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Prior to the main SEAMS School Programme, a 3-Day Pre-SEAMS School Workshop will be held for the participants. This is to equip them with sufficient essential knowledge, so as to enable them to follow the contents of the SEAMS School Programme. The topics for discussion in the Pre-School Workshop are as follows:
Commutative Algebras
Algebraic Geometry
Lie Algebras
Number Theory
Quantum Theory
SEAMS School (7 days)
Quantum Physics
Recent advances in quantum information has begun to explore deeper into the structure of complex projective space as space of quantum states. Among the tools employed are algebraic varieties, symplectic quotients and projective ring geometries.
Elliptic Curve Cryptography
Elliptical curve cryptography (ECC) is a public key encryption technique based on elliptic curve theory that can be used to create faster, smaller, and more efficient cryptographic keys.
Algebraic Statistics
Algebraic statistics advocates the use of algebraic geometry, commutative algebra, and geometric combinatorics as tools for making statistical inferences. The starting point for this connection is the observation that most statistical models for discrete random variables are, in fact, algebraic varieties. | 677.169 | 1 |
The concepts of mathematics for students outside of the fields of mathematics,
business, and the sciences. Placement according to COMPASS math score
or successful completion of DVSM 102 (Developmental Algebra).
Since this is an online course, there are some minimum hardware and
software requirements to complete the course. For recommended operating
system requirements and web browser compatibility, see
click on My Online Courses
For all browsers, JavaScript and cookies must be enabled. To use Chat
and Whiteboard, Java must also be enabled. Please see
click on My Online Courses for details on internet browser setups.
To complete this course, you will need the following software:
a. Microsoft Word, PowerPoint (the 'free' Microsoft PowerPoint
Viewer may be used to view .ppt files) and Excel (2000 or newer). There
is a possibility that if you have Microsoft Office 2007 you will have
to save your Excel documents in a previous file; (i.e.: .xls). Instructions
on doing this will be given to you during the first week of class. Adobe
Acrobat Reader® (free download at
b. WinZip® (download at ) or similar product.
This may not be required.
c. Use of WNMU's Bb tool (
Textbooks and software may be purchased at the WNMU bookstore, in person
or online via : Or you may purchase via other online
resources such as or
Upon completion of this course, the student will have proficiency in
the following:
a. Be able to explore a few of the more common voting methods use in
elections--- how they work, what their implications are, and how they
stack up when we put them to some basic test of fairness. In so doing,
we will also gain some insight into the meaning and significance of Arrow's
impossibility theorem.
b. We will learn that in a diverse society--- it is in the very nature
of things that voters; be they individuals or institutions---are not
equal, and sometimes it is actually desirable to recognize their differences
by giving them different amounts of say over the outcome of an election.
That a principle best described as one voter-x votes, is more formally
known as weighted voting.
c. Dividing things fairly using reason and logic, instead of bullying
our way to a solution, is one of the great achievements of social science,
and, once again, we can trace the roots of this achievement to simple
mathematics.
d. Realize that Article 1, Section 2 of the Constitution of the United
States is continually on a collision course with a mathematical iceberg
known today as the apportionment problem.
e. We will solve problems involving the organization and management
of complex activities, such as the mathematical study of how things are
interconnected. Learn that efficiency is of prime importance in solving
these problems.
f. Understand the type of problem known as the traveling-salesman problem
(TSP).
g. To understand the problem of finding efficient networks connecting
a set of points.
h. Understand that limited or precious resource must be managed to minimize
waste.
Attendance in an online class is evaluated by your attendance to the
material. You will get out of the course what you put into the course.
You will need to be a "self-starter", and control your own
calendar in order to meet the deadlines for the course.
Informed Consent:
Some individuals may choose to disclose personal information during
class. Therefore, it is important that all classmates agree not to discuss
or write about what others have discussed in class.
Professionalism:
Students are learning professional skills and are expected to engage
in classroom discussions, complete reading assignments and turn in assignments
in a timely fashion as befitting professional behavior.
Scholarly Writing:
Use clear college level writing with correct spelling and grammar for
all assignments. If you need help in writing, check with the WNMU Online
Writing Center.
Special Needs:
Students with disabilities in need of accommodation should register
with the Special Needs Office (JUANCB 210, Ext. 6498) at the beginning
of the semester. With student permission, that office will notify instructors
of any special equipment or services a student requires..
Integrated Use of Technology:
Because this is an online course, I am making the assumption that you
are comfortable utilizing a computer, and navigating various software
programs like Blackboard Vista (Bb), Microsoft Word, Powerpoint. If you
have any questions about computer requirements see the "Student
Resoures" course in Bb.
Need Help?
1. Post a question to the Discussion Board. There is no such thing as
a dumb question.
2. Post a question as a Bb email to your instructor.
3. If the Bb system goes down or you have other technical questions,
contact the WNMU Help Desk: helpdesk@wnmu.edu or (505) 574-4357.
Special Needs Students: Students with disabilities in need of accommodation
should register with the Special Needs Office (JUANCB 210, Ext. 6498)
at the beginning of the semester. With student permission, that office
will notify instructors of any special equipment or services a student
requires.
Communication Policy Statement regarding official email :WNMU's
policy requires that all official communication be sent via Mustang Express.
As a result, all emails related to your enrollment at WNMU and class
communication – including changes in assignments and grades – will
be sent to your wnmu.edu email address. It is very important that you
access your Mustang Express e-mail periodically to check for correspondence
from the University. If you receive most of your email at a different
address you can forward your messages from Mustang Express to your other
address.
Example: Martin Classmember was assigned a WNMU email address of classmemberm12@wnmu.edu
but Martin would rather receive his emails at his home email address
of martinclass@yahoo.com.
WNMU Policy on Email Passwords: WNMU requires that passwords for access
to all of the protected software, programs, and applications will be
robust, including complexity in the number of characters required, the
combination of characters required, and the frequency in which passwords
are required to be changed. Minimum complexity shall include:
Academic Integrity Policy and Procedures: Each student shall observe
standards of honesty and integrity in academic work completed at WNMU.
Students may be penalized for violations of the Academic Integrity policy.
Please refer to pages 60 and 61 of the 2008-2009 Catalog. (Clearly specify
what you consider to be violations of academic honesty.)
Caveats: The schedule and procedures in this course are subject to change
in the event of extenuating circumstances.
This is an intensive, undergraduate-level course with regular and firm
deadlines.
Weekly Homework Assignments: You will be assigned homework every other
week, except for the week of Thanksgiving, and the final. Details on
the homework can be found under the assignments icon. Homework format
will normally utilize Microsoft Excel workbooks (sheets). Examples will
be posted on the Bb course for you to study. You must submit your module
(week) assignments by the end of each module (week) period to be considered
for grading. Solutions will be posted in Bb following the submission
deadline. Normally, this will be in the following module (week). Homework
is worth 80 total points or about 27% of the final grade.
Chapter Tests: You will be required to submit chapter tests about every
other week. These tests are worth 20 points each with a total of 140
points or about 47% of your final grade. Chapter tests are available
under the assignment icon 40 points or about 13% of the
final grade module readings and assignments.
However, submitted homework assignments are expected to be your own work.
Do not work together on graded homework problems or exams Your homework points plus your discussion points
through chapter 4 (Module/Week 8) will determine your mid-term grade.
Since each chapter is unique there is no 'building up of knowledge' as
you would expect in a normal mathematics course. Completing each homework
and discussion session is very important if you want the best grade for
yourself.
Final Examination: The final four chapters (chapters 5 - 8) will also
count as a second 'mid-term' examination. The points assigned
will be similar to those of the first 4 chapters (chapters 1 - 4). In
addition, a final exam covering the whole course will allow you to recap
some points that you may have missed in the homework and discussion area.
No collaboration is allowed. Further details will be announced prior
to this period using our Bb course. The weight of the final examination
is 40 points or about 13% of the final grade.
In order to promote a positive, professional atmosphere among students,
faculty and staff, the following Code of Civility has been developed:
Respect: Treat all students, faculty, staff and property with respect
and in a courteous and professional manner. This includes all communications,
whether verbal or written. Let your actions reflect pride in yourself,
your university, and your profession.
Kindness: A kind word and gentle voice go a long way. Refrain from using
profanity, insulting slang remarks, or making disparaging comments. Consider
another person's feelings. Be nice. | 677.169 | 1 |
84%
Marketplace Item
Description
This book explores the standard problem-solving techniques of multivariable mathematics -- integrating vector algebra ideas with multivariable calculus and differential equations. Provides many routine, computational exercises illuminating both theory and practice. Offers flexibility in coverage -- topics can be covered in a variety of orders, and subsections (which are presented in order of decreasing importance) can be omitted if desired. Provides proofs and includes the definitions and statements of theorems" to show how the subject matter can be organized around a few central ideas. Includes new sections on: flow lines and flows; centroids and moments; arc-length and curvature; improper integrals; quadratic surfaces; infinite series--with application to differential equations; and numerical methods. Presents refined method for solving linear systems using exponential matrices. | 677.169 | 1 |
Student Textbook purchase includes access to resource pages, eTools, homework help, and the Parent Guide with Extra Practice**. Each middle school student text includes one copy of the consumable Toolkit . Student book comes in hardbound with eBook license, softbound with eBook license, or you may purchase an eBook license without* a print book. Most titles are also available in Spanish.
Setting up a CPM Core Connections classroom? Here's what you need to order from CPM:
Teacher Edition (1)
Student book for each student*
Toolkit, one per Student: CC1, CC2, or CC3only
Integer Tiles, one set per class of 36, CC2only
Algebra Tiles, one set per class of 36: CC1, CC2, CC3, CCAlgebra, CCIntegrated I, or CCIntegrated II only | 677.169 | 1 |
Essential Mathematics provides mathematically sound and comprehensive coverage of the topics considered essential in a basic college math course. The Aufmann Interactive Method ensures that students master concepts by actively practicing them as they are introduced. This approach is ideal for traditional and returning students in both classroom and distance-learning environments.
For the Sixth Edition, topics from geometry have been integrated into the text, using verbal explanations. In addition, coverage of simple interest (Chapter 6) has been expanded.
The Aufmann Interactive Method helps students learn and understand math concepts by doing the math. Every objective contains one or more sets of matched-pair examples. Students first walk through a worked-out example and then solve a similar "You Try It" example. Complete solutions to these examples are available in an appendix.
An Integrated Learning System organized by objectives helps students understand what they're learning and why as they apply new concepts throughout the chapter. Each chapter begins with a list of goals that form the framework for a complete learning system. These objectives are woven throughout the text, in Exercises, Chapter Tests, Cumulative Reviews, as well as the print and multimedia ancillaries.
An Instructor's Annotated Edition provides reduced pages from the Student Edition to leave space for the following features: Instructor Notes; In-Class Examples; Concept Checks; Discuss the Concepts; Special presentation of new Vocabulary/ Symbols/Formulas/Rules/Properties/Equations; Special review of these same features; Optional Student Activities; Quick Quizzes; Answers to Writing Exercises; Suggested Assignments; and Answers to all exercises.
AIM for Success, a special student preface, offers techniques and support for student success.
Prep Tests at the beginning of each chapter assess students' prerequisite skills. Students may check answers in an appendix, which refers them back to a previous objective for review, if necessary.
Updated data problems, designed to show students the relevance of mathematics across the disciplines and in daily life, reflect current data and trends.
Additional and revised Projects and Group Activities enable students to see the connections between abstract concepts and real-life situations.
Strong emphasis on applications demonstrates the value of mathematics as a real-life tool. Chapter openers have been updated with new photos and captions illustrating a specific application from the chapter.
Unlike most textbooks, this series simultaneously introduces verbal phrases for mathematical operations and the operations themselves. Exercises then prompt students to make a connection between a phrase and a mathematical process.
"synopsis" may belong to another edition of this title.
About the Author:
Vernon Barker has retired from Palomar College where he was Professor of Mathematics. He is a co-author on the majority of Aufmann texts, including the best-selling developmental paperback series.
Richard Aufmann is the lead author of two bestselling developmental math series and a bestselling college algebra and trigonometry series, as well as several derivative math texts. He received a BA in mathematics from the University of California, Irvine, and an MA in mathematics from California State University, Long Beach. Mr. Aufmann taught math, computer science, and physics at Palomar College in California, where he was on the faculty for 28 years. His textbooks are highly recognized and respected among college mathematics professors. Today, Mr. Aufmann's professional interests include quantitative literacy, the developmental math curriculum, and the impact of technology on curriculum development.
Joanne Lockwood received a BA in English Literature from St. Lawrence University and both an MBA and a BA in mathematics from Plymouth State University. Ms. Lockwood taught at Plymouth State University and Nashua Community College in New Hampshire, and has over 20 years' experience teaching mathematics at the high school and college level. Ms. Lockwood has co-authored two bestselling developmental math series, as well as numerous derivative math texts and ancillaries. Ms. Lockwood's primary interest today is helping developmental math students overcome their challenges in learning math. | 677.169 | 1 |
Getting started > Welcome to the course > Welcome to the course
Hello there! Wherever you are in the world, we welcome you very much to this course.
We, the three of us, are your course team! I am Alexander de Haan.
For many years already I am teaching an elaborate off-line version of this course on the Delft University of Technology campus for well over three hundred students each year.
I have a broad interest in education, developed many different courses and taught at many different institutions, both universities as well as commercial companies.
In this online course I will introduce you to new analytical techniques in this course using situations, animations and examples.
I have been in Alexander's course as a student a few years ago.
I worked on an elaborate evaluation of the first online version of this course.
Currently I am involved as a student assistant, organizing the off line on campus version of this course and supervising students in their group work.
My role in this online course is providing you in each video with clear day to day examples of the analytical techniques Alexander introduces.
For more than five years I have been working with Alexander in the on campus version of this course, by organizing it and supervising students.
My role in this course is to help you apply the techniques introduced in each video to a case so you learn them faster.
Over the years we have seen everybody who starts with this course making comparable mistakes.
In this video we give you a quick overview of how the course works.
All three of us, of course, wish you a very useful and pleasant time.
When you have enrolled in this course via EdX, take a look at the top bar in your screen.
Here you see all the important resources for this course.
Starting from left to right, you see courseware, course info, facebook, literature: solving complex problems, glossary, discussion and progress.
The progress page shows you your progression in the course.
The discussion page is categorized; it gives an overview of all the discussion topics of the course.
The glossary provides the definitions of commonly used terms and concepts in the course.
It is our experience that working together makes this course so much easier and more fun to do.
In the course info we post updates and other news about the course.
You can see that the course is divided in five steps.
At the end of every video Elianne introduces some questions that guide you in applying the course content on a case.
They experience their problems as we speak and you can really make a contribution to solving their complex problem by working on it in this course.
We have selected these five cases for you, but you can apply the content of this course on any case you like.
Maybe you have a situation in your work, your country or your private life you want to analyse and solve using the techniques of this course.
You discuss your work and the questions you have on the discussion fora and on facebook.
The content of this course is described more elaborately and very practically in the book called Solving Complex Problems.
It is not necessary for this course to have the book, but it will of course be an advantage to have more elaborate reading material and examples available.
The only way to learn a course like this is to try and apply.
We have provided you with some cases in this course.
We are really looking forward to work with you in this course.
If you, just like us, can't wait, go check out the course site.
Getting started > Cases to practise > Case Schiphol
One of the leading and most innovative airports in the world.
At the airport we are dealing with ongoing growth and we are planning a new terminal at the moment.
I am looking for new ideas and innovative solutions from a new generation of students who are doing this course.
Currently I am doing a project at Amsterdam Airport Schiphol about the arriving experience of passengers.
So what actors might be important here? Think for instance about the airport itself, the airlines or the airport security.
Getting started > Cases to practise > Case electrical vehicles
One of the major questions the government is faced with is 'how environmentally friendly is an electric car?' And that of course depends upon the questions 'where does the electricity come from?' From coal? From solar? From wind.
