can be activated, interconnected and organized accordingly. Although both perspectives continue to demonstrate empirical support, a third point of view focuses on the role of the particular topic and suggests that the stability of the in situ formed conceptions varies under different circumstances being also affected by the researchers' interpretation of data. Since methodol. issues are by far fundamental and crucial for the above theor. premises, in the present research an attempt is made to test the above hypotheses by applying a robust statistical tool, the Latent Class Anal. (LCA). Data were taken from students' responses (N = 329, aged 14-15) using an instrument designed to access students' understanding of the structure of matter and its changes of states that had previously been used to identify students' progression through a series of mental models reported in the literature. However, the results did not support the hypothesis that participants would hold these as coherent mental models. The present research adds to the ongoing dialogue on such a crucial issue for its pedagogical implications by contributing to the methodol. pathways for an advancement of science education research through theory building.
42. 42
Natividad, L. R. Understanding Alternative Conceptions: Constructivism and Nature of Science Approach. Lukad An Online J. Pedagog. 2022, 2 (1), 21– 30
Google Scholar
There is no corresponding record for this reference.
43. 43
le
Maire, N. V.; Verpoorten, D. P.; Fauconnier, M.-L. S.; Colaux-Castillo, C. G. Clash of Chemists: A Gamified Blog to Master the Concept of Limiting Reagent Stoichiometry. J. Chem. Educ. 2018, 95 (3), 410– 415, DOI:
10.1021/acs.jchemed.7b00256
Google Scholar
43
Clash of Chemists: A Gamified Blog To Master the Concept of Limiting Reagent Stoichiometry
le Maire, Nathalie V.; Verpoorten, Dominique Ph.; Fauconnier, Marie-Laure S.; Colaux-Castillo, Catherine G.
Journal of Chemical Education
(2018),
95
(3),
410-415CODEN:
JCEDA8;
ISSN:0021-9584.
(American Chemical Society and Division of Chemical Education, Inc.)
In a first-year university course, students experienced a new learning activity ("Clash of Chemists") prompting them to create and share personal analogies explaining the difference between stoichiometric and nonstoichiometric reaction conditions, also known in the literature as "limiting reagent stoichiometry". To support students' commitment to this unusual assignment, the instructional design drew on a blog enriched with game mechanics (tournament, video rewards, and leaderboard), as found in popular mini-games. The paper reports on the activity's outputs and on participants' perceptions of its usability, usefulness, and generated satisfaction. Overall students' reception of this mini-game was pos. A significant difference between players and nonplayers' end-of-term exam results was highlighted.
44. 44
Stott, A. E. The Efficacy of Instruction in Application of Mole Ratios and Submicro- and Macro-Scopic Equivalent Forms of the Mole within the Unit Factor Method. Chem. Educ. Res. Pract. 2023, 24, 551– 566, DOI:
10.1039/D2RP00245K
Google Scholar
There is no corresponding record for this reference.
Cited By
Click to copy section linkSection link copied!
This article is cited by 2 publications.
Jon-Marc G. Rodriguez. Using Analytic Autoethnography to Characterize the Variation in the Application of the Resources Framework: What Is a Resource?. Journal of Chemical Education 2024, 101
(9)
, 3676-3690.
Angela Elisabeth Stott, Annemarie Hattingh. Epistemological Reflections Arising from Using a Scientific Method Heuristic in a Preservice Teacher Module. Journal of Science Teacher Education 2025, , 1-21.
Figures
References
Recommended Articles
Get e-Alerts
Get e-Alerts
Journal of Chemical Education
Cite this: J. Chem. Educ. 2023, 100, 7, 2548–2557
Click to copy citationCitation copied!
Published June 15, 2023
Publication History
Received
Revised
Published
online
Published
in issue
Copyright © 2023 The Author. Published by American Chemical Society and Division of Chemical Education, Inc. This publication is licensed under
CC-BY 4.0.
License Summary
You are free to share (copy and redistribute) this article in any medium or format and to adapt (remix, transform, and build upon) the material for any purpose, even commercially within the parameters below:
Creative Commons (CC): This is a Creative Commons license.
Attribution (BY): Credit must be given to the creator.
View full license
Disclaimer
This summary highlights only some of the key features and terms of the
actual license. It is not a license and has no legal value. Carefully
review the actual license before using these materials.
Article Views
2730
Altmetric
-
Citations
2
Learn about these metrics
Article Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.
Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.
The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated.
Recommended Articles
Figures
References
References
This article references
44 other publications.
1
Ramful, A.; Narod, F. B. Proportional Reasoning in the Learning of Chemistry: Levels of Complexity. Math. Educ. Res. J. 2014, 26 (1), 25– 46, DOI:
10.1007/s13394-013-0110-7
There is no corresponding record for this reference.
2. 2
Ralph, V. R.; Lewis, S. E. Chemistry Topics Posing Incommensurate Difficulty to Students with Low Math Aptitude Scores. Chem. Educ. Res. Pract. 2018, 19 (3), 867– 884, DOI:
10.1039/C8RP00115D
There is no corresponding record for this reference.
3. 3
Miheso, J. M.; Mavhunga, E. The Retention of Topic Specific Pck: A Longitudinal Study with Beginning Chemistry Teachers. Chem. Educ. Res. Pract. 2020, 21 (3), 789– 805, DOI:
10.1039/D0RP00008F
There is no corresponding record for this reference.
4. 4
BouJaoude, S.; Barakat, H. Students’ Problem Solving Strategies in Stoichiometry and Their Relationships to Conceptual Understanding and Learning Approaches. Electron. J. Sci. Educ. 2003, 7 (3), 1– 42
There is no corresponding record for this reference.
5. 5
Stott, A. E. South African Physical Sciences Teachers’ Use of Formulae and Proportion When Answering Reaction-Based Stoichiometry Calculation Questions. Chem. Educ. Res. Pract. 2021, 22 (2), 443– 456, DOI:
10.1039/D0RP00291G
There is no corresponding record for this reference.
6. 6
Brodie, K. The Power of Professional Learning Communities. Educ. as Chang. 2013, 17 (1), 5– 18, DOI:
10.1080/16823206.2013.773929
There is no corresponding record for this reference.
7. 7
Gardee, A.; Brodie, K. Relationships Between Teachers’ Interactions with Learner Errors and Learners’ Mathematical Identities. Int. J. Sci. Math. Educ. 2022, 20, 193– 214, DOI:
10.1007/s10763-020-10142-1
There is no corresponding record for this reference.
8. 8
Wong, S. S. H.; Lim, S. W. H. Prevention–Permission–Promotion: A Review of Approaches to Errors in Learning. Educ. Psychol. 2019, 54 (1), 1– 19, DOI:
10.1080/00461520.2018.1501693
There is no corresponding record for this reference.
9. 9
Taber, K. S. The Significance of Implicit Knowledge for Learning and Teaching Chemistry. Chem. Educ. Res. Pract. 2014, 15 (4), 447– 461, DOI:
10.1039/C4RP00124A
9
The significance of implicit knowledge for learning and teaching chemistry
Taber, Keith S.
Chemistry Education Research and Practice
(2014),
15
(4),
447-461CODEN:
CERPCE;
ISSN:1756-1108.
(Royal Society of Chemistry)
This article discusses the nature of implicit knowledge, something which is considered to be highly influential in learning. The notion of implicit knowledge is important in conceptualising studies exploring student thinking and learning in chem., and in considering how the results of such studies should be interpreted to inform teaching. Research on cognition suggests that a good deal of the knowledge that people call upon in interpreting their world and making decisions is not accessible to conscious introspection. This has consequences in chem. education research as individuals are not able to directly report implicit knowledge - so it can only be elicited indirectly. A corollary is that the results of many research studies reporting student conceptions in chem. need to be understood as reflecting - at least in part - cognition drawing upon implicit knowledge. The distinction between explicit and implicit knowledge is an important one in understanding chem. learning given that implicit knowledge operates automatically in cognition without deliberation. This suggests that strategies designed to counter students' alternative conceptions may need to be quite different when such ideas derive from the operation of implicit knowledge rather than students' explicit knowledge. The importance of implicit knowledge elements sometimes labeled as p-prims has been widely recognized in physics education research, and it is argued here that research into student thinking and learning in chem. needs to take more account of the distinction between explicit and implicit knowledge elements if it is to better inform teaching. Research is needed to understand the repertoire and action of implicit knowledge elements active in chem. learning. This will then facilitate the design of studies to test out teaching approaches that can recruit the most suitable implicit knowledge elements to support learning of canonical chem. ideas.
10. 10
Posner, G. J.; Strike, K. A.; Hewson, P. W.; Gertzog, W. A. Accommodation of a Scientific Conception: Towards a Theory of Conceptual Change. Sci. Educ. 1982, 66 (2), 211– 227, DOI:
10.1002/sce.3730660207
There is no corresponding record for this reference.
11. 11
Smith, J. P.; Disessa, A. A.; Roschelle, J. Misconceptions Reconceived: A Constructivist Analysis of Knowledge in Transition. J. Learn. Sci. 1994, 3 (2), 115– 163, DOI:
10.1207/s15327809jls0302_1
There is no corresponding record for this reference.
12. 12
diSessa, A. A. A Friendly Introduction to “Knowledge in Pieces”: Modeling Types of Knowledge and Their Roles in Learning. In Invited lectures from the 13th international congress on mathematical education; Springer, 2018; pp 65– 84. DOI:
10.1007/978-3-319-72170-5_5 .
There is no corresponding record for this reference.
13. 13
Cooper, M. M.; Stowe, R. L. Chemistry Education Research─From Personal Empiricism to Evidence, Theory, and Informed Practice. Chem. Rev. 2018, 118 (12), 6053– 6087, DOI:
10.1021/acs.chemrev.8b00020
13
Chemistry Education Research-From Personal Empiricism to Evidence, Theory, and Informed Practice
Cooper, Melanie M.; Stowe, Ryan L.
Chemical Reviews (Washington, DC, United States)
(2018),
118
(12),
6053-6087CODEN:
CHREAY;
ISSN:0009-2665.
(American Chemical Society)
This Review of Chem. Education Research (CER) provides an overview of the development of research in chem. education from the early days, when ideas about how to teach chem. and help students learn were guided by practitioner wisdom, to current research that is based on theories of learning and provides evidence from which to make arguments about improving teaching and learning. We introduce the dominant learning theories that have guided CER over the years and attempt to show how they have been integrated into modern research in chem. education. We also provide examples of how this research can be used to inform the development and use of educational materials. Because CER literature is vast, we chose to limit the research we reviewed to those studies that help us answer three driving questions: (1) What should students know and be able to do with that knowledge (2) How will we know that students have developed a coherent and useful understanding of chem. (3) What evidence do we have about how to help students develop a deep and robust understanding of chem.
14. 14
Hewson, P. W.; Lemberger, J. Status as the Hallmark of Conceptual Learning. In Improving science education: The contribution of research; Millar, R., Leach, J., Osborne, J., Eds.; Open University Press: Buckingham, 2001; pp 110– 125.
There is no corresponding record for this reference.
15. 15
Brown, D. E. Students’ Conceptions as Dynamically Emergent Structures. Sci. Educ. 2014, 23 (7), 1463– 1483, DOI:
10.1007/s11191-013-9655-9
There is no corresponding record for this reference.
16. 16
Taber, K. S. Alternative Conceptions and the Learning of Chemistry. Isr. J. Chem. 2019, 59 (6–7), 450– 469, DOI:
10.1002/ijch.201800046
16
Alternative Conceptions and the Learning of Chemistry
Taber, Keith S.
Israel Journal of Chemistry
(2019),
59
(6-7),
450-469CODEN:
ISJCAT;
ISSN:0021-2148.
(Wiley-VCH Verlag GmbH & Co. KGaA)
A review. A great deal of research has indicated that teaching is rarely a matter of introducing learners to material that simply replaces previous ignorance, but is more often a matter of presenting ideas that are somewhat at odds with existing understanding. In subjects such as chem., learners at school and university come to their studies already holding misconceptions or 'alternative conceptions' of subject matter. This has implications for subsequent learning, and so for teaching. This article reviews a no. of key issues: (i), the origins of these alternative conceptions; (ii), the nature of these ideas; and, (iii), how they influence learning of the chem. curriculum. These issues are in turn significant for guidance on (a) how curriculum should be selected and sequenced, and (b) on the pedagogy likely to be most effective in teaching chem. A specific concern reported in chem. education is that one source of alternative conceptions seems to be instruction itself.
17. 17
Nakhleh, M. B. Theories or Fragments? The Debate over Learners’ Naive Ideas about Science. J. Chem. Educ. 2001, 78 (8), 1107– 1113, DOI:
10.1021/ed078p1107.5
17
Theories or fragments? The debate over learners' naive ideas about science
Nakhleh, Mary B.
Journal of Chemical Education
(2001),
78
(8),
1107CODEN:
JCEDA8;
ISSN:0021-9584.
(Division of Chemical Education of the American Chemical Society)
There is no expanded citation for this reference.
18. 18
Taber, K. S.; García-Franco, A. Learning Processes in Chemistry: Drawing upon Cognitive Resources to Learn about the Particulate Structure of Matter. J. Learn. Sci. 2010, 19 (1), 99– 142, DOI:
10.1080/10508400903452868
There is no corresponding record for this reference.
19. 19
Zarkadis, N.; Papageorgiou, G.; Stamovlasis, D. Studying the Consistency between and within the Student Mental Models for Atomic Structure. Chem. Educ. Res. Pract. 2017, 18 (4), 893– 902, DOI:
10.1039/C7RP00135E
19
Studying the consistency between and within the student mental models for atomic structure
Zarkadis, Nikolaos; Papageorgiou, George; Stamovlasis, Dimitrios
Chemistry Education Research and Practice
(2017),
18
(4),
893-902CODEN:
CERPCE;
ISSN:1756-1108.
(Royal Society of Chemistry)
Science education research has revealed a no. of student mental models for at. structure, among which, the one based on Bohr's model seems to be the most dominant. The aim of the current study is to investigate the coherence of these models when students apply them for the explanation of a variety of situations. For this purpose, a set of six tasks describing different everyday situations was given to 225 students of the 10th and 11th grades of secondary schools from Northern Greece. Quant. anal. of the students' responses using Latent Class Anal. (LCA) showed that there is no consistency between models across the tasks and that the context of the task affects the distribution of students' responses across models. Qual. anal. showed a variety of pieces of knowledge from different models that students combine when manipulating the tasks, which possibly causes a lack of consistency within each one of the models. The findings are discussed in terms of between and within model consistency, and the conclusions contribute to the debate concerning the coherent vs. fragmented knowledge hypotheses. The empirical evidence provided by the anal. clearly demonstrates that student mental models for at. structure were not coherent when applied in different everyday situations. Implications for theory and practice are discussed.
20. 20
Guthrie, G. Classroom Change in Developing Countries: From Progressive Cage to Formalistic Frame; Routledge:: London, 2018. DOI:
10.4324/9781351130479 .
There is no corresponding record for this reference.
21. 21
Rollnick, M.; Mavhunga, E. The Place of Subject Matter Knowledge in Teacher Education. In International handbook of teacher education; Loughran, J., Hamilton, M., Eds.; Springer: Singapore, 2016; pp 423– 452. DOI:
10.1007/978-981-10-0366-0_11 .
There is no corresponding record for this reference.
22. 22
Makhechane, M.; Qhobela, M. Understanding How Chemistry Teachers Transform Stoichiometry Concepts at Secondary Level in Lesotho. South African J. Chem. 2019, 72 (1), 59– 66, DOI:
10.17159/0379-4350/2019/v72a9
22
Understanding how chemistry teachers transform stoichiometry concepts at secondary level in Lesotho
Makhechane, Mamohato; Qhobela, Makomosela
South African Journal of Chemistry
(2019),
72
(1),
59-66CODEN:
SAJCDG;
ISSN:0379-4350.
(South African Chemical Institute)
A review. This qual. study sought to investigate how secondary level chem. teachers transform and represent subject matter knowledge (SMK). The study drew its theor. framework from the notion of Pedagogical Content Knowledge (PCK) that foregrounds the teachers' ability to transform and represent SMK for students to understand the content presented. Seven qualified chem. teachers participated in this study which used questionnaires, follow-up interviews, and lesson observations to collect data. The teachers were obsd. teaching the topic of stoichiometry to grade 11 students. Data anal. started with a coding system that resulted in themes of meanings suggested by teachers. The results show that chem. teachers participating in the study were aware of different teaching strategies but their choice of those strategies was not informed by an attempt to transform the concepts. It is argued that teachers' failure to transform concepts will have a neg. impact on students' understanding of these concepts.
23. 23
Reina, M.; This, H.; Reina, A. Improving the Understanding of Chemistry by Using the Right Words: A Clear-Cut Strategy to Avoid Misconceptions When Talking about Elements, Atoms, and Molecules. J. Chem. Educ. 2022, 99 (8), 2999– 3006, DOI:
10.1021/acs.jchemed.2c00411
23
Improving the understanding of chemistry by using the right words: A clear-cut strategy to avoid misconceptions when talking about elements, atoms and molecules
Reina, Miguel; This, Herve; Reina, Antonio
Journal of Chemical Education
(2022),
99
(8),
2999-3006CODEN:
JCEDA8;
ISSN:0021-9584.
(American Chemical Society and Division of Chemical Education, Inc.)
A language is a system of communication, consisting of a set of sounds or written symbols that enable people to communicate. In chem., a particular language is required in order to represent the phenomenol. world by means of symbols. Choosing the right words and knowing the precise definitions for chem. concepts is needed for avoiding misconceptions. Despite the creation of International Union of Pure and Applied Chem. (IUPAC), fundamental notions are still undergoing const. debate, in particular concerning the conceptual distinction between element, atom, mol., compd., chem. species and substance. In this article, we propose to better distinguish categories of chem. objects and their element, and to define a mol. as a stable and elec. neutral chem. entity that exists with a finite no. of chem. bonds (0 to n, with n finite no.). This new definition implies that noble gases, i.e., reduced to one isolated atom, are made of mols., that all bonded atoms are atoms, and enables distinguishing both of these from compds. We insist that chem. education makes a clear difference between objects and categories, emphasizing the importance of understanding that elements and compds. represent categories of nuclei, whereas chem. entities and substances are a sample of matter. We also point out a no. of nuances in chem. language, primarily in order to avoid misconceptions regarding the macroscopic phys. properties relating to chem. substances and chem. objects.
24. 24
Bierenstiel, M.; Snow, K. Periodic Universe: A Teaching Model for Understanding the Periodic Table of the Elements. J. Chem. Educ. 2019, 96 (7), 1367– 1376, DOI:
10.1021/acs.jchemed.8b00740
24
Periodic Universe: A Teaching Model for Understanding the Periodic Table of the Elements
Bierenstiel, Matthias; Snow, Kathy
Journal of Chemical Education
(2019),
96
(7),
1367-1376CODEN:
JCEDA8;
ISSN:0021-9584.
(American Chemical Society and Division of Chemical Education, Inc.)
The Periodic Table of the Elements (PTE) is arguably one of the most central topics in chem. This article provides a crit. review of current teaching practices of the PTE to high-school students and undergraduate university students. It also provides a new teaching model, the "periodic universe", for understanding of the PTE through building on its foundational periodicity. Students learn through a guided re-creation of the PTE from simple periodic simulations and interdisciplinary processes. The periodic-universe teaching model directs students to conceptualize, rather than memorize, relationships between the elements by predicting patterns in progressively challenging simulated worlds. This mixed-method research project consists of both quant. (n = 58) and qual. (n = 15) responses from university undergraduate students, demonstrating the utility of inquiry-based learning of the PTE for novice chem. students. Herein, we position the periodic-universe teaching model in the current educational literature of inquiry and interdisciplinary studies by outlining the function of the model and illustrating its effectiveness on the basis of empirical anal. with undergraduate students in a postsecondary-education setting. The results indicate that knowledge of the PTE increases after instruction with the model, and that nonchem.-major students have increased engagement, understanding, and appreciation of the importance of periodicity to chem. when presented with the periodic-universe teaching model.
25. 25
Jusniar, J.; Effendy, E.; Budiasih, E.; Sutrisno, S. Eliminating Misconceptions on Reaction Rate to Enhance Conceptual Understanding of Chemical Equilibrium Using EMBE-R Strategy. Int. J. Instr. 2021, 14 (1), 85– 104, DOI:
10.29333/iji.2021.1416a
There is no corresponding record for this reference.
26. 26
Cai, S. Harry Potter Themed Digital Escape Room for Addressing Misconceptions in Stoichiometry. J. Chem. Educ. 2022, 99 (7), 2747– 2753, DOI:
10.1021/acs.jchemed.2c00178
26
Harry Potter Themed Digital Escape Room for Addressing Misconceptions in Stoichiometry
Cai, Shiying
Journal of Chemical Education
(2022),
99
(7),
2747-2753CODEN:
JCEDA8;
ISSN:0021-9584.
(American Chemical Society and Division of Chemical Education, Inc.)
Chem. is often seen as an abstr. and content-heavy subject. Many students struggle with chem. because they attempt to memorize the content without understanding the concepts. As a result, students often have misconceptions. COVID-19 has driven teaching and learning online, and an escape room teaching method, which is a way to enhance student engagement, has gained popularity among educators in higher education. This study examines the effectiveness of teaching through a digital escape room as compared to a typical online lesson with a collaborative learning method to address misconceptions in stoichiometry. A Harry Potter themed digital escape room is created to spark the students' interest in chem. and address misconceptions. Thirty-eight students from the Nanyang Polytechnic Foundation Program participated in this study. The students completed a pretest, a post-test, and a survey, in addn. to participating in the digital escape room and a typical online lesson. Four topics were covered in this study: balancing chem. equations, calcg. empirical formulas, identifying the type of chem. bonding, and interpreting element symbols. Out of these four topics, it was discovered that students tended to have difficulty calcg. empirical formulas. It was found that, on av., students showed a 10% improvement in test scores after being taught through the digital escape room. This result is similar to results obtained from a typical online lesson with a collaborative learning method (9% improvement). This implies that a digital escape room is equally as effective as a typical online lesson with a collaborative learning method at addressing misconceptions. Teaching through a digital escape room has shown potential addnl. benefits of enhancing soft skills, promoting teamwork, the ability to work under time pressure, communication skills, innovation competency, and increasing student motivation. The researcher recommends the use of a digital escape room to complement typical lessons for these addnl. benefits.
27. 27
Tsaparlis, G. Teaching and Learning Electrochemistry. Isr. J. Chem. 2019, 59 (6–7), 478– 492, DOI:
10.1002/ijch.201800071
27
Teaching and Learning Electrochemistry
Tsaparlis, Georgios
Israel Journal of Chemistry
(2019),
59
(6-7),
478-492CODEN:
ISJCAT;
ISSN:0021-2148.
(Wiley-VCH Verlag GmbH & Co. KGaA)
A review. Numerous conceptual difficulties and misconceptions have been reported in the science and chem. education literature about electrochem. concepts such as electrolytes, redox equations, and about electrochem. (galvanic and electrolytic) cells. Other studies have considered teaching approaches aimed at improving learning and at overcoming misconceptions. This paper reviews these studies and considers strategies and approaches for the effective teaching of electrochem. Then, the review focuses on problem solving in equil. electrochem., esp. problems involving the Nernst equation. The main findings of four studies with Greek university chem. students, dealing with: (i) algorithmic problem-solving ability; (ii) practice on problem solving; (iii) the effect of the format of a problem; and (iv) the construction and evaluation of a website devoted to electrochem. problem solving, are reported.
28. 28
Islamiyah, K. K.; Rahayu, S.; Dasna, I. W. The Effectiveness of Remediation Learning Strategy in Reducing Misconceptions on Chemistry: A Systematic Review. Tadris J. Kegur. dan Ilmu Tarb. 2022, 7 (1), 63– 77, DOI:
10.24042/tadris.v7i1.11140
There is no corresponding record for this reference.
29. 29
Vosniadou, S. The Development of Students’ Understanding of Science. In Frontiers in Education; Frontiers Media SA, 2019; Vol. 4, p 32. DOI:
10.3389/feduc.2019.00032 .
There is no corresponding record for this reference.
30. 30
DiSessa, A. A. Toward an Epistemology of Physics. Cogn. Instr. 1993, 10 (2–3), 105– 225, DOI:
10.1080/07370008.1985.9649008
There is no corresponding record for this reference.
31. 31
Talanquer, V. Commonsense Chemistry: A Model for Understanding Students’ Alternative Conceptions. J. Chem. Educ. 2006, 83 (5), 811, DOI:
10.1021/ed083p811
31
Commonsense chemistry: a model for understanding students' alternative conceptions
Talanquer, Vicente
Journal of Chemical Education
(2006),
83
(5),
811-816CODEN:
JCEDA8;
ISSN:0021-9584.
(Journal of Chemical Education, Dept. of Chemistry)
The recognition and characterization of student beliefs and prior knowledge seems to be crucial to helping them build scientific understandings. The paper discusses the results of a research project guided by the assumption that a common explanatory framework does exist and it can be described by analyzing the research literature on alternative conceptions in chem.
32. 32
Talanquer, V. Chemistry Education: Ten Heuristics to Tame. J. Chem. Educ. 2014, 91 (8), 1091– 1097, DOI:
10.1021/ed4008765
32
Chemistry Education: Ten Heuristics To Tame
Talanquer, Vicente
Journal of Chemical Education
(2014),
91
(8),
1091-1097CODEN:
JCEDA8;
ISSN:0021-9584.
(American Chemical Society and Division of Chemical Education, Inc.)
Students in our chem. classes often generate shallow responses to our questions and problems. They fail to recognize relevant cues in making judgments and decisions about the properties of chem. substances and processes, and make hasty generalizations that frequently lead them astray. Results from research in the psychol. of decision making can help us better understand how students approach chem. tasks under conditions of limited knowledge, time, or motivation. In this contribution, I describe 10 cognitive heuristics that are often responsible for biases in student thinking. Helping students tame these heuristics may allow us to foster more meaningful learning in chem. classrooms.
33. 33
Macrie-Shuck, M.; Talanquer, V. Exploring Students’ Explanations of Energy Transfer and Transformation. J. Chem. Educ. 2020, 97 (12), 4225– 4234, DOI:
10.1021/acs.jchemed.0c00984
33
Exploring students' explanations of energy transfer and transformation
Macrie-Shuck, Michael; Talanquer, Vicente
Journal of Chemical Education
(2020),
97
(12),
4225-4234CODEN:
JCEDA8;
ISSN:0021-9584.
(American Chemical Society and Division of Chemical Education, Inc.)
The ability to build mechanistic explanations of natural phenomena is highly valued in all scientific disciplines, including chem. In this paper, we summarize the results of a qual. study designed to characterize the nature of the mechanistic explanations built by college General Chem. students when analyzing a phys. process involving energy transfer between parts of a system at different temps., and a chem. process involving energy transformation and transfer due to chem. bonding. Prior studies have mostly focused on eliciting "what" students think happens in terms of energy transfer when chem. bonds break or form (e.g., energy is absorbed or released) and in this investigation we provide insights into their reasoning about "how" and "why" it happens. Our findings reveal that many students struggle to build sound mechanistic explanations for energy transfer between objects at different temps. and for energy transformation and transfer during the formation of chem. bonds.
34. 34
Talanquer, V. How Do Students Reason about Chemical Substances and Reactions? In Concepts of matter in science education; Tsaparlis, G., Sevian, H., Eds.; Springer: Dordrecht, 2013; pp 331– 346. DOI:
10.1007/978-94-007-5914-5_16 .
There is no corresponding record for this reference.
35. 35
Talanquer, V. Progressions in Reasoning about Structure–Property Relationships. Chem. Educ. Res. Pract. 2018, 19 (4), 998– 1009, DOI:
10.1039/C7RP00187H
There is no corresponding record for this reference.
36. 36
Talanquer, V. The Complexity of Reasoning about and with Chemical Representations. JACS Au 2022, 2, 2658, DOI:
10.1021/jacsau.2c00498
36
The complexity of reasoning about and with chemical representations
Talanquer, Vicente
JACS Au
(2022),
2
(12),
2658-2669CODEN:
JAAUCR;
ISSN:2691-3704.
(American Chemical Society)
External visual representations of chem. entities and processes (chem. representations) play a crit. role in chem. thinking and practice. They support reasoning by serving as bridges between the macroscopic world and the chem. models that help us make sense of the properties and behaviors of substances in our surroundings. Consequently, many chem. education research studies have been carried out to explore and foster students representational competency in our discipline. Nevertheless, in this perspective I argue that investigations in this area would benefit from a more in-depth anal. of how the distinctive characteristics of chem. representations affect student reasoning. I identify four dimensions of variation in these representations (iconicity, quantitativeness, granularity, dimensionality) that affect students ability to interpret, connect, generate, and use chem. representations. I discuss how these features influence the unpacking or packing of information during different types of tasks, affecting sense-making and perceptual competency. Implications for chem. education research and practice are considered.
37. 37
Plowright, D. Using Mixed Methods: Frameworks for an Integrated Methodology; SAGE Publications: London, 2011.
There is no corresponding record for this reference.
38. 38
Horvat, S.; Segedinac, M. D.; Milenković, D. D.; Hrin, T. N. Development of Procedure for the Assessment of Cognitive Complexity of Stoichiometric Tasks. Maced. J. Chem. Chem. Eng. 2016, 35 (2), 275– 284, DOI:
10.20450/mjcce.2016.893
38
Development of procedure for the assessment of cognitive complexity of stoichiometric tasks
Horvat, Sasa; Segedinac, Mirjana D.; Milenkovic, Dusica D.; Hrin, Tamara N.
Macedonian Journal of Chemistry and Chemical Engineering
(2016),
35
(2),
275-284CODEN:
MJCCA9;
ISSN:1857-5552.
(Society of Chemists and Technologists of Macedonia)
The aim of this study was the creation of a procedure for detg. the cognitive complexity of stoichiometric tasks and its validation. The created procedure included an assessment of the difficulty of concepts and skills, and an assessment of the concepts' interactivity. There were 82 students who participated in the study, with an educational profile of a pharmaceutical technician. As a research instrument for assessing performance, test of knowledge was used. Each task in the test was followed by a seven-point Likert scale for the evaluation of invested mental effort. The validity of this instrument for the assessment of cognitive complexity was confirmed by a series of linear regression anal. where extremely high values of correlation coeffs. are obtained among the examd. variables: student's performance and invested mental effort (dependent variables) and cognitive complexity (independent variable).
39. 39
Knaus, K.; Murphy, K.; Blecking, A.; Holme, T. A Valid and Reliable Instrument for Cognitive Complexity Rating Assignment of Chemistry Exam Items. J. Chem. Educ. 2011, 88 (5), 554– 560, DOI:
10.1021/ed900070y
39
A Valid and Reliable Instrument for Cognitive Complexity Rating Assignment of Chemistry Exam Items
Knaus, Karen; Murphy, Kristen; Blecking, Anja; Holme, Thomas
Journal of Chemical Education
(2011),
88
(5),
554-560CODEN:
JCEDA8;
ISSN:0021-9584.
(American Chemical Society and Division of Chemical Education, Inc.)
The design and use of a valid and reliable instrument for the assignment of cognitive complexity ratings to chem. exam items is described in this paper. Use of such an instrument provides a simple method to quantify the cognitive demands of chem. exam items. Instrument validity was established in two different ways: statistically significant correlations between expert-based cognitive complexity ratings and student performance (as measured through statistical difficulty of items), and statistically significant correlations between expert-based cognitive complexity ratings and student mental effort ratings. Key benefits assocd. with instrument use include an enhanced understanding of the cognitive complexity of chem. assessment tasks and as a means for characterizing exam content for the measurement of cognitive development.
40. 40
Klepsch, M.; Seufert, T. Understanding Instructional Design Effects by Differentiated Measurement of Intrinsic, Extraneous, and Germane Cognitive Load. Instr. Sci. 2020, 48 (1), 45– 77, DOI:
10.1007/s11251-020-09502-9
There is no corresponding record for this reference.
41. 41
Stamovlasis, D.; Papageorgiou, G.; Tsitsipis, G. The Coherent versus Fragmented Knowledge Hypotheses for the Structure of Matter: An Investigation with a Robust Statistical Methodology. Chem. Educ. Res. Pract. 2013, 14 (4), 485– 495, DOI:
10.1039/C3RP00042G
41
The coherent versus fragmented knowledge hypotheses for the structure of matter: an investigation with a robust statistical methodology
Stamovlasis, Dimitrios; Papageorgiou, George; Tsitsipis, Georgios
Chemistry Education Research and Practice
(2013),
14
(4),
485-495CODEN:
CERPCE;
ISSN:1756-1108.
(Royal Society of Chemistry)
Research in cognitive psychol. and science education has acknowledged two antagonistic theor. perspectives concerning students' knowledge. One perspective considers students' knowledge as coherent or theory-like, explaining the learning process in terms of framework theories and stable mental models, whereas the other considers it fragmented and based on the existence of knowledge elements that can be activated, interconnected and organized accordingly. Although both perspectives continue to demonstrate empirical support, a third point of view focuses on the role of the particular topic and suggests that the stability of the in situ formed conceptions varies under different circumstances being also affected by the researchers' interpretation of data. Since methodol. issues are by far fundamental and crucial for the above theor. premises, in the present research an attempt is made to test the above hypotheses by applying a robust statistical tool, the Latent Class Anal. (LCA). Data were taken from students' responses (N = 329, aged 14-15) using an instrument designed to access students' understanding of the structure of matter and its changes of states that had previously been used to identify students' progression through a series of mental models reported in the literature. However, the results did not support the hypothesis that participants would hold these as coherent mental models. The present research adds to the ongoing dialogue on such a crucial issue for its pedagogical implications by contributing to the methodol. pathways for an advancement of science education research through theory building.