Why should the Dutch government subsidize electric cars, when there are alternatives that are also environmentally friendly? And what is the innovation potential of the traditional car.
Maybe thanks to the introduction of the electric car there will be strong incentives for the producers of the traditional cars to improve the environmental performance of their cars.
Those questions make it extremely difficult for the Dutch government to decide about the question 'what to do?' And my simple question to you is 'get me out of here!' Give me a clear picture of what the problem is.
Getting started > Cases to practise > Case strategy studies
To what extent are the different risks covered, e.g., on employment or poverty? How are the different systems build to distribute resources and to provide services? And how do they differ in terms of achievements and outcomes, in particular, e.g., with respect to life expectancy? As you can imagine, this involves a broad diversity of actors as well as a broad diversity of perspectives and objectives.
What are the different actors that you think should be included in our analysis? What are their perspectives and their different dilemmas? What are the different tensions that you could think of? And then, could you come up with a well-balanced set of criteria that you think we should include in our analysis for the welfare states research project? Finally, considering that the context would be that of economic crises and rising individualism that threaten the sustainability of the welfare states, could you come up with a number of alternatives in your presentation of the findings.
Getting started > Cases to practise > Case burger restaurant
I am here, with my restaurant, Pure Funky Burgerz.
Why don't you come with me? So here at Burgerz we do about 24 different burgers, ranging from beef, chicken, lamb, vegetarian, but now also vegan.
All the ingredients we use come from local farms within a 100 kilometers.
So let me show you, these are our salads, and, as I said before, we do vegetarian burgers, for example our pumpkin burger or our mushroom burger.
I want to serve real, pure hamburgers from a full service restaurant.
I want to run a successful restaurant based on these principles.
Come up with ideas and suggestions on what I can do.
Getting started > Cases to practise > Case Nile
I am Bert Enserink and I am a policy analist at Delft University of Technology, and I am involved in policy making in the river Nile.
Many people are dependent on the water of the Nile, for their livelihood, for drinking water, for sanitation, for industrial purposes, for energy generation.
It is a very complex and political sensitive issue.
The Nile is a very complex river system as you discovered.
Others want to build this dam for irrigation systems to raise new crops and improve the agricultural system.
If I need to facilitate the policy process, then I need to know what are the real concerns and issues in this river basin.
What do these people want? So I need to know, not the solutions, building a dam, or building an irrigation system, but what are the issues that need to be solved.
The causal diagram, the systems diagram, the problem diagram, and try to get some grip on the issues that are important to the stakeholders in this system. | 677.169 | 1 |
Introduction to Analysis by Irena Swanson
Description: In this course, students learn to write proofs while at the same time learning about binary operations, orders, fields, ordered fields, complete fields, complex numbers, sequences, and series. We also review limits, continuity, differentiation, and integration.
Reader-friendly Introduction to the Measure Theory by Vasily Nekrasov - Yetanotherquant.de This is a very clear and user-friendly introduction to the Lebesgue measure theory. After reading these notes, you will be able to read any book on Real Analysis and will easily understand Lebesgue integral and other advanced topics. (5836 views)
A Course in Mathematical Analysis by E. Goursat, O. Dunkel, E.R. Hedrick - Ginn & company Goursat's three-volume 'A Course in Mathematical Analysis' remains a classic study and a thorough treatment of the fundamentals of calculus. As an advanced text for students with one year of calculus, it offers an exceptionally lucid exposition. (2199 views) | 677.169 | 1 |
Linear Functions and Nonlinear Functions Discovery Lab
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Product Description
Students will get a great introduction to linear and nonlinear functions in this interactive discovery activity (Supports 8th grade standard CCSS 8.F.A.3 ). Using a graphing tool, students will compare linear and nonlinear functions, drawing their own conclusions on the attributes of each.
-Teacher directions and tips
-Powerpoint presentation for teachers to use in introducing the activity to students.
-6 pages for students to record their findings and draw conclusions
-Differentiation included with 2 versions of student conclusion page and 2 additional extensions items for fast finishers.
Please note, in order to complete this activity students will need to have access to a graphing calculator or to the internet and be able to use an online graphing calculator (suggested online tool is given in the teacher directions.) | 677.169 | 1 |
ISBN-10: 0495012726
ISBN-13: 9780495012726
Edition: 11 for liberal arts students and based on the belief that learning to solve problems is the principal reason for studying mathematics, the major theme of this book is problem solving. In the first section, Karl Smith introduces students to Polya's problem-solving techniques and then shows students how to use these techniques throughout the book to solve unfamiliar problems. In addition to the problem solving emphasis, the book is well renowned for its clear writing, coverage of historical topics, selection of topics, level, and exercise sets that feature great applications problems. Since the first edition of Smith's text was published, thousands of students have "experienced" mathematics rather than just doing problems. Smith's writing style gives students the confidence and ability to utilize mathematics in their everyday lives. The emphasis on problem solving and estimation, along with numerous in-text study aids, encourages students to understand the concepts while mastering techniques.
Karl Smith is professor emeritus at Santa Rosa Junior College in Santa Rosa, California. He has written over 36 mathematics textbooks and believes that students can learn mathematics if it is presented to them through the use of concrete examples designed to develop original thinking, abstraction, and problem-solving skills. Over one million students have learned mathematics from Karl Smith's textbooks.
The Nature of Problem Solving
Problem Solving
Inductive and Deductive Reasoning
Scientific Notation and Estimation
The Nature Of Sets
Sets, Subsets, and Venn Diagrams
Operations with Sets
Applications of Sets
Finite and Infinite
The Nature Of Logic
Deductive Reasoning
Truth Tables and the Conditional
Operators and Laws of Logic
The Nature of Proof
Problem Solving Using Logic
Logic Circuits
The Nature Of Numeration Systems
Early Numeration Systems
Hindu-Arabic Numeration Systems
Different Numeration Systems
Binary Numeration System
History of Calculating Devices
The Nature Of Numbers
Natural Numbers
Prime Numbers
Integers
Rational Numbers
Irrational Numbers
Groups, Fields, and Real Numbers
Discrete Mathematics
Cryptography
The Nature Of Algebra
Polynomials
Factoring
Evaluation, Applications, and Spreadsheets
Equations
Inequalities
Algebra in Problem Solving
Ratios, Proportions, and Problem Solving
Percents
Modeling Uncategorized Problems
The Nature Of Geometry
Geometry
Polygons and Angles
Triangles
Similar Triangles
Right Triangle Trigonometry
Golden Rectangles
Projective and Non-Euclidian Geometries
The Nature Of Measurement
Perimeter
Area
Surface Area, Volume and Capacity
Miscellaneous Measurements
The Nature Of Growth
Exponential Equations
Logarithmic Equations
Applications of Growth and Decay
The Nature Of Sequences, Series and Financial Management
Interest
Installment Buying
Sequences
Series
Annuities
Amortization
Summary of Financial Formulas
The Nature Of Counting
Counting Formulas
Permutations
Combinations
Binomial Theorem
Counting without Counting
Rubik's Cube and Instant Insanity
The Nature Of Probability
Introduction to Probability
Mathematical Expectation
Probability Models
Calculated Probabilities
Guest Essay: Extrasensory Perception
The Binomial Distribution
The Nature Of Statistics
Frequency Distributions and Graphs
Measures of Central Tendency
Measures of Dispersion
The Normal Curve
Correlation and Regression
Sampling
The Nature Of Graphs And Functions
Cartesian Coordinates and Graphing Lines
Graphing Half-Planes
Graphing Curves
Conic Sections
Functions
The Nature Of Mathematical Systems
Systems of Linear Equations
Problem Solving with Systems
Matrix Solution of a System of Equations
Inverse Matrices
Systems of Inequalities
Modeling with Linear Programming
The Nature Of Networks And Graph Theory
Euler Circuits and Hamiltonian Cycles
Trees and Minimum Spanning Trees
Topology and Fractals
Guest Essay: What Good are Fractals?
Guest Essay: Chaos
The Nature Of Voting And Apportionment
Voting
Voting Dilemmas
Apportionment
Apportionment Flaws
The Nature Of Calculus
What is Calculus? Limits
Derivatives
Integrals
Epilogue
Why Not Math? Mathematics in the Natural Sciences, Social Sciences, and in the Humanities | 677.169 | 1 |
Chapter 5: Linear Functions
by: BJ Jackson
This 10-day lesson will supplement chapter 5 of the Holt, Rhinehart,
and Winston, Algebra I book. It is desinged as an introductory
Algebra I course for high school freshmen. Click on the Day that
you are interested in to link to a new page that will cover the
day's lesson. | 677.169 | 1 |
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Sleep Deprived: This actually very helpful as I️ am a 7th grader and my parents are making me takes some sat in December and I don't even know what I️t has but I hope it's algebra also if you know what's on the test could you comment please I don't want to over study if I don't have to
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Magic Dragon: Can someone please help me? I'm great at multiplication because it's predictable, but algebra is HAARD. I'm in 10the grade, and the answers to an algebra problem is unpredictable it seems so I have a hard time understanding algebra. Sadly this is something I'm gonna NEED to know so that I can get to where I wanna be in my current situation. Can someone help me, link me? Anything?
K. Zackrisson: listen to dis before ur born it will help u burn it in to your mind
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Bonney Worthington: All those years of text books and now I finally get it.
Decalos Digler: I love ur teaching I never like math so I did not learn it but from witnessing tonight, I am move
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Sporting Saint: I'm still in Primary school and this made sense thnx
Sewing for dolls: I am confused on the whys...why are the rules that way? i really have a lot of work ahead of me. but i am hooked anyway.
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moonlit Night: how much price this cream? Juri: After i found your post on arduino.cc i could manage to use my tft. I don't know why, but i can just use the tft with the lib from 2013. Thank you for still hosting the files! Plissken Armitage: Says Luger. Shows Broomhandle. :P Horus SC: The soundtrack alone turned this game into a perennial classic. Props to Michiru Yamane (and Cynthia Harrell, who sang "I am the Wind" years before singing "Snake Eater"...) kegtime: Great work! ศิริชัย ศรีสัตยเสถียร: สวยคับ เป็นแนวทางการแต่งสวนที่ดีมากเลยคับ BRC: Hello everyone, I made a video showing my collection, I hope you like it :) and I would like your opinion :) Thank you | 677.169 | 1 |
Basic Mathematics | MATH | 106
Topics include fundamental operations fractions and decimals, percents, ratios, and proportions. All students who enroll in this course are expected to complete MATH 0409 in the following consecutive semester before attempting either MATH 0312 or MATH 1332. A comprehensive Departmental Final Exam will be given in this course. | 677.169 | 1 |
Math H1B : Honors single variable calculus, Fall 2015
This course has three parts - integration, sequences and series,
and ordinary differential equations. We will first introduce two basic
techniques of integration - substitution rule and integration by
parts, and then through various examples, we will systematically
develope these into formidable tools. Next, we will move on to
studying infinite sequences, and their summations. The aim is to
introduce Taylor series, which serves as an extension of the
idea of derivatives as first order (linear) approximation to the
function. Differential equations, that is equations involving an
unknown function and its derivatives, are ubiquitous in applications
of mathematics to "real world" problems. Any mathematical model of a
process involving rates of change can usually be formulated in terms
of a differential equation. In the last part of this course, we will
study some natural differential equations that arise in examples
ranging from population models, mixing problems to springs and
electric circuits, and use the techniques developed in the first two
parts of the course to solve such equations. | 677.169 | 1 |
YRR320 - Practical Math 3
Resource students are assigned to classes on the basis of their own choices and with recommendations of esource teachers and parents. The Individualized Education Program (IEP) sets forth a plan of goals and objectives, which specifies accommodations and services necessary to meet the students' needs.
Learning Recommendations: IEP with a qualification in the area of math, math placement test and teacher recommendation
General Description: This course builds on the key principles from Practical Math 2 and prepares the students for pre-algebra. Students begin to develop a more complete understanding of the number system and higher mathematics. Topics include the study of rational numbers (fractions, decimals, and percents), geometry, measurement, and statistics (analyzing data, working with graphs). Continued practice on computational skills, including fractions, mixed numbers, and decimals in all operations, is emphasized. Concepts involving critical thinking and problem solving are taught through guided practice and application of math concepts to everyday living situations. | 677.169 | 1 |
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Description help undergraduates rapidly develop the fundamental knowledge of engineering mathematics.
The book can also be used by graduates to review and refresh their mathematical skills. Step-by-step worked examples will help the students gain more insights and build sufficient confidence in engineering mathematics and problem-solving. The main approach and style of this book is informal, theorem-free, and practical. By using an informal and theorem-free approach, all fundamental mathematics topics required for engineering are covered, and readers can gain such basic knowledge of all important topics without worrying about rigorous (often boring) proofs.
Certain rigorous proof and derivatives are presented in an informal way by direct, straightforward mathematical operations and calculations, giving students the same level of fundamental knowledge without any tedious steps. In addition, this practical approach provides over 100 worked examples so that students can see how each step of mathematical problems can be derived without any gap or jump in steps. Thus, readers can build their understanding and mathematical confidence gradually and in a step-by-step manner.
Key Features
Covers fundamental engineering topics that are presented at the right level, without worry of rigorous proofs
Includes step-by-step worked examples (of which 100+ feature in the work)
Provides an emphasis on numerical methods, such as root-finding algorithms, numerical integration, and numerical methods of differential equations
Balances theory and practice to aid in practical problem-solving in various contexts and applications
Readership
Undergraduates and graduates in all engineering disciplines (mechanical engineering, electrical engineering, civil engineering, geotechnical, water and transport engineering), but also related applied sciences, computer science and management sciences researchers who require an understanding of key mathematical modeling techniques but are not themselves mathematicians
Table of Contents
About the Author
Preface
Acknowledgment
Part I: Fundamentals
Chapter 1: Equations and Functions
Abstract
1.1. Numbers and Real Numbers
1.2. Equations
1.3. Functions
1.4. Quadratic Equations
1.5. Simultaneous Equations
Exercises
Chapter 2: Polynomials and Roots
2.1. Index Notation
2.2. Floating Point Numbers
2.3. Polynomials
2.4. Roots
Exercises
Chapter 3: Binomial Theorem and Expansions
Abstract
3.1. Binomial Expansions
3.2. Factorials
3.3. Binomial Theorem and Pascal's Triangle
Exercises
Chapter 4: Sequences
Abstract
4.1. Simple Sequences
4.2. Fibonacci Sequence
4.3. Sum of a Series
4.4. Infinite Series
Exercises
Chapter 5: Exponentials and Logarithms
Abstract
5.1. Exponential Function
5.2. Logarithm
5.3. Change of Base for Logarithm
Exercises
Chapter 6: Trigonometry
Abstract
6.1. Angle
6.2. Trigonometrical Functions
6.3. Sine Rule
6.4. Cosine Rule
Exercises
Part II: Complex Numbers
Chapter 7: Complex Numbers
Abstract
7.1. Why Do Need Complex Numbers?
7.2. Complex Numbers
7.3. Complex Algebra
7.4. Euler's Formula
7.5. Hyperbolic Functions
Exercises
Part III: Vectors and Matrices
Chapter 8: Vectors and Vector Algebra
Abstract
8.1. Vectors
8.2. Vector Algebra
8.3. Vector Products
8.4. Triple Product of Vectors
Exercises
Chapter 9: Matrices
Abstract
9.1. Matrices
9.2. Matrix Addition and Multiplication
9.3. Transformation and Inverse
9.4. System of Linear Equations
9.5. Eigenvalues and Eigenvectors
Exercises
Part IV: Calculus
Chapter 10: Differentiation
10.1. Gradient and Derivative
10.2. Differentiation Rules
10.3. Series Expansions and Taylor Series
Exercises
Chapter 11: Integration
Abstract
11.1. Integration
11.2. Integration by Parts
11.3. Integration by Substitution
Exercises
Chapter 12: Ordinary Differential Equations
Abstract
12.1. Differential Equations
12.2. First-Order Equations
12.3. Second-Order Equations
12.4. Higher-Order ODEs
12.5. System of Linear ODEs
Exercises
Chapter 13: Partial Differentiation
Abstract
13.1. Partial Differentiation
13.2. Differentiation of Vectors
13.3. Polar Coordinates
13.4. Three Basic Operators
Exercises
Chapter 14: Multiple Integrals and Special Integrals
Abstract
14.1. Line Integral
14.2. Multiple Integrals
14.3. Jacobian
14.4. Special Integrals
Exercises
Chapter 15: Complex Integrals
Abstract
15.1. Analytic Functions
15.2. Complex Integrals
Exercises
Part V: Fourier and Laplace Transforms
Chapter 16: Fourier Series and Transform
Abstract
16.1. Fourier Series
16.2. Fourier Transforms
16.3. Solving Differential Equations Using Fourier Transforms
16.4. Discrete and Fast Fourier Transforms
Exercises
Chapter 17: Laplace Transforms
Abstract
17.1. Laplace Transform
17.2. Transfer Function
17.3. Solving ODE via Laplace Transform
17.4. Z-Transform
17.5. Relationships between Fourier, Laplace and Z-transforms
Exercises
Part VI: Statistics and Curve Fitting
Chapter 18: Probability and Statistics
Abstract
18.1. Random Variables
18.2. Mean and Variance
18.3. Binomial and Poisson Distributions
18.4. Gaussian Distribution
18.5. Other Distributions
18.6. The Central Limit Theorem
18.7. Weibull Distribution
Exercises
Chapter 19: Regression and Curve Fitting
Abstract
19.1. Sample Mean and Variance
19.2. Method of Least Squares
19.3. Correlation Coefficient
19.4. Linearization
19.5. Generalized Linear Regression
19.6. Hypothesis Testing
Exercises
Part VII: Numerical Methods
Chapter 20: Numerical Methods
Abstract
20.1. Finding Roots
20.2. Bisection Method
20.3. Newton-Raphson Method
20.4. Numerical Integration
20.5. Numerical Solutions of ODEs
Exercises
Chapter 21: Computational Linear Algebra
Abstract
21.1. System of Linear Equations
21.2. Gauss Elimination
21.3. LU Factorization
21.4. Iteration Methods
21.5. Newton-Raphson Method
21.6. Conjugate Gradient Method
Exercises
Part VIII: Optimization
Chapter 22: Linear Programming
Abstract
22.1. Linear Programming
22.2. Simplex Method
22.3. A Worked Example
Exercises
Chapter 23: Optimization
Abstract
23.1. Optimization
23.2. Optimality Criteria
23.3. Unconstrained Optimization
23.4. Gradient-Based Methods
23.5. Nonlinear Optimization
23.6. Karush-Kuhn-Tucker Conditions
23.7. Sequential Quadratic Programming
Exercises
Part IX: Advanced Topics
Chapter 24: Partial Differential Equations
Abstract
24.1. Introduction
24.2. First-Order PDEs
24.3. Classification of Second-Order PDEs
24.4. Classic Mathematical Models: Some Examples
24.5. Solution Techniques
Exercises
Chapter 25: Tensors
Abstract
25.1. Summation Notations
25.2. Tensors
25.3. Hooke's Law and Elasticity
Exercises
Chapter 26: Calculus of Variations
Abstract
26.1. Euler-Lagrange Equation
26.2. Variations with Constraints
26.3. Variations for Multiple Variables
Exercises
Chapter 27: Integral Equations
Abstract
27.1. Integral Equations
27.2. Solution of Integral Equations
Exercises
Chapter 28: Mathematical Modeling
Abstract
28.1. Mathematical Modeling
28.2. Model Formulation
28.3. Different Levels of Approximations
28.4. Parameter Estimation
28.5. Types of Mathematical Models
28.6. Brownian Motion and Diffusion: A Worked Example
Exercises
Appendix A: Mathematical Formulas
A.1. Differentiation and Integration
A.2. Complex Numbers
A.3. Vectors and Matrices
A.4. Fourier Series and Transform
A.5. Asymptotics
A.6. Special Integrals
Appendix B: Mathematical Software Packages
B.1. Matlab
B.2. Software Packages Similar to Matlab
B.3. Symbolic Computation Packages
B.4. R and Python
Appendix C: Answers to Exercises
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapteribliography
Index
Details
About the Author
Xin-She Yang
Xin-She Yang obtained his DPhil in Applied Mathematics from the University of Oxford. He then worked at Cambridge University and National Physical Laboratory (UK) as a Senior Research Scientist. He is currently a Reader at Middlesex University London, Adjunct Professor at Reykjavik University (Iceland) and Guest Professor at Xi'an Polytechnic University (China). He is an elected Bye-Fellow at Downing College, Cambridge University. He is also the IEEE CIS Chair for the Task Force on Business Intelligence and Knowledge Management, and the Editor-in-Chief of International Journal of Mathematical Modelling and Numerical Optimisation (IJMMNO).
Affiliations and Expertise
School of Science and Technology, Middlesex University, UK
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Abstract Algebra A Geometric Approach
ISBN-10: 0133198316
ISBN-13: 9780133198317
Edition: 1st a 1 or 2 term course in Abstract Algebra at the Junior level. This book explores the essential theories and techniques of modern algebra, including its problem-solving skills, basic proof techniques, many unusual applications, and the interplay between algebra and geometry. It takes a concrete, example-oriented approach to the subject matter | 677.169 | 1 |
Search
MATH 8.9
Math – Middle School
Grade(s):
8
Theme:
Expressions, equations, and relationships
Description:
Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to develop foundational concepts of simultaneous linear equations. The student is expected to identify and verify the values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed | 677.169 | 1 |
Complementary Mathematics
Objectives
The course aims to introduce plane Euclidean geometry in the Euclid perspectives and in the modern Hilbert's view, to develop competences for problem solving and for an epistemological and didactical analysis of Euclidean plane geometry.
Teaching methods
Frontal lessons, discussions, problem solving.
Examination
Written and oral examination.
Prerequisites
Main concepts studied in Mathematics Bachelor Degree Courses.
Syllabus
Euclid plane geometry. Books I ? VI of Euclid's Elements. Common notions, postulates, definitions, propositions. The fifth postulate and the theory of parallel lines. Introduction to Non-Euclidean Geometries. Classical problems of compass and ruler constructions.
Geometry as the study of invariants: the Erlangen Program.
Geometry as formal system: Hilbert's axioms. The problems of continuity and completeness of line. Issues of consistency, independence and categoricity.
The study will be combined with an analysis from epistemological, cognitive and didactical points of view. | 677.169 | 1 |
Mathematics: A Discrete Introduction
Browse related Subjects ...
Read More directly applicable to computer science and engineering, but it is presented from a mathematician's perspective. While algorithms and analysis appear throughout, the emphasis is on mathematics. Students will learn that discrete mathematics is very useful, especially those whose interests lie in computer science and engineering, as well as those who plan to study probability, statistics, operations research, and other areas of applied mathematics.
Read Less
Very Good. 0840049420 Book is in VERY GOOD condition-may show minor signs of use, may NOT contain supplemental materials-Free tracking information on all orders! (Hawaii, Alaska, Puerto Rico and APO's, allow additional time for delivery. )
Customer Reviews
quick ship; great condition
Shipping was quick and very neat. The condition of book was excellent. This edition is quite nice, the author's done a nice job building into the more complex sections later in the book. I used it for a Discrete class, however I can see thumbing through it after | 677.169 | 1 |
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Summary
The cornerstone of ELEMENTARY LINEAR ALGEBRA is the authors' clear, careful, and concise presentation of material--written so that readers can fully understand how mathematics works. This program balances theory with examples, applications, and geometric intuition for a complete, step-by-step learning system. Featuring a new design that highlights the relevance of the mathematics and improves readability, the Seventh Edition also incorporates new conceptual Capstone exercises that reinforce multiple concepts in each section. Data and applications reflect current statistics and examples to engage users and demonstrate the link between theory and practice. | 677.169 | 1 |
Courses In Charge
mathematical programming
Mathematical programming problems are the mathematical problems which ask for minimal or maximal values of an objective function subject to some constraints. This cource provides a theory to solve these problems.
Main Research
To find algorithms for solving optimization problems in networks
A network is defined as a graph with functions where the graph is a pair of a set of vertices and a set of arcs. Optmization problems are the mathematical problems which ask for minimal or maximal values of an objective function subject to some constraints.
Introduction of Laboratory
Our lab has been doing activity to find algorithms for solving optimization problems in networks. A network is defined as a graph with functions where the graph consists of a set of vertices and a set of arcs. Optmization problems are the mathematical problems which ask for minimal or maximal values of an objective function subject to some constraints. Current themes are to find an optimal generalized network flow efficiently and/or to develop new methods for solving loglinear optimization problems. | 677.169 | 1 |
PhySyCalc - Scientific and Engineering Calculator
By PhySy Ltd
Description
PhySyCalc is the next generation scientific and engineering calculator with units—it makes the others seem like slide rules.
How is PhySyCalc different from other calculators? It allows you to include unit symbols in your calculations, obtaining the answer in the desired unit without those extra unit conversion steps. On top of this great simplification, PhySyCalc knows every fundamental physical constant. It even knows physical properties for elements and isotopes in the periodic table and can calculate formula weights. This allows you to get numerical answers in the desired unit in a fraction of the time you'd spend on a conventional calculator. PhySyCalc is quick to learn and easy to use. Can't remember a unit symbol? PhySyCalc helps you find and append commonly used units onto a number. PhySyCalc uses a natural infix notation for calculations. This means you can enter and read through the entire expression in full before calculating the result, helping you quickly identify and fix any input errors.
Not only does PhySyCalc save you time, it also saves you from mistakes. A conventional calculator will also give you a numerical answer, even if you accidentally multiplied instead of dividing. With PhySyCalc you'll know right away that you entered the calculation wrong because the result will have the wrong units.
As a teaching tool PhySyCalc offers students better insights in the concepts of physical quantity, dimensionality, and units. A student using PhySyCalc soon understands the connections between the base SI quantities and all the derived quantities.
PhySyCalc's sole ambition is to be the world's best scientific "pocket" calculator.
Add 2 centimeters and 2 inches and give the answer in inches Input: "2 cm + 2 in .. in" Output: "2.78740157480315 in" Note: Swipe two fingers across the display to reduce the number of significant figures.
Calculate the area in acres of a circle that has a radius of 1 mile. Recall Area = π•r^2 Input: "π•(1 mi)^2..ac" Output: "2010.619298297468 ac"
Calculate the height in yards of a kite on 200 feet of string and flying at an angle of 40° with the ground. Recall right triangle relationship y = r•sin(angle). Input: "200 ft • sin(40°) .. yd" Output: "42.85250731243595 yd"
PhySyCalc features include: • Quantity units supported in all calculations • Natural infix notation for entering calculations in full before evaluating • Intuitive interface for appending units unto numbers so you can enter your calculation quickly • Minimal buttons in portrait display for ease and speed of use • Complex number support • Press and hold delete button to clear entire display • Left swipe display for unlimited undo makes it easy to find mistakes and correct them • Right swipe display for redo • Swipe up for keyboard • Swipe down and navigate to the right unit for quantity (for iPhone) • Swipe two fingers down on screen for full list of quantities and units • Double-touch slide on display to reduce the number of significant figures • Rotate to landscape for more units and scientific functions • Press and hold any math function to apply to entire display • Press and hold = button to retrieve value stored in memory • Saves and recalls results to memory • Universal app, supporting both iPhone and iPad displays
What's New in Version 2.06
Screenshots
Customer Reviews
Analytical Chemist
by
C Mowry
This is a great app, I use it all the time at work and home. Love that I can select out of many unit choices at the end of calculation! I am a professional chemist using it for volume, mass, moles, and electrical terms. Temp and length too. Super handy! | 677.169 | 1 |
Product Description
▼▲
Accelerated Christian Education (ACE) curriculum has Scripture as its foundation, fully integrating biblical principles, wisdom, and character-building concepts into education. Students move at their own speed through the self-instructional "PACE" workbooks. Following the mastery approach, PACEs are formatted for students to complete the exercises found throughout the workbook, take a practice "self test," and conclude with a "final test" (torn out from the center) to measure understanding.
Students will develop foundational math skills needed for higher education and practical life skills with ACE's Math curriculum. This set includes Math PACEs 1049-1060, which covers:
Our son has always done well at Math, but the most recent homeschool curriculum we were using was frustrating him. I was afraid the ACE program would be too easy, but I was wrong. He has become excited to do his Math work again and is doing very, well!! | 677.169 | 1 |
Course Descriptions
MTH 150 - Survey of Mathematics
3 Credits
A study of various topics including an introduction to estimation, algebra, geometry, consumer mathematics, probability and statistics, with an emphasis on critical thinking and interpreting results. Other topics may be covered at the discretion of the instructor. Three class hours. MTH 150 is a common selection by Liberal Arts students with fewer than three years of high school mathematics. MTH 150 is not a prerequisite course for MTH 160 or higher. Although this course can satisfy your mathematics requirement for some MCC programs and transfer to some baccalaureate institutions, if you are planning to transfer please speak with an academic advisor or Career and Transfer to ensure that this course meets your goals. (SUNY-M)
Prerequisite: TRS 094 with a grade of C or better, or MCC Level 4 Mathematics placement.
Course Learning Outcomes
1. Use estimation to approximate an answer. 2. Use geometric formulas to solve applied problems using unit conversions where necessary. 3. Compute interest for common loans. 4. Compute interest rates for common loans. 5. Compute payments for common loans. 6. Compute probabilities, odds, or expected value. 7. Interpret results from probability based calculations. 8. Interpret standard statistical graphs. 9. Calculate simple descriptive statistics for a given set of data. 10. Use common properties of a normal distribution to draw basic conclusions about the underlying data | 677.169 | 1 |
Explorations in College Algebra, 5th Edition is designed to make algebra interesting and relevant to the student. The text adopts a problem-solving approach that motivates readers to grasp abstract ideas by solving real-world problems. The problems lie on a continuum from basic algebraic drills to open-ended, non-routine questions. The focus is shifted from learning a set of discrete mathematical rules to exploring how algebra is used in the social, physical, and life sciences. The goal of Explorations in College Algebra, 5th Edition is to prepare students for future advanced mathematics or other quantitatively based courses, while encouraging them to appreciate and use the power of algebra in answering questions about the world around us.
Table of Contents
An Introduction to Data and Functions
Describing Single-Variable Data
Visualizing Single-Variable Data
Numerical Descriptors: What is "Average" Anyway?
An Introduction to Algebra Aerobics
An Introduction to Explore and Extend
Describing Relationships between Two Variables
Visualizing Two-Variable Data
Constructing a "60-Second Summary"
Using Equations to Describe Change
An Introduction to Functions
What is a Function?
Representing Functions: Words, Tables, Graphs and Equations
Input and Output: Independent and Dependent Variables
When is a Relationship Not a Function?
The Language of Functions
Function Notation
Domain and Range
Visualizing Functions
Is There a Maximum or Minimum Value?
When is the Output of the Function Positive, Negative or Zero?
Is the Function Increasing or Decreasing?
Is the Graph Concave Up or Concave Down?
Getting the Big Idea
Chapter Summary
Check Your Understanding
Chapter 1 Review: Putting it all Together
Exploration 1.1 Collecting, Representing, and Analyzing Data
Rates of Change and Linear Function
Average Rates of Change
Describing Change in the U.S. Population over Time
Defining the Average Rate of Change
Limitations of the Average Rate of Change
Change in the Average Rate of Change
The Average Rate of Change is a Slope
Calculating Slopes
Putting a Slant on Data
Slanting the Slope: Choosing Different End Points
Slanting the Data with Words and Graphs
Linear Functions: When Rates of Change are Constant
What if the U.S. Population Had Grown at a Constant Rate?