42. 42
Natividad, L. R. Understanding Alternative Conceptions: Constructivism and Nature of Science Approach. Lukad An Online J. Pedagog. 2022, 2 (1), 21– 30
There is no corresponding record for this reference.
43. 43
le
Maire, N. V.; Verpoorten, D. P.; Fauconnier, M.-L. S.; Colaux-Castillo, C. G. Clash of Chemists: A Gamified Blog to Master the Concept of Limiting Reagent Stoichiometry. J. Chem. Educ. 2018, 95 (3), 410– 415, DOI:
10.1021/acs.jchemed.7b00256
43
Clash of Chemists: A Gamified Blog To Master the Concept of Limiting Reagent Stoichiometry
le Maire, Nathalie V.; Verpoorten, Dominique Ph.; Fauconnier, Marie-Laure S.; Colaux-Castillo, Catherine G.
Journal of Chemical Education
(2018),
95
(3),
410-415CODEN:
JCEDA8;
ISSN:0021-9584.
(American Chemical Society and Division of Chemical Education, Inc.)
In a first-year university course, students experienced a new learning activity ("Clash of Chemists") prompting them to create and share personal analogies explaining the difference between stoichiometric and nonstoichiometric reaction conditions, also known in the literature as "limiting reagent stoichiometry". To support students' commitment to this unusual assignment, the instructional design drew on a blog enriched with game mechanics (tournament, video rewards, and leaderboard), as found in popular mini-games. The paper reports on the activity's outputs and on participants' perceptions of its usability, usefulness, and generated satisfaction. Overall students' reception of this mini-game was pos. A significant difference between players and nonplayers' end-of-term exam results was highlighted.
44. 44
Stott, A. E. The Efficacy of Instruction in Application of Mole Ratios and Submicro- and Macro-Scopic Equivalent Forms of the Mole within the Unit Factor Method. Chem. Educ. Res. Pract. 2023, 24, 551– 566, DOI:
10.1039/D2RP00245K
There is no corresponding record for this reference.
Citation
Citation and abstract
Citation and references
More citation options
Share on
Facebook
X
Wechat
LinkedIn
Reddit
Email
Bluesky
Abstract
Introduction
Epistemology and Pedagogy
Empirical Discrimination between Coherent and Fragmented Knowledge
Aims
Research Questions
Method
Results and Discussion
Suggestions for Practice
Limitations and Suggestions for Further Investigation
Author Information
Appendix A
References
Cited By
Wednesday, August 13, 2025
Notice: ACS is aware of a recent issue that may have caused some users to be temporarily blocked from accessing content. The issue has been resolved, and we sincerely apologize for any inconvenience caused. If you continue to experience access problems, please contact us at acs_pubs_assist@acs.org. |
13090 | https://papers.ssrn.com/sol3/Delivery.cfm/0290f391-b325-4ca2-9bac-c35ab329561c-MECA.pdf?abstractid=5340151&mirid=1 | A Hamiltonian Approach for Point Vortices on Non-Orientable Surfaces by Nataliya Balabanova, James Montaldi :: SSRN
Skip to main content
Make use of personalized features like alerts and saved searches
Create accountSign in
Product & Services
Research Paper Series
Site Subscriptions
Sponsored Services
Jobs & Announcements
Conference Papers
Partners in Publishing
First Look
Subscribe
Submit a paper
Browse
Rankings
Top Papers
Top Authors
Top Organizations
Blog↗
Contact
Product & Services
Research Paper Series
Site Subscriptions
Sponsored Services
Jobs & Announcements
Conference Papers
Partners in Publishing
First Look
Subscribe
Submit a paper
Browse
Rankings
Top Papers
Top Authors
Top Organizations
Blog↗
Contact
Create accountSign in
This is a preprint article, it offers immediate access but has not been peer reviewed.
Download This Paper
Open PDF in Browser
Add Paper to My Library
Share:
Permalink
Using these links will ensure access to this page indefinitely
Copy URL
Copy DOI
A Hamiltonian Approach for Point Vortices on Non-Orientable Surfaces
52 Pages Posted: 5 Jul 2025
See all articles by Nataliya Balabanova
Nataliya Balabanova
University of Birmingham
James Montaldi
affiliation not provided to SSRN
Abstract
We investigate the motion of point vortices on the M¨obius band and Klein bottle. Since these are non-orientable surfaces, the standard Hamiltonian approach does not apply. We therefore begin by establishing a modified Hamiltonian approach which works for arbitrary non-orientable surfaces, through describing the phase space, the Hamiltonian and the local equations of motion. To this end, we adapt some of the known notions of vortex dynamics to non-orientable surfaces. For both of the surfaces of interest, we writeHamiltonian-type equations of vortex motion explicitly and follow that by the description of relative equilibria and an investigation of the motion of one and two vortices.
Keywords: Point vortices, Hamiltonian systems, Non-orientable surfaces, Dynamical Systems
Suggested Citation:Suggested Citation
Balabanova, Nataliya and Montaldi, James, A Hamiltonian Approach for Point Vortices on Non-Orientable Surfaces. Available at SSRN: or
Nataliya Balabanova (Contact Author)
University of Birmingham ( email )
Edgbaston, B15 2TT
United Kingdom
James Montaldi
affiliation not provided to SSRN ( email )
Download This Paper
Open PDF in Browser
42 References
R Abraham , J E Marsden , T Ratiu , Manifolds Tensor Analysis, and Applications, volume 75 Posted: 2012
H Aref Motion of three vortices The Physics of Fluids, volume 22, issue 3, p. 393- 400 Posted: 1979
H Aref Point vortex dynamics: a classical mathematics playground Journal of Mathematical Physics, volume 48, issue 6 Posted: 2007
H Aref , J B Kadtke , I Zawadzki , L J Campbell , B Eckhardt Point vortex dynamics: recent results and open problems Fluid Dynamics Research, volume 3, issue 1-4 Posted: 1988
Load more
0 Citations
Fetch Citations
Do you have a job opening that you would like to promote on SSRN?
Place Job Opening
Paper statistics
Downloads
13
Abstract Views
178
42 References
PlumX Metrics
Usage
Abstract Views: 159
Downloads: 12
see details
Usage
Abstract Views: 159
Downloads: 12
see details
Related eJournals
Mathematical Physics eJournal
Follow #### Mathematical Physics eJournal
Subscribe to this fee journal for more curated articles on this topic
FOLLOWERS 45 PAPERS 2,876
Topology eJournal
Follow #### Topology eJournal
Subscribe to this fee journal for more curated articles on this topic
FOLLOWERS 21 PAPERS 295
Dynamical Systems eJournal
Follow #### Dynamical Systems eJournal
Subscribe to this fee journal for more curated articles on this topic
FOLLOWERS 21 PAPERS 681
Nonlinear Dynamics eJournal
Follow #### Nonlinear Dynamics eJournal
Subscribe to this fee journal for more curated articles on this topic
FOLLOWERS 15 PAPERS 1,420
Feedback
Feedback to SSRN
Feedback(required)
Email(required)
Submit
Submit a Paper
Section 508 Text Only Pages
SSRN Quick Links
SSRN Solutions
Research Paper Series
Conference Papers
Partners in Publishing
Jobs & Announcements
Special Topic Hubs
SSRN Rankings
Top Papers
Top Authors
Top Organizations
About SSRN
Network Directors
Announcements
Contact us
FAQs
CopyrightTerms and ConditionsPrivacy Policy
All content on this site: Copyright © 2024 Elsevier Inc., its licensors, and contributors. All rights are reserved, including those for text and data mining, AI training, and similar technologies. For all open access content, the relevant licensing terms apply.
We use cookies to help provide and enhance our service and tailor content.
To learn more, visit Cookie settings | Your Privacy Choices. This page was processed by aws-apollo-l100 in 0.060 seconds
We use cookies that are necessary to make our site work. We may also use additional cookies to analyze, improve, and personalize our content and your digital experience. For more information, see ourCookie Policy
Cookie Settings Accept all cookies
Cookie Preference Center
We use cookies which are necessary to make our site work. We may also use additional cookies to analyse, improve and personalise our content and your digital experience. For more information, see our Cookie Policy and the list of Google Ad-Tech Vendors.
You may choose not to allow some types of cookies. However, blocking some types may impact your experience of our site and the services we are able to offer. See the different category headings below to find out more or change your settings.
You may also be able to exercise your privacy choices as described in our Privacy Policy
Allow all
Manage Consent Preferences
Strictly Necessary Cookies
Always active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookie Details List
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookie Details List
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site.
Cookie Details List
Targeting Cookies
[x] Targeting Cookies
These cookies may be set through our site by our advertising partners. They may be used by those companies to build a profile of your interests and show you relevant adverts on other sites. If you do not allow these cookies, you will experience less targeted advertising.
Cookie Details List
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Confirm my choices |
13091 | https://www.quora.com/Where-can-I-find-all-the-variations-of-stars-and-bars-balls-and-bins-problem-of-combinatorics | Something went wrong. Wait a moment and try again.
Balls into Bins
Combinatorics of Words
Computational Problems
Mathematics Word Problems
Stars and Bars (combinato...
Algorithms, Combinatorics
Combinatorics C
5
Where can I find all the variations of stars and bars / balls and bins problem of combinatorics?
Joe Blitzstein
Professor in the Harvard Statistics Department
·
Upvoted by
Justin Rising
, PhD in statistics and
Alon Amit
, Lover of math. Also, Ph.D. · Author has 262 answers and 4M answer views
·
10y
A general classification of balls and boxes problems in combinatorics is the
Twelvefold way
. There are 12 categories since
Balls may be distinguishable or indistinguishable;
Boxes may be distinguishable or indistinguishable;
We can choose from the following 3 options: Boxes must have at least one ball; no box can have more than one ball; or neither of these constraints.
So by the multiplication rule there are 223=12 categories. The results for some categories are easy to derive (especially using the multiplication rule), while others are much trickier. See the Wikipedia article I linked to a
A general classification of balls and boxes problems in combinatorics is the
Twelvefold way
. There are 12 categories since
Balls may be distinguishable or indistinguishable;
Boxes may be distinguishable or indistinguishable;
We can choose from the following 3 options: Boxes must have at least one ball; no box can have more than one ball; or neither of these constraints.
So by the multiplication rule there are 223=12 categories. The results for some categories are easy to derive (especially using the multiplication rule), while others are much trickier. See the Wikipedia article I linked to and the books I recommend in
What are the best books about combinatorics?
, especially Stanley's
Enumerative Combinatorics (Volume 1)
, for further information.
Assistant
Bot
·
1y
The "stars and bars" or "balls and bins" problems in combinatorics deal with distributing indistinguishable objects (stars or balls) into distinguishable boxes (bars or bins). There are several variations of this problem, each with different constraints and conditions. Here are some common variations and where you can find more information about them:
Common Variations
Basic Stars and Bars:
Problem: Distributing indistinguishable objects into distinguishable boxes.
Formula: The number of ways is given by .
Stars and Bars with Constraints:
Problem: Distributing
The "stars and bars" or "balls and bins" problems in combinatorics deal with distributing indistinguishable objects (stars or balls) into distinguishable boxes (bars or bins). There are several variations of this problem, each with different constraints and conditions. Here are some common variations and where you can find more information about them:
Common Variations
Basic Stars and Bars:
Problem: Distributing indistinguishable objects into distinguishable boxes.
Formula: The number of ways is given by .
Stars and Bars with Constraints:
Problem: Distributing indistinguishable objects into distinguishable boxes with constraints on the minimum or maximum number of objects in each box.
Approach: Use generating functions or inclusion-exclusion principles to account for constraints.
Balls and Bins with Distinguishable Balls:
Problem: Distributing distinguishable balls into distinguishable bins.
Formula: The number of ways is .
At Least One in Each Bin:
Problem: Distributing indistinguishable balls into distinguishable bins with at least one ball in each bin.
Formula: This can be solved by first placing one ball in each bin and then using stars and bars for the remaining balls: .
Negative Solutions:
Problem: Finding the number of ways to distribute indistinguishable objects into distinguishable boxes, allowing for negative counts (which can be interpreted via combinatorial identities).
Approach: Use generating functions or the principle of inclusion-exclusion.
Resources for Further Study
Textbooks:
Concrete Mathematics by Ronald Graham, Donald Knuth, and Oren Patashnik.
Introduction to Combinatorial Mathematics by C.L. Liu.
Generatingfunctionology by Herbert Wilf.
Online Courses:
Websites like Coursera or EdX offer combinatorics courses that often cover these topics.
MIT OpenCourseWare has free resources on combinatorics.
Research Papers and Articles:
Look for combinatorics journals or publications that focus on discrete mathematics for more advanced variations.
Online Forums and Communities:
Math Stack Exchange and other mathematics forums can be great places to ask specific questions about variations you encounter.
YouTube Channels:
Educational channels such as 3Blue1Brown or Mathologer often cover combinatorial topics.
By exploring these resources, you should be able to find a wide variety of "stars and bars" problems and their solutions.
Arpit Shukla
Programmer
·
Author has 113 answers and 455.6K answer views
·
Updated 7y
Related
How do I use the stars and bars combinatorics?
First of all let me clarify that This explanation was given by Neal Wu in some other answer.
ANS. :
How many ways can you give 8 apples to 4 people? Not everyone has to receive the same number of apples, and someone could even receive no apples.
Sounds pretty simple... there couldn't be that many different ways, right? Let's give the people names: Alice, Bob, Carol, and Dave. So you could give Alice 8 apples, or you could give her 7 apples and give someone else 1 apple, or you could give Alice 6 apples and then... OK, maybe this is more complicated than we thought.
Let's just start listing example
First of all let me clarify that This explanation was given by Neal Wu in some other answer.
ANS. :
How many ways can you give 8 apples to 4 people? Not everyone has to receive the same number of apples, and someone could even receive no apples.
Sounds pretty simple... there couldn't be that many different ways, right? Let's give the people names: Alice, Bob, Carol, and Dave. So you could give Alice 8 apples, or you could give her 7 apples and give someone else 1 apple, or you could give Alice 6 apples and then... OK, maybe this is more complicated than we thought.
Let's just start listing examples at random and see if we can find a pattern.
Apples given to (Alice, Bob, Carol, Dave):
(8, 0, 0, 0)
(7, 1, 0, 0)
(7, 0, 1, 0)
(6, 2, 0, 0)
(2, 2, 2, 2)
(5, 1, 0, 2)
So what's going on? We really just need to pick four whole numbers that add up to 8. But how many ways can we do that?
Hmm, let's try representing the apples with stars instead. Same examples as before:
1
2
3
4
5
6
7
Alice | Bob | Carol | Dave
| | |
| | |
| | |
| | |
| | |
| | |
But wait. What if we squished down each of the rows? It just becomes a sequence of stars and dividers:
1
2
3
4
5
6
7
Alice|Bob|Carol|Dave
|||
|||
|||
|||
|||
|||
So now these are sequences of 11 symbols that always contain 8 's and 3 |'s. But can we get any sequence like this? ... Yes, any sequence with 8 's and 3 |'s can be turned uniquely into a way to divide up the apples. Just count the number of stars between dividers.
What about the other way around? Does every way to divide the apples give a different sequence? Yes, that's true too. So we have a 1-to-1 correspondence, also called a bijection. That means the number of ways to divide up the apples is the exact same as the number of 11-symbol sequences that have 8 's and 3 |'s.
But that number is simple. It's exactly 11 choose 3, or the number of ways to get from the top of Pascal's triangle to row 11, entry 3. So we know the answer is 165.
Related questions
How do I use the stars and bars combinatorics?
What are the best books about combinatorics?
How is the stars and bars (combinatorics) concept in mathematics different from the coin change problem in dynamic programming?
How can I get better at combinatorics?
How do I approach the following combinatorics problems?
Jo Suis Foutu
Upvoted by
Anurag Bishnoi
, Ph.D. Mathematics, Ghent University (2016)
·
12y
Related
How should I study combinatorics?
This is a common problem for people learning mathematics, and it's even more of a problem with combinatorics because there are so many ad hoc methods used in the field. It's possible to know how to solve a million problems in combinatorics and feel like it's the first day of class when you're looking at number 1,000,001.
Part of the answer is that, like any skill, it takes experience and a lot of practice to get a feeling for which methods might apply to a particular problem in combinatorics. You'll have to keep doing problems from the book using the methods in the particular section, and at
This is a common problem for people learning mathematics, and it's even more of a problem with combinatorics because there are so many ad hoc methods used in the field. It's possible to know how to solve a million problems in combinatorics and feel like it's the first day of class when you're looking at number 1,000,001.
Part of the answer is that, like any skill, it takes experience and a lot of practice to get a feeling for which methods might apply to a particular problem in combinatorics. You'll have to keep doing problems from the book using the methods in the particular section, and at some point you'll start to find that you'll have ideas of which methods might be effective independent of the book.
My other suggestion has to do with the way you study. You talk about it, but in my experience, a lot of people who are learning mathematics from a textbook use a procedure something like this to do exercises: (1) Find a worked example in the section that's kind of like the exercise. (2) Go back and forth between the exercise and the worked example until they get an answer. (3) Check in the back of the book to see if it's right. (4) If it's right, stop. (5) If it's not right go back to step (1).
The trouble with this method is that it's possible to get the right answer by transforming steps from the worked exercise without ever really internalizing the reasons that those steps are effective. This leaves students with the feeling of having accomplished something (the right answer!!) but without actually having learned anything from the experience.
The worked examples function like training wheels on a bike. At some point with bikes, though, it's necessary to remove them and then fall down a lot. In order to avoid this pitfall, I recommend that you treat the worked examples like exercises for you to do. Cover up the work with a piece of paper. Try to solve the problem. If after an honest effort you can't think of what to do, slide the paper down a little bit and read the first sentence of the solution. Treat it as a hint and keep trying. Repeat as often as necessary.
You also don't say whether you're in school or not. If you are I recommend that you find some people to discuss problems with, your teacher if you have one, or other people who are interested in the same material. Mathematics is a very social activity. Also, you might get some additional books. Rosen is a very comprehensive introductory text, but it's also very dry and confusing in places. Most discrete math books at that introductory level have the same material in them, and having a few others at hand to see how different authors explain things can be useful. You can usually pick up old editions quite cheaply on amazon.
It's a fact about learning mathematics that you have to struggle with problems, not just understand how solutions that other people have done work. My analogy with bike riding is pretty accurate in that way. Don't feel bad if you're having trouble, though. Despite all the bad publicity she got for saying it, Barbie was right when she said that math is hard.
Pranav Shankar
Mathematician by nature, Software Engineer by profession.
·
Author has 114 answers and 2.9M answer views
·
9y
Related
What are the unsolved problems of combinatorics?
There was a particular problem that we had received on a combinatorics assignment in University.
Without divulging too much information about the question (for copyright policies), it said something on the lines of "Such and such, find a simple combinatorial proof for solving this answer". I racked my brains to try and find an easy proof but I could never seem to even come close, every proof that I thought of was shot down by a case that failed. I was insanely stressed, an easy proof?! Why can't I think of an easy proof?! I submitted it without completing the last question. I went forward at t
There was a particular problem that we had received on a combinatorics assignment in University.
Without divulging too much information about the question (for copyright policies), it said something on the lines of "Such and such, find a simple combinatorial proof for solving this answer". I racked my brains to try and find an easy proof but I could never seem to even come close, every proof that I thought of was shot down by a case that failed. I was insanely stressed, an easy proof?! Why can't I think of an easy proof?! I submitted it without completing the last question. I went forward at the end of the next week to collect my assignment, I received a perfect score. I thought it was a mistake because I had left the answer blank. The comment was made by the professor in the next class, that there was no easy proof for it and if you found one, we were to let him know.
That being said, there was a lot of crazy problems in Combinatorics and Optimization that have come up over the last couple of years, I have heard of a couple of the really popular ones like Babai Graph Optimization. I have not, however, heard of plenty of the ones given in the lists below. I would recommend going through them to find a (slightly exhaustive) list of problems.
Page on mit.edu
Problems in Graph Theory and Combinatorics
Unsolved Combinatorial Problems, Part I
Unsolved Problems
Most of them are in Graph theory.
Hope this helped!
Good luck!
Mark Gritter
working computer scientist and amateur mathematician
·
Author has 5.7K answers and 11.7M answer views
·
9y
Related
How is the stars and bars (combinatorics) concept in mathematics different from the coin change problem in dynamic programming?
Work through some small examples and you will see how the problems are distinct. The coin-change problem is "harder" so the methods for the coin-change problem will also work for simpler stars-and-bars problems.
Stars-and-bars problems typically have a fixed number of "bars" and a fixed number of "stars". For example: how many ways can we pick 3 positive integers that add up to 21? The diagram is 21 stars:
and we must place two bars --- no more or no fewer --- in the gaps between, with no two bars in the same gap, for example:
| |
Work through some small examples and you will see how the problems are distinct. The coin-change problem is "harder" so the methods for the coin-change problem will also work for simpler stars-and-bars problems.
Stars-and-bars problems typically have a fixed number of "bars" and a fixed number of "stars". For example: how many ways can we pick 3 positive integers that add up to 21? The diagram is 21 stars:
and we must place two bars --- no more or no fewer --- in the gaps between, with no two bars in the same gap, for example:
| |
That gives us the combinatoric answer [math]{20 \choose 2}[/math].
Coin-change problems differ in two ways. The number of "stars" in each group must match one of the coin sizes. And, the number of coins is not fixed. For example, given the problem of "how can we make 21 cents using pennies, nickels, and dimes" the following is not a valid solution:
| |
but this is:
| | | | | | | | | | | | | | | | | | | |
However, the programming technique which can solve this more general problem obviously also works for the simpler one. That is, we can solve the stars-and-bars counting problem with a recurrence (or a generating function) just like the coin-changing problem.
N( bars, stars ) = N( bars - 1, stars - 1 ) + N( bars, stars - 1 )
Related questions
What is the best site for combinatorics problems with solutions?
How should I study combinatorics?
Are people in combinatorics distinguishable or indistinguishable (combinatorics, math)?
Is there still active research in pure combinatorics or just combinatorics applied to other areas (such as geometry)?
Where is combinatorics used? Who invented combinatorics?
David Joyce
Ph.D. in Mathematics, University of Pennsylvania (Graduated 1979)
·
Upvoted by
Terry Moore
, M.Sc. Mathematics, University of Southampton (1968) and
Horst H. von Brand
, PhD Computer Science & Mathematics, Louisiana State University (1987) · Author has 9.9K answers and 68.1M answer views
·
Updated 11y
Related
What exactly is combinatorics?
For someone who isn't familiar with mathematics, the central problem of combinatorics is to determine how many things there are of a certain kind without actually counting them.
You're familiar with that every time you use addition or multiplication to determine how many things there are. For example, suppose you want to know how many trees there are in an apple orchard. If you find that there are 11 rows of trees with 18 trees in each row, you can multiply 11 times 18 to determine how many apple trees there are. You don't have to count them all.
There are lots of old, traditional methods to
For someone who isn't familiar with mathematics, the central problem of combinatorics is to determine how many things there are of a certain kind without actually counting them.
You're familiar with that every time you use addition or multiplication to determine how many things there are. For example, suppose you want to know how many trees there are in an apple orchard. If you find that there are 11 rows of trees with 18 trees in each row, you can multiply 11 times 18 to determine how many apple trees there are. You don't have to count them all.
There are lots of old, traditional methods to determine how many things there are in certain situations without counting them. An example: If you have five apples in a bowl–a Golden Delicious, a Granny Smith, a Honeycrisp, a Macintosh, and a Winesap—you might ask how many different combinations you can pick two of the apples from the bowl. For instance, you could pick the Winesap and the Granny Smith, that would be one way. Or the Granny Smith and the Honeycrisp, that would be another way. The number of ways, denoted [math]\binom{5}{2}[/math] and read "5 choose 2", is 10.
There's more to combinatorics than just counting things without counting, but that's where it started.
Anurag Bishnoi
Ph.D. in Mathematics, Ghent University (Graduated 2016)
·
Author has 313 answers and 2.1M answer views
·
10y
Related
What are some open problems in algebraic combinatorics?
A nice list by Godsil can be found here: Problems in Algebraic Combinatorics
Here are some that I like:
Is there a regular graph with valency 57, diameter two and girth five?
Is there a set of [math]{m+1 \choose 2}[/math] equiangular lines in [math]\mathbb{R}^m[/math] when [math]m > 23[/math]?
Another list of open problems in this field compiled by Stanley can be found here.
Though it should be noted that this area is quite vast and not so well defined. It can contain many topics from Coding theory, Finite geometry, Representation theory, and Group theory. Here's what the Journal of Algebraic Combinatorics says about its scope:
T
A nice list by Godsil can be found here: Problems in Algebraic Combinatorics
Here are some that I like:
Is there a regular graph with valency 57, diameter two and girth five?
Is there a set of [math]{m+1 \choose 2}[/math] equiangular lines in [math]\mathbb{R}^m[/math] when [math]m > 23[/math]?
Another list of open problems in this field compiled by Stanley can be found here.
Though it should be noted that this area is quite vast and not so well defined. It can contain many topics from Coding theory, Finite geometry, Representation theory, and Group theory. Here's what the Journal of Algebraic Combinatorics says about its scope:
The Journal of Algebraic Combinatorics publishes papers in which combinatorics and algebra interact in a significant and interesting fashion. This interaction might occur through the study of combinatorial structures using algebraic methods, or the application of combinatorial methods to algebraic problems. The combinatorics might be enumerative, or involve matroids, posets, polytopes, codes, designs, or finite geometries. The algebra could be group theory, representation theory, lattice theory or commutative algebra, to mention just a few possibilities.
Aalok Thakkar
math undergrad
·
Author has 72 answers and 498.5K answer views
·
9y
Related
How do I approach the following combinatorics problems?
There should be some limit on the number of letters in the given word. Let us define this limit as l. Therefore, the question becomes: "How many words of length l can be formed from n characters such that the the conditions a holds?"
Let us chuck the condition that first and last characters should be same for a while. We shall consider it at last. We will construct a recursive formula to solve the problem.
Let us consider all words with just one letter. All such words have only one letter hence the first and the last letter is the same. For this case, we can have n possible words, each with eith
There should be some limit on the number of letters in the given word. Let us define this limit as l. Therefore, the question becomes: "How many words of length l can be formed from n characters such that the the conditions a holds?"
Let us chuck the condition that first and last characters should be same for a while. We shall consider it at last. We will construct a recursive formula to solve the problem.
Let us consider all words with just one letter. All such words have only one letter hence the first and the last letter is the same. For this case, we can have n possible words, each with either of the n characters.
Let us consider all words with just two letters. All such words shall have the first letter and the second letter different, as the condition forces that no two adjacent letters should be the same. Hence, the first and the last letters shall be different, which is against the condition. Hence, no such words exist.
Let us consider all words with just three letters. The first letter of such words can have one of the n letters. The second can have all but the one used for the first, hence (n-1) letters. The third is the last hence there is only one possibility, which is the first letter. Hence total ways to do it is n(n-1)
Let us consider all words with just four letters. It is just inserting another letter at the 2nd position in every three letter word. However, the character to be inserted should not be the first letter, or the second letter for the 3 letter word case. Hence the number of possible characters are (n-2) and the number of possible words are n(n-1)(n-2)
Let us consider all words with just five letters. It is again inserting another letter at the second position in every three letter word. Alike the previous step, the number of possible characters are (n-2) and total ways possible are n(n-1)(n-2)(n-2). But there is another thing to consider. We should also count the possibilities where the third and first letters are the same. The number of ways to do that is same as the number of ways to count all three letter words. Now in those words we can insert one of the (n-1) different characters. Hence, possible ways are: n(n-1)(n-1) + n(n-1)(n-2)(n-2).
The same logic works for all cases. Hence we get the formula:
[math]f(1) = n[/math]
[math]f(2) = 0[/math]
[math]f(3) = (n)(n-1)[/math]
[math]f(4) = n(n-1)(n-2)[/math]
[math]f(l) = (n-2)f(n-1) + (n-1)f(n-2) \forall l >4[/math]
Now conditions b and c are easy.
For condition c, we have the first letter fixed. Hence, the number of ways to make three letter words becomes (n-1) and for four letter words becomes (n-1)(n-2). The rest of the logic holds. This gives us:
[math]f(1) = 1[/math]
[math]f(2) = 0[/math]
[math]f(3) = (n-1)[/math]
[math]f(4) = (n-1)(n-2)[/math]
[math]f(l) = (n-2)f(n-1) + (n-1)f(n-2) \forall l >4[/math]
Actually, by fixing the first and the last letters, we have just divided the number of possibilities by n, for all cases.
For condition b we have that the last letter can be anything but the first letter. If we chop of the last letter of each of the words we got by condition a, condition b shall be satisfied. Hence, we have to use the same formula that we got for a, but with the little adjustment that the number of letters have decreased by one. Hence:
[math]f(1) = 0[/math]
[math]f(2) = n(n-1)[/math]
[math]f(3) = n(n-1)(n-2)[/math]
[math]f(l) = (n-2)f(n-1) + (n-1)f(n-2) \forall l >3[/math].
I hope this suffices and answers your question.
Alon Amit
PhD in Mathematics; Mathcircler.
·
Author has 8.7K answers and 172.8M answer views
·
3y
Related
Where can I find problems like the one from Michael Penn’s recent “an intricate combinatorics problem” video?
I suppose you mean this video. Penn starts by saying that the problem is from “USA Math Olympiad 1998”, and I assumed he means USAMO 1998, but unfortunately none of the problems at USAMO 1998 was this one. Nor is it from IMO 1998 or Putnam 1998.
So I’m not sure where the problem is from, but it’s indeed a combinatorics problem in the style of a fairly advanced math Olympiad. If you’re looking for more problems like this, I would look at the Putnam Archive, at AoPS collections such as the IMO, and at problem solving books such as those by Arthur Engel, Paul Zeitz and this list.
Shubham Jain
8y
Related
How do I use the stars and bars combinatorics?
star and bars is used to distribute or to solve equations like
x1 + x2 +x3 = N
and x all >= 0
means distribute n things among the 3 people such that either they get nothing or all
also if x has constraints >=1 means atleast one get 1 thing
so to slove this we use star and stick
as
example let take 10 stars and 3 sticks
………..
now to distribute them among 3 you need only 2 stick
ex. ||
so
lets do it
at first you have 10 stars you can place a stick in between them or in at or at starting
so total 11 choices
now for second you have
12 choice why ?
| you have total 11 elements and you
star and bars is used to distribute or to solve equations like
x1 + x2 +x3 = N
and x all >= 0
means distribute n things among the 3 people such that either they get nothing or all
also if x has constraints >=1 means atleast one get 1 thing
so to slove this we use star and stick
as
example let take 10 stars and 3 sticks
………..
now to distribute them among 3 you need only 2 stick
ex. ||
so
lets do it
at first you have 10 stars you can place a stick in between them or in at or at starting
so total 11 choices
now for second you have
12 choice why ?
| you have total 11 elements and you can place 2nd stick at 12 places so 12 choices
but we repeated some thing as sticks are interchangeable therefore 2! common one
so solution is (11C1 12C1 ) /2!
Anurag Bishnoi
Ph.D. in Mathematics, Ghent University (Graduated 2016)
·
Upvoted by
Aanchal Gupta
, B.Sc. (Research) Mathematics & English Literature, Shiv Nadar University (2018) · Author has 313 answers and 2.1M answer views
·
10y
Related
How do I learn combinatorics?
Solving well defined, easily stated problems using ingenious arguments that don't involve much background in "advanced" mathematics is still a part of doing research in combinatorics. In fact combinatorial problems show up in many forms in other areas of mathematics too, but usually as a small part of a long argument after a lot of reduction (see this for example:
Vipul Naik's answer to International Mathematical Olympiad (IMO): How relevant have you found your contest math experience to math study and research?
). So, it can at times be helpful to have some training in solving olympiad type pr
Solving well defined, easily stated problems using ingenious arguments that don't involve much background in "advanced" mathematics is still a part of doing research in combinatorics. In fact combinatorial problems show up in many forms in other areas of mathematics too, but usually as a small part of a long argument after a lot of reduction (see this for example: Vipul Naik's answer to International Mathematical Olympiad (IMO): How relevant have you found your contest math experience to math study and research?). So, it can at times be helpful to have some training in solving olympiad type problems. But, you should know that research in combinatorics/discrete mathematics is much, much more than just solving such problems.
Firstly, a lot of the research in combinatorics today is done using tools and ideas from other areas of mathematics (and the demarcation between different areas of mathematics is quite fuzzy). For example, you need a good background in topology (both general and algebraic) to tackle problems in Topological combinatorics, and you need a good background in group theory, linear algebra, representation theory to get into Algebraic combinatorics. See the wikipedia entry on various other such subfields of Combinatorics.
Second, even if the research problems that you are attacking do not need a lot of background, they would need some mathematical maturity on your part which is what you are (ideally) trying to gain during your undergrad years. Learning algebra, analysis, topology, logic etc. are really helpful for that. Also, it's not always that clear what background in mathematics is required to attack a given problem.
So, I would strongly recommend that you build a strong background in the traditional areas of mathematics, while exposing yourself to different aspects of combinatorics (see this for some book recommendations: Anurag Bishnoi's answer to What are the best books about combinatorics?). You could also try taking up some additional projects in areas of combinatorics that excite you (since you mentioned it: Erdős–Ko–Rado Theorems: Algebraic Approaches), so that you can get some idea about what research is like in those areas. Spending too much time on olympiad style problem during your undergrad years could be a big hindrance to that. Of course, if you enjoy doing it then go ahead and pursue it as a hobby. But, in my opinion, your time would be better spent in doing other things that I have advised.
Tim Farage
I teach Combinatorics in my Discrete Math Courses.
·
Author has 4.8K answers and 17.6M answer views
·
9y
Related
What is the difference between these two combinatorics problems (in the details)?
It is often true that problems in combinatorics that look different can actually be asking the same question.