Real Examples of a Constant Rate of Change
The General Equation for a Linear Function
Visualizing Linear Functions
The Effect of b
The Effect of m
Finding Graphs and Equations of Linear Functions
Finding the Graph
Finding the Equation
Special Cases
Direct Proportionality
Horizontal and Vertical Lines
Parallel and Perpendicular Lines
Breaking the Line: Piecewise Linear Functions
Piecewise Linear Functions
The absolute value function
Step functions
Constructing Linear Models for Data
Fitting a Line to Data: The Kalama Study
Reinitializing the Independent Variable
Interpolation and Extrapolation: Making Predictions
Looking for Links between Education and Earnings: Using Regression Lines | 677.169 | 1 |
Groebner Bases is a method that offers algorithmic ideas to numerous difficulties in Commutative Algebra and Algebraic Geometry. during this introductory instructional the fundamental algorithms in addition to their generalization for computing Groebner foundation of a suite of multivariate polynomials are provided.
D. 001), that comes closest to producing the 30-year growth experienced. e. Use your answer to 14d to estimate the population in 1985. How does this compare with the average of the populations of 1970 and 2000? Why is that? htm 15. Taoufik looks at the second problem of his wet homework that had fallen in a puddle. a. What is the common ratio? How did you find it? b. What are the missing terms? c. What is the answer he needs to find? Review 16. 20% from 1990 to 2000. 4 million. What population was reported in 1990?
Step 2 Write a recursive formula that generates the sequence in your table. How many days will pass before there is less than 1 mL of medicine in the blood? Is the medicine ever completely removed from the blood? Why or why not? Sketch a graph and describe what happens in the long run. Step 3 Step 4 Step 5 A single dose of medicine is often not enough. Doctors prescribe regular doses to produce and maintain a high enough level of medicine in the body. Next you will modify your simulation to look at what happens when a patient takes medicine daily over a period of time. | 677.169 | 1 |
This book develops the theory of ordinary differential equations (ODEs), starting from an introductory level (with no prior experience in ODEs assumed) through to a graduate-level treatment of the qualitative theory,
In addition to creating a beautiful matte painting shot, this course takes you through all the best practice basics and unspoken rules of matte painting and helps you develop good professional working habits. | 677.169 | 1 |
College Geometry: Using the Geometer's Sketchpad, 1st Edition
From two authors who embrace technology in the classroom and value the role of collaborative learning comes College Geometry Using The Geometer's Sketchpad, a book that is ideal for geometry courses for both mathematics and math education majors. The book's truly discovery-based approach guides students to learn geometry through explorations of topics ranging from triangles and circles to transformational, taxicab, and hyperbolic geometries. In the process, students hone their understanding of geometry and their ability to write rigorous mathematical proofs.
Former Chapter 1 has been re-written as two chapters:
Chapter 1: Using the Geometer's Sketchpad and Chapter 2:
Constructing à Proving. The authors
split these chapters into two in order to provide better
explanation and deeper coverage of each topic.
The introduction and development of proof skills in
Chapters 3 and 4 has been revised in order to make the concept of
proof more accessible to students.
New Chapter 7: Finite Geometries has been added based on
feedback from instructors who cover this topic in their geometry
courses.
Chapter 11: Hyperbolic Geometry has been expanded with
additional problems and more in-depth coverage by the authors.
Coverage of the Real Projective Plane in chapter 11 has
been re-written to be more clear to | 677.169 | 1 |
Jumpstart to College Algebra
Video Transcription
[MUSIC PLAYING] Let's go over our objectives for today. We'll start by introducing our course goals for Jumpstart to College Algebra. We'll then go over what you can expect from me, the instructor. And finally, we'll look at what you can expect upon completion of this course.
Let's start by looking at our course goals. This course will provide fundamental knowledge and skills necessary to be successful in a college algebra course. The content covered will be applied to everyday as well as professional and academic situations. The knowledge and skills you develop in this course will be helpful in a variety of fields, including science, economics, and business. Algebraic thinking is also used in everyday situations, such as when comparing different vehicles to purchase or whether or not you should refinance your home loan.
Now, let's talk about what you can expect from me, the instructor. I've been a teacher and a tutor of mathematics for 10 years. I've also created other online mathematics tutorial videos.
I will provide quality math instruction using short but direct tutorials. Some of the topics in these tutorials include finding a percentage to analyze survey data, solving linear inequalities to determine how much income you need to cover your bills, writing an equation to analyze a cellphone plan, and finding the intersection point of two lines to determine the optimal quantity and price for a certain product produced. The goal of these examples in each tutorial is that they are meaningful and relatable and will show you how to apply the content to different situations.
Finally, let's go over at the expectations for completing the course. Upon completion of this course, you should be able to apply problem-solving techniques to everyday and professional problems. You should also be able to perform algebraic processes and tasks. And finally, you should be able to confidently enter a credit-bearing college algebra course.
So I hope that this brief tutorial gave you a good idea of what to expect from Jumpstart College Algebra. When you watch each tutorial, make sure to take careful notes so that you can refer to them later. So again, welcome to the course, and thanks for watching. | 677.169 | 1 |
ISBN 13: 9781111988289
Precalculus: A Make it Real Approach
This new text provides a contemporary approach to college algebra, ideal for the many skeptical or apprehensive students who ask, "When am I ever going to use this?" The key phrase is "Make It Real" since the goal is to make the material relevant and understandable to today's college students. But many books make this claim - so how is "Precalculus: A Make It Real Approach, International Edition" different? In other texts, which simply wrap real-world situations around problems, the context isn't needed to do the mathematics. Written by skilled and passionate teachers, this text uses real-world data sets and situations to draw out mathematical concepts. Students are immersed in familiar contexts - from golf course ratings to Egyptian pyramids - from which concepts emerge naturally, and then guided in using their understanding of those ideas to make sense of the mathematics. The real-world contexts are not only helpful for understanding procedures - they're necessary. The concept of a function, the use of modeling, and the thorough integration of real-world applications are integral to the text. If there's one new precalculus text crafted to stand up to a "reality check" comparison with your current book, this is it.
Book Description Book Condition: Brand New. Brand New, 1 edition, , color Printing, soft 4417N | 677.169 | 1 |
This page will detail several
different types of homework that will be assigned during the semester.
Assigned
Reading and Reading Log. The course will follow the
textbook
closely, and you are expected to read each section before it is
discussed in class. To see what will be discussed at each meeting,
consult the class
schedule.
An important goal for the course will be the development of active
reading
strategies to decode written mathematics. When you set out to
read
a section of the text, keep paper and pen or pencil at hand and plan to
use
them frequently. As you read, you should be formulating and answering
questions,
making up examples, drawing figures, etc. All of this should be
recorded
on paper. This work should be included in the portfolio in the Reading
Log section, at least at
the
start of the course. This is work in progress and is not expected to be
in
polished form. However, when you sit down to read (part of a) section
of
the text, you should begin a new page of the reading log, with a
heading
indicating the section of the text and the page number.
The reading log is not an outline of what you have read. I am
not
asking
that you produce a list with the section headings, definitions, and
theorems.
Instead, you should record ideas that expand on what is in the text.
For
example, one of the topics we will read about is a kind of number
system
called a ring. After you read that definition, you might ask,
in
the log, what familiar algebraic systems satisfy the definition of a
ring.
Are the integers a ring? Are the real numbers? Are the 2 by 2 matrices?
I
would like to see such questions in your reading log, as well as any
answers
you find.
Your goal is to gain a deep understanding of each section of the text
before it is discussed in class. Often, class time will be
devoted to discussing homework problems and student reactions (as
documented in your reading logs) to the text. Do not expect each
class to include a lecture that repeats what is already presented in
the text.
Regular homework. This
is traditional homework from problems in the text, to be written up and
handed in. Most of these will be statements you are
asked to prove, though there may also be problems that ask you to work
out properties of a specific example, or to construct an example of
your own. But even for such exercises, you are expected to
"prove" your conclusions. As a prerequisite for this course,
students are assumed to know how to write proofs. For students
who would like a brief review, see this webpage.
The
assignments are posted in this Assignment
Sheet.
The
required format for regular homework is illustrated here.
Please staple the pages of each problem set together.
Polished Work. One
or more problems in each problem set will be marked with an asterisk
(*).
These problems are to be polished into a final form that meets the same
standards
for form and neatness that you would expect for a term paper. Generally
these
will be proofs, and will require written out explanations of your
reasoning.
The recommended approach to these problems is to treat the
solutions
handed
in with homework as rough drafts. Based on further thought, class
discussion,
or comments on the homework paper, you should then prepare a second,
polished
draft. Periodically, I will collect all the second drafts of starred
problems,
and give you a second set of comments, and a grade. Format
and writing style requirements for polished work are detailed here.
Optional & Masters
Problems. Some problems on the assignment sheet are
designated with two
asterisks, like so: 37**. Undergraduates may consider these to be
optional; masters students should consider them to be required.
Typically these problems will either be more difficult than the normal
problems, or will emphasize extensions of the material covered in a
section.
Exam Solutions. After
each exam, students will work in groups to correct the errors of all
group members. The objective will be to compile, as a group, a
completely correct set of solutions to the exam questions. Each
group will submit one set of solutions for grading, and each member of
the group will receive the same grade for this assignment.
Class worksheets. Class
time may sometimes involve working alone or in groups to complete
worksheets. If you do not finish a worksheet before the end of
class, you should plan on completing all or most of it before the next
class meeting. These worksheets will not be collected or graded,
but should be kept in the appropriate section of your portfolio. | 677.169 | 1 |
Scientific Computing for Scientists and Engineers is designed to teach undergraduate students relevant numerical methods and required fundamentals in scientific computing. Most problems in science and engineering require the solution of mathematic............
Gain confidence with this overview of the basics in numerical reasoning tests, followed by a step-by-step guide to the skills you need to master before taking such a test.
Author: Smith, Heidi
Publisher: Kogan Page
Illustration: n
Language: ENG
T......
If you're among the many hobbyists and designers who came to electronics through Arduino and Raspberry Pi, this cookbook will help you learn and apply the basics of electrical engineering without the need for an EE degree. Through a series o......
Kubota V1205-b V1205-t-b V1305-bSubaru Legacy & OutbackRange Rover L322Terex Ta40 Ocdb Articulated DumpTerex Atlas 1704 1804Terex Ta35 & Ta40 Articulated Dumptruck | 677.169 | 1 |
2017-18 Senior Math Syllabus
Course Description/Overview/Welcome Statement
Mathematical Decision Making for Life (Senior Math) is a four-quarter course for seniors. The course includes mathematical decision making in finance, modeling, probability and statistics, and making choices. Students will make sense of authentic problems and persevere in solving them. They will reason abstractly and quantitatively while communicating mathematics to others. Students will use appropriate tools, including technology, to model mathematics. Students will use structure and regularity of reasoning to describe mathematical situations and solve problems.
Learning Expectations
Grade Scale:
Grade
Percentage
Grade
Percentage
Grade
Percentage
A
93% – 100%
A-
90% – 92%
B+
87% – 89%
B
83% – 86%
B-
80% – 82%
C+
77% – 79%
C
73 % – 76%
C-
70% – 72%
D+
67% – 69%
D
63% – 66%
D-
60% – 62%
F
0% – 59%
Grading Categories:
Category
Items
% of Grade
Assessments
Tests & Quizzes
40%
Assignments
Class work / Homework / Study Guides / Notes
30%
Citizenship
Attendance, Behavior, Preparation and Participation, Workbook Checks
30%
Assessment of Progress
We will have a daily quiz on the work we did the previous day. At the end of each unit there will be a test. There will be a Cumulative Review for the information taught during the Semester at the end of each Semester. All of these scores will be added to Power School. Grades will be entered on Thursdays.
Course Materials
Required Items:
Math Workbook which is given to the student. If the workbook is lost or damaged (i.e. with inappropriate drawings or writings) student will need to pay $5 for a replacement. Senior Math will have assignments on the computer, classwork found in the workbook and a few projects also found in the workbook.
Working Pencil and Pen
Calculator (optional)
Classroom Procedures
Class Rules:
Be Respectful (Respect everyone in the classroom and the facilities in the room. No food, candy, gum, drinks, hats, electronic devices, cell phones, notes to friends, are allowed in the classroom. Also hoodies can be worn but should not have the hoods up on the head. Any of these items that are visible will be confiscated. No additional warnings will be given. Water in a closed container is allowed)
Be Prepared (Come on time, be in your seat when the bell starts to ring, prepared with paper, pencil and math folder ready).
Be a Self Advocate. Ask for help when needed, be self directed when possible, try your best at all times.
Be a Team Player
Dream Big!
Follow school/district policies.
Tardies:
If students are not in their seat prepared to work with required items when the bell rings then they are tardy for class. It is the student's responsibility to sign the Tardy Log, to ensure that their attendance gets updated appropriately. Students will be marked with a 3 if they are over 5 minutes late for class without an approved written excuse. Students who bring food or drinks into my class will be asked to throw it away or put it in their locker; they will be marked tardy if they leave to put it away.
Hall Passes:
Students will be allowed 3 emergency hall passes during class time. Any activity that has the student going into the hall will require that the student carry the yellow hall pass sign. Only one student at a time may leave. Students will need to wait for the hall pass to return before another may leave. Excessive time out of the room (more than 5 minutes) will count as using more than one hall pass. Students who were late to class will not be allowed additional time out of class on the same day.
Non-Participation and Cheating:
Students are given weekly participation points, point reductions can occur when they break rules or do not participate with the class. Non-Participation (N) can be given to students, who are unprepared, not working, causing disruptions, or otherwise not participating with the class. No participation points will be awarded for that day. Students may also be sent to the office if they are not willing to work or participate.
If a student is found to be cheating (giving or receiving) by deliberately using unauthorized materials, information, technology, study-aids or giving/receiving improper assistance, the district procedure (4485 P1) will be followed.
Class Work:
There will be class work every day. Students who finish in class do not have homework. So it is up the student if there is homework. Missing work can be found at the website: canvas.instructure.com . It is the student's responsibility to make-up any notes, take any quiz/test, or get help on any assignments for the days they miss. Students, who regularly attend class, responsibly do their work in class, or after class as needed; who feel they need more time to finish assignments, may talk to me about ways that they may makeup the work and get a grade for the class. Note that attendance is very important as it may become difficult to pass the class if a student misses or refuses to work for 5 or more days in any given term. Students may spend time working with me after school to make-up activities that were missed during regular class time.
Late and Make-up Work: (all work must be completed within the term assigned)
I want the work done, so I will accept late work. Work will be graded and a score entered into Power School on Thursdays. The students will need to make sure their work has been submitted on the computer or in their Workbook (turned into the correct location). Work will be accepted after that time but may receive a 5% deduction for each day late. Re-do's on assignments and assessments will also be accepted but only for work that was honestly attempted and ALL missed problems on the assignment or assessment must be re-done to be accepted. Tests and Quizzes can only be made up in my room and arrangements can be made to do this before school, after school, or during provided remediation times. Attendance points missed for doctor's or pre-approved absents from my class will need to be made up in my classroom within a week of returning.
Remediation:
I will be available for help sessions everyday before school and after school, except on Fridays when we have Teacher Meetings. If more time is needed, contact me and I will make accommodations accordingly | 677.169 | 1 |
Description
The 100+ Series, Algebra II, offers in-depth practice and review for challenging middle school math topics such as factoring and polynomials; quadratic equations; and trigonometric functionsSimilar
Integrate TI Graphing Calculator technology into your algebra instruction with this award-winning resource book. Perfect for grades 6-12, this resource includes lessons, problem-solving practice, and step-by-step instructions for using graphing calculator technology. 238pp plus Teacher Resource CD with PDF files of the tables, templates, activity sheets, and student guides for TI-83/84 Plus Family and TI-73 Explorer™. This resource is correlated to the Common Core State Standards, is aligned to the interdisciplinary themes from the Partnership for 21st Century Skills, and supports core concepts of STEM instruction.
All About Decimals: Math for CCSS focuses on basic instruction in adding, subtracting, multiplying, and dividing decimals. The book is arranged in a systematic way with each lesson focusing on one new skill or concept that builds on those learned previously. The content is aligned with the Common Core State Standards for MathematicsEach page in the Common Core Math Workouts for grade 7 6 8The 100+ Series, Algebra, offers in-depth practice and review for challenging middle school math topics such as radicals and exponents; factoring; and solving and graphing equationsThis work is based on a series of thematic workshops on the theory of wavelets and the theory of splines. Important applications are included. The volume is divided into four parts: Spline Functions, Theory of Wavelets, Wavelets in Physics, and Splines and Wavelets in Statistics. Part one presents the broad spectrum of current research in the theory and applications of spline functions. Theory ranges from classical univariate spline approximation to an abstract framework for multivariate spline interpolation. Applications include scattered-data interpolation, differential equations and various techniques in CAGD. Part two considers two developments in subdivision schemes; one for uniform regularity and the other for irregular situations. The latter includes construction of multidimensional wavelet bases and determination of bases with a given time frequency localization. In part three, the multifractal formalism is extended to fractal functions involving oscillating singularites. There is a review of a method of quantization of classical systems based on the theory of coherent states. Wavelets are applied in the domains of atomic, molecular and condensed-matter physics. In part four, ways in which wavelets can be used to solve important function estimation problems in statistics are shown. Different wavelet estimators are proposed in the following distinct cases: functions with discontinuities, errors that are no longer Gaussian, wavelet estimation with robustness, and error distribution that is no longer stationary. Some of the contributions in this volume are current research results not previously available in monograph form. The volume features many applications and interesting new theoretical developments. Readers will find powerful methods for studying irregularities in mathematics, physics, and statistics.
This book deals with problems of approximation of continuous or bounded functions of several variables by linear superposition of functions that are from the same class and have fewer variables. The main topic is the space of linear superpositions D considered as a sub-space of the space of continous functions C(X) on a compact space X. Such properties as density of D in C(X), its closedness, proximality, etc. are studied in great detail. The approach to these and other problems based on duality and the Hahn-Banach theorem is emphasized. Also, considerable attention is given to the discussion of the Diliberto-Straus algorithm for finding the best approximation of a given function by linear superpositions | 677.169 | 1 |
Grade 11 - Mathematics
Algebra II (Credit: 1.00)
More exciting than a traditional textbook, this interactive course allows students to work at an individualized pace as they delve into the complexities of higher-level math. Students will learn to combine terms in algebraic expressions and explore compound sentences. Other topics include polynomial functions, data analysis, logarithms, function graphing, joint and combined variation, algorithms, conic sections, and probability. The course units include text-based lessons, on and off-computer exercises, special projects, learning games, quizzes, and tests that appeal to today's digital generation and help teachers to evaluate progress and mastery of the materials. Ignitia Algebra II enriches the educational experience for Christian school students and sparks a passion for learning. | 677.169 | 1 |
MATH205: ELEM ALGEBRA
Course Description
This course is a standard beginning algebra course, including algebraic expressions, linear equations and inequalities in one variable, graphing, equations and inequalities in two variables, integer exponents, use of a scientific calculator, polynomials, rational expressions and equations, radicals and rational exponents, and quadratic equations. Mathematics 205, 205A and 205B, and 206 have similar course content. This course may not be taken by students who have completed Mathematics 205B or 206 with a grade of "C" or better. This course may be taken for Mathematics 205B credit (2.5 units) by those students who have successfully completed Mathematics 205A with a grade of "C" or better. PREREQUISITE: MATH 402 with a grade of 'Pass' or with a 'C' or better, or assessment test recommendation.
Learning Outcomes
Simplify and evaluate expressions. Solve linear equations and inequalities in one variable and their applications.
Evaluate and solve formulas.
Graph linear equations and inequalities in two variables.
Solve systems of equations and inequalities in two variables and their applications.
Apply the laws of exponents to algebraic expressions. Use scientific notation and a scientific calculator.
Define a polynomial and perform the operations of addition, subtraction, multiplication, and division of polynomials.
Factor polynomials and solve polynomial equations in one variable.
Simplify and add, subtract, multiply, and divide with rational expressions. | 677.169 | 1 |
In Quest Of Calculus Homework Solutions 10 Helpful Hints
It is very natural to get problem while doing homework. The students get confused what to do. So to help them we provide 10 helpful hints. Here they are:
Ask to the class teacher
The students are asked to take help of their class teacher. They will surely make you understand if you ask. It is not necessary that you understand everything at one time.
Include friends in doing homework
When you do your coursework then it would be good if you do it with your school friend. Your level of stress will come down by taking help from friend.
Study regularly
Some students sit to read only to complete coursework. Make a time schedule to read and follow it regularly. This is a very good habit and if you follow then it will be good for you. You will learn more.
Make plan for study
Every day make a plane for the next day what you will read at the end of you study. It helps you reduce panic. If you do not have plan then your mind get puzzled. So make plan to avoid puzzled.
Do not watch TV for long
This is very important to you to keep distance from watching TV. Many students watch TV for long time and therefore they cannot do their coursework.
Be sure that all work is done
Students should always make their work up to date. If coursework is done then try to do study other lesson and keep yourself advanced.
Motivate yourself to do coursework
Do not break down if you find coursework difficult. Try to read it again and again to motivate and ask your parent if cannot solve coursework.
Follow the best student in your class
Student should always follow to the topper. Follow them how they read, how they ask with teacher, how they solve question in the class. There are many things to learn from them. So do not have ego to follow them.
Make friendship with best students
Those who are topper they know more than you do. So make friendship with them and share your coursework or anything relating to study.
Take help anybody around you if get difficulty
While doing coursework if you find any kind of problem then you should ask anybody around you. It may be your mother, father, sister, brother and whoever present. | 677.169 | 1 |
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The Humongous Book of Trigonometry Problems
This was a good book and I recommend it highly. Our knowledge of special triangles tells us that this angle may be either p /3 or 2p /3. Care was taken to remove any potential stumbling blocks. --And if algebra and trig are old hat to you, rest assured that you will still find here some unexpected goodies. Now we can use the identities we already know to find tan (x + y) and all the other functions we need. Another book that is quite formalistic and dry and reflects pre-computer science and yet I come back to again and again and is simply a favorite is: It also has an excellent treatment of Polya's counting theory.
Higher geometry and trigonometry; being the third part of a series on elementary and higher geometry, trigonomentary and mensuration: containing many ... mathematical science, especially in relation
Elementary Trigonometry, Plane and Spherical
However, I was not at all happy with the result. The book itself does not explain anything. It makes learning a chore and kills every ounce of motivation to learn math. My two recently purchased Calculus books (Calculus Made Easy and A First Course in Calculus) made the subject fun and motivating Fourier Analysis in Several Complex Variables (Dover Books on Mathematics). MathBoard will make learning math fun. - Number ranges are configurable from -1000 to 1000, including the ability to require certain numbers to be in each Strengthen reasoning strategies for whole number addition and multiplication by helping monsters make a target sum or product. Earn points with each correct answer… but watch out for distractions Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry! This app gives you just a sample (over 600) of the many (over 1,300) helpful formulas, figures, tips, and examples that are included in the full version of Math Ref. So, if you're a student, teacher, or need math for work and want to try before you buy, this app is for you. SAT Mathematics Trainer will greatly help you get ready for the SAT Math Exam by gradually bringing you to the required skills and speed Trigonometry (Speedy Study Guides: Academic). In Mean Girls, Cady pretends to hate math to fit in with everyone else, even though she actually enjoys it. She asks the boy she likes to tutor her, even though he's even worse at it than she pretends to be. Averted in Better Off Dead, which has the most enthusiastic class ever. In The Mirror Has Two Faces, Jeff Bridges is a math professor trying to figure out how to keep people interested in his class Student Solutions Manual for Stewart/Redlin/Watson's College Algebra 4e. It also included aids for interpolating chords for minutes of angle. Ptolemy used a different large fixed radius than Hipparchus. The advantage of a large radius is that fractions can be avoided. In contrast, our present-day trigonometric functions are based on a unit circle, that is, a circle of radius 1. Of course using a unit circle doesn't avoid fractions, but we have decimal fractions which are easy to work with An elementary treatise on the application of trigonometry to orthographic and stereographic projection, dialling, mensuration of heights and ... of the university at Cambridge, New England.
Outline of the method of conducting a trigonometrical survey, for the formation of geographical and topographical maps and plans; military reconnaissance, levelling, etc
A Treatise On Spherical Trigonometry: With Applications To Spherical Geometry And Numerous Examples, Part 1 - Primary Source Edition
Modern Geometry and Trigonometry
The Complete Idiot's Guide to Trigonometry (Idiot's Guides)
Cosine to the 4th becomes u^4, and sin x dx becomes not quite du, watch for the signum, watch for this minus sign here. And I can integrate both of these powers, so I get -u^3 / 3. And then this 4th power gives me a 5th power, when I integrate. I have to back substitute and get rid of my choice of variable, u, and replace it with yours. I forgot this minus sign when I came down here College Algebra With Trigonometry. Even the simplest calculator often has at least a square root (~) key. Scientific calculators will also have a reciprocal (lIX) key, a yX or x} key for finding powers and roots of numbers, keys for finding the trigonometric functions of SIN, COS and TAN and their inverses together with a natural logarithm (LN) key and a base 10 logarithm (LOG) key Constructive text-book of practical mathematics Volume 4. Topics are added as they become necessary for the Calculus and Vector Calculus portions of the text Introduction to Trigonometry and Geometry: With Applications and Connections (Pilot Edition). Drag the labels on the triangle into the formula and mark your answer: 3. Formula Triangles - these have the added advantage of giving you the different arrangements of the formulas you need! You may have used them before in Physics. All you do is cover what you want to calculate and what's left is the formula to calculate it TI-83 Graphing Calculator Manual for Trigonometry, 4th. Using observational techniques like heliacal rising, which occurs when a planet, star or other body first becomes visible above the eastern horizon at dawn, it was discovered that: transits of planets (e.g Eleventh year mathematics: Intermediate algebra and trigonometry. The Far Side once showed us "Hell's Library", filled with nothing but books full of story problems. Another showed a math phobic's worst nightmare: Heaven having a complicated "Train Leaves at X Miles Per Hour" story problem as an entry requisite. Several arcs of Peanuts showed Sally struggling with the "new math" and driving both Charlie Brown and Linus up the wall as they tried tutoring her Trigonometry: A Complete Introduction: Teach Yourself. Is any method more accurate or efficient than others? [E] [E] (1) An upper bound recursive equation for Pi using regular polygons circumscribed about a circle to approximate its circumference. (2) An Algebraic Polynomial of which one root is Pi itself. [E] A recursive equations for Pi by estimating the area and circumference of a circle in terms of squares and triangles. [E] (1) An expression for Pi using the concept of centripetal acceleration, (2) investigate the nature of the Pi Associates. (3) expressions for Pi by approximating the areas of definite integrals. [E] What is the effect of putting different variable values in the fractal "Mandel's" equation? [E] [P] Circles, Tangent Lines and Triangles Proofs with the Geometry Applet. [E] [E] Prove that the sum of the perimeters of the inscribed semicircles is equal to the perimeter of the outside semicircle. [E] Fractals: 1 The Humongous Book of Trigonometry Problems online. Learn the properties of the interior and exterior angles of polygons including triangles. Learn the three different types of triangles i.e. isosceles, equilateral, and scalene. Start with studying right-angled triangles. Right angled triangles are easy to study and will give you a good grasp of basic trigonometry and the three trigonometric ratios. Familiarize yourself with the three sides of a right-angled triangle download The Humongous Book of Trigonometry Problems pdf. | 677.169 | 1 |
College Algebra Advice
Showing 1 to 3 of 9
The lessons were visual and helped you learn more in depth about Algebra
Hours per week:
6-8 hours
Advice for students:
Pay attention to the lectures and work hard on your homework so you know what to do on the test
Course Term:Fall 2016
Professor:Nancy Jones
Course Required?Yes
Course Tags:Great Intro to the SubjectMany Small AssignmentsGreat Discussions
May 30, 2017
| No strong feelings either way.
Not too easy. Not too difficult.
Course Overview:
You gotta get thru it was okay I'm not a fan of math so it was pretty challenging for me
Course highlights:
Passing and never having to deal with collage algebra again
Hours per week:
3-5 hours
Advice for students:
Stay on top of your work never give up and alway ask for help
Course Term:Spring 2017
Professor:Artemas Holloway
Course Required?Yes
Course Tags:Math-heavyGo to Office HoursParticipation Counts
Apr 11, 2017
| Would highly recommend.
Not too easy. Not too difficult.
Course Overview:
As a future business student, I believe it is important for me to have a basic knowledge of math. Algebra is one of the more common maths that is more likely to show up in my future of business. This class also takes math problems and puts them into real world examples which help give the math a real purpose further than simply passing the class!
Course highlights:
I gained a much better understanding of algebra that I can use in the real world throughout my business career!
Hours per week:
3-5 hours
Advice for students:
When you learn a new topic, do the homework that same afternoon or night. This helps engrave the new information in your brain compared to if you had waited until the night before the big test. | 677.169 | 1 |
Integrated Math II
This course is designed to further expose students in the area of algebra in a way that provides logical connections and practical application. This course is designed to follow Integrated Mathematics 1, but could also be taken by a student who has algebra experience. Specific content that will be addressed in this curriculum include: Foundations of Algebra, expressions, sentences, equalities and inequalities. Graphing functions will be stressed as well as systems of equations and inequalities. Lastly, a unit involving introductory geometric concepts and definitions will be included. | 677.169 | 1 |
Math Trigonometry sin cos Trigonometric Hand Trick This is an easy way to remember the values of common values of trigonometric functions in the first quadrant.
Calculus for Beginners and Artists is an online textbook that provides an overview of Calculus in clear, easy to understand language designed for the non-mathematician. Thank you MIT OpenCourseWare for awesome, free resources and classes! | 677.169 | 1 |
Calculus bc calculator programs
It is a good idea to bring extra batteries. Exploration Versus Mathematical Solution, a graphing calculator is a powerful tool for exploration, but please remember that exploration is not a mathematical serial number cs5 mac keygen solution.This is an example of a polar graph called the four-leaf rose.For example, if you are asked to find a relative minimum value of a function, you are expected to use calculus and show the mathematical steps that lead to the answer.I use only past released exam formulas published by Collegeboard, which makes AP Pass the most accurate and up-to-date calculator available.Unfortunately, private tutors usually charge quite a bit.While taking the Advanced Placement (AP) Calculus BC exam is not required, this course prepares students to succeed on the AP Calculus BC exam and subsequent courses that draw on material from this course.Magoosh blog comment policy: To create the best experience for our readers, we will approve and respond to comments that are relevant to the article, general enough to be helpful to other students, concise, and well-written!Furthermore, each section consists of two parts: calculator and no-calculator. Copyright, Parker Shepherd, All Rights Reserved. Differential equations, sequences and series, applications of series, for a detailed list of topics, click the List of Topics tab.