In this case, these two questions are not asking for the same thing.
Let's look at the first question. Since the pens are identical, and each student must get at least one, you can start by giving each student one pen. There's only one way to do this, and that's nice. Because we now only have to distribute 4 identical pens to 4 students.
You can pretty easily count this 'by hand' if you wish. For instance, one student could get all 4 pens; or one student could get 3 pens and one of the r
It is often true that problems in combinatorics that look different can actually be asking the same question.
In this case, these two questions are not asking for the same thing.
Let's look at the first question. Since the pens are identical, and each student must get at least one, you can start by giving each student one pen. There's only one way to do this, and that's nice. Because we now only have to distribute 4 identical pens to 4 students.
You can pretty easily count this 'by hand' if you wish. For instance, one student could get all 4 pens; or one student could get 3 pens and one of the remaining three students could get the 4th pens, etc. If you do this you get 35.
(Here's a general way to do this. If you have n identical items and you want to know how many ways you can distribute them to r people, the answer is: [math]\binom{n+r-1}{r}[/math].
(In your example this would give [math]\binom{4+4-1}{4}=\binom{7}{4}=35[/math]).
In your second problem, when you first distribute one apple to each box, you can think of it this way. Box 1 can get any of the 6 different apples, and Box 2 can get any of the 5 remaining apples, and Box 3 can get any of the 4 remaining apples. So there are 654 = 120 ways just in your first step of putting one apple into the three boxes.
Now you have to multiply this by the number of ways to distribute the remaining apples into the boxes. We don't need to compute this to see that these are two different problems.
Related questions
How do I use the stars and bars combinatorics?
What are the best books about combinatorics?
How is the stars and bars (combinatorics) concept in mathematics different from the coin change problem in dynamic programming?
How can I get better at combinatorics?
How do I approach the following combinatorics problems?
What is the best site for combinatorics problems with solutions?
How should I study combinatorics?
Are people in combinatorics distinguishable or indistinguishable (combinatorics, math)?
Is there still active research in pure combinatorics or just combinatorics applied to other areas (such as geometry)?
Where is combinatorics used? Who invented combinatorics?
Can the "Stars and Bars" technique shortcut the need for the partition function in a certain type of Combinatorics problem? If so, how does it work?
What are the unsolved problems of combinatorics?
Combinatorics: What does it mean that order matters or does not matter?
How does the Stars-and-Bars technique in Combinatorics obviate the need to calculate integer partitions?
How do you resolve a 5-digit combinatoric problem (linear algebra, combinatorics, math)?
Related questions
How do I use the stars and bars combinatorics?
What are the best books about combinatorics?
How is the stars and bars (combinatorics) concept in mathematics different from the coin change problem in dynamic programming?
How can I get better at combinatorics?
How do I approach the following combinatorics problems?
What is the best site for combinatorics problems with solutions?
How should I study combinatorics?
Are people in combinatorics distinguishable or indistinguishable (combinatorics, math)?
Is there still active research in pure combinatorics or just combinatorics applied to other areas (such as geometry)?
Where is combinatorics used? Who invented combinatorics?
About
·
Careers
·
Privacy
·
Terms
·
Contact
·
Languages
·
Your Ad Choices
·
Press
·
© Quora, Inc. 2025 |
13092 | https://www.dreambox.com/math/skills/decimals/multiplying-decimals | Skip to content
Math Resources
Multiplying Decimals
How to Multiply Decimals
All you need to know about multiplying decimals by each other, whole numbers, and exponents. Boost your skills and confidence with this step-by-step guide.
Author
Tess Loucka
Published:
Oct 2024
Key takeaways
• Knowing how to multiply decimals is a crucial skill that you’ll use throughout your daily life.
• To multiply decimals, convert them into whole numbers first. Multiply the whole numbers together, then place the decimal points back so the final answer has the same number of decimal spaces as the two factors.
• When multiplying decimals by 10, 100, 1,000, or other exponents, move the decimal point to the right the same number of spaces as there are zeros.
Multiplying decimals is a simple math calculation that you’ll be using for the rest of your life. Calculating money, time, weight, length, and distance are all likely to involve decimals, and knowing how to multiply decimalsconfidently will save you plenty of trouble and time in the future.
For example, you multiply decimals when calculating how much gas you can afford at a rate of $3.67 per gallon, or how far you can walk at a rate of 2.5 miles per hour.
Decimal multiplicationis easy once you know the trick!
How to Multiply Decimals Step by Step
The easiest way to multiply decimalsis to convert them to whole numbers first, multiply them, then convert them back to decimals.
To explain how to multiply with decimalsusing this technique, let’s look at a few methods with examples.
Table of contents
How to multiply decimals
Practice problems
Access more math practice with DreamBox
Start Free Trial
Turn math into playtime with DreamBox Math
DREAMBOX MATH
Get started for FREE today!
Purchase DreamBox Math
Try DreamBox for 14 Days
Multiplying decimals by whole numbers
To start, let’s learn how to multiply decimals by whole numbers.
Take 0.15 x 4 as an example.
Step 1: To solve this problem, let’s turn the decimal number into a whole number by moving the decimal point to the right two places so that it falls after the digits.
In this example, 0.15 becomes 15 when we move the decimal point over 2 spaces.
Step 2: Now, multiply 15 x 4 to get 60.
Step 3: Next, convert this number back into a decimal number. Since we moved the decimal point over two spaces to the right in Step 1, we now have to move the decimal point over two spaces to the left.
The final answer is 0.15 x 4 = 0.60 or 0.6
Multiplying two decimals
Now let’s take a look at how we can multiply two decimal numbers together.
We will use 0.15 x 0.4 as our example.
Step 1: First, turn both numbers into whole numbers by moving their decimal points to the right so they fall after the digits.
Step 2: Next, we multiply our new factors, 15 and 4, to get 60.
Step 3: Lastly, convert the whole numbers back into decimals. We moved the decimal point over two spaces to the right for our first number, 0.15, and one space to the right for our second number, 0.4. That’s three spaces in total.
So, we have to move the decimal space back to the left by three spaces to get our final answer.
0.15 x 0.4 = 0.060 or 0.06
Multiplying decimals by 10, 100, and 1,000
To multiply a decimal by 10, 100, or 1,000, all we have to do is move the decimal point over to the right the same number of spaces as there are zeros.
Let’s look at an example:
11.7 x 10 = 117
There is only one zero in 10, so we move the decimal point over to the right by one to get our answer.
But how would we multiply this number by 100?
11.7 x 100 = 1,170
In this case, we have to move the decimal point over to the right by two spaces because there are two zeros in 100. That means we’ll have to move the decimal point a space past the 7. We can fill in that space with a 0.
11.7 x 1,000 is 11,700
Move the decimal point to the right by three spaces since there are three zeros in 1,000. The empty spaces can be replaced with zeros.
For more practice multiplying with decimals, download a math practice app or use a math website that offers practice problems, answers, and explanations to all your questions. The more often you practice multiplying decimals, the more confident you’ll be.
The math program that drives results
Get started today!
DreamBox adapts to your child’s level and learning needs, ensuring they are appropriately challenged and get confidence-building wins.
Purchase DreamBox Math
Try DreamBox for 14 Days
Practice Problems
Click on the boxes below to see the answers!
Question 1: What is 1.45 x 0.13?
Answer: 0.1885
Turn both numbers into whole numbers. Solve for 145 x 13 to get 1,885. Next, add the decimal points back in. You moved the decimal point to the right two spaces for 1.45 and two spaces for 0.13 to turn them into whole numbers, so move the decimal point to the left by 4 spaces for the final answer. That means 1,885 turns into 0.1885.
Question 2: How much money will you spend if you buy 12 pieces of candy that cost $1.25 each?
Answer: $15.00 or $15
Write out your equation for the problem, 1.25 x 12. Turn 1.25 into a whole number. Solve for 125 x 12 to get 1,500. Then, add the decimal points back in by counting two spaces to the left. So, 1,500 becomes 15.00 or 15. Use a dollar sign since the problem refers to money.
Question 3: What is 1.4 x 100?
Answer: 140. or 140
When multiplying by 10, 100, or 1,000, move the decimal point to the right the same number of spaces as there are zeros. Since 100 has two zeros, move the decimal point to the right by two. So, 1.4 becomes 140.
FAQs about Multiplying Decimals
How do I multiply a number by a decimal?
Turn the decimal number into a whole number. Multiply the whole numbers together, then add the decimal point back in to get your final answer.
How do you multiply two decimals by each other?
Turn both numbers into whole numbers and multiply. Put the decimal points back in by counting the decimal spaces you got rid of when converting them into whole numbers. There should be the same number of decimal spaces in the final answer.
How do you quickly multiply decimals?
Turn them into whole numbers, multiply them, and convert them back to decimals.
How many steps are there when multiplying decimals?
Three steps. Step 1: Make the decimal numbers whole numbers. Step 2: Multiply the whole numbers. Step 3: Add the decimal points back in.
Take at home math practice to the next level
Empowering parents and educators to make math practice more impactful. Plus, your kids will love it.
Try DreamBox for Free!
About the Author
Tess Loucka
Tess Loucka discovered her passion for writing in high school and has not stopped writing since. Combined with her love of numbers, she became a math and English tutor, focusing on middle- and high-school-level topics. Since graduating from Hunter College, her goal has been to use her writing to spread knowledge and the joy of learning to readers of all ages. |
13093 | https://brainly.in/question/14458680 | if two triangles have a common vertex and their basis lie on the same line then their areas are proportional - Brainly.in
Skip to main content
Ask Question
Log in
Join for free
For parents
For teachers
Honor code
Textbook Solutions
Brainly App
oasheen24
03.01.2020
Math
Secondary School
answered • expert verified
if two triangles have a common vertex and their basis lie on the same line then their areas are proportional to the lengths of their bases??
1
See answer
See what the community says and unlock a badge.
Add answer+5 pts
0:00
/
0:15
Read More
oasheen24 is waiting for your help.
Add your answer and earn points.
Add answer +5 pts
Expert-Verified Answer
10 people found it helpful
sk940178
sk940178
The areas of the two triangles are proportional to the lengths of their bases.
Step-by-step explanation:
See the attached diagram.
We have two triangles Δ ABC and Δ ADE having the same vertex A and their bases lie on the same line BE.
The area of the triangle Δ ABC will be equal to
X =
X =
Again, the area of the triangle Δ ADE will be
Y =
Hence,
Hence, the areas of the two triangles are proportional to the lengths of their bases. (Answer)
Explore all similar answers
Thanks 10
rating answer section
Answer rating 4.1
(7 votes)
Find Math textbook solutions?
See all
Class 12
Class 11
Class 10
Class 9
Class 8
Class 7
Class 6
Class 5
Class 4
Class 3
Class 2
Class 1
NCERT Class 9 Mathematics
619 solutions
NCERT Class 8 Mathematics
815 solutions
NCERT Class 7 Mathematics
916 solutions
NCERT Class 10 Mathematics
721 solutions
NCERT Class 6 Mathematics
1230 solutions
Xam Idea Mathematics 10
2278 solutions
ML Aggarwal - Understanding Mathematics - Class 8
2090 solutions
R S Aggarwal - Mathematics Class 8
1964 solutions
R D Sharma - Mathematics 9
2199 solutions
R S Aggarwal - Mathematics Class 7
2222 solutions
SEE ALL
Advertisement
Still have questions?
Find more answers
Ask your question
New questions in Math
1. Cubic Box: A cube has a side length of 5 cm. What is its surface area and volume?
An integer between -3 and -1 is.
Find the quotient and (a) a²+8a-9 ÷ a-
[Q1] @ Find the transpose of Mattix ⑥ Find the conjugate of mattix Ⓒ Find the determinant of Mattix 1 2 -3-2 -4 -3 -3 6 8 po 3+2i -4i S 7-gi 1 0 4 -2 0](
(x) RPT is a tangent at P. angleAPB=? (a) 10° (b) 20° (c) 40° (d) 80°
PreviousNext
Advertisement
Ask your question
Free help with homework
Why join Brainly?
ask questions about your assignment
get answers with explanations
find similar questions
I want a free account
Company
Careers
Advertise with us
Terms of Use
Copyright Policy
Privacy Policy
Cookie Preferences
Help
Signup
Help Center
Safety Center
Responsible Disclosure Agreement
Get the Brainly App
⬈(opens in a new tab)⬈(opens in a new tab)
Brainly.in
We're in the know
(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab) |
13094 | https://d32ogoqmya1dw8.cloudfront.net/files/mathyouneed/yau_uc_practice.pdf | Practice Problems on Unit Conversion Using Dimensional Analysis (Factor Label Method) These are practice problems. It is assumed that you have already been introduced to the method of “dimensional analysis.” Answers are provided at the end of this document. You should look at the question, work it out on paper (not in your head), before checking the answers at the end. The purpose of these problems is not merely to get the right answer, but to practice writing out the dimensional analysis setup. We will be using this method all semester and I will be asking for your setups, so don’t just work out an answer on your calculator without writing out a setup. In these practice problems, I am going to ask you to stick to ONLY the following conversions between the English and metric system (these are the only conversions that I will give you on exams). In some cases you can look up conversions elsewhere, but I would rather you didn’t. I want you to learn how to make conversions that take more than one single step. 1 inch = 2.54 cm exactly 1 lb = 454 g 1 qt = 0.946 L 1 mi = 5280 ft 1 qt = 2 pt 4qt = 1 gal You should also remember that 1 cc = 1 cm3 = 1 mL exactly. (This is a conversion you need to know.) For all problems, please show your dimensional analysis setup and give your answer to the correct significant figures. Remember you can use any of the conversions shown above. 1. Convert 3598 grams into pounds. 2. Convert 231 grams into ounces. 3. A beaker contains 578 mL of water. What is the volume in quarts? 4. How many ng are there in 5.27x10−13 kg? 5. What is 7.86 x 10−2 kL in dL? 6. What is 0.0032 gallons in cL? 7. A box measures 3.12 ft in length, 0.0455 yd in width and 7.87 inches in height. What is its volume in cubic centimeters? 8. A block occupies 0.2587 ft3. What is its volume in mm3? 9. If you are going 55 mph, what is your speed in nm per second? 10. If the density of an object is 2.87 x 10−4 lbs/cubic inch, what is its density in g/mL? Answers are on the next page. Answers: 1. 1 lb x lb = 3598 g = 7.93 lb 454 g 2. x oz = 1 lb 16 oz 231 g = 8.14 g 454 g 1 lb 3. x qt = 3 1 L 1 qt 578 mL =0.611 qt 10 mL 0.946 L 4. x ng = 12 13 10 ng 5.27x10 kg = 0.527 ng 1 kg − 5. x dL = 4 2 10 dL 7.86x10 kL = 786 dL 1 kL − 6. x cL = 2 4 qt 0.946 L 10 cL 0.0032 gal = 1.2 cL 1 gal 1 qt 1 L 7. You should the volume of a box is calculated thus; V = L x W x H. First you have to convert all the dimensions to the same unit such as inches. x in = 3.12 ft 12 in 37.4 in 1 ft = x in = 0.0455 yd 3 ft 12 in 1.64 in 1 yd 1 ft = V = 37.4 in x 1.64 in x 7.87 in = 483 in3 Note the question is asking for cm3. We know the conversion from in to cm. We can easily convert in3 to cm3 thus: 3 3 3 3 2.54 cm 2.54 cm = 1 in 1 in Thus, we can convert 483 in3 into cm3 as follows: x cm3 = 3 3 3 3 3 3 2.54 cm 483 in = 7.91 x 10 cm 1 in 8. x mm3 = 3 3 3 3 3 3 3 6 3 3 3 3 12 in 2.54 cm 10 mm 0.2587 ft = 7.326 x 10 mm 1 ft 1 in 1 cm 9. 7 10 nm 55 mi 5280 ft 12 in 2.54 cm 10 nm 1 h 1 min x = 2.5 x 10 nm/s s 1 h 1 mi 1 ft 1 in 1 cm 60 min 60 s = 10. x 4 3 3 -3 3 3 3 g 2.87x10 lb 454 g 1 in 1 cm = =7.95x10 g/mL mL 1 in 1 lb 2.54 cm 1 mL − |
13095 | https://math.stackexchange.com/questions/1760696/locus-of-centers-of-circles-tangent-to-a-given-line-and-a-given-circle | Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Locus of centers of circles tangent to a given line and a given circle
Ask Question
Asked
Modified 7 years, 6 months ago
Viewed 2k times
0
$\begingroup$
I'm trying to find locus of centers of circles tangent to $y$ axis and tangent to the unit circle ($x^2 + y^2 = 1$).
My try:
Call $(x,y)$ the center of the circle. We know $$d( (x,y), y\text{-axis}) = |x| $$
Pick point $(a,b)$ on the unit circle. Then $$\begin{align} d( (a,b), (x,y) ) &= \sqrt{ (a-x)^2 + (b-y)^2 } \[4pt] &= \sqrt{ a^2 - 2ax + x^2 + b^2 - 2by + y^2 } \[4pt] &= \sqrt{ x^2+y^2 - 2ax-2by + 1 } \end{align}$$
We have that $d( (x,y), \text{y-axis}) = d( (a,b), (x,y) ) $ so
$$\begin{align} |x| &= \sqrt{x^2+y^2 - 2ax-2by + 1} \[4pt] \implies \qquad\qquad\quad x^2 &= x^2 + y^2 - 2ax-2by+1 \[4pt] \implies \qquad 2ax+2by &=y^2 + 1 \end{align}$$
My question is:
What should I do with $a,b$? I mean, they are not constants, they are changing as $(x,y)$ changes so what am I missing here?
geometry
Share
edited Apr 27, 2016 at 6:58
Blue
84.4k1515 gold badges128128 silver badges266266 bronze badges
asked Apr 27, 2016 at 6:28
user203867user203867
$\endgroup$
Add a comment |
3 Answers 3
Reset to default
2
$\begingroup$
Let $O = (0,0)$ be the center of the circle, and $\ell$ be the line $x = 0$. Consider a point $P$ on the locus; by construction, $P$ is the center of a circle tangent to circle $O$ and line $\ell$, therefore $P$ is equidistant from the point of tangency $T$ to circle $O$ and line $\ell$. That is to say, $PT = PR$, where $R$ is the foot of the perpendicular from $P$ to $\ell$. But $O$ is collinear with $P$ and $T$, and $OT = 1$, a constant. So this suggests that we construct two lines $\ell'$ and $\ell''$ parallel on either side of $\ell$ such that the distance between $\ell$ and $\ell'$ is $1$, and similarly, the distance between $\ell$ and $\ell''$ is $1$. Then $\ell'$ and $O$ are the directrix and focus of a parabola which is one half of the desired locus of points, and $\ell''$ and $O$ are the directrix and focus of the other half of the locus. Therefore, the complete locus is the union of two parabolas as described.
Share
answered Apr 27, 2016 at 7:20
heropupheropup
145k1515 gold badges114114 silver badges201201 bronze badges
$\endgroup$
2
$\begingroup$ IS my approach correct? what am I missing ? $\endgroup$
user203867
– user203867
2016-04-27 07:35:27 +00:00
Commented Apr 27, 2016 at 7:35
$\begingroup$ @Learner Your approach is incomplete, for the reason that there are additional constraints on $(a,b)$ in terms of the choice of $(x,y)$; namely, $ay = bx$, because the point of the circle's tangency with the unit circle is collinear with the centers of the two circles; and $a^2+b^2=1$ (as given in the other answer), because $(a,b)$ lies on the unit circle. That said, the answer I provided should suggest to you a more elegant approach: if you study the answer carefully, you should conclude that $1+|x| = \sqrt{x^2+y^2}$, thus eliminate the need to introduce $(a,b)$ in the first place. $\endgroup$
heropup
– heropup
2016-04-27 16:15:52 +00:00
Commented Apr 27, 2016 at 16:15
Add a comment |
0
$\begingroup$
$a^2+b^2=1$ . As they lie on the unit circle. So now you can replace them with $x,y$ to get a second degree equation which is the required locus.
Share
answered Apr 27, 2016 at 6:49
Archis WelankarArchis Welankar
16.1k88 gold badges3737 silver badges6868 bronze badges
$\endgroup$
1
$\begingroup$ I dont get it. I get the equation $2ax + 2by = y^2 +1 $... how can we use $a^2+ b^2 =1$ in this euqation? $\endgroup$
user203867
– user203867
2016-04-27 07:14:27 +00:00
Commented Apr 27, 2016 at 7:14
Add a comment |
0
$\begingroup$
The locus of centers of the circle have the coordinates (x,y). Since this circle is tangent to the y axis, the radius of the circle will be the value of the x coordinate, (sketch on graph if you are unsure why). The circle x^2+y^2=1 has a radius of 1. Since the two circles are tangent to each other, then the distance between the centers of the circles are the sum of the different radii (x+1). The distance between the centers of the circles can also be denoted using distance formula. Thus the distance can also be written as the square root of: (x-0)^2+(y-0)^2, being that the center of the first circle is (0,0). You set the distance formula equal to the sum of the radii, and solve. The answer should be y^2 = 2x+1. This becomes the locus of the centers of circles tangent to the y-axis, and tangent to the unit circle.
Share
answered Mar 5, 2018 at 0:23
LoriLori
1
$\endgroup$
1
1
$\begingroup$ Please use MathJax. $\endgroup$
Ѕᴀᴀᴅ
– Ѕᴀᴀᴅ
2018-03-05 00:27:04 +00:00
Commented Mar 5, 2018 at 0:27
Add a comment |
You must log in to answer this question.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Related
How can the centers of these 5 related circles be specified as a formula?
Construct a circle tangent to sides $BC$ and $CD$ and s.t. its meetings with the diagonal $BD$ are tangent points from tangents draw from point $A$
1 Placing n equally-sized circles and one larger circle inside the circumference of larger circle
Relationships between $2$ mutually tangent circles in a semi-circle
1 Unit Circle Internally Tangent to Ellipse
0 What is the area of intersection of two unit circles whose centers are $d$ units apart?
2 The midpoint of distance between the centers of similarity of two circles is the focal point of their tangent parabola
2 The locus of the center of a circle that intercepts a chord of length $2a$ on the $x$-axis and touches $y$-axis at $(0,b)$.
Hot Network Questions
Calculating the node voltage
I'm having a hard time intuiting throttle position to engine rpm consistency between gears -- why do cars behave in this observed way?
How to home-make rubber feet stoppers for table legs?
I have a lot of PTO to take, which will make the deadline impossible
Overfilled my oil
Verify a Chinese ID Number
Spectral Leakage & Phase Discontinuites
How to rsync a large file by comparing earlier versions on the sending end?
Can I go in the edit mode and by pressing A select all, then press U for Smart UV Project for that table, After PBR texturing is done?
Suggestions for plotting function of two variables and a parameter with a constraint in the form of an equation
alignment in a table with custom separator
What meal can come next?
Do sum of natural numbers and sum of their squares represent uniquely the summands?
Why include unadjusted estimates in a study when reporting adjusted estimates?
Why do universities push for high impact journal publications?
Can a cleric gain the intended benefit from the Extra Spell feat?
Can Monks use their Dex modifier to determine jump distance?
Can a state ever, under any circumstance, execute an ICC arrest warrant in international waters?
How can the problem of a warlock with two spell slots be solved?
Find non-trivial improvement after submitting
With with auto-generated local variables
What is the feature between the Attendant Call and Ground Call push buttons on a B737 overhead panel?
Where is the first repetition in the cumulative hierarchy up to elementary equivalence?
What is a "non-reversible filter"?
more hot questions
Question feed |
13096 | https://www.youtube.com/watch?v=FBCs95Co_oU | Compound Interest - Corbettmaths
corbettmaths
234000 subscribers
Description
641702 views
Posted: 29 Apr 2019
This video explains how to answer compound interest questions. It covers how to approach calculator and non-calculator questions.