Either way, youre going to need these five AP calculus BC exam study tips!Answers pdf to word farsi support should show enough work so that the reasoning process can be followed throughout the solution.The following tips represent a few general test-taking strategies as well as specific info about the Calculus BC exam.Justifications must include mathematical reasons, not merely calculator results.The Adobe Connect Add-in, Adobe Flash plugin, and Adobe Connect Mobile app are available for free download.Start your 1 Week Free Trial of Magoosh SAT Prep or your 1 Week Free Trial of Magoosh ACT Prep today! You can expect to see polar functions on the AP Calculus BC exam. From the Oberlin Conservatory in the same year, with a major in music composition. | 677.169 | 1 |
MATH6: CALCULUS BUS/SOC SC
Course Description
This course applies the fundamental principles and techniques of calculus to problems in business, economics, the life sciences and the social sciences. Topics will include limits, and differentiation and integration of linear, quadratic, polynomial, exponential and logarithmic functions. This course is not intended for students majoring in engineering, the physical sciences or math. Using a calculator is required. Graphing calculator is recommended. PREREQUISITE: Mathematics 235 or Mathematics 240 with a grade of 'C' or better.
Learning Outcomes
Students will be able to analyze properties of quadratic functions and their graphs. Applications in business, social sciences and life sciences will be chosen to demonstrate knowledge in polynomial and rational functions.
Students will be able to find derivatives of polynomials, rational, exponential, and logarithmic functions They will use the rules for sums as well as use product and quotient and chain rules to solve problems involving complex equations.
Students will be able to use calculus to sketch the graph of functions using horizontal and vertical asymptotes, intercepts, and first and second derivatives to determine intervals where the function is increasing and decreasing, maximum and minimum values, intervals of concavity and points of inflection.
Students will analyze the marginal cost, profit and revenue when given the appropriate function; determine maxima and minima in optimization problems using the derivative; use derivatives to find rates of change and tangent lines; and use calculus to analyze revenue, cost, and profit. They will apply this to applications in business, economics, and the life sciences.
Students will be able to work with the definite integral as a limit of a sum and how it relates to the fundamental theorem of calculus.
Students will be able to find definite and indefinite integrals by using general formulas, substitution, integration by parts, integral tables, and other integration techniques. They will use integration techniques and apply them to business and economic applications and to the life sciences. | 677.169 | 1 |
9 Most-Wanted Instruments for a Perfect Algebra Paper to Ease Your Student Life
The performance of different academic assignments may run pretty easy or it may go the hard way. Each student has to find out different methods and strategies to overcome definite impediments. The students who deal with mathematics face great challenges associated with different kinds of calculations. It is necessary to find out the final results counting different figures. This is a difficult task and it takes some time to add, deduct, multiply, and divide.
Fortunately, there are special instruments that ease this way and help receive the results in a blink of an eye. These are specially designed online tools that may come to aid whenever they are needed. Let's consider the most-wanted instruments that will help you write a quick and effective algebra paper.
1. BMI Calculator
This is a simple tool, which helps define the body mass index. You only have to fill the necessary fields with your parameters of height and weight. It will quickly calculate your BMI and will give you the meaning.
2. Scientific Calculator
This very calculator will help you find out sinuses, cosines, tangents, and other meanings. Choose radians, gradients, or degrees and calculate what is needed.
3. Grade Calculator
This calculator will help you to figure out your final grades. You should mention your scores and the total percentage for each assignment. The results will be 100% correct if mentioning all your scores.
4. Function Grapher Online
This is an utterly helpful tool, which understands almost all functions. Type the function you want to plot, and you will receive the optimal x-minimum and optimal y-maximum.
5. Quadratic Equation Solver
This smart tool will help you find the roots. Simply put the coefficients of the quadratic equation in the required fields, and you will get the solution.
6. System of Equations Solver
Using this great solver, you will figure out any intersection point between two straight lines. Simply type two equations and two lines, and you will get the answer.
7. Greatest Common Divisor Calculator
Typing the necessary information, you will find out the needed n1 and n2 values. It will take only a few seconds.
8. Fraction Operations Calculator
You can easily define the required result using this smart tool. You are to specify the fractions and conduct the necessary operation. Use the required notation, use "%" for the division and receive the results.
9. Algebraic Expression Calculator
This tool will sufficiently ease your tasks. The only thing you are to perform is the sought algebraic expression. The rest will be done by the calculator, which will figure out the final outcome.
As you can see, these tools are extremely beneficial and will help you save a lot of time. But if you still have no time to handle the assignment on your own, you can use the custom writing services of such online agencies like DoMyCaseStudy.com. Such resources provide a student with the needed sample that helps them complete the plot of any algebra paper. Use this sample, calculate all the needed items using the mentioned online tools and you will easily create a piece on your own in no time. | 677.169 | 1 |
In some courses, all it's going to take to pass an examination is observe taking, memorization, and recall. Having said that, exceeding in a very math course will take a different type of exertion. You can't simply just demonstrate up for any lecture and enjoy your teacher "talk" about math and . You discover it by undertaking: being attentive in class, actively researching, and resolving math problems – even though your instructor has not assigned you any. Should you find yourself battling to carry out well as part of your math class, then go to most effective web page for fixing math challenges to learn how you can become an improved math scholar.
Affordable math authorities on the net
Math abstract algebra problems courses adhere to a organic development – each builds on the know-how you have acquired and mastered through the preceding program. For those who are acquiring it challenging to abide by new concepts at school, pull out your previous math notes and assessment former substance to refresh you. Be sure that you satisfy the stipulations right before signing up for your class.
Assessment Notes The Night time In advance of Course
Hate when a instructor calls on you and you've forgotten ways to remedy a specific challenge? Stay clear of this instant by reviewing your math notes. This may allow you to identify which concepts or thoughts you'd like to go over at school another day.
The thought of doing homework each night time could seem bothersome, but if you desire to achieve , it is actually important that you continually follow and grasp the problem-solving solutions. Use your textbook or online guides to operate by way of top rated math troubles on the weekly foundation – even when you have got no homework assigned.
Make use of the Nutritional supplements That include Your Textbook
Textbook publishers have enriched present day publications with more materials (including CD-ROMs or on the web modules) that could be accustomed to help pupils achieve added exercise in . A few of these products may additionally include things like a solution or explanation information, which might assist you with doing the job by means of math problems on your own.
Read through Ahead To stay Ahead
If you want to reduce your in-class workload or maybe the time you invest on homework, use your free time soon after school or within the weekends to browse forward for the chapters and ideas that can be covered the following time you're in class.
Overview Previous Exams and Classroom Illustrations
The do the job you need to do in school, for homework, and on quizzes can supply clues to what your midterm or remaining examination will appear like. Make use of your old checks and classwork to make a personalized analyze manual on your forthcoming exam. Glimpse on the way your trainer frames questions – this is often in all probability how they will show up in your exam.
Discover how to Operate From the Clock
It is a popular research tip for men and women using timed exams; primarily standardized tests. In case you only have forty minutes for the 100-point examination, then you can optimally expend four minutes on just about every 10-point concern. Get details regarding how extensive the examination is going to be and which sorts of queries will likely be on it. Then program to assault the easier concerns to start with, leaving your self sufficient the perfect time to shell out to the a lot more tough ones.
Improve your Resources to acquire math research support
If you are possessing a hard time understanding ideas in school, then you should definitely get support beyond class. Question your mates to create a research team and pay a visit to your instructor's office several hours to go about tricky complications one-on-one. Go to examine and evaluation periods once your teacher announces them, or retain the services of a personal tutor if you want one particular.
Converse To Yourself
Once you are examining problems for an exam, test to clarify out loud what strategy and techniques you accustomed to obtain your remedies. These verbal declarations will arrive in helpful throughout a take a look at any time you really need to remember the methods it is best to just take to locate a option. Get added apply by hoping this tactic with a good friend.
Use Examine Guides For Added Follow
Are your textbook or course notes not supporting you have an understanding of what you really should be learning in class? Use examine guides for standardized exams, including the ACT, SAT, or DSST, to brush up on aged product, or . Review guides commonly occur outfitted with thorough explanations of how you can address a sample difficulty, , and also you can usually discover wherever is definitely the far better get mathcomplications. | 677.169 | 1 |
2017-18 Undergraduate and Graduate Catalogs
Search Results
MATH 146. Applied Calculus. 3 Hours.
Introduction to differential and integral calculus with applications from areas such as social science and business. Topics include limits, derivatives, integrals, exponential and logarithmic functions, and applications. Prerequisite(s): MATH 103 with a C or better, or qualifying math placement test score, or ACT math subtest of 25 or higher. | 677.169 | 1 |
I read a book, written in 1949, by a retiring teacher that focused on how to be an effective teacher. The four points of the book:
Know your students.
Love your students.
Know your subject.
Love your subject.
The last two are really the answer to your question. It is not enough to take a few math courses in college, you have to live math.
You can get experience in many fields: Business, architecture, engineering, science. If you do not have this experience, use your summer holidays to gain it. Professionals will happily let you tag along on projects, to learn the math applications to relate to your students. Summer workshops are another source of learning, having taught many of these, it is surprising what one can learn, and how to demonstrate/teach it.
factorise: 2x^2+6x-16, and 14x^2-3x-2. Solve: 6y^2-5y-6=0, and 8x^2-2x-1=0. Rearrange: p=5t-u/u to make u the subject, and p(q+r)=(q-p). please help i've done about 40 questions and im just stuck on these. thanks for the help If you take this aesthetic view, you believe that how the artist used the elements and principles of art is the most important part of an artwork. -Civic View -Composition View 2. If you take this aesthetic view, you believe that … | 677.169 | 1 |
Math
A resource website containing fully worked problem sets, examination advice, personally written topic summaries for the A level math student. Everything made freely accesible at no cost to the learner.
Mathguru is an innovative math-help program designed to provide solutions to the student's problems in a step by step fashion using a pen and virtual notebook. Mathguru has a repository that contains all math problems and their solutions, from the NCERT Math textbooks, for Classes VI to XII. There are over 10,000 solutions available on the Mathguru website.
Author has taught mathematics in Australian secondary schools (grades 7 to 12), TAFE (Technical and Further Education), and at university level (at Griffith University and in Japan). He has also taught lots of other things, including music, English as a second language, computers and more recently, teachers. | 677.169 | 1 |
Certificate in Women's Education - Everyday Maths
Pricing
Description
This resource will provide the student with knowledge and skills to perform calculations for routine business related tasks with confidence. Students learn to use both manual and electronic means and well cover topics such as division,subtraction,multiplication,fractions,decimals,percentages,calculators and the interpretation of graphs and diagrams.
Catalogue Item
This diary has been written specifically to be used by Deaf students or mathematics, within the Education and Training - Deaf Program at TAFE SA Adelaide City Campus. These students are enrolled in the Deaf Stream of the nationally recognised introductory Vocational Education Certificate 1 and 2 (IVEC 1 and IVEC 2). However, the diary has also been designed so that it may be applicable to other Australian or international programs for Deaf adults/young people. This set includes: A Reference Guide for Deaf Students, a Journal for Deaf Students and a Teachers Guide.
This reference guide presents a wealth of information and handy hints to the beginning writer of adult learning materials - information that is based on extensive experience in the VET sector. It is expected that those who refer to this guide will have some knowledge of the VET arena. | 677.169 | 1 |
The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.
A.2
The student will perform operations on polynomials, including
a) applying the laws of exponents to perform operations on expressions;
b) adding, subtracting, multiplying, and dividing polynomials; and
c) factoring completely first- and second-degree binomials and trinomials in one or two variables. Graphing calculators will be used as a tool for factoring and for confirming algebraic factorizations.
A.3
The student will express the square roots and cube roots of whole numbers and the square root of a monomial algebraic expression in simplest radical form.
A.4
The student will solve multistep linear and quadratic equations in two variables, including
a) solving literal equations (formulas) for a given variable;
b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets;
c) solving quadratic equations algebraically and graphically;
d) solving multistep linear equations algebraically and graphically;
e) solving systems of two linear equations in two variables algebraically and graphically; and
f) solving real-world problems involving equations and systems of equations.
Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.
A.5
The student will solve multistep linear inequalities in two variables, including
a) solving multistep linear inequalities algebraically and graphically;
b) justifying steps used in solving inequalities, using axioms of inequality and properties of order that are valid for the set of real numbers and its subsets;
c) solving real-world problems involving inequalities; and
d) solving systems of inequalities.
A.6
; and
b) writing the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line.
A.7
The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including
a) determining whether a relation is a function;
b) domain and range;
c) zeros of a function;
d) x- and y-intercepts;
e) finding the values of a function for elements in its domain; and
f) making connections between and among multiple representations of functions including concrete, verbal, numeric, graphic, and algebraic.
A.8
The student, given a situation in a real-world context, will analyze a relation to determine whether a direct or inverse variation exists, and represent a direct variation algebraically and graphically and an inverse variation algebraically.
A.9
The student, given a set of data, will interpret variation in real-world contexts and calculate and interpret mean absolute deviation, standard deviation, and z-scores.
A.10
The student will compare and contrast multiple univariate data sets, using box-and-whisker plots.
A.11
The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve real-world problems, using mathematical models. Mathematical models will include linear and quadratic functions. | 677.169 | 1 |
Note that the test dates given here are for informational purposes only. Test dates
announced in class supercede those given here. As the information becomes available,
this page will include a list of topics covered by each exam.
covering chapters 5 through 8.
Specific topics list available
as html or Postscript or
PDF.
A solved copy of one of the practice tests (so, for example, you can get a
sense of *how* answers can/should be typed) may be found here.
What you should note is that I made the *computer* do all of the arithmetic; you don't need to add or
multiply or divide any number before you *write* your answer. The only hard part is that this makes the
placement of parentheses ( ) very important... Note that "sqrt( )" means the square root of the stuff
inside the parentheses. | 677.169 | 1 |
The Mutually Beneficial Relationship of Graphs and Matrices(Paperback)
Synopsis
Graphs and matrices enjoy a fascinating and mutually beneficial relationship. This interplay has benefited both graph theory and linear algebra. In one direction, knowledge about one of the graphs that can be associated with a matrix can be used to illuminate matrix properties and to get better information about the matrix. Examples include the use of digraphs to obtain strong results on diagonal dominance and eigenvalue inclusion regions and the use of the Rado-Hall theorem to deduce properties of special classes of matrices. Going the other way, linear algebraic properties of one of the matrices associated with a graph can be used to obtain useful combinatorial information about the graph. The adjacency matrix and the Laplacian matrix are two well-known matrices associated to a graph, and their eigenvalues encode important information about the graph. Another important linear algebraic invariant associated with a graph is the Colin de Verdiere number, which, for instance, characterises certain topological properties of the graph. This book is not a comprehensive study of graphs and matrices.
The particular content of the lectures was chosen for its accessibility, beauty, and current relevance, and for the possibility of enticing the audience to want to learn | 677.169 | 1 |
This course is an introduction to linear algebra, intended for students who are interested in
mathematics. Although there will be a significant amount of computation, the focus is on
understanding the ideas, what they mean, and how we know what we know about them.
There will also be an emphasis on learning how to think mathematically, and the process of exploring
our initially hazy understanding of a new concept and refining it into clear, concise arguments.
In short, we are interested in understanding ideas as completely and throughly as possible, and the
course will reflect that goal.
Note: New editions of the book are expensive. Any edition of the book (e.g., an older, cheaper
version) would be fine.
Classes (slot 15)
Tutorials
Tues. 12:30–13:30
Wed. 17:30–18:30
Thurs. 11:30–12:30
Fri. 14:30–15:30
Fri. 13:30–14:30
All classes are in Kingston 201.
All tutorials are in Jeff 201/202.
Grading Scheme
Fall
Winter
Homework
30%
30%
In-class exams (two each term)
20%
30%
End-of-term exam
50%
40%
There are twelve homework assignments each term. The lowest two of these twelve grades
will be dropped when computing the homework grade for that term.
The final course grade will consist of the average of the final grades from the fall and winter terms.
All grades in the course will be computed as numbers, and the final grade (out of 100) will be converted
into a Queen's letter grade using the standard conversion. | 677.169 | 1 |
Introduction to Function Rules; Challenge activity with rules and tables
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The first two pages are guided notes for introducing the concepts of variable relations and function rules.
The Stumped worksheet is a challenge activity for advanced 7th grade students (only 5/24 of my students finished it) | 677.169 | 1 |
Math 110-02, Spring 2014
Information on the Final Exam
The last test for this course will be given during the scheduled
final exam period: Sunday, May 11, at 1:30 PM. The exam will be in
our regular classroom. The final exam counts for 15% of the
overall grade for the course.
The exam will be five or six pages long, and it will be only a little longer than the
three in-class tests from earlier in the semester. There will be one essay
question at the end covering general ideas from the course, and you will know that
essay question in advance. Aside from that essay question, the exam is not
cumulative. It will cover material from the last part of the course, since the
third test. This includes: dimension and the fourth dimension;
graphs, including Euler circuits, the Euler characteristic, and regular polyhedra;
and voting, including various voting methods and Arrow's Impossibility Theorem.
The reading from the textbook includes sections 4.7, 6.1, 6.2, and 10.4; however,
we did a few things that are not in the book.
Here is the general essay question for the end of the test: "Over the course of the semester,
we have discussed many mathematical ideas, but one that came up over and over is the idea of
infinity. Write an essay discussing infinity and what you have learned about it,
including some of the specific ways that infinity has come up in the course. What in the end
do you think about the mathematical idea of infinity?"
Here are some other terms and ideas that might be on the test:
the fourth dimension
understanding the fourth dimension by analogy
what a 2D object looks like passing through a 1D world (that is, a line)
what a 3D object looks like passing through a 2D world (that is, a plane)
what a 4D object might look like passing through a 3D world (that is, space)
imagining living on the surface of a torus or in the 3D analog of a torus
how a torus can be modeled as a rectangle with edges identified
how a 3D torus can be modeled as a brick or fishtank with sides identified.
hypercube (also known as tesseract)
graph (in the sense of vertices plus edges)
vertex (plural is "vertices")
edge
understanding diagrams of graphs
how a graph can be used to model connections by bridges between land masses
Euler circuit
degree of a vertex (number of edges that have the vertex as an endpoint)
connected graph
a graph has an Euler circuit if and only if
it is connected and all vertices have even degree
finding an Euler circuit in a graph
Euler path
a graph has an Euler path if and only if it has an Euler circuit
OR is connected and has exactly two vertices that have odd degree
finding an Euler path in a graph
regular polygon
regular polyhedron (plural is "polyhedra")
Platonic solids: tetrahedron, cube, octahedron, dodecahedron, icosahedron
the formula V - E + F for graphs and polyhedra: How to count V, E, F
planar graph (drawn in the plane so that edges don't cross)
for any planar graph, V - E + F = 2
for any polyhedron (without "holes"), V - E + F = 2
the only regular polyhedra are the five Platonic solids
social choice (how a group can make a choice among alternatives)
voting
ranking (each voter lists alternatives in order of preference)
voting methods:
plurality voting
plurality voting with runoff
IRV (Instant Runoff Voting)
Borda count
approval voting
Condorcet winner: wins every one-on-one matchup with other alternatives
Condorcet paradox
desirable properties of voting methods
1. No dictator:
The winner is not simply the choice of some particular voter.
2. Unanimity:
If one alternative is the first choice of every voter,
then that alternative wins.
3. Ignore the irrelevant:
The result doesn't change if a losing alternative drops out.
4. Better is better:
If some voters raise their ranking of the winning alternative, that
will not cause that alternative to lose.
Arrow's Impossibility Theorem:
There is no voting method that satisfies the four above properties. | 677.169 | 1 |
Aleks Pie Mat222 Intermediate Algebra Answers
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On pages 345-6 and 353 of Elementary and Intermediate Algebra,. because the math we use throughout MAT 222 will depend upon.Provides a complete web based educational environment for K-12 and Higher-Education mathematics, accounting, statistics, and chemistry.Related Book Ebook Pdf Aleks Pie Mat222 Intermediate Algebra Answers: - Home - Parenting From The Inside Out - Parenting Gifted Children Authoritative Association.Here is the best resource for homework help with MAT 222: algebra at.
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From Wikipedia, the free encyclopedia Contents 1 2 Accessibility relation 1 1.1 Description of Terms.Aleks Pie Mat222 Intermediate Algebra Answers Scouting for Do you really need this respository of It takes me 72 hours just to get the right download link, and. | 677.169 | 1 |
Geometry & Topology:
Geometry deals with quantitative properties of space, such as distance and curvature on manifolds. Topology deals with more qualitative properties of space, namely those that remain unchanged under bending and stretching. (For this reason, topology is often called "the geometry of rubber sheets".) The two subjects are closely related and play a central role in many other fields such as Algebraic Geometry, Dynamical Systems, and Physics. At McMaster research focuses on Algebraic Topology (homotopy theory, K-theory, surgery), Geometric Topology (group actions on manifolds, gauge theory, knot theory), and Differential Geometry (curvature, Dirac operators, Einstein equations, and general relativity).
Discover More
McMaster Academic Planner (MAP)
The MCMASTER ACADEMIC PLANNER (M.A.P.) is a multipurpose tool used to educate prospective and current Science students on the admission requirements, course/program options, research, experiential education & co-op opportunities available in the Faculty of Science . | 677.169 | 1 |
SimilarMany colleges and universities require students to take at least one math course, and Calculus I is often the chosen option. Calculus Essentials For Dummies provides explanations of key concepts for students who may have taken calculus in high school and want to review the most important concepts as they gear up for a faster-paced college course. Free of review and ramp-up material, Calculus Essentials For Dummies sticks to the point with content focused on key topics only. It provides discrete explanations of critical concepts taught in a typical two-semester high school calculus class or a college level Calculus I course, from limits and differentiation to integration and infinite series. This guide is also a perfect reference for parents who need to review critical calculus concepts as they help high school1001 Calculus Practice Problems For Dummies takes you beyond the instruction and guidance offered in Calculus For Dummies, giving you 1001 opportunities to practice solving problems from the major topics in your calculus course. Plus, an online component provides you with a collection of calculus problems presented in multiple-choice format to further help you test your skills as you go.
Gives you a chance to practice and reinforce the skills you learn in your calculus course Helps you refine your understanding of calculus Practice problems with answer explanations that detail every step of every problem
The practice problems in 1001 Calculus Practice Problems For Dummies range in areas of difficulty and style, providing you with the practice help you need to score high at exam time.
A solutions manual to accompany An Introduction to Numerical Methods and Analysis, Second Edition
An Introduction to Numerical Methods and Analysis, Second Edition reflects the latest trends in the field, includes new material and revised exercises, and offers a unique emphasis on applications. The author clearly explains how to both construct and evaluate approximations for accuracy and performance, which are key skills in a variety of fields. A wide range of higher-level methods and solutions, including new topics such as the roots of polynomials, spectral collocation, finite element ideas, and Clenshaw-Curtis quadrature, are presented from an introductory perspective, and theSecond Edition also features: Chapters and sections that begin with basic, elementary material followed by gradual coverage of more advanced material Exercises ranging from simple hand computations to challenging derivations and minor proofs to programming exercises Widespread exposure and utilization of MATLAB® An appendix that contains proofs of various theorems and other material differential calculus of the real line, the Riemann-Stieltjes integral, sequences and series of functions, transcendental functions, inner product spaces and Fourier series, normed linear spaces and the Riesz representation theorem, and the Lebesgue integral. Supplementary materials include an appendix on vector spaces and more than 750 exercises of varying degrees of difficulty. Hints and solutions to selected exercises, indicated by an asterisk, appear at the back of the book.
Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan, Second Edition provides an accessible approach for conducting Bayesian data analysis, as material is explained clearly with concrete examples. Included are step-by-step instructions on how to carry out Bayesian data analyses in the popular and free software R and WinBugs, as well as new programs in JAGS and Stan. The new programs are designed to be much easier to use than the scripts in the first edition. In particular, there are now compact high-level scripts that make it easy to run the programs on your own data sets.
The book is divided into three parts and begins with the basics: models, probability, Bayes' rule, and the R programming language. The discussion then moves to the fundamentals applied to inferring a binomial probability, before concluding with chapters on the generalized linear model. Topics include metric-predicted variable on one or two groups; metric-predicted variable with one metric predictor; metric-predicted variable with multiple metric predictors; metric-predicted variable with one nominal predictor; and metric-predicted variable with multiple nominal predictors. The exercises found in the text have explicit purposes and guidelines for accomplishment.
This book is intended for first-year graduate students or advanced undergraduates in statistics, data analysis, psychology, cognitive science, social sciences, clinical sciences, and consumer sciences in business.
Accessible, including the basics of essential concepts of probability and random samplingExamples with R programming language and JAGS softwareComprehensive coverage of all scenarios addressed by non-Bayesian textbooks: t-tests, analysis of variance (ANOVA) and comparisons in ANOVA, multiple regression, and chi-square (contingency table analysis)Coverage of experiment planningR and JAGS computer programming code on websiteExercises have explicit purposes and guidelines for accomplishment
Provides step-by-step instructions on how to conduct Bayesian data analyses in the popular and free software R and WinBugs
"This is quite a well-done book: very tightly organized, better-than-average exposition, and numerous examples, illustrations, and applications." —Mathematical Reviews of the American Mathematical Society
An Introduction to Linear Programming and Game Theory, Third Edition presents a rigorous, yet accessible, introduction to the theoretical concepts and computational techniques of linear programming and game theory. Now with more extensive modeling exercises and detailed integer programming examples, this book uniquely illustrates how mathematics can be used in real-world applications in the social, life, and managerial sciences, providing readers with the opportunity to develop and apply their analytical abilities when solving realistic problems.
This Third Edition addresses various new topics and improvements in the field of mathematical programming, and it also presents two software programs, LP Assistant and the Solver add-in for Microsoft Office Excel, for solving linear programming problems. LP Assistant, developed by coauthor Gerard Keough, allows readers to perform the basic steps of the algorithms provided in the book and is freely available via the book's related Web site. The use of the sensitivity analysis report and integer programming algorithm from the Solver add-in for Microsoft Office Excel is introduced so readers can solve the book's linear and integer programming problems. A detailed appendix contains instructions for the use of both applications.
Additional features of the Third Edition include:
A discussion of sensitivity analysis for the two-variable problem, along with new examples demonstrating integer programming, non-linear programming, and make vs. buy models
Revised proofs and a discussion on the relevance and solution of the dual problem
A section on developing an example in Data Envelopment Analysis
An outline of the proof of John Nash's theorem on the existence of equilibrium strategy pairs for non-cooperative, non-zero-sum games
Providing a complete mathematical development of all presented concepts and examples, Introduction to Linear Programming and Game Theory, Third Edition is an ideal text for linear programming and mathematical modeling courses at the upper-undergraduate and graduate levels. It also serves as a valuable reference for professionals who use game theory in business, economics, and management science.
This book presents a twenty-first century approach to classical polynomial and rational approximation theory. The reader will find a strikingly original treatment of the subject, completely unlike any of the existing literature on approximation theory, with a rich set of both computational and theoretical exercises for the classroom. There are many original features that set this book apart: the emphasis is on topics close to numerical algorithms; every idea is illustrated with Chebfun examples; each chapter has an accompanying Matlab file for the reader to download; the text focuses on theorems and methods for analytic functions; original sources are cited rather than textbooks, and each item in the bibliography is accompanied by an editorial comment. This textbook is ideal for advanced undergraduates and graduate students across all of applied mathematics.
For over three decades, this best-selling classic has been used by thousands of students in the United States and abroad as a must-have textbook for a transitional course from calculus to analysis. It has proven to be very useful for mathematics majors who have no previous experience with rigorous proofs. Its friendly style unlocks the mystery of writing proofs, while carefully examining the theoretical basis for calculus. Proofs are given in full, and the large number of well-chosen examples and exercises range from routine to challenging.
The second edition preserves the book's clear and concise style, illuminating discussions, and simple, well-motivated proofs. New topics include material on the irrationality of pi, the Baire category theorem, Newton's method and the secant method, and continuous nowhere-differentiable functions.
Review from the first edition:
"This book is intended for the student who has a good, but naïve, understanding of elementary calculus and now wishes to gain a thorough understanding of a few basic concepts in analysis.... The author has tried to write in an informal but precise style, stressing motivation and methods of proof, and ... has succeeded admirably."
Designed for students familiar with abstract mathematical concepts but possessing little knowledge of physics, this text focuses on generality and careful formulation rather than problem-solving. Its author, a member of the distinguished National Academy of Science, based this graduate-level text on the course he taught at Harvard University. Opening chapters on classical mechanics examine the laws of particle mechanics; generalized coordinates and differentiable manifolds; oscillations, waves, and Hilbert space; and statistical mechanics. A survey of quantum mechanics covers the old quantum theory; the quantum-mechanical substitute for phase space; quantum dynamics and the Schrödinger equation; the canonical "quantization" of a classical system; some elementary examples and original discoveries by Schrödinger and Heisenberg; generalized coordinates; linear systems and the quantization of the electromagnetic field; and quantum-statistical mechanics. The final section on group theory and quantum mechanics of the atom explores basic notions in the theory of group representations; perturbations and the group theoretical classification of eigenvalues; spherical symmetry and spin; and the n-electron atom and the Pauli exclusion principle thatA coherent introduction to the techniques for modeling dynamic stochastic systems, this volume also offers a guide to the mathematical, numerical, and simulation tools of systems analysis. Suitable for advanced undergraduates and graduate-level industrial engineers and management science majors, it proposes modeling systems in terms of their simulation, regardless of whether simulation is employed for analysis. Beginning with a view of the conditions that permit a mathematical-numerical analysis, the text explores Poisson and renewal processes, Markov chains in discrete and continuous time, semi-Markov processes, and queuing processes. Each chapter opens with an illustrative case study, and comprehensive presentations include formulation of models, determination of parameters, analysis, and interpretation of results. Programming language–independent algorithms appear for all simulation and numerical procedures.
This incredibly useful guide book to mathematics contains the fundamental working knowledge of mathematics which is needed as an everyday guide for working scientists and engineers, as well as for students. Now in its fifth updated edition, it is easy to understand, and convenient to use. Inside you'll find the information necessary to evaluate most problems which occur in concrete applications. In the newer editions emphasis was laid on those fields of mathematics that became more important for the formulation and modeling of technical and natural processes. For the 5th edition, the chapters "Computer Algebra Systems" and "Dynamical Systems and Chaos" have been revised, updated and expanded.
Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The exposition maintains a balance between theoretical, algorithmic and applied aspects. This second edition has been extensively updated, and includes new chapters on emerging subject areas: geometric numerical integration, spectral methods and conjugate gradients. Other topics covered include multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; and a variety of algorithms to solve large, sparse algebraic systems.
In the late 1950s, many of the more refined aspects of Fourier analysis were transferred from their original settings (the unit circle, the integers, the real line) to arbitrary locally compact"This book covers many interesting topics not usually covered in a present day undergraduate course, as well as certain basic topics such as the development of the calculus and the solution of polynomial equations. The fact that the topics are introduced in their historical contexts will enable students to better appreciate and understand the mathematical ideas involved...If one constructs a list of topics central to a history course, then they would closely resemble those chosen here."
(David Parrott, Australian Mathematical Society)
"The book...is presented in a lively style without unnecessary detail. It is very stimulating and will be appreciated not only by students. Much attention is paid to problems and to the development of mathematics before the end of the nineteenth century... This book brings to the non-specialist interested in mathematics many interesting results. It can be recommended for seminars and will be enjoyed by the broad mathematical community."
(European Mathematical Society)
"Since Stillwell treats many topics, most mathematicians will learn a lot from this book as well as they will find pleasant and rather clear expositions of custom materials. The book is accessible to students that have already experienced calculus, algebra and geometry and will give them a good account of how the different branches of mathematics interact."