Practice Questions:
Textbook Exercises:
137 comments
Transcript:
hi welcome to this corporate math video on compound interest in this video we're going to look at four examples the first one being a non-calculator example and then three questions which are then calculator questions whenever you're using compound interest or solving compound interest questions this formula can be very important and it's initial multiplied by the multiplier to the power of time or you could just apply the percentage repeatedly in the question and i'll show you what i mean now so let's have a look at the first question so let's have a look at our first question so david invests 3 pound in the bank for two years and it earns compound interest of ten percent per year calculate the total amount of money that david has in the bank after two years so whenever you're doing a non-calculated compound interest question what we're going to do is we're going to work out 10 percent to begin with and add that on and i'll tell you how much money david has in the bank after one year and then you're gonna work at 10 of how much he has in the bank after one year and add that on see how much he has in after two years so first of all after one year so we've got three thousand pounds to begin with that's how much he invests in the bank and we're gonna be adding on ten percent so ten percent of three thousand is three hundred so ten percent is equal to three hundred pound and that's how much interest he earns after one year so we add that on and so after one year in the bank his balance will say three thousand three hundred now he's going to earn another ten percent over the next year hearings another ten percent so we're gonna work at ten percent of three thousand three hundred so ten percent of three thousand three hundred is equal to three hundred and thirty pound we then add that on to the 3 300 to see how much he has in the bank after two years so if we add that on we're gonna get three thousand six hundred and thirty pound so after two years david will have in the bank three thousand six hundred and thirty pound he's earned six hundred and thirty pound interest so whenever you don't compound interest questions without a calculator we just just keep on applying the percentage each time so you work at 10 add it on another 10 percent add it on 10 add it on and so on um it could be that it's decreasing and then in that case you would be taking to get the percentage and taken off getting the new percentage taken off and so on okay another formula this formula is very useful this initial times multiplier to the power of time and this is really useful whenever you're dealing with calculator questions so let's have a look at a calculator question now it just makes it really quick and simple to solve compound interest questions so the question says emily invests eight thousand pound in the bank for four years it earns compound interest of three percent per year calculate the total amount of money that emily has in the bank after four years so the formula is initial multiplied by the multiplier to the power of time so first of all her initial amount of money well that's going to be the 8000 pound then times by the multiplier so it's a it's going to be earning three percent interest that's it's going to be increasing by three percent so what's the core mass video multipliers if you need to but whenever you want to increase something by three percent you multiply it by 1.03 that's the multiplier it's a three percent increase because it's gone from a hundred percent up three percent so that's one hundred and three percent and as a decimal that's one point zero three so a multiplier is multiplied by one point zero three if it was five percent interest it would be one point zero five if it was twenty percent interest it'd be one point two zero and so on now in terms of the time because we need to do the multiplier to the power of time the question says it's three percent interest per year and it's been invested in the money for four years because it's been invested for four years we're going to do to the power of four so now we just need to do 8000 multiply by 1.03 to the power 4 in your calculator and when you do that we get the answer of 9004.07048 but because this is to do with money we're just going to run that to two decimal places so answer will be nine thousand and four pound and seven pence so if emily invests eight thousand pound in the bank at three percent interest per year four years she would have nine thousand and four pound and seven pounds in the bank at the end of it so it's just much quicker than having to do it each year individually so now we're gonna do is we're gonna do a different question we're gonna do one where actually it's decreasing so here we've got a question that says a full tank of water a full water tank has sprung a leak and it's going to lose five percent of its water every hour what percentage of water will be left after three hours now there's two differences with this question first of all it's decreasing it's losing five percent each hour the other thing is it didn't actually remember the formula is initial multiplied by the multiplier to the power of time now in this question we know the multiplier because it's losing five percent but it's going down by five percent it's going to be multiplied by no point nine five because it's going down by five percent if you had 100 to begin with you're losing five percent it'll be 95 percent so that's no 0.95 and to the power of time it's three hours so it's to the power of three the only difference in this question is that we don't actually know what the initial is uh because we've been told it's a full water tank and the question is what percentage of water will be left after three hours we're going to say the initial is 100 it was full so we're going to then type that into the calculator we're going to type 100 multiplied by 0.95 to the power of 3 and we get that after 3 hours there would be 85.7375 of the water left in the water tank and that's it now in terms of this question the question said what percentage of the water will be left after three hours so that's 85.7375 if the question said what percentage of water was lost after three hours we would have taken that percentage away from the 100 to see how much had been lost last question says a tree's height increases by 30 percent a year a tree is two meters tall so the tree is two meters tall to begin with after how many years will it take the tree to exceed 12 meters in height so in this question well we're going to write down the formula initial times multiplier to the power of time and we know the initial is equal to 2 meters and we know the multiplier well it's an increase of 30 percent so that's going to be multiple of 1.30 or just 1.3 to the power of time so in this question we haven't been given the time but we're going to try different values of time to find when the height of the tree will exceed 12 meters so and whenever you're doing this in the like on a test or for a homework make sure you're showing your method to reach them don't just do it in the calculator and write the final one down write your different attempts down on the page to begin with so i'm going to try to begin with five years so i've typed in two times one point three to the power five and i've got a height for the tree of seven point four two five eight six meters well that hasn't reached the twelve meters that we wanted and we want to exceed one height of greater than two meters so that's clearly not greater than twelve meters so that down on the page next i'm going to try my six hours six years so i'm gonna try six years and i've got that's equal to nine point six five three six one eight so again that's not quite source it's not actually close to the 12 meters we need to try and over here so we're not going to try seven years and when we try seven years we get that the height of the tree whenever we do two times one point three to the power of seven we get twelve point five four nine seven zero three four so after seven years the height of the tree has exceeded 12 meters so the answer would be seven years and that's it so on a question like that make sure you're showing your method so for compound interest it's very important you learn the formula initial times the multiplier to the power of time unknown calculator question what you would want to do is to just keep on applying the percentage after each year or each week or each month or whatever the time scale the question says that's it |
13097 | https://openaccess.sgul.ac.uk/id/eprint/112723/1/PlacentaPraeviaandAccreta+GTG27A+Accepted+version.pdf | Page 1 of 47 Green-top Guideline No. 27a 1 Final Draft – Winter 2017 2 3 Placenta Praevia and Placenta Accreta: Diagnosis and Management 4 5 This is the fourth edition of this guideline. The first, published in 2001, was entitled Placenta Praevia: 6 Diagnosis and Management; the second, published in 2005, was entitled Placenta Praevia and 7 Placenta Praevia Accreta: Diagnosis and Management; and the third, published in 2011, was entitled 8 Placenta Praevia, Placenta Praevia Accreta and Vasa Praevia: Diagnosis and Management. 9 10 Executive summary 11 12 Antenatal diagnosis and management of placenta praevia or a low-lying placenta 13 14 What are the risk factors for placenta praevia or a low-lying placenta? 15 16 Caesarean delivery is associated with an increased risk of placenta praevia in subsequent 17 pregnancies. This risk rises as the number of prior caesarean sections increases. [B] 18 19 Assisted reproductive technology and maternal smoking increase the risk of placenta praevia. [B] 20 21 Should we screen for placenta praevia or a low-lying placenta, if so, at what gestation and with what 22 follow-up? 23 24 The midpregnancy routine fetal anomaly scan should include placental localisation thereby 25 identifying women at risk of persisting placenta praevia or a low-lying placenta. [GPP] 26 27 The term placenta praevia should be used when the placenta lies directly over the internal os. For 28 pregnancies at more than 16 weeks of gestation the term low-lying placenta should be used when 29 the placental edge is 20 mm or less from the internal os on transabdominal or transvaginal 30 scanning (TVS). [D] 31 32 If the placenta is thought to be low lying (less than 20 mm from the internal os) or praevia 33 (covering the os) at the routine fetal anomaly scan, a follow-up ultrasound examination including 34 a TVS is recommended at 32 weeks of gestation to diagnose persistent low-lying placenta and/or 35 placenta praevia. [D] 36 37 What is the role and what are the risks of TVS? 38 39 Clinicians should be aware that TVS for the diagnosis of placenta praevia or a low-lying placenta is 40 superior to transabdominal and transperineal approaches and is safe. [GPP] 41 42 In women with a persistent low-lying placenta or placenta praevia at 32 weeks of gestation who 43 remain asymptomatic, an additional TVS is recommended at around 36 weeks of gestation to 44 inform discussion about mode of delivery. [D] 45 46 Cervical length measurement may help facilitate management decisions in asymptomatic women 47 with placenta praevia. A short cervical length on TVS before 34 weeks of gestation increases the 48 risk of preterm emergency delivery and massive haemorrhage at caesarean section. [D] 49 50 Where should women with a low-lying placenta or placenta praevia be cared for in the third 51 trimester? 52 53 Page 2 of 47 Women with a low-lying placenta or placenta praevia with recurrent bleeding 54 55 Tailor antenatal care, including hospitalisation, to individual patient need and social 56 circumstances, e.g. distance between home and hospital and availability of transportation, 57 previous bleeding episodes, haematology laboratory results and acceptance of receiving donor 58 blood or blood products. [GPP] 59 60 It should be made clear to any woman being treated at home in the third trimester that she should 61 attend the hospital immediately if she experiences any bleeding, including spotting, contractions 62 or pain (including vague suprapubic period-like aches). [GPP] 63 64 Women with asymptomatic placenta praevia or a low-lying placenta 65 66 Women with asymptomatic placenta praevia or a low-lying placenta in the third trimester should 67 be counselled about the risks of preterm delivery and obstetric haemorrhage, and their care 68 should be tailored to their individual needs. [GPP] 69 70 Women with asymptomatic placenta praevia confirmed at the 32-week follow-up scan and 71 managed at home should be encouraged to ensure they have safety precautions in place, including 72 having someone available to help them as necessary and ready access to the hospital. [GPP] 73 74 Is there a place for cervical cerclage in placenta praevia or a low-lying placenta? 75 76 The use of cervical cerclage to reduce bleeding and prolong pregnancy is not supported by 77 sufficient evidence to recommend its use outside of a clinical trial. [GPP] 78 79 In what circumstances, and at what gestation, should women be offered antenatal corticosteroids? 80 81 A single course of antenatal corticosteroid therapy is recommended between 34+0 and 36+0 weeks 82 of gestation for pregnant women with a low-lying placenta or placenta praevia and is appropriate 83 prior to 34+0 weeks of gestation in women at higher risk of preterm birth. [GPP] 84 85 Is there a place for the use of tocolytics in women presenting with a low-lying placenta or placenta 86 praevia and preterm labour? 87 88 Tocolysis for women presenting with symptomatic placenta praevia or a low-lying placenta may 89 be considered for 48 hours to facilitate administration of antenatal corticosteroids. [C] 90 91 Should delivery be indicated based on maternal or fetal concerns, tocolysis should not be used in 92 attempt to prolong gestation. [C] 93 94 At what gestation should planned delivery occur? 95 96 Late preterm (35+1 to 36+6 weeks of gestation) delivery should be considered for women 97 presenting with complicated placenta praevia or a low-lying placenta. [C] 98 99 In what situations is vaginal delivery appropriate for women with a low-lying placenta? 100 101 In women with a third trimester asymptomatic low-lying placenta, the mode of delivery should be 102 based on the clinical background, the woman’s preferences supplemented by ultrasound findings, 103 including the distance between the placental edge and the fetal head position relative to the 104 leading edge of the placenta on TVS. [D] 105 106 Page 3 of 47 Optimising the delivery of placenta praevia 107 108 Prior to delivery, all women with placenta praevia and their partners should have a discussion 109 regarding delivery. Indications for blood transfusion and hysterectomy should be reviewed and 110 concerns or plans to decline blood or blood products should be discussed openly and documented. 111 [GPP] 112 113 Placenta praevia and anterior low-lying placenta carry a higher risk of massive obstetric 114 haemorrhage and hysterectomy. Delivery should be arranged in a maternity unit with on-site 115 blood transfusion services and access to critical care. [D] 116 117 Women with atypical antibodies form a particularly high-risk group and the care of these women 118 should involve discussions with the local haematologist and blood bank. [D] 119 120 Prevention and treatment of anaemia during the antenatal period is recommended for women 121 with placenta praevia or a low-lying placenta as for any pregnant woman. [D] 122 123 Delivery for women with placenta praevia or a low-lying placenta 124 125 What grade of obstetrician and anaesthetist should attend the caesarean delivery for a placenta 126 praevia? 127 128 As a minimum requirement for a planned caesarean section for placenta praevia, the surgical 129 procedure should be carried out by an appropriately experienced operator. [GPP] 130 131 In cases of planned caesarean section for placenta praevia or a low-lying placenta, a senior 132 obstetrician (usually a consultant) and senior anaesthetist (usually a consultant) should be present 133 within the delivery or theatre suite where the surgery is occurring. [GPP] 134 135 When an emergency arises, the senior obstetrician and senior anaesthetist should be alerted 136 immediately and attend urgently. [GPP] 137 138 What anaesthetic procedure is most appropriate for caesarean section in placenta praevia or a low-139 lying placenta? 140 141 Regional anaesthesia is considered safe and associated with lower risks of haemorrhage than 142 general anaesthesia for caesarean delivery in women with placenta praevia or a low-lying 143 placenta. Women with anterior placenta praevia or a low-lying placenta should be advised that it 144 may be necessary to convert to general anaesthesia if required and asked to consent. [D] 145 146 What blood products should be available? 147 148 Close liaison with the hospital transfusion laboratory is essential for women presenting with 149 placenta praevia or a low-lying placenta. [GPP] 150 151 Rapid infusion and fluid warming devices should be immediately available. [GPP] 152 153 Cell salvage is recommended for patients where the anticipated blood loss is great enough to 154 induce anaemia, in particular, in women who would decline blood products. [D] 155 156 What surgical approach should be used for placenta praevia or a low-lying placenta? 157 158 Consider vertical skin and/or uterine incisions when the fetus is in a transverse lie to avoid the 159 Page 4 of 47 placenta, particularly below 28 weeks of gestation. [GPP] 160 161 Consider using preoperative and/or intraoperative ultrasonography to precisely determine 162 placental location and the optimal place for uterine incision. [D] 163 164 If the placenta is transected during the uterine incision, immediately clamp the umbilical cord 165 after fetal delivery to avoid excessive fetal blood loss. [D] 166 167 If pharmacological measures fail to control haemorrhage, initiate intrauterine tamponade and/or 168 surgical haemostatic techniques sooner rather than later. Interventional radiological techniques 169 should also be urgently employed where possible. [C] 170 171 Early recourse to hysterectomy is recommended if conservative medical and surgical interventions 172 prove ineffective. [D] 173 174 Antenatal diagnosis and outcome of placenta accreta spectrum 175 176 What are the risk factors for placenta accreta spectrum? 177 178 The major risk factors for placenta accreta spectrum are history of accreta in a previous 179 pregnancy, previous caesarean delivery and other uterine surgery, including repeated endometrial 180 curettage. This risk rises as the number of prior caesarean sections increases. [B] 181 182 Women requesting elective caesarean delivery for non-medical indications should be informed of 183 the risk of placenta accreta spectrum and its consequences for subsequent pregnancies. [GPP] 184 185 How can a placenta accreta spectrum be suspected and diagnosed antenatally? 186 187 Antenatal diagnosis of placenta accreta spectrum is crucial in planning its management and has 188 been shown to reduce maternal morbidity and mortality. [D] 189 190 Previous caesarean delivery and the presence of an anterior low-lying placenta or placenta praevia 191 should alert the antenatal care team of the higher risk of placenta accreta spectrum. [D] 192 193 Ultrasound screening and diagnosis of placenta accreta spectrum 194 195 Ultrasound imaging is highly accurate when performed by a skilled operator with experience in 196 diagnosing placenta accreta spectrum. [C] 197 198 Refer women with any ultrasound features suggestive of placenta accreta spectrum to a specialist 199 unit with imaging expertise. [B] 200 201 Women with a history of previous caesarean section seen to have an anterior low-lying placenta 202 or placenta praevia at the routine fetal anomaly scan should be specifically screened for placenta 203 accreta spectrum. [D] 204 205 Is there a role for magnetic resonance imaging (MRI) in the diagnosis of placenta accreta spectrum? 206 207 Clinicians should be aware that the diagnostic value of MRI and ultrasound imaging in detecting 208 placenta accreta spectrum is similar when performed by experts. [C] 209 210 MRI may be used to complement ultrasound imaging to assess the depth of invasion and lateral 211 extension of myometrial invasion, especially with posterior placentation and/or in women with 212 Page 5 of 47 ultrasound signs suggesting parametrial invasion. [GPP] 213 214 Where should women with placenta accreta spectrum be cared for? 215 216 Women diagnosed with placenta accreta spectrum should be cared for by a multidisciplinary team 217 in a specialist centre with expertise in diagnosing and managing invasive placentation. [GPP] 218 219 Delivery for women diagnosed with placenta accreta spectrum should take place in a specialist 220 centre with logistic support for immediate access to blood products, adult intensive care unit and 221 neonatal intensive care unit (NICU) by a multidisciplinary team with expertise in complex pelvic 222 surgery. [D] 223 224 When should delivery be planned for women with placenta accreta spectrum? 225 226 In the absence of risk factors for preterm delivery and evidence of invasive placentation, planned 227 delivery at 35+0 to 36+6 weeks of gestation provides the best balance between fetal maturity and 228 the risk of unscheduled delivery. [GPP] 229 230 Planning delivery of a suspected placenta accreta spectrum 231 232 Once the diagnosis of placenta accreta spectrum is made, a contingency plan for emergency 233 delivery should be developed, including the use of an institutional protocol for the management of 234 maternal haemorrhage. [GPP] 235 236 What should be included in the consent form for caesarean section in cases of suspected placenta 237 accreta spectrum? 238 239 Any woman giving consent for caesarean section should understand the risks associated with 240 caesarean section in general, and the specific risks of placenta accreta spectrum in terms of 241 massive obstetric haemorrhage, increased risk of lower urinary tract damage, the need for blood 242 transfusion and the risk of hysterectomy. [GPP] 243 244 Additional possible interventions in the case of massive haemorrhage should also be discussed, 245 including cell salvage and interventional radiology where available. [D] 246 247 What healthcare professionals should be involved? 248 249 The elective delivery of women with placenta accreta spectrum should be managed by a 250 multidisciplinary team, which should include senior anaesthetists, obstetricians and 251 gynaecologists with appropriate experience in managing the condition and other surgical 252 specialties if indicated. In an emergency, the most senior clinicians available should be involved. 253 [GPP] 254 255 What anaesthetic is most appropriate for delivery? 256 257 The choice of anaesthetic technique for caesarean section for placenta accreta spectrum should be 258 made by the anaesthetist conducting the procedure in consultation with the patient in advance. 259 [GPP] 260 261 The woman should be informed that the surgical procedure can be performed safely with regional 262 anaesthesia but should be advised that it may be necessary to convert to general anaesthesia if 263 required and asked to consent. [D] 264 265 Page 6 of 47 Optimising the delivery of placenta accreta spectrum 266 267 What surgical approach should be used for placenta accreta spectrum? 268 269 Caesarean section hysterectomy with the placenta left in situ is preferable to attempting to 270 separate it from the uterine wall. [C] 271 272 When the extent of the placenta accreta is limited in depth and surface area, and the entire 273 placental implantation area is accessible and visualised (i.e. completely anterior, fundal or 274 posterior without deep pelvic invasion), uterus-preserving surgery may be appropriate, including 275 partial myometrial resection. [GPP] 276 277 Uterus-preserving surgical techniques should only be attempted by surgeons working in teams 278 with appropriate expertise to manage such cases and after appropriate counselling regarding risks 279 and with informed consent. [D] 280 281 There are currently insufficient data to recommend the routine use of ureteric stents in placenta 282 creta and increta. [C] 283 284 What surgical approach should be used for placenta percreta? 285 286 There is limited evidence to support uterus-preserving surgery in placenta percreta and women 287 should be informed of the high risk of peripartum and secondary complications, including the 288 need for secondary hysterectomy. [D] 289 290 Expectant management (leaving the placenta in situ) 291 292 Elective peripartum hysterectomy may be unacceptable to women desiring uterine preservation 293 or considered inappropriate by the surgical team. In such cases, leaving the placenta in situ should 294 be considered. [D] 295 296 When the placenta is left in situ, local arrangements need to be made to ensure regular review, 297 ultrasound examination and access to emergency care should the woman experience 298 complications, such as bleeding or infection. [D] 299 300 Methotrexate (MTX) adjuvant therapy should not be used for expectant management as it is of 301 unproven benefit and has significant adverse effects, including a reported maternal death. [C] 302 303 When is interventional radiology indicated? 304 305 Larger studies are necessary to determine the safety and efficacy of interventional radiology 306 before this technique can be advised in the routine management of placenta accreta spectrum. [D] 307 308 Women diagnosed with placenta accreta spectrum who decline donor blood transfusion should be 309 managed in a unit with an interventional radiology service. [D] 310 311 How is unsuspected placenta accreta spectrum at delivery best managed? 312 313 If at the time of an elective repeat caesarean section, where both mother and baby are stable, it is 314 immediately apparent that placenta percreta is present on opening the abdomen, the caesarean 315 section should be delayed until the appropriate staff and resources have been assembled and 316 adequate blood products are available. This may involve closure of the maternal abdomen and 317 urgent transfer to a specialist unit for delivery. [GPP] 318 Page 7 of 47 319 In case of unsuspected placenta accreta spectrum diagnosed after delivery of the baby, the 320 placenta should be left in situ and an emergency hysterectomy performed. [D] 321 322 1. Purpose and scope 323 324 The purpose of this guideline is to describe the diagnostic modalities and review the evidence-based 325 approach to the clinical management of pregnancies complicated by placenta praevia and placenta 326 accreta. 327 328 2. Introduction and background epidemiology 329 330 Placenta praevia and placenta accreta are associated with high maternal and neonatal morbidity and 331 mortality.1–5 The rates of placenta praevia and accreta have increased and will continue to do so as a 332 result of rising rates of caesarean deliveries, increased maternal age and use of assisted reproductive 333 technology (ART), placing greater demands on maternity-related resources. The highest rates of 334 complication for both mother and newborn are observed when these conditions are only diagnosed 335 at delivery. 336 337 2.1 Placenta praevia 338 339 Determining placental location is one of the first aims of routine midpregnancy (18+6–21+6 weeks of 340 gestation) transabdominal obstetric ultrasound examination.6,7 Placenta praevia was originally 341 defined using transabdominal scan as a placenta developing within the lower uterine segment and 342 graded according to the relationship and/or the distance between the lower placental edge and the 343 internal os of the uterine cervix. Grade I or minor praevia was defined as a lower edge inside the 344 lower uterine segment; grade II or marginal praevia as a lower edge reaching the internal os; grade 345 III or partial praevia when the placenta partially covers the cervix; and grade IV or complete praevia 346 when the placenta completely covers the cervix. Grades I and II are also often defined as ’minor’ 347 placenta praevia whereas grades III and IV are referred to as ’major’ placenta praevia. 348 349 The introduction of transvaginal scanning (TVS) in obstetrics in the 1980s has allowed for a more 350 precise evaluation of the distance between the placental edge and the internal os. A recent 351 multidisciplinary workshop of the American Institute of Ultrasound in Medicine (AIUM)8 has 352 recommended discontinuing the use of the terms ‘partial’ and ‘marginal’, suggesting that the term 353 ‘placenta praevia’ is used when the placenta lies directly over the internal os. For pregnancies 354 greater than 16 weeks of gestation, the placenta should be reported as ‘low lying’ when the 355 placental edge is less than 20 mm from the internal os and as normal when the placental edge is 20 356 mm or more from the internal os on transabdominal or TVS. This new classification could better 357 define the risks of perinatal complications, such as antepartum haemorrhage and major postpartum 358 haemorrhage (PPH),9,10 and has the potential of improving the obstetric management of placenta 359 praevia. Recent articles reviewed in this guideline refer to the AIUM classification. 360 361 The estimated incidence of placenta praevia at term is 1 in 200 pregnancies.5,9 However, this is 362 dependent on the definition used and is likely to change with the introduction of the AIUM 363 classification described above and with the rising incidence of the main risk factors, i.e. prior 364 caesarean delivery and pregnancies resulting from ART. The relationship between a low-lying 365 placenta or placenta praevia and a velamentous insertion of the umbilical cord is presented and 366 discussed in sister Green-top Guideline no. 27b: Vasa Praevia: Diagnosis and Management. 367 368 2.2 Placenta accreta 369 370 Placenta accreta is a histopathological term first defined by Irving and Hertig in 1937, as the 371 Page 8 of 47 “abnormal adherence of the afterbirth in whole or in parts to the underlying uterine wall in the 372 partial or complete absence of decidua”.11 Irving and Hertig did not include abnormally invasive 373 placentation in their series and thus in their description was limited to abnormally adherent 374 placenta. Depending on the depth of villous tissue invasiveness, placenta accreta was subsequently 375 subdivided by modern pathologists into ‘creta’ or ‘adherenta’ where the villi adheres superficially to 376 the myometrium without interposing decidua; ‘increta’ where the villi penetrate deeply into the 377 uterine myometrium down to the serosa; and ‘percreta’ where the villous tissue perforates through 378 the entire uterine wall and may invade the surrounding pelvic organs, such as the bladder.12-14 Cases 379 of placenta accreta are also often subdivided into total, partial or focal according to the amount of 380 placental tissue involved and the different depths of accreta placentation have been found to co-381 exist in the same case.12,15 Thus placenta accreta is a spectrum disorder ranging from abnormally 382 adherent to deeply invasive placental tissue. 383 384 Detailed data on clinical findings and, where possible, on histopathological examination are essential 385 when describing different diagnostic or management techniques.16,17 The diagnostic conundrum is 386 obvious at the abnormally adherent end of the spectrum where the differential diagnosis between a 387 difficult manual removal and an abnormally adherent or placenta accreta may be impossible in the 388 absence of histopathological confirmation. These diagnostic difficulties probably explain the current 389 wide variation in reported prevalence of placenta accreta ranging between 1 in 300 and 1 in 2000 390 pregnancies1–5 and highlight the need for a standardised approach to imaging, clinical and 391 histopathological descriptions. In the last decade, even the condition itself has begun to be known 392 by many different names, with ‘morbidly adherent placenta’ becoming particularly popular. This 393 terminology was originally used in the 19th century to describe the clinical complications associated 394 with a retained placenta. This terminology is misleading as ‘morbidly adherent’ does not encompass 395 the abnormally invasive end of the accreta spectrum (increta and percreta), which usually have the 396 worst clinical outcomes.16,17 In order to overcome these difficulties, the terms ‘placenta accreta 397 spectrum’ or ’abnormally adherent and invasive placenta’ should be used to include both the 398 abnormally adherent and invasive forms of accreta placentation.18 In this guideline, the term 399 placenta accreta spectrum will be used. 400 401 In the 1990s, the maternal mortality of placenta percreta was reported to be as high as 7% of 402 cases.19 More recent large series have reported lower rates of maternal death and this is likely to be 403 further improved by screening for placenta accreta spectrum in women at high risk and in planning 404 the delivery in specialist centres.20–22 405 406 3. Identification and assessment of evidence 407 408 This guideline was developed in accordance with standard methodology for producing Royal College 409 of Obstetricians and Gynaecologists (RCOG) Green-top Guidelines. The Cochrane Library (including 410 the Cochrane Database of Systematic Reviews and the Database of Abstracts of Reviews of Effects 411 [DARE]), EMBASE, Trip, MEDLINE and PubMed (electronic databases) were searched for relevant 412 randomised controlled trials (RCT), systematic reviews and meta-analyses. The search was restricted 413 to articles published between May 2009 and December 2017 (the search for the previous Guideline 414 was up to May 2009). The databases were searched using the relevant Medical Subject Headings 415 (MeSH) terms, including all subheadings, and this was combined with a keyword search. Search 416 words included ‘placenta praevia’, ‘low lying placenta’, ‘placenta accreta’, ‘placenta increta’ 417 ‘placenta percreta’, ‘abnormally adherent placenta’ and ‘abnormally invasive placenta’. The search 418 was restricted to humans and the English language. The National Library for Health and the National 419 Guideline Clearinghouse were also searched for relevant guidelines and reviews. 420 421 Where possible, recommendations are based on available evidence. In the absence of published 422 evidence, these have been annotated as ‘good practice points’. Further information about the 423 assessment of evidence and the grading of recommendations may be found in Appendix I. 424 Page 9 of 47 425 4. Antenatal diagnosis and management of placenta praevia or a low-lying placenta 426 427 4.1 What are the risk factors for placenta praevia or a low-lying placenta? 428 429 Caesarean delivery is associated with an increased risk of placenta praevia in subsequent 430 pregnancies. This risk rises as the number of prior caesarean sections increases. [B] 431 432 ART and maternal smoking increase the risk of placenta praevia. [B] 433 434 In 1997, a meta-analysis of the association of placenta praevia with history of caesarean delivery 435 found a dose-response pattern for the relative risk (RR) of placenta praevia of 4.5 (95% CI 3.6–5.5) 436 for one, 7.4 (95% CI 7.1–7.7) for two, 6.5 (95% CI 3.6–11.6) for three, and 44.9 (95% CI 13.5–149.5) 437 for four or more prior caesarean deliveries compared with vaginal delivery.23 [Evidence level 2++] 438 439 A systematic review and meta-analysis of 22 studies including over 2 million deliveries indicated that 440 the incidence of placenta praevia increases from 10 in 1000 deliveries with one previous caesarean 441 delivery to 28 in 1000 with three or more caesarean deliveries.24 A 2014 meta-analysis confirmed 442 these findings and reported an overall odds ratio (OR) of 1.47 (95% CI 1.44–1.51) for placenta 443 praevia after caesarean section.25 [Evidence level 1+] 444 445 Cohort studies have also reported that a second pregnancy within 1 year of a caesarean section is 446 associated with an increased risk of placenta praevia (RR 1.7, 95% CI 0.9–3.1).26 Compared with 447 vaginal birth, a previous prelabour caesarean section is associated with an increased risk of placenta 448 praevia in the second delivery (adjusted OR [aOR] 2.62, 95% CI 1.24–5.56).27 [Evidence level 2++] 449 450 There have been contradictory reports regarding the incidence of placenta praevia in multiple 451 pregnancies. A retrospective cohort study of 1 172 405 twin live births and stillbirths in the USA 452 between 1989 and 1998 found no increased risk in twins.28 A retrospective cohort of 67 895 453 singleton and twin pregnancies found that dichorionic (aOR 1.54, 95% CI 1.15–2.06) and 454 monochorionic (RR 3.29, 95% CI 1.32–8.21) twin pregnancies had an increased risk of placenta 455 praevia compared with singletons.29 [Evidence level 2+] 456 457 ART is associated with a higher incidence of placenta praevia independently of the high rate of 458 multiple pregnancies generated by the technique used.30,31 A 2016 meta-analysis of ART singleton 459 pregnancies reported a RR of 3.71 (95% CI 2.67–5.16) for placenta praevia32 that was confirmed by a 460 2017 meta-analysis (OR 2.67, 95% CI 2.01–3.34).33 Furthermore, a 2017 meta-analysis of the impact 461 of maternal smoking on placental position34 (OR 1.42, 95% CI 1.30–1.50) has found an increased risk 462 of placenta praevia. [Evidence level 1+] 463 464 Advanced maternal age has been also associated with a slight increase in the risk of placenta praevia 465 (OR 1.08, 95% CI 1.07–1.09) but this effect may be due to parity 35. [Evidence level 2–] 466 467 4.2 Should we screen for placenta praevia or a low-lying placenta, if so, at what gestation and with 468 what follow-up? 469 470 The midpregnancy routine fetal anomaly scan should include placental localisation thereby 471 identifying women at risk of persisting placenta praevia or a low-lying placenta. [GPP] 472 473 The term placenta praevia should be used when the placenta lies directly over the internal os. For 474 pregnancies at more than 16 weeks of gestation the term low-lying placenta should be used when 475 the placental edge is 20 mm or less from the internal os on transabdominal or TVS. [D] 476 477 Page 10 of 47 If the placenta is thought to be low lying (less than 20 mm from the internal os) or praevia 478 (covering the os) at the routine fetal anomaly scan, a follow-up ultrasound examination including 479 a TVS is recommended at 32 weeks of gestation to diagnose persistent low-lying placenta and/or 480 placenta praevia. [D] 481 482 Placenta praevia is a well-established complication of pregnancy associated with high maternal and 483 perinatal complication rates.4–9 The UK National Screening Committee (UK NSC) does not 484 recommend a national screening program for placenta praevia, but it has supported current local 485 practices of identifying it at the routine midpregnancy (18+6–21+6 weeks of gestation) antenatal 486 screening ultrasound examination in women whose placenta extends onto the internal cervical os 487 (www.screening.nhs.uk/policies).36 An update published in 2014 that included a literature search 488 covering the period between January 2008 and November 2012 concluded that this practice was not 489 supported by new evidence, but that the placental site is routinely reported at the time of the 490 routine fetal anomaly scan. In turn, this routine study has become the main screening test for 491 placenta praevia.37 [Evidence level 4] 492 493 Apparent placental ‘migration’ following the development of the lower uterine segment during the 494 third trimester of pregnancy results in the resolution of the low-lying placenta in 90% of the cases 495 before term.38–46 This is less likely to occur in women with a previous caesarean delivery.39 [Evidence 496 level 4] 497 498 In twin pregnancies, the likelihood of persistence of placenta praevia is also dependent on the 499 gestational age at sonographic detection. Among those with placenta praevia diagnosed in the 500 second trimester the majority of cases resolve by 32 weeks of gestation.29,47 After 32 weeks of 501 gestation around 50% of the remaining placenta praevia will resolve, with no further changes after 502 36 weeks of gestation.29 [Evidence level 3] 503 504 The timing of a confirmatory ultrasound examination in the third trimester has varied between 32 505 and 36 weeks of gestation depending on the extent of the placenta praevia over the internal cervical 506 os. It is based on the perceived risk of antenatal haemorrhage, but there is no strong evidence that it 507 makes a difference in the care of asymptomatic women.37 The timing of the follow-up ultrasound 508 examination should also be tailored according to a previous history of caesarean delivery to exclude 509 an associated placenta accreta spectrum. [Evidence level 4] 510 511 4.3 What is the role and what are the risks of TVS? 512 513 Clinicians should be aware that TVS for the diagnosis of placenta praevia or a low-lying placenta is 514 superior to transabdominal and transperineal approaches and is safe. [GPP] 515 516 In women with a persistent low-lying placenta or placenta praevia at 32 weeks of gestation who 517 remain asymptomatic, an additional TVS is recommended at around 36 weeks of gestation to 518 inform discussion about mode of delivery. [D] 519 520 Cervical length measurement may help facilitate management decisions in asymptomatic women 521 with placenta praevia. A short cervical length on TVS before 34 weeks of gestation increases the 522 risk of preterm emergency delivery and massive haemorrhage at caesarean section. [D] 523 524 TVS improves the accuracy of placental localisation particularly when the placenta is posterior or if 525 the transabdominal ultrasound is unclear, for example, due to maternal obesity or the presence of 526 large uterine fibroids.5 [Evidence level 4] 527 528 There is only one small (n = 38) RCT comparing transabdominal scan and TVS for placenta praevia, 529 which supports this safety profile and reports superior views, especially for posterior placentas.48 530 Page 11 of 47 [Evidence level 1+] 531 532 If the distance between the internal os and the placental edge is 20 mm or more on TVS, the 533 placental location should be recorded as normal and managed as per routine. Studies have not 534 demonstrated an increased risk for caesarean section due to haemorrhage in these cases.4,5 By 535 contrast, if the placenta extends beyond the internal os on TVS during the second trimester, it is 536 likely to be confirmed as placenta praevia at 32 weeks of gestation.48–50 However, ‘migration’ is still 537 possible after 32 weeks of gestation.50,51 [Evidence level 2+] 538 539 TVS will reclassify 26–60% of placentas diagnosed as low lying at the routine fetal anomaly scan.52–54 540 Overall, TVS has a high accuracy (positive predictive value of 93.3%, negative predictive value of 541 97.6% and false-negative rate of 2.33%) in predicting placenta praevia in women suspected of having 542 a low-lying placenta on transabdominal scan in the second and early third trimester, with a 543 sensitivity of 87.5% and a specificity of 98.8%.55 [Evidence level 2+] 544 545 TVS has also been used to measure the cervical length to predict preterm birth56 and cohort studies 546 with low risks of confounding bias have shown that cervical length is a predictor of antepartum 547 bleeding and emergency preterm caesarean section in placenta praevia.57–60 A prospective cohort 548 study of 59 women presenting with placenta praevia covering the internal os has shown that the 549 best cut-off point for the identification of women at risk of haemorrhage requiring a caesarean 550 delivery before 34 weeks of gestation is a cervical length of 31 mm or less (sensitivity of 83.3% and 551 specificity of 76.6%). Women with a cervical length of less than 31 mm have a 16 times (OR 16.4; 552 95% CI 3.4–75.9) higher risk of emergency caesarean section due to massive haemorrhage.57 553 Similarly, a prospective cohort study of 54 women with placenta praevia covering the internal os has 554 shown that combining a cervical length of less than 30 mm and measurement of the lower placental 555 edge thickness of more than 10 mm has a sensitivity of 83.3% and a specificity of 78.4%.58 More 556 prospective studies using a standardised ultrasound definition of placental edge thickness are 557 required before this sign can be used in clinical practice. [Evidence level 2+] 558 559 Compared with women with a long cervical length, women with a short cervical length (less than 25 560 mm) have a RR of 7.2 (95% CI 2.3–22.3) for massive haemorrhage during caesarean section for 561 placenta praevia.59 [Evidence level 2+] 562 563 Serial TVS cervical length measurements from 26 weeks of gestation have indicated that when the 564 length of the cervix decreases rapidly to 35 mm or less there is an increased risk of preterm 565 caesarean section due to massive haemorrhage.60 [Evidence level 2–] 566 567 4.4 Where should women with a low-lying placenta or placenta praevia be cared for in the third 568 trimester? 569 570 4.4.1 Women with a low-lying placenta or placenta praevia with recurrent bleeding 571 572 Tailor antenatal care, including hospitalisation, to individual patient need and social 573 circumstances, e.g. distance between home and hospital and availability of transportation, 574 previous bleeding episodes, haematology laboratory results and acceptance of receiving donor 575 blood or blood products. [GPP] 576 577 Where hospital admission has been decided, a documented assessment of risk factors for VTE in 578 pregnancy should be performed as outlined in RCOG GTG No. 37a. [D] 579 580 It should be made clear to any woman being treated at home in the third trimester that she should 581 attend the hospital immediately if she experiences any bleeding, including spotting, contractions 582 or pain (including vague suprapubic period-like aches). [GPP] 583 Page 12 of 47 584 The Cochrane systematic review by Nielson on the impact of an intervention in women diagnosed as 585 having, or being likely to have a placenta praevia, which has not been updated since October 2002, 586 includes only one small RCT (n = 53) comparing hospital versus home care for symptomatic placenta 587 praevia.61 This trial found little evidence of any clear advantage or disadvantage to a policy of home 588 versus hospital care, and the only significant difference was a reduction in length of hospital stay.62 589 [Evidence level 1–] 590 591 Two large retrospective studies of women presenting with placenta praevia at the routine fetal 592 anomaly scan have proposed scores to predict the risk of emergency caesarean section. The first 593 study (n = 250) found that the risk is increased if the first (sentinel) vaginal bleeding episode occurs 594 before 29 weeks of gestation (OR 2.64, 95% CI 1.17–5.98), and with the occurrence of three or more 595 episodes of antepartum haemorrhage (OR 2.53, 95% CI 1.1–5.86).63 The second (n = 214) found that 596 independent predictors for emergency delivery are a history of caesarean section (OR 4.7, 95 CI 1.2– 597 12); antepartum haemorrhage on one (OR 7.5, 95% CI 2.5–23), two (OR 14, 95% CI 4.3–47), and 598 three or more occasions (OR 27, 95% CI 8.3–90); and need for antenatal blood transfusion (OR 6.4, 599 95% CI 1.7–23).10 A retrospective study of 214 women with singleton pregnancies found that the risk 600 of preterm emergency caesarean delivery increases with the number of antepartum bleeding 601 episodes with one (OR 7.5, 95% C, 2.5-23), two (OR 14, 95% CI 4.3-47), and three or more (OR 27, 602 95% CI 8.3-90), as well as need for blood transfusion (OR 6.4, 95% C, 1.7-23).64 The results of these 603 studies suggest that predictors for emergency delivery in women with placenta praevia can be used 604 for individualised antenatal care regarding need for hospital admission, corticosteroids 605 administration and timing of delivery. [Evidence level 2–] 606 607 4.4.2 Women with asymptomatic placenta praevia or a low-lying placenta 608 609 Women with asymptomatic placenta praevia or a low-lying placenta in the third trimester should 610 be counselled about the risks of preterm delivery and obstetric haemorrhage, and their care 611 should be tailored to their individual needs. [GPP] 612 613 Women with asymptomatic placenta praevia confirmed at the 32-week follow-up scan and 614 managed at home should be encouraged to ensure they have safety precautions in place, including 615 having someone available to help them as necessary and ready access to the hospital. [GPP] 616 617 Most women with asymptomatic placenta praevia (no bleeding or contractions) can be cared for as 618 outpatients with similar outcomes compared with hospitalisation and at lower cost.5 Numerous 619 factors influence the chances of the placenta praevia persisting until delivery, such as prior 620 caesarean section,43 the distance between the placental edge and the internal os, and the thickness 621 of the placental edge.4 These parameters can be useful in tailoring individual patient needs. 622 [Evidence level 4] 623 624 4.5 Is there a place for cervical cerclage in placenta praevia or a low-lying placenta? 625 626 The use of cervical cerclage to reduce bleeding and prolong pregnancy is not supported by 627 sufficient evidence to recommend its use outside of a clinical trial. [GPP] 628 629 The Cochrane systematic review by Nielson61 on the impact of cerclage in women diagnosed as 630 having, or being likely to have, placenta praevia included two small RCTs (n = 25 and 36) comparing 631 cervical cerclage versus no cerclage. There may be a reduction in preterm births before 34 weeks of 632 gestation (RR 0.45, 95% CI 0.23–0.87), but this evidence is not robust enough to recommend its use 633 outside of clinical trials. [Evidence level 1–] 634 635 There have been no new trials looking at this issue since the last update of this guideline. 636 Page 13 of 47 637 4.6 In what circumstances, and at what gestation, should women be offered antenatal 638 corticosteroids? 639 640 A single course of antenatal corticosteroid therapy is recommended between 34+0 and 36+0 weeks 641 of gestation for pregnant women with a low-lying placenta or placenta praevia and is appropriate 642 prior to 34+0 weeks of gestation in women at higher risk of preterm birth. [GPP] 643 644 A large case–control study found that neonatal morbidities in women with placenta praevia include 645 an increased risk of lower 5-minute Apgar scores, neonatal intensive care unit (NICU) admission, 646 anaemia, respiratory distress syndrome, mechanical ventilation and intraventricular haemorrhage.65 647 Neonates born to mothers with placenta praevia have lower birthweights (2806 versus 3285 g) and 648 lower gestational ages (36+2 versus 38+1 weeks). These differences were not significant after 649 adjusting for confounders such as prematurity.66 [Evidence level 2++] 650 651 Compared with placebo or no treatment with antenatal corticosteroids (betamethasone, 652 dexamethasone or hydrocortisone), antenatal corticosteroids are associated with a reduction in the 653 most serious adverse outcomes related to prematurity, including perinatal death (RR 0.72, 95% CI 654 0.58–0.89), respiratory distress syndrome (average RR 0.66, 95% CI 0.56–0.77), intraventricular 655 haemorrhage (average RR 0.55, 95% CI 0.40–0.76) and necrotising enterocolitis (RR 0.50, 95% CI 656 0.32–0.78).67 [Evidence level 1+] 657 658 The 2016 RCT has found that the administration of betamethasone to women with a singleton 659 pregnancy at risk for late preterm delivery (34+0 to 36+5 weeks of gestation) significantly reduces the 660 rate of neonatal respiratory complications.68 [Evidence level 1+] 661 662 A decision analytic model designed to compare total maternal and neonatal quality-adjusted life 663 years for delivery of women with placenta praevia at 34+0 to 36+6 weeks of gestation indicated that 664 corticosteroids administration at 35+5 weeks of gestation followed by planned delivery at 36 weeks 665 of gestation optimises maternal and neonatal outcomes.69 [Evidence level 4] 666 667 4.7 Is there a place for the use of tocolytics in women presenting with a low-lying placenta or 668 placenta praevia and preterm labour? 669 670 Tocolysis for women presenting with symptomatic placenta praevia or a low-lying placenta may 671 be considered for 48 hours to facilitate administration of antenatal corticosteroids. [C] 672 673 Should delivery be indicated based on maternal or fetal concerns, tocolysis should not be used in 674 attempt to prolong gestation. [C] 675 A systematic review to determine if the prolonged (48 hours or more) use of tocolytics in women 676 with symptomatic preterm placenta praevia improves perinatal outcome identified two 677 retrospective studies (total, n = 217) and one RCT (n = 60).70 The results of the RCT showed that 678 pregnancy can be prolonged for more than 7 days with continued tocolytics (OR 3.10, 95% CI 1.38– 679 6.96). When combined with the data of retrospective studies, the results did not reach significance 680 (OR 1.19, 95% CI 0.63–2.28). The RCT was judged inadequately compliant with the Consolidated 681 Standards of Reporting Trials statement. [Evidence level 1–] 682 683 A recent randomised, double-blind, placebo-controlled multicentre trial including 109 women at 24+0 684 to 33+6 weeks with at least one episode of placenta praevia bleeding and intact membranes has 685 shown that there is no difference in the prolongation of pregnancy between the nifedipine (n = 54) 686 and placebo (n = 55) groups.71 Adverse perinatal outcomes were comparable between groups. 687 [Evidence level 1+] 688 689 Page 14 of 47 4.8 At what gestation should planned delivery occur? 690 691 Delivery timing should be tailored according to antenatal symptoms and for women presenting 692 with uncomplicated placenta praevia delivery should be considered between 36 and 37 weeks of 693 gestation. [C] 694 695 As the risk of major haemorrhage increases rapidly after 36 weeks of gestation, expert opinions have 696 highlighted that decisions regarding timing of delivery must be individualised and suggested that on 697 the basis of limited data available, women with uncomplicated placenta praevia should undergo 698 scheduled birth by caesarean section between 36 and 37 weeks of gestation.69,72,73 [Evidence level 4] 699 700 The risks of bleeding, labour, or bleeding and labour leading to the need for emergency delivery 701 increase with advancing gestational age, whereas the risks of morbidity associated with prematurity 702 decrease.4,5 The risk of an emergent bleed associated with placenta praevia has been reported to be 703 4.7% by 35 weeks of gestation, 15% by 36 weeks of gestation, 30% by 37 weeks of gestation and 59% 704 by 38 weeks of gestation.74 [Evidence level 2–] 705 706 A US population-based cohort study using the Centre for Disease Control and Prevention’s Linked 707 Birth-Infant Death data files has evaluated the effects of delivering placenta praevia at 35, 36 and 37 708 weeks of gestation on the risk of several neonatal outcomes.75 Compared with neonates born at 38 709 weeks of gestation, those delivered at 35, 36 and 37 weeks of gestation have no greater odds of 710 meconium passage, fetal distress, fetal anaemia, neonatal seizures, increased ventilator needs or 711 infant death at 1 year. However, aOR odds of 5-minute Apgar scores of less than 7 are greater at 35 712 and 36 weeks of gestation (aOR 3.33, 95% CI 1.71–6.47; and aOR 2.17, 1.11–4.22, respectively) as 713 are odds of NICU admission rates (aOR 2.25, 95% CI 2.01–2.50; and aOR 1.57, 1.38–1.76, 714 respectively). [Evidence level 2+] 715 716 4.9 In what situations is vaginal delivery appropriate for women with a low-lying placenta? 717 718 In women with a third trimester asymptomatic low-lying placenta, the mode of delivery should be 719 based on the clinical background, the woman’s preferences supplemented by ultrasound findings, 720 including the distance between the placental edge and the fetal head position relative to the 721 leading edge of the placenta on TVS. [D] 722 723 Women presenting with a placental edge less than 20 mm from the internal os in the third trimester 724 are more likely to need delivery by caesarean section when the placental edge is thicker (over 10 725 mm)76,77 and/or contains a sponge-like echo78 or marginal ‘sinus’.79 These additional ultrasound 726 features are poorly defined, not routinely assessed in UK practice and the success rates of vaginal 727 delivery when the placental edge is between 10 and 20 mm from the internal os vary widely (56% 728 and 93%, respectively).80–83 The corresponding studies are small, observational and retrospective, 729 making a recommendation for a specific mode of delivery based on ultrasound findings difficult. 730 [Evidence level 2–] 731 732 5. Optimising the delivery of a placenta praevia 733 734 Prior to delivery, all women with placenta praevia and their partners should have a discussion 735 regarding delivery. Indications for blood transfusion and hysterectomy should be reviewed and 736 concerns or plans to decline blood or blood products should be discussed openly and documented. 737 [GPP] 738 739 Placenta praevia and anterior low-lying placenta carry a higher risk of massive obstetric 740 haemorrhage and hysterectomy. Delivery should be arranged in a maternity unit with on-site 741 blood transfusion services and access to critical care. [D] 742 Deleted: Late preterm (35+1 to 36+6 weeks of gestation) 743 Deleted: for women presenting with complicated placenta 744 praevia or a low-lying placenta 745 Deleted: ¶ 746 Deleted: late preterm (34+0 to 36+6 weeks of gestation) 747 Page 15 of 47 748 Women with atypical antibodies form a particularly high-risk group and the care of these women 749 should involve discussions with the local haematologist and blood bank. [D] 750 751 Prevention and treatment of anaemia during the antenatal period is recommended for women 752 with placenta praevia or a low-lying placenta as for any pregnant woman. [D] 753 754 General procedures for discussing and obtaining consent for caesarean section are described in 755 detail in RCOG Consent Advice No.7: Caesarean section.84 [Evidence level 4] 756 757 Women having a caesarean section for placenta praevia are at increased risk of blood loss of more 758 than 1000 ml compared with women having a caesarean section for other indications (RR 3.97, 95% 759 CI 3.24–4.85).85 Women with anterior placenta regardless of type of placenta praevia are at 760 increased blood loss.86 Placenta praevia covering the internal cervical os and anterior placentation 761 are independent risk factors (OR 4.1 and OR 3.5, respectively) for massive haemorrhage during 762 caesarean section.86 A US case–control study from the National Institute of Child Health (NICH) and 763 Human Development Maternal-Fetal Medicine Units (MFMU) Network Caesarean Section Registry 764 has shown that maternal haemorrhagic morbidity is more common in women with praevia (19% 765 versus 7%, adjusted RR 2.6, 95% CI 1.9-3.5) and the main factors associated with maternal 766 haemorrhage include pre-delivery anaemia, thrombocytopenia, diabetes and magnesium use.87 767 [Evidence level 2++] 768 769 The risk of massive haemorrhage together with the possibility of needing a blood transfusion has 770 been estimated to be approximately 12 times more likely in caesarean section for placenta praevia 771 than in caesarean delivery for other indications.88,89 Similarly to uncomplicated pregnancies, women 772 with placenta praevia should be screened for anaemia and investigated if their haemoglobin levels 773 are outside the normal UK range (110 g/l at first visit and 105 g/l at 28 weeks).36 Iron 774 supplementation should be implemented if indicated. [Evidence level 4] 775 776 For women at high risk of emergency transfusion, such as those presenting with placenta praevia 777 and with no clinically significant alloantibodies, it has been recommended that group and screen 778 samples should be sent once a week to exclude or identify any new antibody formation and to keep 779 blood available if necessary for delivery. However, this should be at the discretion of the team 780 responsible and managed according to local facilitites.89 [Evidence level 4] 781 782 6. Delivery for women with placenta praevia or a low-lying placenta 783 784 6.1 What grade of obstetrician and anaesthetist should attend the caesarean delivery for a placenta 785 praevia? 786 787 As a minimum requirement for a planned caesarean section for placenta praevia, the surgical 788 procedure should be carried out by an appropriately experienced operator. [GPP] 789 790 In cases of planned caesarean section for placenta praevia or a low-lying placenta, a senior 791 obstetrician (usually a consultant) and senior anaesthetist (usually a consultant) should be present 792 within the delivery or theatre suite where the surgery is occurring. [GPP] 793 794 When an emergency arises, the senior obstetrician and senior anaesthetist should be alerted 795 immediately and attend urgently. [GPP] 796 797 Maternal complications at caesarean section increase when the primary surgeon is a 798 trainee/resident rather than an experienced surgeon.90 Placenta praevia is often associated with 799 Page 16 of 47 additional including fetal malpresentation (transverse or breech presentation) requiring complex 800 intraoperative manoeuvres to deliver the baby.91 [Evidence level 4] 801 802 6.2 What anaesthetic procedure is most appropriate for caesarean section in placenta praevia? 803 804 Regional anaesthesia is considered safe and associated with lower risks of haemorrhage than 805 general anaesthesia for caesarean delivery in women with placenta praevia or a low-lying 806 placenta. Women with anterior placenta praevia or a low-lying placenta should be advised that it 807 may be necessary to convert to general anaesthesia if required and asked to consent. [D] 808 809 There is insufficient evidence to support one technique over another and there have been no new 810 trials since the previous version of this guideline. 811 812 An RCT of regional versus general anaesthesia for placenta praevia, including women with placenta 813 accreta, has indicated that blood transfusion requirements (although not estimated blood loss) are 814 greater in the general anaesthetic group.92 [Evidence level 1–] 815 816 A 4-year observational study at 19 US academic centres of women undergoing caesarean delivery 817 found that the risk factors for haemorrhage-related morbidity are increased in those undergoing 818 general anaesthesia.93 [Evidence level 2–] 819 820 The recent case–control study from the NICHD/MFMU Network Cesarean Section Registry found 821 general anaesthesia to be one of the main factors associated with maternal haemorrhage in women 822 with placenta praevia.87 [Evidence level 2++] 823 824 6.3 What blood products should be available? 825 826 Close liaison with the hospital transfusion laboratory is essential for women presenting with 827 placenta praevia or a low-lying placenta. [GPP] 828 829 Rapid infusion and fluid warming devices should be immediately available. [GPP] 830 831 Cell salvage is recommended for patients where the anticipated blood loss is great enough to 832 induce anaemia, in particular, in women who would decline blood products. [D] 833 834 Red cells, fresh frozen plasma, and cryoprecipitate or fibrinogen concentrate are all kept by blood 835 banks supplying obstetric units. If the haemoglobin is less than 70 g/l in the postoperative period, 836 where there is no ongoing or threat of bleeding, the decision to transfuse should be made on an 837 informed individual basis.89 In an extreme situation and when the blood group is unknown, group O 838 rhesus D-negative red cells should be given.89 Further recommendations are provided in Green-top 839 Guideline No.52: Prevention and Management of Postpartum Haemorrhage. 88 [Evidence level 4] 840 841 There is no evidence to support the use of autologous blood transfusion for placenta praevia.90 842 [Evidence level 4] 843 844 Cell salvage was not often used previously in obstetrics because of the perceived risk of amniotic 845 fluid embolism or induction of maternal alloimmunisation. No definite cases of amniotic fluid 846 embolism have been reported so far and the risks of cell salvage in the obstetric population parallel 847 those in the non-pregnant population.94,95 [Evidence level 4] 848 849 6.4 What surgical approach should be used for placenta praevia or a low-lying placenta? 850 851 Consider vertical skin and/or uterine incisions when the fetus is in a transverse lie to avoid the 852 Page 17 of 47 placenta, particularly below 28 weeks of gestation. [GPP] 853 854 Consider using preoperative and/or intraoperative ultrasonography to precisely determine 855 placental location and the optimal place for uterine incision. [D] 856 857 If the placenta is transected during the uterine incision, immediately clamp the umbilical cord 858 after fetal delivery to avoid excessive fetal blood loss. [D] 859 860 If pharmacological measures fail to control haemorrhage, initiate intrauterine tamponade and/or 861 surgical haemostatic techniques sooner rather than later. Interventional radiological techniques 862 should also be urgently employed where possible. [C] 863 864 Early recourse to hysterectomy is recommended if conservative medical and surgical interventions 865 prove ineffective. [D] 866 867 In cases of anterior placenta praevia, cutting through the placenta is often associated with increased 868 maternal bleeding. A retrospective cohort study found that avoiding incision of the anterior placenta 869 praevia after 24 weeks of gestation reduces the need for maternal blood transfusion during or after 870 caesarean delivery.96 [Evidence level 2–] 871 872 A ‘J’-shaped uterine incision has been evaluated in women presenting with placenta praevia in a 873 small retrospective study and shown to decrease intraoperative blood loss and facilitate the delivery 874 of the fetus.97 [Evidence level 2–] 875 876 Intrauterine balloon tamponade, different types of compression sutures and uterine artery occlusion 877 techniques have been increasingly used since the previous version of the guideline in women with 878 placenta praevia to control, reduce or stop intraoperative bleeding and PPH. Case series on the use 879 of intrauterine hydrostatic balloon catheters, including the Bakri balloon,98–102 the BT-Cath® 880 balloon103 or the Sengstaken–Blakemore tube,104 in women with placenta praevia have reported 881 success in controlling PPH ranging from 75% to 88%. [Evidence level 3] 882 883 Factors associated with the failure of Bakri balloon tamponade for placenta praevia include prior 884 caesarean section, anterior placentation, thrombocytopenia and/or coagulopathy at the time of 885 insertion, and a PPH volume of more than 500 ml within the first hour of placement.100 [Evidence 886 level 2++] 887 888 Uterine compressive and endouterine sutures are well established techniques for the control of 889 haemorrhage following atonic PPH. The best known suture technique was described by B-Lynch in 890 1997.105 A combined method of B-Lynch suture and the intrauterine balloon has also been 891 successfully used in preventing PPH in placenta praevia.106 [Evidence level 3] 892 893 Intraoperative interventional radiological techniques, including transarterial embolisation107 and 894 temporary balloon occlusion108 of the internal iliac arteries, have also been successfully used to 895 prevent and control haemorrhage in placenta praevia and should be considered when available. 896 Follow-up studies of women who have undergone arterial embolisation for control of PPH suggest 897 that the intervention does not impair subsequent menstruation and fertility.109–111 [Evidence level 3] 898 899 7. Antenatal diagnosis and outcome of placenta accreta spectrum 900 901 7.1 What are the risk factors for placenta accreta spectrum? 902 903 The major risk factors for placenta accreta spectrum are history of accreta in a previous 904 pregnancy, previous caesarean delivery and other uterine surgery, including repeated endometrial 905 Page 18 of 47 curettage. This risk rises as the number of prior caesarean sections increases. [B] 906 907 Women requesting elective caesarean delivery for non-medical indications should be informed of 908 the risk of placenta accreta spectrum and its consequences for subsequent pregnancies. [GPP] 909 910 All epidemiological studies of the last 2 decades have shown a direct association between the 911 increase in caesarean deliveries and the incidence of placenta accreta spectrum (abnormally 912 adherent and invasive placenta) in subsequent pregnancies worldwide.112–122 The 2016 Nordic 913 Obstetric Surveillance Study found that the risk of invasive placentation increases seven-fold after 914 one prior caesarean section.118 [Evidence level 2+] 915 916 A meta-analysis of five cohorts and 11 case–control studies reported a summary OR of 1.96 (95% CI 917 1.41–2.74) for placenta accreta spectrum after a caesarean section.24 [Evidence level 2++] 918 919 The risk of placenta accreta spectrum increases with the number of previous caesarean sections. A 920 systematic review reported an increase in the incidence of accreta placentation from 3.3–4.0% in 921 women with placenta praevia and no previous caesarean delivery, to 50–67% in women with three 922 or more caesarean deliveries.25 When stratified for the number of previous caesarean sections, the 923 OR for placenta accreta spectrum in a subsequent pregnancy ranges between 8.6 (95% CI 3.536– 924 21.078)112 and 17.4 (95% CI 9.0–31.4) for two previous caesarean sections, and 55.9 (95% CI 25.0– 925 110.3) for three or more caesarean sections.121 [Evidence level 2++] 926 927 Placenta praevia is another important risk factor for placenta accreta spectrum (see Appendix II). A 928 large multicentre US cohort study noted that for women presenting with placenta praevia and prior 929 caesarean section the risk of accreta placentation was 3%, 11%, 40%, 61% and 67% for one, two, 930 three, four, and five or more caesarean deliveries, respectively.113 The national case–control study 931 using the UK Obstetric Surveillance System found that the incidence of placenta accreta spectrum 932 increases from 1.7 per 10 000 women overall to 577 per 10 000 in women with both a previous 933 caesarean section and placenta praevia.114 [Evidence level 2+] 934 935 Other additional risk factors include maternal age111,114,118,121 and ART, in particular in vitro 936 fertilisation.114,121,123–126 Advanced maternal age (35 years or more) in women without a previous 937 caesarean section increases the aOR by 1.30 (95%CI 1.13-1.50) for every 1-year increase in age.114 938 [Evidence level 2-] 939 940 Placenta accreta spectrum is not exclusively a consequence of caesarean delivery. Other surgical 941 trauma to the integrity of the uterine endometrium and/or superficial myometrium, such as those 942 following uterine curettage, manual removal of the placenta, postpartum endometritis or 943 myomectomy, has been associated with accreta placentation in subsequent pregnancies.1,12,13 944 Overall, the aOR for placenta accreta spectrum after previous uterine surgery is 3.40 (95% CI 1.30– 945 8.91).114 [Evidence level 2+] 946 947 The development of placenta accreta spectrum has also been reported in women with no surgical 948 history but presenting with a uterine pathology, such as bicornuate uterus, adenomyosis, 949 submucous fibroids and myotonic dystrophy.1,12,13 [Evidence level 3] 950 951 More recently there has been an increase in reports describing implantation into deficient caesarean 952 section scars and mounting evidence that a caesarean scar pregnancy diagnosed in early pregnancy 953 can evolve into an abnormally adherent or invasive placenta in the second half of pregnancy.127–131 A 954 caesarean scar pregnancy can be diagnosed using TVS from the second month of pregnancy using 955 specific ultrasound criteria.130,131 In the last decade, the number of reported cases of caesarean scar 956 pregnancy has increased due to improved awareness of the condition, widespread use of ultrasound 957 scanning in early pregnancy and an increase in the number of prior caesarean sections. The outcome 958 Page 19 of 47 of caesarean scar pregnancy depends on the amount of definitive placenta developing inside the 959 scar and depth of villous invasion. Further data are required to establish the relationship between a 960 first trimester scar pregnancy and the development of invasive placentation. [Evidence level 3] 961 962 7.2 How can a placenta accreta spectrum be suspected and diagnosed antenatally? 963 964 Antenatal diagnosis of placenta accreta spectrum is crucial in planning its management and has 965 been shown to reduce maternal morbidity and mortality. [D] 966 967 Previous caesarean delivery and the presence of an anterior low-lying placenta or placenta praevia 968 should alert the antenatal care team of the higher risk of placenta accreta spectrum. [D] 969 970 Maternal complications in placenta accreta spectrum are primarily the result of massive 971 haemorrhage.5 Median estimated blood loss in cohorts of placenta accreta spectrum ranges from 972 2000 to 7800 ml and the median number of units of blood transfused is 5 units.132 Antenatal 973 diagnosis of placenta accreta spectrum reduces maternal peripartum haemorrhage and 974 morbidity.20,133–136 [Evidence level 4] 975 976 Population studies have shown that placenta accreta spectrum remains undiagnosed before delivery 977 in one-half137 to two-thirds of cases.121 In a series from specialist centres, approximately one-third of 978 cases of placenta accreta were not diagnosed during pregnancy.138 [Evidence level 2+] 979 980 Multidisciplinary management in a maternity unit with access to maternal and neonatal intensive 981 care is often required for women with placenta accreta spectrum.21,22,136,139 For such care to be 982 organised, the diagnosis must be made antenatally. [Evidence level 4] 983 984 7.2.1 Ultrasound screening and diagnosis of placenta accreta spectrum 985 986 Ultrasound imaging is highly accurate when performed by a skilled operator with experience in 987 diagnosing placenta accreta spectrum. [C] 988 989 Refer women with any ultrasound features suggestive of placenta accreta spectrum to a specialist 990 unit with imaging expertise. [B] 991 992 Women with a history of previous caesarean section seen to have an anterior low-lying placenta 993 or placenta praevia at the routine fetal anomaly scan should be specifically screened for placenta 994 accreta spectrum. [D] 995 996 Numerous ultrasound imaging techniques have been reported over the years, including greyscale 997 imaging and colour Doppler imaging (CDI), and/or three-dimensional power Doppler 998 sonography.16,17,140–142 In 2016, the European Working Group on Abnormally Invasive Placenta 999 proposed a standardised description of ultrasound signs (see Appendix III) used for the prenatal 1000 diagnosis of placenta accreta141 and the International Abnormally Invasive Placenta Expert Group 1001 produced a proforma protocol for the ultrasound assessment.142 [Evidence level 4] 1002 1003 A systematic review and meta-analysis of 23 ultrasound studies including 3707 pregnancies at risk of 1004 placenta accreta found that the overall performance of ultrasound when performed by skilled 1005 operators was very good with a sensitivity of 90.72% (95% CI 87.2–93.6), specificity of 96.94% (95% 1006 CI 96.3–97.5) and diagnostic OR of 98.59 (95%CI 48.8–199.0). Among the different ultrasound signs, 1007 abnormality of the uterus–bladder interface had the best specificity of 99.75% (95% CI 99.5–99.9) 1008 for the prediction of placenta accreta. Abnormal vasculature on CDI had the best predictive accuracy 1009 with a sensitivity of 90.74% (95% CI 85.2–94.7), specificity of 87.68% (95% CI 84.6–90.4) and 1010 diagnostic OR of 69.02 (95% CI 22.8–208.9).143 [Evidence level 2++] 1011 Page 20 of 47 1012 A 2017 systematic review and meta-analysis using the standardised ultrasound signs (see Appendix 1013 III) has shown that in women presenting with placenta praevia and history of prior caesarean 1014 section, the performance of ultrasound for the antenatal detection of placenta accreta spectrum is 1015 even higher with a sensitivity of 97.0% (95% CI 93.0–99.0), specificity of 97.0% (95% CI 97.0–98.0) 1016 and diagnostic OR of 228.5 (95% CI 67.2–776.9) in prospective studies.144 Placental lacunae give the 1017 placenta a ‘moth-eaten’ appearance on greyscale imaging and the increased vascularity of the 1018 placental bed with large feeder vessels entering the lacunae are the most common ultrasound signs 1019 associated with placenta accreta spectrum.16,17,143,144 [Evidence level 2++] 1020 1021 Determining the depth and lateral extension of placental invasion is helpful for planning the 1022 individual care of women diagnosed with placenta accreta spectrum.16,17,145 No ultrasound sign or a 1023 combination of ultrasound signs have so far been found to be specific to the depth of placenta 1024 accreta spectrum and thus to provide with an accurate differential diagnosis between adherent and 1025 invasive accreta placentation.16 This may be due to the wide heterogeneity in terminology used to 1026 describe the grades of placenta accreta spectrum, differences in the study design with most studies 1027 not reporting detailed data on clinical diagnosis at birth and/or on histopathology examination, and 1028 many studies having included cases of placental retention in their cohort with no evidence of 1029 abnormal villous adherence or invasion. [Evidence level 2++] 1030 1031 As the vast majority of placenta accreta spectrum are now the consequence of low placentation into 1032 a previous caesarean section scar, TVS has an important role in the early diagnosis, follow-up, 1033 differential diagnosis between adherent and invasive accreta placentation and management of 1034 placenta accreta spectrum.144 [Evidence level 4] 1035 1036 7.2.2 Is there a role for magnetic resonance imaging (MRI) in the diagnosis of placenta accreta 1037 spectrum? 1038 1039 Clinicians should be aware that the diagnostic value of MRI and ultrasound imaging in detecting 1040 placenta accreta spectrum is similar when performed by experts. [C] 1041 1042 MRI may be used to complement ultrasound imaging to assess the depth of invasion and lateral 1043 extension of myometrial invasion, especially with posterior placentation and/or in women with 1044 ultrasound signs suggesting parametrial invasion. [GPP] 1045 1046 MRI has been increasingly used for the prenatal diagnosis of placenta accreta.146–150 The main MRI 1047 features of placenta accreta include abnormal uterine bulging, dark intraplacental bands on T2-1048 weighted imaging, heterogeneous signal intensity within the placenta, disorganised vasculature of 1049 placenta and disruption of the uteroplacental zone. A systematic review has found that most studies 1050 are of a small sample size and thus, sensitivity and specificity of MRI in diagnosing placenta accreta 1051 varies widely between 75% and 100%, and 65% and 100%, respectively.149 [Evidence level 2++] 1052 1053 Two systematic reviews and meta-analyses have found that the diagnostic value of ultrasound 1054 imaging and MRI in detecting placenta accreta spectrum is similar. The first review148 included 13 1055 studies and reported a sensitivity of 83% (95% CI 77–88), specificity of 95% (95% CI 93–96) and 1056 detection OR of 63.41 (95% CI 29.04–138.48) for ultrasound, compared with a sensitivity of 82% 1057 (95% CI 72–90), specificity of 88% (95% CI 81–94) and detection OR of 22.95 (95% CI 3.19–165.11) 1058 for MRI. The second review (2014)149 included 18 studies and found that the overall diagnostic 1059 accuracy of MRI has a sensitivity of 94.4% (95% CI 86.0–97.9), specificity of 84.0% (95% CI 76.0–89.8) 1060 and diagnostic OR of 89.0 (95% CI 22.8–348.1). The latter review also found that MRI has high 1061 predictive accuracy in assessing both the depth and topography of placental invasion. [Evidence level 1062 2++] 1063 1064 Page 21 of 47 The use of intravenous gadolinium injection may increase the sensitivity and specificity of MRI in the 1065 diagnosis of the invasive forms of placenta accreta spectrum but the evidence on long-term fetal 1066 safety is limited.150 Furthermore, the experience of the radiologists remains an independent factor in 1067 the diagnostic accuracy of MRI and access to expert radiologists is highly variable. [Evidence level 4] 1068 1069 7.3 Where should women with placenta accreta spectrum be cared for? 1070 1071 Women diagnosed with placenta accreta spectrum should be cared for by a multidisciplinary team 1072 in a specialist centre with expertise in diagnosing and managing invasive placentation. [GPP] 1073 1074 Delivery for women diagnosed with placenta accreta spectrum should take place in a specialist 1075 centre with logistic support for immediate access to blood products, adult intensive care unit and 1076 NICU by a multidisciplinary team with expertise in complex pelvic surgery. [D] 1077 1078 More data have become available since the last version of this guideline on the specific management 1079 of placenta accreta spectrum. Overall, women with accreta placentation should be cared for 1080 according to the risks of severe maternal bleeding and premature delivery. Placenta percreta can be 1081 associated with major prenatal complications from early in pregnancy, such as uterine rupture151–153 1082 and bladder involvement with associated life-threatening haemorrhage.154–156 [Evidence level 4] 1083 1084 A 2015 expert review has suggested that caesarean delivery of women at high risk and/or diagnosed 1085 prenatally with placenta accreta spectrum, in particular its invasive forms, should occur in a 1086 specialist centre with multidisciplinary expertise and experience in managing complex pelvic surgery, 1087 and with access to an adult intensive care unit and NICU.136 [Evidence level 4] 1088 1089 A retrospective cohort study of 77 women with suspected placenta accreta found that women who 1090 delivered prior to a planned delivery date were significantly more likely to have had vaginal bleeding 1091 and uterine activity when compared with women who had a scheduled delivery.20 Each episode of 1092 antenatal vaginal bleeding is associated with an increased risk of unscheduled delivery (aOR 3.8, 95% 1093 CI 1.8–7.8) and the risk increases when associated with preterm prelabour rupture of membranes. 1094 [Evidence level 2–] 1095 1096 Considering the higher frequency of placenta praevia in the accreta group,144,157 these results are 1097 likely to be influenced by the perinatal complications of placenta praevia. Surveys of healthcare 1098 providers in the US and Canada have highlighted widely varied approaches to virtually every aspect 1099 of care for placenta accreta spectrum.158–161 Similarly, a recent online survey completed by members 1100 of the expert panel for the perinatal management of placenta accreta spectrum disorders for the 1101 International Federation of Gynecology and Obstetrics (FIGO) has found wide variation in global 1102 practices.162 [Evidence level 4] 1103 1104 There is increasing evidence from retrospective cohort studies from the USA that women with 1105 placenta accreta spectrum diagnosed prenatally, cared for by a specialist multidisciplinary team, are 1106 less likely to require large volume blood transfusion and reoperation within 7 days of delivery for 1107 bleeding complications compared with women cared for by non-multidisciplinary standard obstetric 1108 care without a specific protocol.21,22,136,139,163,164 Women admitted at 34 weeks of gestation and 1109 delivered between 34 and 35 weeks of gestation by a specialist multidisciplinary team have a 1110 significantly lower emergency surgery rate than those not cared for by such a team (23% versus 64%; 1111 P = 0.001) despite a similar median gestational age at delivery (34 [16–39] weeks versus 34 [19-40] 1112 weeks; P = 0.50, respectively).21 In addition, maternal outcomes are improved over time with 1113 increasing experience within a well-established multidisciplinary team performing 2–3 cases per 1114 month.22 Very few of these studies provide with data on the differential clinical diagnosis between 1115 abnormally adherent and abnormally invasive accreta and detailed pathologic confirmation of the 1116 depth and lateral extension of villous myometrial invasion. [Evidence level 2–] 1117 Page 22 of 47 1118 7.4 When should delivery be planned for women with placenta accreta spectrum? 1119 1120 In the absence of risk factors for preterm delivery and evidence of invasive placentation, planned 1121 delivery at 35+0 to 36+6 weeks of gestation provides the best balance between fetal maturity and 1122 the risk of unscheduled delivery. [GPP] 1123 1124 Similarly to placenta praevia, clinical factors should be considered when determining the timing of 1125 administration of antenatal corticosteroids and the optimal gestational age for delivery in women 1126 with placental accreta.165,166 There are currently no RCTs or well-controlled observational studies to 1127 guide best practice in delivery timing of placenta accreta spectrum. [Evidence level 4] 1128 1129 In cases of suspected placenta accreta spectrum, where significant blood loss and caesarean 1130 hysterectomy is anticipated, delivery at between 34 and 35 weeks of gestation has been proposed in 1131 order to avoid emergency delivery, which still occurs about 20% of the time even in scheduled 1132 cases.165,167 A 2010 decision analysis supports this approach based on the increasing likelihood of 1133 emergency delivery as pregnancy goes beyond 34 weeks of gestation.168 [Evidence level 4] 1134 1135 The data of three recent single institution retrospective cohort studies of women with prior 1136 caesarean delivery diagnosed prenatally with placenta accreta have indicated that in the absence of 1137 risk factors for preterm delivery, it is safe to plan the delivery at 36 weeks of gestation. The first 1138 study included 103 women delivered between 1982 and 2002 and found that the mean gestational 1139 age at delivery is 33+5 weeks of gestation in cases of deep placental invasion (increta and percreta) 1140 compared with 35+2 weeks of gestation in the superficial adherent group.169 The second study of 216 1141 women found that urgent delivery for bleeding decreased significantly with advancing gestation.170 1142 Most women were delivered at 36 weeks of gestation or greater, with nearly 90% in the absence of 1143 bleeding complications. The third study of 84 women who had reached 34+0 weeks of gestation with 1144 a suspected praevia accreta found that those with no risk factors for preterm birth are at low risk for 1145 an unscheduled delivery prior to 36 weeks of gestation.171 [Evidence level 2+] 1146 1147 8. Planning delivery of a suspected placenta accreta spectrum 1148 1149 Once the diagnosis of placenta accreta spectrum is made, a contingency plan for emergency 1150 delivery should be developed, including the use of an institutional protocol for the management of 1151 maternal haemorrhage. [GPP] 1152 1153 Due to a lack of RCTs or well-controlled observational studies, the optimal management of placenta 1154 accreta spectrum remains undefined and is determined by the expertise available, the depth and 1155 lateral extension of the accreta portion of the placenta, the presence of an associated placenta 1156 praevia, radiological findings, the medical and surgical comorbidities, and finally, the accessibility of 1157 a regional team focused on these patients. 1158 1159 The main risk associated with the delivery of placenta accreta spectrum is massive haemorrhage and 1160 its associated complications, such as coagulopathy, multisystem organ failure and death. Many 1161 women with placenta accreta spectrum require massive blood transfusion (8 units or more) and 1162 their median platelet count is lowest compared with other causes of massive PPH.172,173 [Evidence 1163 level 2+] 1164 1165 A review of 34 studies published between 1977 and 2012, including a total number of 508 617 1166 deliveries and 865 cases of confirmed placenta accreta, found that the most significant maternal 1167 risks associated with delivery are the need for postpartum transfusion due to haemorrhage and 1168 peripartum hysterectomy. Maternal mortality remains rare, but significantly higher than among 1169 matched postpartum controls.123 [Evidence level 4] 1170 Page 23 of 47 1171 Transfusions in placenta accreta spectrum should be guided by a national and/or institutional 1172 protocol for management of PPH.88,89 [Evidence level 4] 1173 1174 8.1 What should be included in the consent form for caesarean section in cases of suspected placenta 1175 accreta spectrum? 1176 1177 Any woman giving consent for caesarean section should understand the risks associated with 1178 caesarean section in general, and the specific risks of placenta accreta spectrum in terms of 1179 massive obstetric haemorrhage, increased risk of lower urinary tract damage, the need for blood 1180 transfusion and the risk of hysterectomy. [GPP] 1181 1182 Additional possible interventions in the case of massive haemorrhage should also be discussed, 1183 including cell salvage and interventional radiology where available. [D] 1184 1185 Any woman with suspected placenta accreta spectrum should meet with a senior obstetrician in the 1186 antenatal period. The different risks and treatment options should have been discussed and a plan 1187 agreed, which should be reflected clearly in the consent form and medical record. This should 1188 include standard discussion for the caesarean section procedure84 and whether conservative 1189 management of the placenta or proceeding straight to hysterectomy is preferred in the situation 1190 where increta or percreta is confirmed at surgery. [Evidence level 4] 1191 1192 Where available, cell salvage should be considered. If the woman refuses donor blood transfusion, it 1193 is recommended89 that she be transferred to a unit with a cell saver. [Evidence level 4] 1194 1195 8.2 What healthcare professionals should be involved? 1196 1197 The elective delivery of women with placenta accreta spectrum should be managed by a 1198 multidisciplinary team, which should include senior anaesthetists, obstetricians and 1199 gynaecologists with appropriate experience in managing the condition and other surgical 1200 specialties if indicated. In an emergency, the most senior clinicians available should be involved. 1201 [GPP] 1202 1203 Following the previous version of the guideline, the National Patient Safety Agency in collaboration 1204 with the RCOG and the Royal College of Midwives set up an expert working group to develop a care 1205 bundle for placenta accreta.174 Six elements of good care were agreed upon. The care bundle was 1206 then tested in six units over a 5-month pilot study period and it was found to be both achievable and 1207 practical. Clinical outcomes were monitored, confirming the high morbidity associated with this 1208 condition. [Evidence level 4] 1209 1210 The six elements considered to be reflective of good care are: 1211 1212 • Consultant obstetrician planning and directly supervising delivery. 1213 • Consultant anaesthetist planning and directly supervising anaesthesia at delivery. 1214 • Blood and blood products available. 1215 • Multidisciplinary involvement in preoperative planning. 1216 • Discussion and consent, including possible interventions (such as hysterectomy, leaving the 1217 placenta in situ, cell salvage and interventional radiology). 1218 • Local availability of a level 2 critical care bed. 1219 1220 The 2015 MBRRACE report from the Confidential Enquiry into Maternal Deaths in the UK has 1221 indicated that despite increasing numbers of women at risk from placenta accreta spectrum 1222 following previous caesarean section, only one death occurred in a woman who had a placenta 1223 Page 24 of 47 praevia percreta and history of two previous caesarean sections.175 There were no deaths from 1224 unexpected placenta accreta found at caesarean section, suggesting that previous recommendations 1225 regarding imaging and preparations for women with placenta praevia and a previous caesarean 1226 section have been followed.176 [Evidence level 2++] 1227 1228 A 2015 single centre retrospective cohort study of the effectiveness of a standardised operative 1229 approach in 98 cases of histologically confirmed placenta accreta supports the early presence of a 1230 gynaecological surgeon and oncologist at delivery and demonstrates that a ‘call if needed’ approach 1231 is not acceptable for these complex cases.177 [Evidence level 2+] 1232 1233 The American College of Obstetricians and Gynecologists (ACOG) guidelines highlight that to 1234 enhance patient safety, it is important that the delivery be performed by an experienced obstetric 1235 team that includes an obstetric surgeon, with other surgical specialists, such as urologists, general 1236 surgeons, and gynaecological surgeons and oncologists, available if necessary.166 [Evidence level 4] 1237 1238 8.3 What anaesthetic is most appropriate for delivery? 1239 1240 The choice of anaesthetic technique for caesarean section for placenta accreta spectrum should be 1241 made by the anaesthetist conducting the procedure in consultation with the patient in advance. 1242 [GPP] 1243 1244 The woman should be informed that the surgical procedure can be performed safely with regional 1245 anaesthesia but should be advised that it may be necessary to convert to general anaesthesia if 1246 required and asked to consent. [D] 1247 1248 Both general and regional anaesthetic techniques have been shown to be safe for surgical 1249 procedures required for the delivery of placenta accreta spectrum; the judgment of which type of 1250 technique to be used should be made on an individual basis.167 [Evidence level 4] 1251 1252 There is insufficient evidence to support one technique over another and there have been no new 1253 trials since the previous version of this guideline. 1254 1255 8.4 Optimising the delivery of placenta accreta spectrum 1256 1257 There are no RCTs comparing different surgical approaches for placenta accreta spectrum suspected 1258 antenatally. Both conservative and radical surgical approaches can be associated with a high 1259 maternal morbidity although the value of an experienced team in a specialist centre decreases the 1260 risk significantly.21,22,136,139,163,164 [Evidence level 4] 1261 1262 8.4.1 What surgical approach should be used for placenta accreta spectrum? 1263 1264 Caesarean section hysterectomy with the placenta left in situ is preferable to attempting to 1265 separate it from the uterine wall. [C] 1266 1267 When the extent of the placenta accreta is limited in depth and surface area, and the entire 1268 placental implantation area is accessible and visualised (i.e. completely anterior, fundal or 1269 posterior without deep pelvic invasion), uterus-preserving surgery may be appropriate, including 1270 partial myometrial resection. [GPP] 1271 1272 Uterus-preserving surgical techniques should only be attempted by surgeons working in teams 1273 with appropriate expertise to manage such cases and after appropriate counselling regarding risks 1274 and with informed consent. [D] 1275 1276 Page 25 of 47 There are currently insufficient data to recommend the routine use of ureteric stents in placenta 1277 creta and increta. [C] 1278 1279 The choice of surgical technique will depend on the position of the placenta, the depth of invasion, 1280 and the parametrial extension of the placenta accreta spectrum as assessed by ultrasound and/or 1281 MRI before delivery, the visual assessment of the uterus at the time of surgery and the presenting 1282 clinical symptoms, i.e. bleeding or no bleeding.5 [Evidence level 4] 1283 1284 The ACOG recommends planned, preterm caesarean section hysterectomy with the placenta left in 1285 situ as removal of a placenta accreta spectrum is associated with significant haemorrhagic 1286 morbidity.166 In cases of high suspicion for accreta during caesarean delivery, the majority of 1287 members of the US Society of Maternal-Fetal Medicine (SMFM) and FIGO expert panel proceed with 1288 hysterectomy.158–162 [Evidence level 4] 1289 1290 Similarly, in a 2017 systematic review and meta-analysis on the diagnosis and outcome of placenta 1291 accreta, an elective or emergency caesarean hysterectomy was performed in 208 out of 232 (89.7%) 1292 cases.144 [Evidence level 2++] 1293 1294 A retrospective study of 57 cases of suspected accreta demonstrated significantly reduced short-1295 term morbidity if the placenta is left in place and hysterectomy performed electively compared with 1296 attempting to remove the placenta first.178 Attempting placental separation risks hysterectomy in up 1297 to 100% of cases as also confirmed by other authors.178,179 [Evidence level 2++] 1298 1299 A case–control study of 49 women requiring a peripartum hysterectomy for massive haemorrhage, 1300 including 20 women presenting with placenta accreta, reported that the use of a vessel sealing 1301 device during surgery decreases the estimated blood loss, the need for massive blood transfusions, 1302 and does not increase operative time or complication rates.180 [Evidence level 2+] 1303 1304 A systematic review found that uterus-preserving surgery resulted in a secondary hysterectomy in 1305 24/77 women (31%), maternal mortality in 2/55 women (4%), subsequent menstruation in 28/34 1306 women (82%) and subsequent pregnancy in 19/26 women (73%).181 A more recent systematic 1307 review showed that uterus-preserving surgery is associated with a success rate of 48/76 women 1308 (63.2%), a secondary hysterectomy in 23/76 women (30.0%), maternal mortality in 2/54 women 1309 (3.7%), subsequent menstruation in 20/37 women (81.1%) and subsequent pregnancy in 21/27 1310 women (77.8%).182 [Evidence level 2++] 1311 1312 A small cohort study has shown that the introduction of the Triple-P procedure (perioperative 1313 placental localization, pelvic devascularization and placental non-separation) involving delivery of 1314 the fetus via transverse uterine incision above the upper border of the placenta, myometrial excision 1315 and reconstruction of the uterine wall reduces the rate of hysterectomy, PPH and duration of 1316 hospital stay in women with placenta accreta.183 The incidence of post-operative complications of 1317 the Triple-P procedure depends on comorbidities and in particular, the placental position and the 1318 depth of villous invasion.184 Small case series have also reported on the successful use of 1319 compression sutures and on using the cervix as a natural tamponade by inverting it into the uterine 1320 cavity, and suturing the anterior and/or the posterior cervical lips into the anterior and/or posterior 1321 walls of the lower uterine segment.185–188 [Evidence level 3] 1322 1323 A systematic review of peripartum surgical techniques used in placenta accreta spectrum has found 1324 that methotrexate (MTX) and uterus-preserving surgical techniques are associated with a 16% 1325 unintentional urinary tract injury rate as opposed to 57% for standard hysterectomy and that use of 1326 ureteric stents reduces the risk of urologic injury.189 [Evidence level 2++] 1327 1328 There are no RCTs on the use of ureteric stents in placenta accreta spectrum. Ureteric stents or 1329 Commented [EJ1]: Too Commented [EJ2]: Page 26 of 47 catheters are more commonly used pre-operatively in the USA where around 26% of the members 1330 of both the SMFM159 and ACOG fellows161 are using them in the management of suspected of 1331 abnormally invasive placenta. [Evidence level 4] 1332 1333 8.4.2 What surgical approach should be used for placenta percreta? 1334 1335 There is limited evidence to support uterus-preserving surgery in placenta percreta and women 1336 should be informed of the high risk of peripartum and secondary complications, including the 1337 need for secondary hysterectomy. [D] 1338 1339 The following four approaches have been described.137,159–161,165,167,190 1340 1341 1. Primary hysterectomy following delivery of the fetus, without attempting placental separation. 1342 2. Delivery of the fetus avoiding the placenta, with repair of the incision leaving the placenta in 1343 situ. 1344 3. Delivery of the fetus without disturbing the placenta, followed by partial excision of the uterine 1345 wall (placental implantation site) and repair of the uterus. 1346 4. Delivery of the fetus without disturbing the placenta, and leaving it in situ, followed by elective 1347 secondary hysterectomy 3–7 days following the primary procedure. 1348 1349 There are no well-controlled observational studies, and therefore, no firm recommendations can be 1350 made. 1351 1352 Women with placenta percreta are more likely to require additional blood products and intensive 1353 care admission than women with placenta creta or increta.190 The incidence of urological 1354 complications is also increased, including cystotomy and ureteric injury.191 [Evidence level 4] 1355 1356 When the urinary bladder is invaded by placental tissue, preoperative cystoscopy and the placement 1357 of ureteric stents have been recommended.161,192 Planned cystotomy can prevent extensive 1358 muscularis damage and bleeding from attempts at dissection.192 [Evidence level 4] 1359 1360 Filling the bladder to identify the bladder separation site, opening the bladder to identify percreta 1361 villous tissue and removal of the involved bladder area have also been recommended by different 1362 authors.161,165,193 [Evidence level 4] 1363 1364 Uterus-preserving surgery is possible in placenta percreta as demonstrated in a cohort study of 71 1365 women. A multidisciplinary stepwise surgical approach, including bilateral ligations of the anterior 1366 division of the iliac arteries before removing the placenta, was shown to be successful in controlling 1367 the bleeding and preserving the patient's uterus in around 90% of the cases, with 14% of urinary 1368 tract complications, most of which can be identified and repaired during caesarean section.194 1369 [Evidence level 3] 1370 1371 A review of 119 placenta percreta cases published in the international literature has shown that 1372 expectant management with the placenta left in situ is associated with severe long-term 1373 complications of haemorrhage and infections, including a 58% risk of secondary hysterectomy up to 1374 9 months after the delivery. Local resection appears to be associated with fewer complications 1375 within 24 hours postoperatively compared with hysterectomy or leaving the placenta in situ. 1376 However, a selection bias in the direction of less severe cases for the local resection technique may 1377 in part explain the lower complication rates with that approach.195 [Evidence level 4] 1378 1379 8.5 Expectant management (leaving the placenta in situ) 1380 1381 Elective peripartum hysterectomy may be unacceptable to women desiring uterine preservation 1382 Page 27 of 47 or considered inappropriate by the surgical team. In such cases, leaving the placenta in situ should 1383 be considered. [D] 1384 1385 When the placenta is left in situ, local arrangements need to be made to ensure regular review, 1386 ultrasound examination and access to emergency care should the woman experience 1387 complications, such as bleeding or infection. [D] 1388 1389 MTX adjuvant therapy should not be used for expectant management as it is of unproven benefit 1390 and has significant adverse effects, including a reported maternal death. [C] 1391 1392 Conservative management in placenta accreta spectrum, including in cases of placenta increta and 1393 percreta, is an option in women who desire to preserve their fertility. However, it is not 1394 recommended in women presenting with major bleeding as it is unlikely to be successful and risks 1395 delaying definitive treatment and increasing morbidity.5 [Evidence level 4] 1396 1397 A retrospective multicentre study examined 167 women treated conservatively for placenta accreta 1398 in tertiary university hospital centres in France between 1993 and 2007. Conservative expectant 1399 management with part of the placenta left in situ was successful in 131 out of 167 cases (78.4%; 95% 1400 CI 71.4–84.4).196 One woman died of myelosuppression and nephrotoxicity related to MTX 1401 administration through the umbilical cord. Spontaneous placental resorption occurred in 87 out of 1402 116 cases (75.0%; 95% CI 66.1–82.6), with a median delay from delivery of 13.5 weeks (range 4–60 1403 weeks).196 [Evidence level 2+] 1404 1405 The patient should be warned of the risks of chronic bleeding, sepsis, septic shock, peritonitis, 1406 uterine necrosis, fistula, injury to adjacent organs, acute pulmonary oedema, acute renal failure, 1407 deep venous thrombosis or pulmonary embolism.196 Prophylactic antibiotics may be helpful in the 1408 immediate postpartum period to reduce the risk of infective complications.197 [Evidence level 4] 1409 1410 An observational case series, including 24 women with placenta accreta left in situ after delivery and 1411 treated with MTX, reported placental delivery in 33.3% of the cases (spontaneously in 55%, and in 1412 45% following dilatation and surgical evacuation).199 There was no control group of patients who did 1413 not receive MTX and so it is unknown whether or not the MTX was clinically helpful. One patient did 1414 suffer liver damage and the risks of this therapy must be balanced against the unproven benefit. 1415 [Evidence level 3] 1416 1417 The pattern of follow-up for the conservative management of placenta accreta spectrum is not 1418 supported by RCTs and not stratified for the according to the depth and lateral extension of villous 1419 myometrial invasion. Some authors have reported cases where retained villous tissues were 1420 removed after conservative management using hysteroscopic resection200,201 or high-intensity 1421 focused ultrasound.202 In rare cases, a disseminated intravascular coagulation may develop requiring 1422 a secondary hysterectomy. 203 [Evidence level 3] 1423 1424 8.6 When is interventional radiology indicated? 1425 1426 Larger studies are necessary to determine the safety and efficacy of interventional radiology 1427 before this technique can be advised in the routine management of placenta accreta spectrum. [D] 1428 1429 Women diagnosed with placenta accreta spectrum who decline donor blood transfusion should be 1430 managed in a unit with an interventional radiology service. [D] 1431 1432 Since the publication of the last version of this guideline there have been several cohort studies 1433 describing the use of interventional radiology in assisting surgical and conservative management of 1434 placenta accreta with variable success. The main aim of this procedure is to reduce the risks of 1435 Page 28 of 47 intraoperative haemorrhage during the caesarean delivery of pregnancies diagnosed antenatally 1436 with praevia increta or percreta. Various combinations have been proposed, including intraoperative 1437 internal iliac artery and/or postoperative uterine artery embolisation204,205 and internal iliac artery206– 1438 209 or abdominal balloon occlusion210–215. The latter technique has been increasingly used in China 1439 but the methodology of these studies is very heterogeneous with no data on the diagnosis of the 1440 different grades of villous invasion and variable confounding factors such as placental position and 1441 number of previous caesarean deliveries. Small cohort studies have also been published on the use 1442 of a tourniquet216,217 and of surgical artery ligation.218 [Evidence level 3] 1443 1444 A single institution observational cohort study of 45 cases of placenta accreta describes the use of 1445 prophylactic lower abdominal aorta balloon occlusion and found a reduced need for blood 1446 transfusion.212 One of the cases was complicated by lower extremity arterial thrombosis and another 1447 by ischaemic injury to the femoral nerve. A comparative study of abdominal aortic occlusion versus 1448 internal iliac artery occlusion found that aortic balloon occlusion resulted in better clinical outcomes 1449 with less blood loss, blood transfusion, balloon insertion time, fluoroscopy time and fetal radiation 1450 dose.215 [Evidence level 2–] 1451 1452 A systematic review reported success rates of 159/177 (89.8%) for arterial embolisation, with 1453 secondary hysterectomy being necessary in 20/177 (11.3%) and subsequent menstruation occurring 1454 in 74/85 (87.1%). In 3/10 women (30%) a subsequent pregnancy occurred. Arterial balloon occlusion 1455 catheters have been associated with a success rate of 33/42 (78.6%) and the need for a secondary 1456 hysterectomy in 8/42 (19%).182 [Evidence level 2++] 1457 1458 The value of prophylactic placement of balloon catheters in the iliac arteries in cases of placenta 1459 accreta has been more controversial. This is mainly because of the higher risks of complications than 1460 embolisation, including iliac artery thrombus or rupture, and ischaemic nerve injury.219–222 [Evidence 1461 level 3] 1462 1463 A small RCT of women presenting with a prenatal diagnosis of placenta accreta was published in 1464 2015.223 The women were randomised to either preoperative prophylactic balloon catheters (n = 13) 1465 or to a control group (n = 14). No difference was observed for the number of women with blood loss 1466 greater than 2500 ml, number of plasma products transfused, duration of surgery, peripartum 1467 complications and hospitalisation length. Reversible adverse effects related to prophylactic balloon 1468 catheter insertion were noted in 2/13 (15.4%) cases. [Evidence level 1+] 1469 1470 8.7 How is unsuspected placenta accreta spectrum at delivery best managed? 1471 1472 If at the time of an elective repeat caesarean section, where both mother and baby are stable, it is 1473 immediately apparent that placenta percreta is present on opening the abdomen, the caesarean 1474 section should be delayed until the appropriate staff and resources have been assembled and 1475 adequate blood products are available. This may involve closure of the maternal abdomen and 1476 urgent transfer to a specialist unit for delivery. [GPP] 1477 1478 In case of unsuspected placenta accreta spectrum diagnosed after delivery of the baby, the 1479 placenta should be left in situ and an emergency hysterectomy performed. [D] 1480 1481 If the placenta fails to separate with the usual measures, leaving it in place and closing, or leaving it 1482 in place, closing the uterus and proceeding to a hysterectomy are both associated with less blood 1483 loss than trying to separate it. Attempts at removing placenta accreta at caesarean section can lead 1484 to massive haemorrhage, high maternal morbidity and possible maternal death. These risks are 1485 particularly high when the caesarean section takes place in an environment with no emergency 1486 access to blood bank products and expertise in managing placenta accreta.20,21,123,136 [Evidence level 1487 4] 1488 Page 29 of 47 1489 9. Clinical governance 1490 1491 9.1 Debriefing 1492 1493 Postnatal follow-up should include debriefing with an explanation of what happened, why it 1494 happened and any implications for future pregnancy or fertility. In particular, women where 1495 conservative treatment of placenta accreta spectrum has been successful should be informed of the 1496 risk of recurrence. 1497 1498 9.2 Training 1499 1500 Raising the awareness about the clinical risk factors of placenta accreta spectrum should be pursued 1501 locally, including organising policies or guidelines for flagging up women at risk and arranging for 1502 them to see a specialist consultant when suspected. 1503 1504 There should be appropriate training for ultrasound staff in the antenatal diagnosis of placenta 1505 accreta spectrum. 1506 1507 9.3 Clinical incident reporting 1508 1509 Any lack of compliance with the care bundle by the clinical team for a woman with either placenta 1510 praevia or accreta should be investigated. 1511 1512 There should be written protocols for identification of and planning further care of women 1513 suspected to have placenta accreta spectrum. 1514 1515 10. Recommendations for future research 1516 1517 • A large prospective study comparing the impact on the management of the use of the ’low-lying 1518 placenta or placenta praevia’ classification with the traditional grades 1–4 classification at 1519 different gestations is needed. 1520 • Prospective studies are needed to assess the role of third trimester ultrasound in evaluating the 1521 risks of haemorrhage and emergency caesarean section in low-lying placenta and determining 1522 the mode of delivery. 1523 • Large prospective population-based studies are needed in order to assess whether ultrasound is 1524 a cost-effective screening tool for placenta accreta spectrum in women with a history of 1525 caesarean section(s) presenting with a low-lying placenta or placenta praevia in the second 1526 trimester of pregnancy. 1527 • Prospective comparative ultrasound imaging including transvaginal ultrasound and MRI studies 1528 are needed to evaluate the diagnostic accuracy for evaluation of the depth and topography of 1529 villous invasion in adjacent organs. 1530 • RCTs of optimal timing of delivery for both conditions (placenta praevia and placenta accreta) 1531 are needed. 1532 • RCTs of surgical and nonsurgical management strategies for placenta accreta spectrum 1533 (including interventional radiology) and comparing conventional versus conservative 1534 management, stratified according to the depth and lateral extension of villous myometrial 1535 invasion, are needed. 1536 • Future studies on the diagnosis and management of placenta accreta spectrum should use 1537 standardised evidence-based approach including a systematic correlation between ultrasound 1538 signs and detailed clinical diagnosis at delivery and pathologic confirmation of grades of villous 1539 invasiveness when possible. 1540 1541 Page 30 of 47 11. Auditable topics 1542 1543 Placenta praevia 1544 1545 • Antenatal diagnosis of placenta praevia (100%). 1546 • Antenatal detection and treatment of anaemia (100%). 1547 • Antenatal imaging performed according to hospital policy (100%). 1548 • Appropriate antenatal delivery plan made and documented, to include discussion with woman 1549 and her partner, documentation that the risks and indications for blood transfusion and 1550 hysterectomy have been discussed and that concerns, queries or refusals of treatments have 1551 been addressed (100%). 1552 • Involvement of local blood bank and haematologist in the care of women with placenta praevia 1553 and atypical antibodies (100%). 1554 • Appropriate personnel present at delivery (100%). 1555 • Appropriate site for delivery (100%). 1556 • Appropriate surgical approaches performed (100%). 1557 • Early-term elective delivery between 37+0 and 37+6 weeks of gestation for asymptomatic women 1558 with placenta praevia and no other risk factors (100%). 1559 • Antenatal steroid administration between 34+0 and 36+0 weeks of gestation (100%). 1560 • Women requesting elective caesarean section for nonmedical reasons are informed of the risk of 1561 placenta praevia and accreta spectrum, and its consequences in future deliveries (100%). 1562 1563 Placenta accreta spectrum 1564 1565 • Antenatal imaging performed according to hospital policy with diagnosis confirmed at birth 1566 (100%). 1567 • Appropriate antenatal delivery plan made and documented, to include discussion with woman 1568 and her partner, documentation that the risks and indications for blood transfusion and 1569 hysterectomy have been discussed and that concerns, queries or refusals of treatments have 1570 been addressed (100%). 1571 • All elements of the care bundle satisfied before elective surgery in women with placenta accreta 1572 spectrum (100%): 1573 o consultant obstetrician planned and directly supervising delivery 1574 o consultant anaesthetist planned and directly supervising anaesthetic at delivery 1575 o blood and blood products available 1576 o multidisciplinary involvement in preoperative planning 1577 o discussion and consent includes possible interventions (such as hysterectomy, leaving the 1578 placenta in place, cell salvage and interventional radiology) and local availability of a level 2 1579 critical care bed. 1580 1581 12. Useful links and support groups 1582 1583 • Royal College of Obstetricians and Gynaecologists. Low-lying placenta after 20 weeks (placenta 1584 praevia). Information for you. London: RCOG; 20XX [insert web address]. 1585 • National Childbirth Trust. Placenta praevia – low-lying placenta 1586 [ 1587 1588 1589 1590 1591 1592 1593 Page 31 of 47 References 1594 1595 1. Jauniaux E, Jurkovic D. Placenta accreta: pathogenesis of a 20th century iatrogenic uterine 1596 disease. Placenta 2012;33:244–51. 1597 2. Solheim KN, Esakoff TF, Little SE, Cheng YW, Sparks TN, Caughey AB. The effect of cesarean 1598 delivery rates on the future incidence of placenta previa, placenta accreta, and maternal 1599 mortality. J Matern Fetal Neonatal Med 2011;24:1341–6. 1600 3. Bowman ZS, Eller AG, Bardsley TR, Greene T, Varner MW, Silver RM. Risk factors for placenta 1601 accreta: a large prospective cohort. Am J Perinatol 2014;31:799–804. 1602 4. Vintzileos AM, Ananth CV, Smulian JC. Using ultrasound in the clinical management of placental 1603 implantation abnormalities. Am J Obstet Gynecol 2015;213:S70–7. 1604 5. Silver RM. Abnormal placentation: Placenta previa, vasa previa and placenta accreta. Obstet 1605 Gynecol 2015;126:654–68. 1606 6. Jauniaux E, Campbell S. Ultrasonographic assessment of placental abnormalities. Am J Obstet 1607 Gynecol 1990;163:1650–8. 1608 7. Ballas S, Gitstein S, Jaffa AJ, Peyser MR. Midtrimester placenta previa: normal or pathologic 1609 finding. Obstet Gynecol 1979;54:12–4. 1610 8. Reddy UM, Abuhamad AZ, Levine D, Saade GR; Fetal Imaging Workshop Invited Participants. Fetal 1611 imaging: Executive summary of a joint Eunice Kennedy Shriver National Institute of Child Health 1612 and Human Development, Society for Maternal-Fetal Medicine, American Institute of Ultrasound 1613 in Medicine, American College of Obstetricians and Gynecologists, American College of Radiology, 1614 Society for Pediatric Radiology, and Society of Radiologists in Ultrasound Fetal Imaging 1615 Workshop. J Ultrasound Med 2014;33:745–57. 1616 9. Vahanian SA, Lavery JA, Ananth CV, Vintzileos A. Placental implantation abnormalities and risk of 1617 preterm delivery: a systematic review and metaanalysis. Am J Obstet Gynecol 2015;213:S78–90. 1618 10. Ruiter L, Eschbach SJ, Burgers M, Rengerink KO, van Pampus MG, Goes BY, et al. Predictors for 1619 emergency cesarean delivery in women with placenta previa. Am J Perinatol 2016;33:1407–14. 1620 11. Irving C, Hertig AT. A study of placenta accreta. Surg Gynec Obst 1937;64:178–200. 1621 12.Luke RK, Sharpe JW, Greene RR. Placenta accreta: The adherent or invasive placenta. Am J Obstet 1622 Gynecol 1966;95:660–8. 1623 13. Fox H, Sebire NJ, editors. Pathology of the Placenta 3rd edition. Philadelphia: Saunders-Elsevier; 1624 2007. 1625 14. Benirschke K, Burton GJ, Baergen RN, editors. Pathology of the Human Placenta. 6th ed. Berlin: 1626 Springer-Verlag; 2012. 1627 15. Zosmer N, Jauniaux E, Bunce C, Panaiotova J, Shaikh H, Nicholaides KH. Interobserver agreement 1628 on standardized ultrasound and histopathologic signs for the prenatal diagnosis of placenta 1629 accreta spectrum disorders. Int J Gynaecol Obstet 2018;140:326-31. 1630 16. Jauniaux E, Collins SL, Jurkovic D, Burton GJ. Accreta placentation: a systematic review of 1631 prenatal ultrasound imaging and grading of villous invasiveness. Am J Obstet Gynecol 1632 2016;215:712–21. 1633 17. Jauniaux E, Collins S, Burton GJ. Placenta accreta spectrum: pathophysiology and evidence-based 1634 anatomy for prenatal ultrasound imaging. Am J Obstet Gynecol 2018;218:75-87. 1635 18. Collins SL, Chantraine F, Morgan TK, Jauniaux E. Abnormally adherent and invasive placenta: A 1636 spectrum disorder in need of a name. Ultrasound Obstet Gynecol. 2018;51:165-6. 1637 19.O’Brien JM, Barton JR, Donaldson ES. The management of placenta percreta: conservative and 1638 operative strategies. Am J Obstet Gynecol 1996;175:1632–8. 1639 20. Bowman ZS, Manuck TA, Eller AG, Simons M, Silver RM. Risk factors for unscheduled delivery in 1640 patients with placenta accreta. Am J Obstet Gynecol 2014;210:241.e1–6. 1641 21. Shamshirsaz AA, Fox KA, Salmanian B, Diaz-Arrastia CR, Lee W, Baker BW et al. Maternal 1642 morbidity in patients with morbidly adherent placenta treated with and without a 1643 standardized multidisciplinary approach. Am J Obstet Gynecol. 2015;212:218.e1-9. 1644 Page 32 of 47 22. Shamshirsaz AA, Fox KA, Erfani H, Clark SL, Salmanian B, Baker BW, et al. Multidisciplinary team 1645 learning in the management of the morbidly adherent placenta: outcome improvements 1646 over time. Am J Obstet Gynecol. 2017;216:612.e1-612.e5. 1647 23. Ananth CV, Smulian JC, Vintzileos AM. The association of placenta previa with history of cesarean 1648 delivery and abortion: a meta-analysis. Am J Obstet Gynecol 1997;177:1071–8. 1649 24. Marshall NE, Fu R, Guise JM. Impact of multiple cesarean deliveries on maternal morbidity: a 1650 systematic review. Am J Obstet Gynecol 2011;205:262.e1–8. 1651 25. Klar M, Michels KB. Cesarean section and placental disorders in subsequent pregnancies--a meta-1652 analysis. J Perinat Med 2014;42:571–83. 1653 26. Getahun D, Oyelese Y, Salihu HM, Ananth CV. Previous cesarean delivery and risks of placenta 1654 previa and placental abruption. Obstet Gynecol 2006;107:771–8. 1655 27. Downes KL, Hinkle SN, Sjaarda LA, Albert PS, Grantz KL. Previous prelabor or intrapartum 1656 cesarean delivery and risk of placenta previa. Am J Obstet Gynecol 2015;212:669.e1–6. 1657 28. Ananth CV, Demissie K, Smulian JC, Vintzileos AM. Placenta praevia in singleton and twin births in 1658 the United States, 1989 through 1998: a comparison of risk factor profiles and associated 1659 conditions. Am J Obstet Gynecol 2003;188:275–81. 1660 29. Weis MA, Harper LM, Roehl KA, Odibo AO, Cahill AG. Natural history of placenta previa in twins. 1661 Obstet Gynecol 2012;120:753–8. 1662 30. Grady R, Alavi N, Vale R, Khandwala M, McDonald SD. Elective single embryo transfer and 1663 perinatal outcomes: a systematic review and meta-analysis. Fertil Steril 2012;97:324–31. 1664 31. Korosec S, Ban Frangez H, Verdenik I, Kladnik U, Kotar V, Virant-Klun I, et al. Singleton pregnancy 1665 outcomes after in vitro fertilization with fresh or frozen-thawed embryo transfer and incidence of 1666 placenta praevia. Biomed Res Int 2014;2014:431797. 1667 32. Qin J, Liu X, Sheng X, Wang H, Gao S. Assisted reproductive technology and the risk of pregnancy-1668 related complications and adverse pregnancy outcomes in singleton pregnancies: a meta-analysis 1669 of cohort studies. Fertil Steril 2016;105:73–85.e1–6. 1670 33. Karami M, Jenabi E, Fereidooni B. The association of placenta previa and associated reproductive 1671 techniques: a meta-analysis. J Matern Fetal Neonatal Med 2017;284:47–51. 1672 34. Shobeiri F, Jenabi E. Smoking and placenta previa: a meta-analysis. J Matern Fetal Neonatal Med 1673 2017;30:2985–90. 1674 35. Rosenberg T, Pariente G, Sergienko R, Wiznitzer A, Sheiner E. Critical analysis of risk factors and 1675 outcome of placenta previa. Arch Gynecol Obstet. 2011;284:47-51. 1676 36. National Institute of Health and Care Excellence. Antenatal care for uncomplicated pregnancies. 1677 Clinical guideline 62. Manhester: NICE; 2017. 1678 37. UK National Screening Committee. Screening for vasa praevia in the second trimester of 1679 pregnancy. External review against programme appraisal criteria for the UK National Screening 1680 Committee (UK NSC). London: UK NSC; 2017 [ 1681 38. Dashe JS, McIntire DD, Ramus RM, Santos-Ramos R, Twickler DM. Persistence of placenta previa 1682 according to gestational age at ultrasound detection. Obstet Gynecol 2002;99:692–7. 1683 39. Cho JY, Lee YH, Moon MH, Lee JH. Difference in migration of placenta according to the location 1684 and type of placenta previa. J Clin Ultrasound 2008;36:79-84. 1685 40. Eichelberger KY, Haeri S, Kessler DC, Swartz A, Herring A, Wolfe HM. Placenta previa in the 1686 second trimester: sonographic and clinical factors associated with its resolution. Am J Perinatol 1687 2011;28:735–9. 1688 41. Copland JA, Craw SM, Herbison P. Low-lying placenta: who should be recalled for a follow-up 1689 scan? J Med Imaging Radiat Oncol 2012;56:158–62. 1690 42. Robinson AJ, Muller PR, Allan R, Ross R, Baghurst PA, Keirse MJ. Precise mid-trimester placenta 1691 localisation: does it predict adverse outcomes? Aust N Z J Obstet Gynaecol 2012;52:156–60. 1692 43. Lal AK, Nyholm J, Wax J, Rose CH, Watson WJ. Resolution of complete placenta previa: does prior 1693 cesarean delivery matter? J Ultrasound Med 2012;31:577–80. 1694 44. Kapoor S, Thomas JT, Petersen SG, Gardener GJ. Is the third trimester repeat ultrasound scan for 1695 placental localisation needed if the placenta is low lying but clear of the os at the mid-trimester 1696 morphology scan? Aust N Z J Obstet Gynaecol 2014;54:428–32. 1697 Page 33 of 47 45. Quant HS, Friedman AM, Wang E, Parry S, Schwartz N. Transabdominal ultrasonography as a 1698 screening test for second-trimester placenta previa. Obstet Gynecol 2014;123:628–33. 1699 46. Heller HT, Mullen KM, Gordon RW, Reiss RE, Benson CB. Outcomes of pregnancies with a low-1700 lying placenta diagnosed on second-trimester sonography. J Ultrasound Med 2014;33:691–6. 1701 47.Kohari KS, Roman AS, Fox NS, Feinberg J, Saltzman DH, Klauser CK, et al. Persistence of placenta 1702 previa in twin gestations based on gestational age at sonographic detection. J Ultrasound Med 1703 2012;31:985–9. 1704 48. Sherman SJ, Carlson DE, Platt LD, Medearis AL. Transvaginal ultrasound: does it help in the 1705 diagnosis of placenta previa? Ultrasound Obstet Gynecol 1992;2:256–60. 1706 49. Mustafá SA, Brizot ML, Carvalho MH, Watanabe L, Kahhale S, Zugaib M. Transvaginal 1707 ultrasonography in predicting placenta previa at delivery: a longitudinal study. Ultrasound Obstet 1708 Gynecol 2002;20:356–9. 1709 50. Becker RH, Vonk R, Mende BC, Ragosch V, Entezami M. The relevance of placental location at 20-1710 23 gestational weeks for prediction of placenta previa at delivery: evaluation of 8650 cases. 1711 Ultrasound Obstet Gynecol 2001;17:496–501. 1712 51. Oppenheimer L, Holmes P, Simpson N, Dabrowski A. Diagnosis of low-lying placenta: can 1713 migration in the third trimester predict outcome? Ultrasound Obstet Gynecol 2001;18:100–2. 1714 52. Taipale P, Hiilesmaa V, Ylöstalo P. Transvaginal ultrasonography at 18-23 weeks in predicting 1715 placenta praevia at delivery. Ultrasound Obstet Gynecol 1998;12:422–5. 1716 53. Lauria MR, Smith RS, Treadwell MC, Comstock CH, Kirk JS, Lee W, et al. The use of second-1717 trimester transvaginal sonography to predict placenta previa. Ultrasound Obstet Gynecol 1718 1996;8:337–40. 1719 54. Smith RS, Lauria MR, Comstock CH, Treadwell MC, Kirk JS, Lee W, et al. Transvaginal 1720 ultrasonography for all placentas that appear to be low-lying or over the internal cervical os. 1721 Ultrasound Obstet Gynecol 1997;9:22–4. 1722 55. Leerentveld RA, Gilberts EC, Arnold MJ, Wladimiroff JW. Accuracy and safety of transvaginal 1723 sonographic placental localization. Obstet Gynecol 1990;76:759–62. 1724 56. Conde-Agudelo A, Romero R. Predictive accuracy of changes in transvaginal sonographic cervical 1725 length over time for preterm birth: a systematic review and metaanalysis. Am J Obstet Gynecol 1726 2015;213:789–801. 1727 57. Ghi T, Contro E, Martina T, Piva M, Morandi R, Orsini LF, et al. Cervical length and risk of 1728 antepartum bleeding in women with complete placenta previa. Ultrasound Obstet Gynecol 1729 2009;33:209–12. 1730 58. Zaitoun MM, El Behery MM, Abd El Hameed AA, Soliman BS. Does cervical length and the lower 1731 placental edge thickness measurement correlates with clinical outcome in cases of complete 1732 placenta previa? Arch Gynecol Obstet 2011;284:867–73. 1733 59. Mimura T, Hasegawa J, Nakamura M, Matsuoka R, Ichizuka K, Sekizawa A, et al. Correlation 1734 between the cervical length and the amount of bleeding during cesarean section in placenta 1735 previa. J Obstet Gynaecol Res 2011;37:830–5. 1736 60. Sekiguchi A, Nakai A, Okuda N, Inde Y, Takeshita T. Consecutive cervical length measurements as 1737 a predictor of preterm cesarean section in complete placenta previa. J Clin Ultrasound 1738 2015;43:17–22. 1739 61. Neilson JP. Interventions for suspected placenta praevia. Cochrane Database Syst Rev 1740 2003;(2):CD001998. 1741 62. Wing DA, Paul RH, Millar LK. Management of the symptomatic placenta previa: a randomized, 1742 controlled trial of inpatient versus outpatient expectant management. Am J Obstet Gynecol 1743 1996;175:806–11. 1744 63. Pivano A, Alessandrini M, Desbriere R, Agostini A, Opinel P, d'Ercole C, et al. A score to predict the 1745 risk of emergency caesarean delivery in women with antepartum bleeding and placenta praevia. 1746 Eur J Obstet Gynecol Reprod Biol 2015;195:173–6. 1747 64. Ruiter L, Eschbach SJ, Burgers M, Rengerink KO, van Pampus MG, Goes BY, et al. Predictors for 1748 emergency cesarean delivery in women with placenta previa. Am J Perinatol 2016;33:1407–14. 1749 Page 34 of 47 65. Lal AK, Hibbard JU. Placenta previa: an outcome-based cohort study in a contemporary obstetric 1750 population. Arch Gynecol Obstet 2015;292:299–305. 1751 66. Nørgaard LN, Pinborg A, Lidegaard Ø, Bergholt T. A Danish national cohort study on neonatal 1752 outcome in singleton pregnancies with placenta previa. Acta Obstet Gynecol Scand 2012;91:546– 1753 51. 1754 67. Roberts D, Brown J, Medley N, Dalziel SR. Antenatal corticosteroids for accelerating fetal lung 1755 maturation for women at risk of preterm birth. Cochrane Database Syst Rev 2017;3:CD004454. 1756 68. Gyamfi-Bannerman C, Thom EA, Blackwell SC, Tita AT, Reddy UM, Saade GR, et al.; NICHD 1757 Maternal–Fetal Medicine Units Network. Antenatal betamethasone for women at risk for late 1758 preterm delivery. N Engl J Med 2016;374:1311–20. 1759 69. Zlatnik MG, Little SE, Kohli P, Kaimal AJ, Stotland NE, Caughey AB. When should women with 1760 placenta previa be delivered? A decision analysis. J Reprod Med 2010;55:373–81. 1761 70. Bose DA, Assel BG, Hill JB, Chauhan SP. Maintenance tocolytics for preterm symptomatic placenta 1762 previa: a review. Am J Perinatol 2011;28:45–50. 1763 71. Verspyck E, de Vienne C, Muszynski C, Bubenheim M, Chanavaz-Lacheray I, Dreyfus M, et al. 1764 Maintenance nifedipine therapy for preterm symptomatic placenta previa: A randomized, 1765 multicenter, double-blind, placebo-controlled trial. PLoS One 2017;23;12:e0173717. 1766 72. American College of Obstetricians and Gynaecologists. ACOG committee opinion no. 560: 1767 Medically indicated late-preterm and early-term deliveries. Obstet Gynecol 2013;121:908–10. 1768 73. Spong CY, Mercer BM, D'alton M, Kilpatrick S, Blackwell S, Saade G. Timing of indicated late-1769 preterm and early-term birth. Obstet Gynecol 2011;118:323–33. 1770 74. Zlatnik MG, Cheng YW, Norton ME, Thiet MP, Caughey AB. Placenta previa and the risk of 1771 preterm delivery. J Matern Fetal Neonatal Med 2007;20:719–23. 1772 75. Balayla J, Wo BL, Bédard MJ. A late-preterm, early-term stratified analysis of neonatal outcomes 1773 by gestational age in placenta previa: defining the optimal timing for delivery. J Matern Fetal 1774 Neonatal Med 2015;28:1756–61. 1775 76. Bhide A, Prefumo F, Moore J, Hollis B, Thilaganathan B. Placental edge to internal os distance in 1776 the late third trimester and mode of delivery in placenta praevia. BJOG 2003;110:860–4. 1777 77. Ghourab S. Third-trimester transvaginal ultrasonography in placenta previa: does the shape of 1778 the lower placental edge predict clinical outcome? Ultrasound Obstet Gynecol 2001;18:103–8. 1779 78. Saitoh M, Ishihara K, Sekiya T, Araki T. Anticipation of uterine bleeding in placenta previa based 1780 on vaginal sonographic evaluation. Gynecol Obstet Invest 2002;54:37–42. 1781 79. Taga A, Sato Y, Sakae C, Satake Y, Emoto I, Maruyama S, et al. Planned vaginal delivery versus 1782 planned cesarean delivery in cases of low-lying placenta. J Matern Fetal Neonatal Med 1783 2017;30:618–22. 1784 80. Vergani P, Ornaghi S, Pozzi I, Beretta P, Russo FM, Follesa I, et al. Placenta previa: distance to 1785 internal os and mode of delivery. Am J Obstet Gynecol 2009;201:266.e1–5. 1786 81. Nakamura M, Hasegawa J, Matsuaka R, Mimura T, Ichizuka K, Sekizawa A, et al. Amount of 1787 hemorrhage during vaginal delivery correlates with length from placental edge to external os in 1788 cases with low-lying placenta whose length between placental edge and internal os was 1-2 cm. J 1789 Obstet Gynaecol Res 2012;38:1041–5. 1790 82. Al Wadi K, Schneider C, Burym C, Reid G, Hunt J, Menticoglou S. Evaluating the safety of labour in 1791 women with a placental edge 11 to 20 mm from the internal cervical Os. J Obstet Gynaecol Can 1792 2014;36:674–7. 1793 83. Wortman AC, Twickler DM, McIntire DD, Dashe JS. Bleeding complications in pregnancies with 1794 low-lying placenta. J Matern Fetal Neonatal Med 2016;29:1367–71. 1795 84. Royal College of Obstetricians and Gynaecologists. Caesarean section. Consent Advice No. 7. 1796 London: RCOG; 2009. 1797 85. Thomas J, Paranjothy S, editors. The National Sentinel Caesarean Section Audit Report. London: 1798 RCOG Press; 2001. 1799 86. Baba Y, Matsubara S, Ohkuchi A, Usui R, Kuwata T, Suzuki H, et al. Anterior placentation as a risk 1800 factor for massive hemorrhage during cesarean section in patients with placenta previa. J Obstet 1801 Gynaecol Res 2014;40:1243–8. 1802 Page 35 of 47 87. Gibbins KJ, Einerson BD, Varner MW, Silver RM. Placenta previa and maternal haemorrhagic 1803 morbidity. J Matern Fetal Neonatal Med 2018;31:494–9. 1804 88. Mavrides E, Allard S, Chandraharan E, Collins P, Green L, Hunt BJ, et al. on behalf of the Royal 1805 College of Obstetricians and Gynaecologists. Prevention and management of postpartum 1806 haemorrhage. BJOG 2016;124:e106–49. 1807 89. Royal College of Obstetricians and Gynaecologists. Blood Transfusions in Obstetrics. Green-top 1808 Guideline No. 47. London: RCOG; 2015. 1809 90. Madsen K, Grønbeck L, Rifbjerg Larsen C, Østergaard J, Bergholt T, Langhoff-Roos J, Sørensen JL. 1810 Educational strategies in performing cesarean section. Acta Obstet Gynecol Scand 2013;92:256-1811 63. 1812 91. Pelosi MA, Apuzzio J, Fricchione D, Gowda VV. The "intra-abdominal version technique" for 1813 delivery of transverse lie by low-segment cesarean section. Am J Obstet Gynecol 1979;135:1009- 1814 11. 1815 92. Hong JY, Jee YS,Yoon HJ, Kim SM. Comparison of general and epidural anesthesia in elective 1816 cesarean section for placenta pevia totalis: maternal hemodynamics, blood loss and neonatal 1817 outcome. Int J Obstet Anesth 2003:12:12–6. 1818 93. Butwick AJ, Carvalho B, El-Sayed YY. Risk factors for obstetric morbidity in patients with uterine 1819 atony undergoing caesarean delivery. Br J Anaesth 2014;113:661–8. 1820 94. Goucher H, Wong CA, Patel SK, Toledo P. Cell salvage in obstetrics. Anesth Analg 2015;121:465–8. 1821 95. Morikawa M, Kuramoto A, Nakayama M, Oguchi H, Hasegawa M, Funakoshi T, et al. 1822 Intraoperative red cell salvage during obstetric surgery in 50 Japanese women. Int J Gynaecol 1823 Obstet 2015;128:256–9. 1824 96. Verspyck E, Douysset X, Roman H, Marret S, Marpeau L. Transecting versus avoiding incision of 1825 the anterior placenta previa during cesarean delivery. Int J Gynaecol Obstet 2015;128:44–7. 1826 97. Zou L, Zhong S, Zhao Y, Zhu J, Chen L. Evaluation of "J"-shaped uterine incision during caesarean 1827 section in patients with placenta previa: a retrospective study. J Huazhong Univ Sci Technolog 1828 Med Sci 2010;30:212–6. 1829 98. Kumru P, Demirci O, Erdogdu E, Arısoy R, Ertekin AA, Tugrul S, et al. The Bakri balloon for the 1830 management of postpartum hemorrhage in cases with placenta previa. Eur J Obstet Gynecol 1831 Reprod Biol 2013;167:167–70. 1832 99. Beckmann MM, Chaplin J. Bakri balloon during cesarean delivery for placenta previa. Int J 1833 Gynaecol Obstet 2014;124:118–22. 1834 100. Cho HY, Park YW, Kim YH, Jung I, Kwon JY. Efficacy of intrauterine Bakri balloon tamponade 1835 in cesarean section for placenta previa patients. PLoS One 2015;10:e0134282. 1836 101. Maher MA, Abdelaziz A. Comparison between two management protocols for 1837 postpartum hemorrhage during cesarean section in placenta previa: Balloon protocol 1838 versus non-balloonprotocol. J Obstet Gynaecol Res 2017;43:447–55. 1839 102. Soyama H, Miyamoto M, Sasa H, Ishibashi H, Yoshida M, Nakatsuka M, et al. Effect of 1840 routine rapid insertion of Bakri balloon tamponade on reducing hemorrhage 1841 from placenta previa during and after cesarean section. Arch Gynecol Obstet 2017 Jun 24 1842 [Epub ahead of print]. 1843 103. Uygur D, Altun Ensari T, Ozgu-Erdinc AS, Dede H, Erkaya S, Danisman AN. Successful use of 1844 BT-Cath(®) balloon tamponade in the management of postpartum haemorrhage due to placenta 1845 previa. Eur J Obstet Gynecol Reprod Biol 2014;181:223–8. 1846 104. Ishii T, Sawada K, Koyama S, Isobe A, Wakabayashi A, Takiuchi T, et al. Balloon tamponade 1847 during cesarean section is useful for severe post-partum hemorrhage due to placenta previa. J 1848 Obstet Gynaecol Res 2012;38:102–7. 1849 105. B-Lynch C, Coker A, Lawal AH, Abu J, Cowen MJ. The B-Lynch surgical technique for the 1850 control of massive postpartum haemorrhage: an alternative to hysterectomy? Five cases 1851 reported. Br J Obstet Gynaecol 1997;104:372–5. 1852 Page 36 of 47 106. Yoong W, Ridout A, Memtsa M, Stavroulis A, Aref-Adib M, Ramsay-Marcelle Z, et al. 1853 Application of uterine compression suture in association with intrauterine balloon tamponade 1854 ('uterine sandwich') for postpartum hemorrhage. Acta Obstet Gynecol Scand 2012;91:147–51. 1855 107. Inoue S, Masuyama H, Hiramatsu Y; Multi-Institutional Study Group of Transarterial 1856 Embolization for Massive Obstetric Haemorrhage in Chugoku & Shikoku Area Society of 1857 Obstetrics and Gynecology. Efficacy of transarterial embolisation in the management of post-1858 partum haemorrhage and its impact on subsequent pregnancies. Aust N Z J Obstet Gynaecol 1859 2014;54:541–5. 1860 108. Broekman EA, Versteeg H, Vos LD, Dijksterhuis MG, Papatsonis DN. Temporary balloon 1861 occlusion of the internal iliac arteries to prevent massive hemorrhage during cesarean delivery 1862 among patients with placenta previa. Int J Gynaecol Obstet 2015;128:118–21. 1863 109. Sentilhes L, Gromez A, Clavier E, Resch B, Verspyck E, Marpeau L. Fertility and pregnancy 1864 following pelvic arterial embolisation for postpartum haemorrhage. BJOG 2010;117:84–93. 1865 110. Doumouchtsis SK, Nikolopoulos K, Talaulikar V, Krishna A, Arulkumaran S. Menstrual and 1866 fertility outcomes following the surgical management of postpartum haemorrhage: a systematic 1867 review. BJOG 2014;121:382–8. 1868 111. Soro MP, Denys A, de Rham M, Baud D. Short and long term adverse outcomes after arterial 1869 embolisation for the treatment of postpartum haemorrhage: a systematic review. Eur Radiol 1870 2017;27:749–62. 1871 112. Wu S, Kocherginsky M, Hibbard JU. Abnormal placentation: twenty-year analysis. Am J 1872 Obstet Gynecol 2005;192:1458–61. 1873 113. Silver RM, Landon MB, Rouse DJ, Leveno KJ, Spong CY, Thom EA, et al.; National Institute of 1874 Child Health and Human Development Maternal-Fetal Medicine Units Network. Maternal 1875 morbidity associated with multiple repeat cesarean deliveries. Obstet Gynecol 2006;107:1226– 1876 32. 1877 114. Fitzpatrick KE, Sellers S, Spark P, Kurinczuk JJ, Brocklehurst P, Knight M. Incidence and risk 1878 factors for placenta accreta/increta/percreta in the UK: a national case-control study. PLoS One 1879 2012;7:e52893. 1880 115. Morlando M, Sarno L, Napolitano R, Capone A, Tessitore G, Maruotti GM, et al. Placenta 1881 accreta: incidence and risk factors in an area with a particularly high rate of cesarean section. 1882 Acta Obstet Gynecol Scand 2013;92:457–60. 1883 116. Cook JR, Jarvis S, Knight M, Dhanjal MK. Multiple repeat caesarean section in the UK: 1884 incidence and consequences to mother and child. A national, prospective, cohort study. BJOG 1885 2013;120:85–91. 1886 117. Higgins MF, Monteith C, Foley M, O'Herlihy C. Real increasing incidence of hysterectomy for 1887 placenta accreta following previous caesarean section. Eur J Obstet Gynecol Reprod Biol 1888 2013;171:54–6. 1889 118. Eshkoli T, Weintraub AY, Sergienko R, Sheiner E. Placenta accreta: risk factors, perinatal 1890 outcomes, and consequences for subsequent births. Am J Obstet Gynecol 2013;208:219.e1–7. 1891 119. Kamara M, Henderson JJ, Doherty DA, Dickinson JE, Pennell CE. The risk of placenta accreta 1892 following primary elective caesarean delivery: a case-control study. BJOG 2013;120:879–86. 1893 120. Creanga AA, Bateman BT, Butwick AJ, Raleigh L, Maeda A, Kuklina E, et al. Morbidity 1894 associated with cesarean delivery in the United States: is placenta accreta an increasingly 1895 important contributor? Am J Obstet Gynecol 2015;213:384.e1–11. 1896 121. Thurn L, Lindqvist PG, Jakobsson M, Colmorn LB, Klungsoyr K, Bjarnadóttir RI, et al. 1897 Abnormally invasive placenta-prevalence, risk factors and antenatal suspicion: results from a 1898 large population-based pregnancy cohort study in the Nordic countries. BJOG 2016;123:1348–55. 1899 122. Farquhar CM, Li Z, Lensen S, McLintock C, Pollock W, Peek MJ, et al. Incidence, risk factors 1900 and perinatal outcomes for placenta accreta in Australia and New Zealand: a case-control 1901 study. BMJ Open 2017;7:e017713. 1902 123. Balayla J, Bondarenko HD. Placenta accreta and the risk of adverse maternal and neonatal 1903 outcomes. J Perinat Med 2013;41:141–9. 1904 Page 37 of 47 124. Esh-Broder E, Ariel I, Abas-Bashir N, Bdolah Y, Celnikier DH. Placenta accreta is associated 1905 with IVF pregnancies: a retrospective chart review. BJOG 2011;118:1084–9. 1906 125. Ishihara O, Araki R, Kuwahara A, Itakura A, Saito H, Adamson GD. Impact of frozen-thawed 1907 single-blastocyst transfer on maternal and neonatal outcome: an analysis of 277,042 single-1908 embryo transfer cycles from 2008 to 2010 in Japan. Fertil Steril 2014;101:128–33. 1909 126. Kaser DJ, Melamed A, Bormann CL, Myers DE, Missmer SA, Walsh BW, et al. Cryopreserved 1910 embryo transfer is an independent risk factor for placenta accreta. Fertil Steril 2015;103:1176– 1911 84.e2. 1912 127. Ben-Nagi J, Ofili-Yebovi D, Marsh M, Jurkovic D. First-trimester cesarean scar pregnancy 1913 evolving into placenta previa/accreta at term. J Ultrasound Med 2005;24:1569–73. 1914 128. Zosmer N, Fuller J, Shaikh H, Johns J, Ross JA. Natural history of early first-trimester 1915 pregnancies implanted in Cesarean scars. Ultrasound Obstet Gynecol 2015;46:367–75. 1916 129. Timor-Tritsch IE, Monteagudo A, Cali G, Palacios-Jaraquemada JM, Maymon R, Arslan AA, et 1917 al. Cesarean scar pregnancy and early placenta accreta share common histology. Ultrasound 1918 Obstet Gynecol 2014;43:383–95. 1919 130. Jurkovic D, Knez J, Appiah A, Farahani L, Mavrelos D, Ross JA. Surgical treatment of 1920 Caesarean scar pregnancy: efficacy and safety of ultrasound-guided suction curettage. Ultrasound 1921 Obstet Gynecol 2016;47:511–7. 1922 131. Cali G, Forlani F, Timor-Tritsch IE, Palacios-Jaraquemada J, Minneci G, D'Antonio F. Natural 1923 history of Caesarean scar pregnancy on prenatal ultrasound: the crossover sign. Ultrasound 1924 Obstet Gynecol 2017;50:100–4. 1925 132. Wright JD, Pri-Paz S, Herzog TJ, Shah M, Bonanno C, Lewin SN, et al. Predictors of massive 1926 blood loss in women with placenta accreta. Am J Obstet Gynecol 2011;205:38.e1–6. 1927 133. Tikkanen M, Paavonen J, Loukovaara M, Stefanovic V. Antenatal diagnosis of placenta 1928 accreta leads to reduced blood loss. Acta Obstet Gynecol Scand 2011;90:1140–6. 1929 134. Chantraine F, Braun T, Gonser M, Henrich W, Tutschek B. Prenatal diagnosis of abnormally 1930 invasive placenta reduces maternal peripartum hemorrhage and morbidity. Acta Obstet Gynecol 1931 Scand 2013;92:439–44. 1932 135. Hall T, Wax JR, Lucas FL, Cartin A, Jones M, Pinette MG. Prenatal sonographic diagnosis of 1933 placenta accreta: Impact on maternal and neonatal outcomes. J Clin Ultrasound 2014;42:449–55. 1934 136. Silver RM, Fox KA, Barton JR, Abuhamad AZ, Simhan H, Huls CK, et al. Center of excellence 1935 for placenta accreta. Am J Obstet Gynecol 2015;212:561–8. 1936 137. Bailit JL, Grobman WA, Rice MM, Reddy UM, Wapner RJ, Varner MW, et al.; Eunice Kennedy 1937 Shriver National Institute of Child Health and Human Development (NICHD) Maternal-Fetal 1938 Medicine Units (MFMU) Network. Morbidly adherent placenta treatments and outcomes. Obstet 1939 Gynecol 2015;125:683–9. 1940 138. Bowman ZS, Eller AG, Kennedy AM, Richards DS, Winter TC 3rd, Woodward PJ, et al. 1941 Accuracy of ultrasound for the prediction of placenta accreta. Am J Obstet Gynecol 1942 2014;211:177.e1–7. 1943 139. Smulian JC, Pascual AL, Hesham H, Qureshey E, Bijoy Thomas M, Depuy AM, et al. 1944 Invasive placental disease: the impact of a multi-disciplinary team approach to management. J 1945 Matern Fetal Neonatal Med 2017;30:1423–7. 1946 140. Calì G, Giambanco L, Puccio G, Forlani F. Morbidly adherent placenta: evaluation of 1947 ultrasound diagnostic criteria and differentiation of placenta accreta from percreta. Ultrasound 1948 Obstet Gynecol 2013;41:406–12. 1949 141. Collins SL, Ashcroft A, Braun T, Calda P, Langhoff-Ross J, Morel O, et al.; European Working 1950 Group on Abnormally Invasive Placenta (EW-AIP). Proposal for standardized ultrasound 1951 descriptors of abnormally invasive placenta (AIP). Ultrasound Obstet Gynecol 2016;47:271–5. 1952 142. Alfirevic Z, Tang AW, Collins SL, Robson SC, Palacios-Jaraquemada J; Ad-hoc International AIP 1953 Expert Group. Pro forma for ultrasound reporting in suspected abnormally invasive placenta 1954 (AIP): an international consensus. Ultrasound Obstet Gynecol 2016;47:276–8. 1955 143. D'Antonio F, Iacovella C, Bhide A. Prenatal identification of invasive placentation using 1956 ultrasound: systematic review and meta-analysis. Ultrasound Obstet Gynecol 2013;42:509–17. 1957 Page 38 of 47 144. Jauniaux E, Bhide A. Prenatal ultrasound diagnosis and outcome of placenta previa accreta 1958 after cesarean delivery: a systematic review and meta-analysis. Am J Obstet Gynecol 1959 2017;217:27–36. 1960 145. Chantraine F, Nisolle M, Petit P, Schaaps JP, Foidart JM. Individual decisions in placenta 1961 increta and percreta: a case series. J Perinat Med 2012;40:265–70. 1962 146. Palacios-Jaraquemada JM, Bruno CH, Martín E. MRI in the diagnosis and surgical 1963 management of abnormal placentation. Acta Obstet Gynecol Scand 2013;92:392–7. 1964 147. Rahaim NS, Whitby EH. The MRI features of placental adhesion disorder and their diagnostic 1965 significance: systematic review. Clin Radiol 2015;70:917–25. 1966 148. Meng X, Xie L, Song W. Comparing the diagnostic value of ultrasound and magnetic 1967 resonance imaging for placenta accreta: a systematic review and meta-analysis. Ultrasound Med 1968 Biol 2013;39:1958–65. 1969 149. D'Antonio F, Iacovella C, Palacios-Jaraquemada J, Bruno CH, Manzoli L, Bhide A. Prenatal 1970 identification of invasive placentation using magnetic resonance imaging: systematic review and 1971 meta-analysis. Ultrasound Obstet Gynecol 2014;44:8–16. 1972 150. Millischer AE, Salomon LJ, Porcher R, Brasseur-Daudruy M, Gourdier AL, Hornoy P, et al. 1973 Magnetic resonance imaging for abnormally invasive placenta: the added value of intravenous 1974 gadolinium injection. BJOG 2017;124:88–95. 1975 151. Hornemann A, Bohlmann MK, Diedrich K, Kavallaris A, Kehl S, Kelling K, et al. Spontaneous 1976 uterine rupture at the 21st week of gestation caused by placenta percreta. Arch Gynecol Obstet 1977 2011;284:875–8. 1978 152. Dew L, Harris S, Yost N, Magee K, dePrisco G. Second trimester placenta percreta presenting 1979 as acute abdomen. Proc (Bayl Univ Med Cent) 2015;28:38–40. 1980 153. Sun JN, Zhang BL, Yu HY, Zhang Q. Spontaneous uterine rupture due to placenta percreta 1981 during pregnancy. Am J Emerg Med 2016;34:1918.e1–3. 1982 154. Abbas F, Talati J, Wasti S, Akram S, Ghaffar S, Qureshi R. Placenta percreta with bladder 1983 invasion as a cause of life threatening hemorrhage. J Urol 2000;164:1270–4. 1984 155. Wagaskar VG, Daga SO, Patwardhan SK. Placenta percreta presenting with delayed 1985 haematuria. J Clin Diagn Res 2015;9:PD01–2. 1986 156. Brown JV 3rd, Epstein HD, Laflamme LA, Goldstein BH. First-trimester placenta percreta with 1987 urinary bladder invasion. Int J Gynaecol Obstet 2016;132:102–3. 1988 157. Vinograd A, Wainstock T, Mazor M, Beer-Weisel R, Klaitman V, Dukler D, et al. Placenta 1989 accreta is an independent risk factor for late pre-term birth and perinatal mortality. J Matern 1990 Fetal Neonatal Med 2015;28:1381–7. 1991 158. Mehrabadi A, Hutcheon JA, Liu S, Bartholomew S, Kramer MS, Liston RM, et al; Maternal 1992 Health Study Group of Canadian Perinatal Surveillance System (Public Health Agency of Canada). 1993 Contribution of placenta accreta to the incidence of postpartum hemorrhage and severe 1994 postpartum hemorrhage. Obstet Gynecol 2015;125:814–21. 1995 159. Jolley JA, Nageotte MP, Wing DA, Shrivastava VK. Management of placenta accreta: a survey 1996 of Maternal-Fetal Medicine practitioners. J Matern Fetal Neonatal Med 2012 ;25:756–60. 1997 160. Esakoff TF, Handler SJ, Granados JM, Caughey AB. PAMUS: placenta accreta management 1998 across the United States. J Matern Fetal Neonatal Med 2012;25:761–5. 1999 161. Wright JD, Silver RM, Bonanno C, Gaddipati S, Lu YS, Simpson LL, et al. Practice patterns and 2000 knowledge of obstetricians and gynecologists regarding placenta accreta. J Matern Fetal 2001 Neonatal Med 2013;26:1602–9. 2002 162. Cal M, Ayres-de-Campos D, Jauniaux E. International survey of practices used in the 2003 diagnosis and management of placenta accreta spectrum disorders. Int J Gynaecol Obstet 2017 2004 nov 17 [Epub ahead of publication]. 2005 163. Eller AG, Bennett MA, Sharshiner M, Masheter C, Soisson AP, Dodson M, et al. Maternal 2006 morbidity in cases of placenta accreta managed by a multidisciplinary care team compared with 2007 standard obstetric care. Obstet Gynecol 2011;117:331–7. 2008 Page 39 of 47 164. Al-Khan A, Gupta V, Illsley NP, Mannion C, Koenig C, Bogomol A, et al. Maternal and fetal 2009 outcomes in placenta accreta after institution of team-managed care. Reprod Sci 2014;21:761– 2010 71. 2011 165. Publications Committee, Society for Maternal-Fetal Medicine, Belfort MA. Placenta accreta. 2012 Am J Obstet Gynecol 2010;203:430–9. 2013 166. Committee on Obstetric Practice. Committee opinion no. 529: placenta accreta. Obstet 2014 Gynecol 2012;120:207–11. 2015 167. Belfort MA. Indicated preterm birth for placenta accreta. Semin Perinatol 2011;35:252–6. 2016 168. Robinson BK, Grobman WA. Effectiveness of timing strategies for delivery of individuals with 2017 placenta previa and accreta. Obstet Gynecol 2010;116:835–42. 2018 169. Seet EL, Kay HH, Wu S, Terplan M. Placenta accreta: depth of invasion and neonatal 2019 outcomes. J Matern Fetal Neonatal Med 2012;25:2042–5. 2020 170. Rac MW, Wells CE, Twickler DM, Moschos E, McIntire DD, Dashe JS. Placenta accreta and 2021 vaginal bleeding according to gestational age at delivery. Obstet Gynecol 2015;125:808–13. 2022 171. Perlman NC, Little SE, Thomas A, Cantonwine DE, Carusi DA. Patient selection for later 2023 delivery timing with suspected previa-accreta. Acta Obstet Gynecol Scand 2017;96:1021–8. 2024 172. Green L, Knight M, Seeney FM, Hopkinson C, Collins PW, Collis RE, et al. The epidemiology 2025 and outcomes of women with postpartum haemorrhage requiring massive transfusion with eight 2026 or more units of red cells: a national cross-sectional study. BJOG 2016;123:2164–70. 2027 173. Brookfield KF, Goodnough LT, Lyell DJ, Butwick AJ. Perioperative and transfusion outcomes 2028 in women undergoing cesarean hysterectomy for abnormal placentation. Transfusion 2029 2014;54:1530–6. 2030 174. Paterson-Brown S, Singh C. Developing a care bundle for the management of suspected 2031 placenta accreta. The Obstetrician & Gynaecologist 2010;12:21–7. 2032 175. Knight M, Tuffnell D, Kenyon S, Shakespeare J, Gray R, Kurinczuk JJ, editors on behalf of 2033 MBRRACE-UK. Saving Lives, Improving Mothers’ Care. Surveillance of maternal deaths in the UK 2034 2011-13 and lessons learned to inform maternity care from the UK and Ireland Confidential 2035 Enquiries into Maternal Deaths and Morbidity 2009-13. Oxford: National Perinatal Epidemiology 2036 Unit, University of Oxford; 2015. 2037 176. Paterson-Brown S, Bamber J on behalf of the MBRRACE-UK haemorrhage chapter writing 2038 group. Prevention and treatment of haemorrhage. In Knight M, Kenyon S, Brocklehurst P, Neilson 2039 J, Shakespeare J, Kurinczuk JJ editors on behalf of MBRRACE-UK. Saving Lives, Improving Mothers’ 2040 Care - Lessons learned to inform future maternity care from the UK and Ireland Confidential 2041 Enquiries into Maternal Deaths and Morbidity 2009–12. Oxford: National Perinatal Epidemiology 2042 Unit, University of Oxford; 2014. 2043 177. Brennan DJ, Schulze B, Chetty N, Crandon A, Petersen SG, Gardener G, et al. Surgical 2044 management of abnormally invasive placenta: a retrospective cohort study demonstrating the 2045 benefits of a standardized operative approach. Acta Obstet Gynecol Scand 2015;94:1380–6. 2046 178. Eller AG, Porter TF, Soisson P, Silver RM. Optimal management strategies for placenta 2047 accreta. BJOG 2009;116:648–54. 2048 179. Hudon L, Belfort MA, Broome DR. Diagnosis and management of placenta percreta: a 2049 review. Obstet Gynecol Surv 1998;53:509–17. 2050 180. Rossetti D, Vitale SG, Bogani G, Rapisarda AM, Gulino FA, Frigerio L. Usefulness of vessel-2051 sealing devices for peripartum hysterectomy: a retrospective cohort study. Updates Surg 2052 2015;67:301–4. 2053 181. Steins Bisschop CN, Schaap TP, Vogelvang TE, Scholten PC. Invasive placentation and uterus 2054 preserving treatment modalities: a systematic review. Arch Gynecol Obstet 2011;284:491–502. 2055 182. Mei J, Wang Y, Zou B, Hou Y, Ma T, Chen M, Xie L. Systematic review of uterus-preserving 2056 treatment modalities for abnormally invasive placenta. J Obstet Gynaecol 2015;35:777–82. 2057 183. Teixidor Viñas M, Belli AM, Arulkumaran S, Chandraharan E. Prevention of postpartum 2058 hemorrhage and hysterectomy in patients with morbidly adherent placenta: a cohort study 2059 comparing outcomes before and after introduction of the Triple-P procedure. Ultrasound Obstet 2060 Gynecol 2015;46:350–5. 2061 Page 40 of 47 184. El Tahan M, Carrillo AP, Moore J, Chandraharan E. Predictors of postoperative hospitalisation 2062 in women who underwent the Triple-P Procedure for abnormal invasion of the placenta. J Obstet 2063 Gynaecol 2018;38:71–3. 2064 185. Shazly SA, Badee AY, Ali MK. The use of multiple 8 compression suturing as a novel 2065 procedure to preserve fertility in patients with placenta accreta: case series. Aust N Z J Obstet 2066 Gynaecol 2012;52:395–9. 2067 186. Huang G, Zhou R, Hu Y. A new suture technique for cesarean delivery complicated by 2068 hemorrhage in cases of placenta previa accreta. Int J Gynaecol Obstet 2014;124:262–3. 2069 187. Kaplanoğlu M, Kaplanoğlu DK, Koyuncu O. A different approach to placenta previa accreta: 2070 intrauterine gauze compress combined B-Lynch uterine compression suture. Clin Exp Obstet 2071 Gynecol 2015;42:53–6. 2072 188. El Gelany SA, Abdelraheim AR, Mohammed MM, Gad El-Rab MT, Yousef AM, Ibrahim EM, et 2073 al. The cervix as a natural tamponade in postpartum hemorrhage caused by placenta previa and 2074 placenta previa accreta: a prospective study. BMC Pregnancy Childbirth 2015;15:295. 2075 189. Tam Tam KB, Dozier J, Martin JN Jr. Approaches to reduce urinary tract injury during 2076 management of placenta accreta, increta, and percreta: a systematic review. J Matern Fetal 2077 Neonatal Med 2012;25:329–34. 2078 190. Grace Tan SE, Jobling TW, Wallace EM, McNeilage LJ, Manolitsas T, Hodges RJ. Surgical 2079 management of placenta accreta: a 10-year experience. Acta Obstet Gynecol Scand 2013;92:445– 2080 50. 2081 191. Woldu SL, Ordonez MA, Devine PC, Wright JD. Urologic considerations of placenta accreta: a 2082 contemporary tertiary care institutional experience. Urol Int 2014;93:74–9. 2083 192. Norris BL, Everaerts W, Posma E, Murphy DG, Umstad MP, Costello AJ, et al. The urologist's 2084 role in multidisciplinary management of placenta percreta. BJU Int 2016;117:961–5. 2085 193. Matsubara S, Kuwata T, Usui R, Watanabe T, Izumi A, Ohkuchi A, et al. Important surgical 2086 measures and techniques at cesarean hysterectomy for placenta previa accreta. Acta Obstet 2087 Gynecol Scand 2013;92:372–7. 2088 194. Shabana A, Fawzy M, Refaie W. Conservative management of placenta percreta: a stepwise 2089 approach. Arch Gynecol Obstet 2015;291:993–8. 2090 195. Clausen C, Lönn L, Langhoff-Roos J. Management of placenta percreta: a review of published 2091 cases. Acta Obstet Gynecol Scand 2014;93:138–43. 2092 196. Sentilhes L, Ambroselli C, Kayem G, Provansal M, Fernandez H, Perrotin F, et al. Maternal 2093 outcome after conservative treatment of placenta accreta. Obstet Gynecol 2010;115:526–34. 2094 197. Fox KA, Shamshirsaz AA, Carusi D, Secord AA, Lee P, Turan OM, et al. Conservative 2095 management of morbidly adherent placenta: expert review. Am J Obstet Gynecol 2015;213:755– 2096 60. 2097 198. Ibrahim MA, Liu A, Dalpiaz A, Schwamb R, Warren K, Khan SA. Urological Manifestations of 2098 Placenta Percreta. Curr Urol 2015;8:57–65. 2099 199. Lin K, Qin J, Xu K, Hu W, Lin J. Methotrexate management for placenta accreta: a prospective 2100 study. Arch Gynecol Obstet 2015;291:1259–64. 2101 200. Legendre G, Zoulovits FJ, Kinn J, Senthiles L, Fernandez H. Conservative management of 2102 placenta accreta: hysteroscopic resection of retained tissues. J Minim Invasive Gynecol 2103 2014;21:910–3. 2104 201. Mazzon I, Favilli A, Grasso M, Horvath S, Gerli S. Is the cold loop hysteroscopic technique a 2105 myometrial sparing treatment for placenta accreta residuals in a puerperal uterus? J Matern Fetal 2106 Neonatal Med 2016;29:1613–6. 2107 202. Bai Y, Luo X, Li Q, Yin N, Fu X, Zhang H, et al. High-intensity focused ultrasound treatment of 2108 placenta accreta after vaginal delivery: a preliminary study. Ultrasound Obstet Gynecol 2109 2016;47:492–8. 2110 203. Judy AE, Lyell DJ, Druzin ML, Dorigo O. Disseminated intravascular coagulation complicating 2111 the conservative management of placenta percreta. Obstet Gynecol 2015;126:1016–8. 2112 Page 41 of 47 204. Teixidor Viñas M, Chandraharan E, Moneta MV, Belli AM. The role of interventional 2113 radiology in reducing haemorrhage and hysterectomy following caesarean section for morbidly 2114 adherent placenta. Clin Radiol 2014;69:e345–51. 2115 205. Bouvier A, Sentilhes L, Thouveny F, Bouet PE, Gillard P, Willoteaux S, et al. Planned 2116 caesarean in the interventional radiology cath lab to enable immediate uterine artery 2117 embolization for the conservative treatment of placenta accreta. Clin Radiol 2012;67:1089–94. 2118 206. Dilauro MD, Dason S, Athreya S. Prophylactic balloon occlusion of internal iliac arteries in 2119 women with placenta accreta: literature review and analysis. Clin Radiol 2012;67:515–20. 2120 207. Clausen C, Stensballe J, Albrechtsen CK, Hansen MA, Lönn L, Langhoff-Roos J. Balloon occlusion 2121 of the internal iliac arteries in the multidisciplinary management of placenta percreta. Acta 2122 Obstet Gynecol Scand 2013;92:386–91. 2123 208. D'Souza DL, Kingdom JC, Amsalem H, Beecroft JR, Windrim RC, Kachura JR. Conservative 2124 management of invasive placenta using combined prophylactic internal iliac artery balloon 2125 occlusion and immediate postoperative uterine artery embolization. Can Assoc Radiol J 2126 2015;66:179–84. 2127 209. Chou MM, Kung HF, Hwang JI, Chen WC, Tseng JJ. Temporary prophylactic 2128 intravascular balloon occlusion of the common iliac arteries before cesarean hysterectomy for 2129 controlling operative blood loss in abnormal placentation. Taiwan J Obstet Gynecol 2015;54:493– 2130 8. 2131 210. Duan XH, Wang YL, Han XW, Chen ZM, Chu QJ, Wang L, et al. Caesarean section combined 2132 with temporary aortic balloon occlusion followed by uterine artery embolisation for the 2133 management of placenta accreta. Clin Radiol 2015;70:932–7. 2134 211. Duan XH, Wang YL, Han XW, Chen ZM, Chu QJ, Wang L, et al. Caesarean section combined 2135 with temporary aortic balloon occlusion followed by uterine artery embolisation for the 2136 management of placenta accreta. Clin Radiol 2015;70:932–7. 2137 212. Wei X, Zhang J, Chu Q, Du Y, Xing N, Xu X, et al. Prophylactic abdominal aorta balloon 2138 occlusion during caesarean section: a retrospective case series. Int J Obstet Anesth 2016;27:3–8. 2139 213. Wu Q, Liu Z, Zhao X, Liu C, Wang Y, Chu Q, et al. Outcome of Pregnancies 2140 After Balloon Occlusion of the Infrarenal Abdominal Aorta During Caesarean in 230 Patients 2141 With Placenta Praevia Accreta. Cardiovasc Intervent Radiol 2016;39:1573–9. 2142 214. Xie L, Wang Y, Luo FY, Man YC, Zhao XL. Prophylactic use of an infrarenal abdominal 2143 aorta balloon catheter in pregnancies complicated by placenta accreta. J Obstet Gynaecol 2144 2017;37:557–61. 2145 215. Wang YL, Duan XH, Han XW, Wang L, Zhao XL, Chen ZM, et al. Comparison of temporary 2146 abdominal aortic occlusion with internal iliac artery occlusion for patients with placenta accreta - 2147 a non-randomised prospective study. Vasa 2017;46:53–7. 2148 216. Ikeda T, Sameshima H, Kawaguchi H, Yamauchi N, Ikenoue T. Tourniquet technique prevents 2149 profuse blood loss in placenta accreta cesarean section. J Obstet Gynaecol Res 2005;31:27–31. 2150 217. Meng JL, Gong WY, Wang S, Ni XJ, Zuo CT, Gu YZ. Two-tourniquet sequential blocking as a 2151 simple intervention for hemorrhage during cesarean delivery for placenta previa accreta. Int J 2152 Gynaecol Obstet 2017;138:361–2. 2153 218. Iwata A, Murayama Y, Itakura A, Baba K, Seki H, Takeda S. Limitations of internal iliac 2154 artery ligation for the reduction of intraoperative hemorrhage during cesarean hysterectomy in 2155 cases of placenta previa accreta. J Obstet Gynaecol Res 2010;36:254–9. 2156 219. Bishop S, Butler K, Monaghan S, Chan K, Murphy G, Edozien L. Multiple complications 2157 following the use of prophylactic internal iliac artery balloon catheterisation in a patient 2158 with placenta percreta. Int J Obstet Anesth 2011;20:70–3. 2159 220. Gagnon J, Boucher L, Kaufman I, Brown R, Moore A. Iliac artery rupture related to balloon 2160 insertion for placenta accreta causing maternal hemorrhage and neonatal compromise. Can J 2161 Anaesth 2013;60:1212–7. 2162 221. Teare J, Evans E, Belli A, Wendler R. Sciatic nerve ischaemia after iliac artery occlusion 2163 balloon catheter placement for placenta percreta. Int J Obstet Anesth 2014;23:178–81. 2164 Page 42 of 47 222. Matsueda S, Hidaka N, Kondo Y, Fujiwara A, Fukushima K, Kato K. External iliac artery 2165 thrombosis after common iliac artery balloon occlusion during cesarean hysterectomy for 2166 placenta accreta in cervico-isthmic pregnancy. J Obstet Gynaecol Res 2015;41:1826–30. 2167 223. Salim R, Chulski A, Romano S, Garmi G, Rudin M, Shalev E. Precesarean prophylactic balloon 2168 catheters for suspected placenta accreta: A randomized controlled trial. Obstet Gynecol 2169 2015;126:1022–8. 2170 2171 2172 2173 Page 43 of 47 Appendix I: Explanation of guidelines and evidence levels 2174 2175 Clinical guidelines are: ‘systematically developed statements which assist clinicians and patients in 2176 making decisions about appropriate treatment for specific conditions’. Each guideline is 2177 systematically developed using a standardised methodology. Exact details of this process can be 2178 found in Clinical Governance Advice No.1 Development of RCOG Green-top Guidelines (available on 2179 the RCOG website at These recommendations are 2180 not intended to dictate an exclusive course of management or treatment. They must be evaluated 2181 with reference to individual patient needs, resources and limitations unique to the institution and 2182 variations in local populations. It is hoped that this process of local ownership will help to 2183 incorporate these guidelines into routine practice. Attention is drawn to areas of clinical uncertainty 2184 where further research may be indicated. 2185 2186 The evidence used in this guideline was graded using the scheme below and the recommendations 2187 formulated in a similar fashion with a standardised grading scheme. 2188 2189 Classification of evidence levels 2190 1++ High-quality meta-analyses, systematic reviews of randomised controlled trials or randomised controlled trials with a very low risk of bias 1+ Well-conducted meta-analyses, systematic reviews of randomised controlled trials or randomised controlled trials with a low risk of bias 1– Meta-analyses, systematic reviews of randomised controlled trials or randomised controlled trials with a high risk of bias 2++ High-quality systematic reviews of case–control or cohort studies or high-quality case–control or cohort studies with a very low risk of confounding, bias or chance and a high probability that the relationship is causal 2+ Well-conducted case–control or cohort studies with a low risk of confounding, bias or chance and a moderate probability that the relationship is causal 2– Case–control or cohort studies with a high risk of confounding, bias or chance and a significant risk that the relationship is not causal 3 Non-analytical studies, e.g. case reports, case series 4 Expert opinion 2191 Grades of Recommendation 2192 At least one meta-analysis, systematic reviews or RCT rated as 1++, and directly applicable to 2193 the target population; or a systematic review of RCTs or a body of evidence consisting 2194 principally of studies rated as 1+, directly applicable to the target population and 2195 demonstrating overall consistency of results 2196 2197 A body of evidence including studies rated as 2++ directly applicable to the target 2198 population, and demonstrating overall consistency of results; or 2199 Extrapolated evidence from studies rated as 1++ or 1+ 2200 2201 A body of evidence including studies rated as 2+ directly applicable to the target population, 2202 and demonstrating overall consistency of results; or 2203 Extrapolated evidence from studies rated as 2++ 2204 2205 Evidence level 3 or 4; or 2206 Extrapolated evidence from studies rated as 2+ 2207 2208 Good Practice Points 2209 2210 Recommended best practice based on the clinical experience of the guideline development 2211 A B C ü D Page 44 of 47 group 2212 2213 Page 45 of 47 Appendix II: Flow diagram for ultrasound diagnosis and follow-up of placenta praevia and placenta accreta spectrum 2214 2215 2216 Abbreviations: BMI body mass index; PAS placenta accreta spectrum; TAS transabdominal scan; TVS transvaginal scan. 2217 Page 46 of 47 Appendix III: Ultrasound imaging signs commonly used to diagnose placenta accreta spectrum (modified from Collins SL)141 2218 2219 Ultrasound imaging signs Description 2D greyscale signs Loss of the ‘clear zone’ Loss or irregularity of the hypoechoic plane in the myometrium underneath the placental bed (the ‘clear zone’). Abnormal placental lacunae Presence of numerous lacunae, including some that are large and irregular (Finberg grade 3), often containing turbulent flow visible in greyscale imaging. Bladder wall interruption Loss or interruption of the bright bladder wall (the hyperechoic band or ‘line’ between the uterine serosa and the bladder lumen). Myometrial thinning Thinning of the myometrium overlying the placenta to less than 1 mm or undetectable. Placental bulge Deviation of the uterine serosa away from the expected plane, caused by an abnormal bulge of placental tissue into a neighboring organ, typically the bladder. The uterine serosa appears intact but the outline shape is distorted. Focal exophytic mass Placental tissue seen breaking through the uterine serosa and extending beyond it. Most often seen inside a filled urinary bladder. 2D colour Doppler signs Uterovesical hypervascularity Striking amount of colour Doppler signal seen between the myometrium and the posterior wall of the bladder. This sign probably indicates numerous, closely packed, tortuous vessels in that region (demonstrating multidirectional flow and aliasing artifact). Subplacental hypervascularity Striking amount of colour Doppler signal seen in the placental bed. This sign probably indicates numerous, closely packed, tortuous vessels in that region (demonstrating multidirectional flow and aliasing artifact). Bridging vessels Vessels appearing to extend from the placenta, across the myometrium and beyond the serosa into the bladder or other organs. Often running perpendicular to the myometrium. Placental lacunae feeder vessels Vessels with high velocity blood flow leading from the myometrium into the placental lacunae, causing turbulence upon entry. 3D colour Doppler signs Intraplacental hypervascularity (power Doppler) Complex, irregular arrangement of numerous placental vessels, exhibiting tortuous courses and varying calibers. 2220 Page 47 of 47 2221 This guideline was produced on behalf of the Royal College of Obstetricians and Gynaecologists by: 2222 Professor ERM Jauniaux FRCOG, London (Lead Developer); Professor Z Alfirevic FRCOG, Liverpool, UK; Mr AG 2223 Bhide FRCOG, London, UK; Professor MA Belfort, Baylor College of Medicine, Houston, Texas, USA; Professor GJ 2224 Burton, University of Cambridge, UK; Professor SL Collins MRCOG, Oxford, UK; Dr S Dornan, Royal Jubilee 2225 Maternity Hospital, Belfast, UK; Mr D Jurkovic FRCOG, London, UK; Professor G Kayem, Armand-Trousseau and 2226 Louis-Mourier University Hospitals, Paris, France; Professor John Kingdom, Mont Sinai, Toronto University, 2227 Canada; Professor R Silver, University of Utah, Salt Lake City, Utah, USA; Professor L Sentilhes, University Hospital 2228 Angers, France 2229 2230 and peer reviewed by: Professor ML Brizot, University of São Paulo, São Paulo, Brazil; Dr G Calì MSIEOG, ARNAS 2231 Civico Hospital, Palermo, Italy; Professor J Dashe, University of Texas Southwestern Medical Center, Dallas, TX, USA; 2232 Professor O Erez, Soroka University Medical Center, Beer Sheva, Israel; Dr D Fraser FRCOG, Norwich; Dr F Forlani, 2233 University Hospital "Paolo Giaccone", Palermo, Italy; Dr J Hasegawa, St. Marianna University School of Medicine, 2234 Kawasaki, Kanagawa, Japan; Dr YY Hu, Sichuan University, Chengdu, Sichuan, China; Dr N Lucas, Obstetric 2235 Anaesthetists' Association, London; Professor P Martinelli, Università di Napoli Federico II, Naples, Italy; Princess 2236 Royal Maternity Invasive Placenta Team, London; RCOG Women’s Network; Professor SC Robson MRCOG, FRCP, 2237 Newcastle University; Royal College of Anaesthetists; Dr R Salim, Emek Medical Center, Afula, Israel; Professor RM 2238 Silver, The University of Utah, Salt Lake City, UT, USA; Dr JT Thomas FRANZCOG, CMFM, Mater Mothers' Hospital, 2239 Brisbane, Australia; The UK Vasa Praevia Raising Awareness Trust and the International Vasa Previa Foundation; Mr 2240 N Thomson, Society and College of Radiographers, London; Dr M Tikkanen, Women´s Clinic, Helsinki University 2241 Hospital Finland, Helsinki, Finland 2242 2243 Committee lead reviewers were: Dr A McKelvey MRCOG, Norwich and Mr RJ Fernando FRCOG, London 2244 2245 The chair of the Guidelines Committee was: Dr AJ Thomson MRCOG, Paisley. 2246 2247 All RCOG guidance developers are asked to declare any conflicts of interest. A statement summarising any conflicts of 2248 interest for this guideline is available from: XXX. 2249 2250 The final version is the responsibility of the Guidelines Committee of the RCOG. 2251 2252 The guideline will be considered for update 3 years after publication, with an intermediate assessment of the need 2253 to update 2 years after publication. 2254 2255 2256 DISCLAIMER 2257 2258 The Royal College of Obstetricians and Gynaecologists produces guidelines as an educational aid to good clinical 2259 practice. They present recognised methods and techniques of clinical practice, based on published evidence, for 2260 consideration by obstetricians and gynaecologists and other relevant health professionals. The ultimate judgement 2261 regarding a particular clinical procedure or treatment plan must be made by the doctor or other attendant in the 2262 light of clinical data presented by the patient and the diagnostic and treatment options available. 2263 2264 This means that RCOG Guidelines are unlike protocols or guidelines issued by employers, as they are not intended to 2265 be prescriptive directions defining a single course of management. Departure from the local prescriptive protocols or 2266 guidelines should be fully documented in the patient’s case notes at the time the relevant decision is taken. 2267 |
13098 | https://www.etymonline.com/word/cultivation | Want to remove ads? Log in to see fewer ads, and become a Premium Member to remove all ads.
Origin and history of cultivation
cultivation(n.)
1700, "the devoting of special attention or study to the development of" (a branch of knowledge); by 1716 in the general sense of "promotion of mental growth or development," in both cases a figurative use, from French cultivation (16c.), noun of action from cultiver, from Latin cultivare "to till" (see cultivate). Meaning "the raising of a plant or crop" is from 1719; sense of "act or practice of tilling land and preparing it for crops" is from 1725.
also from 1700
Entries linking to cultivation
by 1650s, of land, "till, prepare for crops;" by 1690s of crops, "raise or produce by tillage;" from Medieval Latin cultivatus, past participle of cultivare "to cultivate," from Late Latin cultivus "tilled," from Latin cultus "care, labor; cultivation," from past participle of colere "to cultivate, to till; to inhabit; to frequent, practice, respect; tend, guard," from PIE root kwel- (1) "revolve, move round; sojourn, dwell."
Figurative sense of "improve by labor or study, devote one's attention to" is from 1680s. Meaning "court the acquaintance of (someone)" is by 1707. Related: Cultivated; cultivating.
Want to remove ads? Log in to see fewer ads, and become a Premium Member to remove all ads.
Trends of cultivation
More to explore
Share cultivation
Want to remove ads? Log in to see fewer ads, and become a Premium Member to remove all ads.
Want to remove ads? Log in to see fewer ads, and become a Premium Member to remove all ads.
Want to remove ads? Log in to see fewer ads, and become a Premium Member to remove all ads.
Quick and reliable accounts of the origin and history of English words. Scholarly, yet simple.
About
Support
Apps |
13099 | https://math.stackexchange.com/questions/1363749/vietas-formula-failed | algebra precalculus - Vieta's Formula failed? - Mathematics Stack Exchange
Join Mathematics
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Mathematics helpchat
Mathematics Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Vieta's Formula failed?
Ask Question
Asked 10 years, 2 months ago
Modified1 year, 11 months ago
Viewed 605 times
This question shows research effort; it is useful and clear
7
Save this question.
Show activity on this post.
Find the possible values of p p for which p p and q q are the solutions of the equation
x 2+p x+q=0,{x,p,q}∈R.x 2+p x+q=0,{x,p,q}∈R.
Plugging p p and q q into the given equation {2 p 2+q=0 q 2+p q+q=0{2 p 2+q=0 q 2+p q+q=0
gives p={0,1,−1 2},p={0,1,−1 2}, which is the answer given by my book.
On the other hand, using Vieta's formula for the sum and product of roots {p+q=−p p q=q{p+q=−p p q=q
gives p={0,1}.p={0,1}. I am highly confused as to why Vieta's formula fails.
algebra-precalculus
quadratics
Share
Share a link to this question
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this question to receive notifications
edited Oct 17, 2023 at 5:48
ryang
1
asked Jul 16, 2015 at 20:27
TAKI TAKITAKI TAKI
165 4 4 bronze badges
8
@ Will Jagy: I didn't understand , can u add context ?TAKI TAKI –TAKI TAKI 2015-07-16 20:33:58 +00:00 Commented Jul 16, 2015 at 20:33
how a degree 2 ploy can have 3 answers ? do you check the feasibility ?Cardinal –Cardinal 2015-07-16 20:35:29 +00:00 Commented Jul 16, 2015 at 20:35
if p is a rout how a degree two linear equation can have 3 roots ? i am confused.Cardinal –Cardinal 2015-07-16 20:40:21 +00:00 Commented Jul 16, 2015 at 20:40
1 @cardinal: It is not showing 3 3 roots, its showing 3 3 possibilities of a single root TAKI TAKI –TAKI TAKI 2015-07-16 20:41:10 +00:00 Commented Jul 16, 2015 at 20:41
1 @Cardinal: The system of equations does in fact have 3 3 possible combinations of (p,q)(p,q). The problem is that why OP cannot arrive into those by vieta's formulas.Eemil Wallin –Eemil Wallin 2015-07-16 20:48:40 +00:00 Commented Jul 16, 2015 at 20:48
|Show 3 more comments
3 Answers 3
Sorted by: Reset to default
This answer is useful
4
Save this answer.
Show activity on this post.
In short, the difference is because the Vieta formulae assume that p p and q q are different roots of the equation (in the sense of corresponding to different factors — not necessarily in the sense of being numerically unequal), while your second system only assumes that p p and q q both solve the equation.
To be more explicit, let's solve your second system to see what's going on. Subtracting the two equations from each other gives
q 2+p q−2 p 2=0(q−p)(q+2 p)=0 q 2+p q−2 p 2=0(q−p)(q+2 p)=0
So either q=p q=p or q=−2 p q=−2 p.
If q=−2 p q=−2 p, we get 2 p 2−2 p=0 2 p 2−2 p=0, so either p=q=0 p=q=0 or p=1 p=1, q=−2 q=−2. Note that both of these are solutions to the Vieta system as well.
If q=p q=p, we get 2 p 2+p=0 2 p 2+p=0, so either p=q=0 p=q=0 (again) or p=q=−1/2 p=q=−1/2. This last solution does not satisfy the Vieta formulae. Why not? Because it means the original equation was
x 2−1 2 x−1 2=0 x 2−1 2 x−1 2=0
which has solutions x=−1 2 x=−1 2 and x=1 x=1. So in this case both p p and q q are roots of x 2+p x+q x 2+p x+q, but they aren't "the" roots of x 2+p x+q x 2+p x+q (since they're both the same root).
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
edited Jul 16, 2015 at 21:15
answered Jul 16, 2015 at 21:01
MicahMicah
38.8k 15 15 gold badges 89 89 silver badges 138 138 bronze badges
2
Nice answer. And to expand on this, from the process that gives the three answer, {2 p 2+q=0 q 2+p q+q=0{2 p 2+q=0 q 2+p q+q=0 q 4 p 4−2 p 3−2 p 2 p 2(2 p+1)(p−1)=−2 p 2=0=0 q=−2 p 2 4 p 4−2 p 3−2 p 2=0 p 2(2 p+1)(p−1)=0 neither of the constraints p+q=−p p+q=−p nor p q=q p q=q are considered.peterwhy –peterwhy 2015-07-16 21:10:40 +00:00 Commented Jul 16, 2015 at 21:10
So the book's answer is wrong, which should be underlined: the question says that “p p and q q are the roots”, not that “p p and q q are roots”. There's a big difference.egreg –egreg 2015-07-16 21:45:57 +00:00 Commented Jul 16, 2015 at 21:45
Add a comment|
This answer is useful
1
Save this answer.
Show activity on this post.
Find the possible values of p p for which p p and q q are the solutions of the equation
x 2+p x+q=0,{x,p,q}∈R.x 2+p x+q=0,{x,p,q}∈R.
Plugging p p and q q into the given equation {2 p 2+q=0 q 2+p q+q=0{2 p 2+q=0 q 2+p q+q=0
gives p={0,1,−1 2},p={0,1,−1 2}, which is the answer given by my book.
Your book's answer is wrong. It is a mistake to translate
p p and q q are (all) the solutions of x 2+p x+q=0 x 2+p x+q=0
as the weaker statement
p p and q q satisfy x 2+p x+q=0 x 2+p x+q=0 (i.e., p p and q q are solutions of x 2+p x+q=0 x 2+p x+q=0).
Since statement 1 implies statement 2 and we don't initially know whether is converse is true, the key is to note that the latter may introduce extraneous possibilities (while capturing at least all the required possibilities); so, we need to check whether any possibility needs to be eliminated from the latter. Here, since p=−1 2 p=−1 2 gives rise to p=−1 2=q p=−1 2=q yet the equation x 2−1 2 x−1 2=0 x 2−1 2 x−1 2=0 has solutions −1 2−1 2 and 1 1 instead of just −1 2,−1 2, we deduce that the value p=−1 2 p=−1 2 fails to satisfy statement 1 (the given statement).
On the other hand, using Vieta's formula for the sum and product of roots {p+q=−p p q=q{p+q=−p p q=q
gives p={0,1}.p={0,1}. I am highly confused as to why Vieta's formula fails.
Vieta's formula has not failed, because p p really can equal only either 0 0 or 1.1.
P.S. And that "{x,p,q}⊂R{x,p,q}⊂R" restriction turns out to have been redundant.
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this answer to receive notifications
edited Oct 17, 2023 at 9:14
answered Oct 17, 2023 at 5:48
ryangryang
1
Add a comment|
This answer is useful
0
Save this answer.
Show activity on this post.
The question asks 'For which values of p,q p,q do we have
x 2+p x+q=(x−p)(x−q)x 2+p x+q=(x−p)(x−q)
?'
A simple expansion of the RHS gives
x 2+p x+q=x 2−(p+q)x+p q x 2+p x+q=x 2−(p+q)x+p q
Hence the application of Vieta's formulas, and gives (p,q)=(0,0),(1,−2)(p,q)=(0,0),(1,−2).
The book's solution finds answers to 'For which values of p,q p,q do we have
x 2+p x+q=0 x 2+p x+q=0
(x−p)(x−q)=0(x−p)(x−q)=0
?'
which is a different question, and gives the extra answer of (p,q)=(−1 2,−1 2)(p,q)=(−1 2,−1 2).
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this answer to receive notifications
answered Oct 17, 2023 at 8:32
JMPJMP
22.7k 51 51 gold badges 37 37 silver badges 55 55 bronze badges
1
1 How can imposing an extra equation (the =0) allow an extra solution to appear? Your restatements of the question are a little off since you don't say what x means. In your first restatement you imply the equation should hold for all x, i.e. that the two sides are equal as polynomials (and while that is equivalent to the original question, that is confusing). In your second restatement, it is unclear what to do with x when we "have" the equations.Jaap Scherphuis –Jaap Scherphuis 2023-10-17 08:48:53 +00:00 Commented Oct 17, 2023 at 8:48
Add a comment|
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
algebra-precalculus
quadratics
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Report this ad
Linked
2Find p p and q q in x 2−p x+q=0 x 2−p x+q=0
Related
3Prove relations between the roots of 3 quadratic equations
2Assume this equation has distinct roots. Prove k=−1/2 k=−1/2 without using Vieta's formulas.
1Using Vieta's formula to solve a Cubic Polynomial
3Vieta's Theorems Clarification
2Vieta's formula for quartic complex polynomial?
2Finding the quadratic equation from its given roots.
2Am I doing Vieta's Formulas correctly?
0Find the value of the parameter L L via Cardano-Vieta's Formula knowing a condition on two roots
2Cardano's formula running empty
Hot Network Questions
Can a cleric gain the intended benefit from the Extra Spell feat?
Do we need the author's permission for reference
What's the expectation around asking to be invited to invitation-only workshops?
Why does LaTeX convert inline Python code (range(N-2)) into -NoValue-?
ConTeXt: Unnecessary space in \setupheadertext
Origin of Australian slang exclamation "struth" meaning greatly surprised
Riffle a list of binary functions into list of arguments to produce a result
Is there a way to defend from Spot kick?
What were "milk bars" in 1920s Japan?
With line sustain pedal markings, do I release the pedal at the beginning or end of the last note?
Who is the target audience of Netanyahu's speech at the United Nations?
Suggestions for plotting function of two variables and a parameter with a constraint in the form of an equation
Why do universities push for high impact journal publications?
Explain answers to Scientific American crossword clues "Éclair filling" and "Sneaky Coward"
Are there any world leaders who are/were good at chess?
Is it possible that heinous sins result in a hellish life as a person, NOT always animal birth?
Bypassing C64's PETSCII to screen code mapping
Do sum of natural numbers and sum of their squares represent uniquely the summands?
Childhood book with a girl obsessed with homonyms who adopts a stray dog but gives it back to its owners
Cannot build the font table of Miama via nfssfont.tex
alignment in a table with custom separator
how do I remove a item from the applications menu
Making sense of perturbation theory in many-body physics
How to home-make rubber feet stoppers for table legs?
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Mathematics
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.