(Denis Bonheure, Bulletin of the Belgian Society)
This third edition includes new chapters on simple groups and combinatorics, and new sections on several topics, including the Poincare conjecture. The book has also been enriched by added exercises
This
The purpose of this book is to isolate and draw attention to the most important problem-solving techniques typically encountered in undergradu ate mathematics and to illustrate their use by interesting examples and problems not easily found in other sources. Each section features a single idea, the power and versatility of which is demonstrated in the examples and reinforced in the problems. The book serves as an introduction and guide to the problems literature (e.g., as found in the problems sections of undergraduate mathematics journals) and as an easily accessed reference of essential knowledge for students and teachers of mathematics. The book is both an anthology of problems and a manual of instruction. It contains over 700 problems, over one-third of which are worked in detail. Each problem is chosen for its natural appeal and beauty, but primarily to provide the context for illustrating a given problem-solving method. The aim throughout is to show how a basic set of simple techniques can be applied in diverse ways to solve an enormous variety of problems. Whenever possible, problems within sections are chosen to cut across expected course boundaries and to thereby strengthen the evidence that a single intuition is capable of broad application. Each section concludes with "Additional Examples" that point to other contexts where the technique is appropriate.
Fast Fourier Transform - Algorithms and Applications presents an introduction to the principles of the fast Fourier transform (FFT). It covers FFTs, frequency domain filtering, and applications to video and audio signal processing.
As fields like communications, speech and image processing, and related areas are rapidly developing, the FFT as one of the essential parts in digital signal processing has been widely used. Thus there is a pressing need from instructors and students for a book dealing with the latest FFT topics.
Fast Fourier Transform - Algorithms and Applications provides a thorough and detailed explanation of important or up-to-date FFTs. It also has adopted modern approaches like MATLAB examples and projects for better understanding of diverse FFTs.
Fast Fourier Transform - Algorithms and Applications is designed for senior undergraduate and graduate students, faculty, engineers, and scientists in the field, and self-learners to understand FFTs and directly apply them to their fields, efficiently. It is designed to be both a text and a reference. Thus examples, projects and problems all tied with MATLAB, are provided for grasping the concepts concretely. It also includes references to books and review papers and lists of applications, hardware/software, and useful websites. By including many figures, tables, bock diagrams and graphs, this book helps the reader understand the concepts of fast algorithms readily and intuitively. It provides new MATLAB functions and MATLAB source codes. The material in Fast Fourier Transform - Algorithms and Applications is presented without assuming any prior knowledge of FFT. This book is for any professional who wants to have a basic understanding of the latest developments in and applications of FFT. It provides a good reference for any engineer planning to work in this field, either in basic implementation or in research and development.
This incisive text deftly combines both theory and practical example to introduce and explore Fourier series and orthogonal functions and applications of the Fourier method to the solution of boundary-value problems. Directed to advanced undergraduate and graduate students in mathematics as well as in physics and engineering, the book requires no prior knowledge of partial differential equations or advanced vector analysis. Students familiar with partial derivatives, multiple integrals, vectors, and elementary differential equations will find the text both accessible and challenging. The first three chapters of the book address linear spaces, orthogonal functions, and the Fourier series. Chapter 4 introduces Legendre polynomials and Bessel functions, and Chapter 5 takes up heat and temperature. The concluding Chapter 6 explores waves and vibrations and harmonic analysis. Several topics not usually found in undergraduate texts are included, among them summability theory, generalized functions, and spherical harmonics. Throughout the text are 570 exercises devised to encourage students to review what has been read and to apply the theory to specific problems. Those preparing for further study in functional analysis, abstract harmonic analysis, and quantum mechanics will find this book especially valuable for the rigorous preparation it provides. Professional engineers, physicists, and mathematicians seeking to extend their mathematical horizons will find it an invaluable reference as well. students. Difficult does not necessarily mean theoretical; often a starred problem is an interesting application that requires insight into what calculus is really about. • The exercises come in groups of two and often four similar ones.
Riemannian geometry is characterized, and research is oriented towards and shaped by concepts (geodesics, connections, curvature,...) andobjectives,inparticularto understand certain classes of (compact) Riemannian manifolds de?ned by curvature conditions (constant or positive or negative curvature,...). Bywayofcontrast,g- metric analysis is a perhaps somewhat less systematic collection of techniques, for solving extremal problems naturally arising in geometry and for investigating and characterizing their solutions. It turns out that the two ?elds complement each other very well; geometric analysis o?ers tools for solving di?cult problems in geometry, and Riemannian geometry stimulates progress in geometric analysis by setting am- tious goals. It is the aim of this book to be a systematic and comprehensive introduction to Riemannian geometry and a representative introduction to the methods of geometric analysis. It attempts a synthesis of geometric and analytic methods in the study of Riemannian manifolds. The present work is the ?fth edition of my textbook on Riemannian geometry and geometric analysis. It has developed on the basis of several graduate courses I taught at the Ruhr-University Bochum and the University of Leipzig. The main new features of the present edition are the systematic inclusion of ?ow equations and a mathematical treatment of the nonlinear sigma model of quantum ?eld theory. These new topics also led to a systematic reorganization of the other material. Naturally, I have also included several smaller additions and minor corrections (for which I am grateful to several readers).
Richard Courant's classic text Differential and Integral Calculus is an essential text for those preparing for a career in physics or applied math. Volume 1 introduces the foundational concepts of "function" and "limit", and offers detailed explanations that illustrate the "why" as well as the "how". Comprehensive coverage of the basics of integrals and differentials includes their applications as well as clearly-defined techniques and essential theorems. Multiple appendices provide supplementary explanation and author notes, as well as solutions and hints for all in-text problems.
"In the world of mathematics, the 1980's might well be described as the "decade of the fractal". Starting with Benoit Mandelbrot's remarkable text The Fractal Geometry of Nature, there has been a deluge of books, articles and television programmes about the beautiful mathematical objects, drawn by computers using recursive or iterative algorithms, which Mandelbrot christened fractals. Gerald Edgar's book is a significant addition to this deluge. Based on a course given to talented high- school students at Ohio University in 1988, it is, in fact, an advanced undergraduate textbook about the mathematics of fractal geometry, treating such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology...the book also contains many good illustrations of fractals (including 16 color plates)."
Mathematics Teaching
"The book can be recommended to students who seriously want to know about the mathematical foundation of fractals, and to lecturers who want to illustrate a standard course in metric topology by interesting examples."
Christoph Bandt, Mathematical Reviews
"...not only intended to fit mathematics students who wish to learn fractal geometry from its beginning but also students in computer science who are interested in the subject. Especially, for the last students the author gives the required topics from metric topology and measure theory on an elementary level. The book is written in a very clear style and contains a lot of exercises which should be worked out."
H.Haase, Zentralblatt
About the second edition: Changes throughout the text, taking into account developments in the subject matter since 1990; Major changes in chapter 6. Since 1990 it has become clear that there are two notions of dimension that play complementary roles, so the emphasis on Hausdorff dimension will be replaced by the two: Hausdorff dimension and packing dimension. 6.1 will remain, but a new section on packing dimension will follow it, then the old sections 6.2--6.4 will be re-written to show both types of dimension; Substantial change in chapter 7: new examples along with recent developments; Sections rewritten to be made clearer and more focused.
The principal aim of this book is to give an introduction to harmonic analysis and the theory of unitary representations of Lie groups. The second edition has been brought up to date with a number of textual changes in each of the five chapters, a new appendix on Fatou's theorem has been added in connection with the limits of discrete series, and the bibliography has been tripled in length.
An in-depth look at real analysis and its applications-now expanded and revised.
This new edition of the widely used analysis book continues to cover real analysis in greater detail and at a more advanced level than most books on the subject. Encompassing several subjects that underlie much of modern analysis, the book focuses on measure and integration theory, point set topology, and the basics of functional analysis. It illustrates the use of the general theories and introduces readers to other branches of analysis such as Fourier analysis, distribution theory, and probability theory.
This edition is bolstered in content as well as in scope-extending its usefulness to students outside of pure analysis as well as those interested in dynamical systems. The numerous exercises, extensive bibliography, and review chapter on sets and metric spaces make Real Analysis: Modern Techniques and Their Applications, Second Edition invaluable for students in graduate-level analysis courses. New features include: * Revised material on the n-dimensional Lebesgue integral. * An improved proof of Tychonoff's theorem. * Expanded material on Fourier analysis. * A newly written chapter devoted to distributions and differential equations. * Updated material on Hausdorff dimension and fractal dimensionThe objective of this book is to analyze within reasonable limits (it is not a treatise) the basic mathematical aspects of the finite element method. The book should also serve as an introduction to current research on this subject.
On the one hand, it is also intended to be a working textbook for advanced courses in Numerical Analysis, as typically taught in graduate courses in American and French universities. For example, it is the author's experience that a one-semester course (on a three-hour per week basis) can be taught from Chapters 1, 2 and 3 (with the exception of Section 3.3), while another one-semester course can be taught from Chapters 4 and 6.
On the other hand, it is hoped that this book will prove to be useful for researchers interested in advanced aspects of the numerical analysis of the finite element method. In this respect, Section 3.3, Chapters 5, 7 and 8, and the sections on "Additional Bibliography and Comments should provide many suggestions for conducting seminars.
Includes sections on the spectral resolution and spectral representation of self adjoint operators, invariant subspaces, strongly continuous one-parameter semigroups, the index of operators, the trace formula of Lidskii, the Fredholm determinant, and more. * Assumes prior knowledge of Naive set theory, linear algebra, point set topology, basic complex variable, and real variables. * Includes an appendix on the Riesz representation theorem.
This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gábor Szego's classic 1915 theorem and its 1920 extension. Barry Simon emphasizes necessary and sufficient conditions, and provides mathematical background that until now has been available only in journals. Topics include background from the theory of meromorphic functions on hyperelliptic surfaces and the study of covering maps of the Riemann sphere with a finite number of slits removed. This allows for the first book-length treatment of orthogonal polynomials for measures supported on a finite number of intervals on the real line.
In addition to the Szego and Killip-Simon theorems for orthogonal polynomials on the unit circle (OPUC) and orthogonal polynomials on the real line (OPRL), Simon covers Toda lattices, the moment problem, and Jacobi operators on the Bethe lattice. Recent work on applications of universality of the CD kernel to obtain detailed asymptotics on the fine structure of the zeros is also included. The book places special emphasis on OPRL, which makes it the essential companion volume to the author's earlier books on OPUCThis book outlines an elementary, one-semester course whichPrinceton University's Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory. His fundamental contributions include the Kunze-Stein phenomenon, the construction of new representations, the Stein interpolation theorem, the idea of a restriction theorem for the Fourier transform, and the theory of Hp Spaces in several variables. Through his great discoveries, through books that have set the highest standard for mathematical exposition, and through his influence on his many collaborators and students, Stein has changed mathematics. Drawing inspiration from Stein's contributions to harmonic analysis and related topics, this volume gathers papers from internationally renowned mathematicians, many of whom have been Stein's students. The book also includes expository papers on Stein's work and its influence.
Engineers and physicists are more and more encountering integrations involving nonelementary integrals and higher transcendental functions. Such integrations frequently involve (not always in immediately re cognizable form) elliptic functions and elliptic integrals. The numerous books written on elliptic integrals, while of great value to the student or mathematician, are not especially suitable for the scientist whose primary objective is the ready evaluation of the integrals that occur in his practical problems. As a result, he may entirely avoid problems which lead to elliptic integrals, or is likely to resort to graphical methods or other means of approximation in dealing with all but the simplest of these integrals. It became apparent in the course of my work in theoretical aero dynamics that there was a need for a handbook embodying in convenient form a comprehensive table of elliptic integrals together with auxiliary formulas and numerical tables of values. Feeling that such a book would save the engineer and physicist much valuable time, I prepared the present volume.
This is a collection of exercises in the theory of analytic functions, with completed and detailed solutions. We wish to introduce the student to applications and aspects of the theory of analytic functions not always touched upon in a first course. Using appropriate exercises we wish to show to the students some aspects of what lies beyond a first course in complex variables. We also discuss topics of interest for electrical engineering students (for instance, the realization of rational functions and its connections to the theory of linear systems and state space representations of such systems). Examples of important Hilbert spaces of analytic functions (in particular the Hardy space and the Fock space) are given. The book also includes a part where relevant facts from topology, functional analysis and Lebesgue integration are reviewed.
The second part of the book concludes with Plancherel's theorem in Chapter 8. This theorem is a generalization of the completeness of the Fourier series, as well as of Plancherel's theorem for the real line. The third part of the book is intended to provide the reader with a ?rst impression of the world of non-commutative harmonic analysis. Chapter 9 introduces methods that are used in the analysis of matrix groups, such as the theory of the exponential series and Lie algebras. These methods are then applied in Chapter 10 to arrive at a clas- ?cation of the representations of the group SU(2). In Chapter 11 we give the Peter-Weyl theorem, which generalizes the completeness of the Fourier series in the context of compact non-commutative groups and gives a decomposition of the regular representation as a direct sum of irreducibles. The theory of non-compact non-commutative groups is represented by the example of the Heisenberg group in Chapter 12. The regular representation in general decomposes as a direct integral rather than a direct sum. For the Heisenberg group this decomposition is given explicitly. Acknowledgements: I thank Robert Burckel and Alexander Schmidt for their most useful comments on this book. I also thank Moshe Adrian, Mark Pavey, Jose Carlos Santos, and Masamichi Takesaki for pointing out errors in the ?rst edition. Exeter, June 2004 Anton Deitmar LEITFADEN vii Leitfaden 1 2 3 5 4 6 . studep,ts. Difficult does not necessarily mean theoretical; often a starred problem is an interesting application that requires insight into what calculus is really about. • The exercises come in groups of two and often four similar onesLogical thinking, the analysis of complex relationships, the recognition of und- lying simple structures which are common to a multitude of problems — these are the skills which are needed to do mathematics, and their development is the main goal of mathematics education. Of course, these skills cannot be learned 'in a vacuum'. Only a continuous struggle with concrete problems and a striving for deep understanding leads to success. A good measure of abstraction is needed to allow one to concentrate on the essential, without being distracted by appearances and irrelevancies. The present book strives for clarity and transparency. Right from the beg- ning, it requires from the reader a willingness to deal with abstract concepts, as well as a considerable measure of self-initiative. For these e?orts, the reader will be richly rewarded in his or her mathematical thinking abilities, and will possess the foundation needed for a deeper penetration into mathematics and its applications. Thisbookisthe?rstvolumeofathreevolumeintroductiontoanalysis.It- veloped from courses that the authors have taught over the last twenty six years at theUniversitiesofBochum,Kiel,Zurich,BaselandKassel.Sincewehopethatthis book will be used also for self-study and supplementary reading, we have included far more material than can be covered in a three semester sequence. This allows us to provide a wide overview of the subject and to present the many beautiful and important applications of the theory. We also demonstrate that mathematics possesses, not only elegance and inner beauty, but also provides e?cient methods for the solution of concrete problems.
There is an explosion of interest in Bayesian statistics, primarily because recently created computational methods have finally made Bayesian analysis tractable and accessible to a wide audience. Doing Bayesian Data Analysis, A Tutorial Introduction with R and BUGS, is for first year graduate students or advanced undergraduates and provides an accessible approach, as all mathematics is explained intuitively and with concrete examples. It assumes only algebra and 'rusty' calculus. Unlike other textbooks, this book begins with the basics, including essential concepts of probability and random sampling. The book gradually climbs all the way to advanced hierarchical modeling methods for realistic data. The text provides complete examples with the R programming language and BUGS software (both freeware), and begins with basic programming examples, working up gradually to complete programs for complex analyses and presentation graphics. These templates can be easily adapted for a large variety of students and their own research needs.The textbook bridges the students from their undergraduate training into modern Bayesian methods.Accessible, including the basics of essential concepts of probability and random samplingExamples with R programming language and BUGS softwareComprehensive coverage of all scenarios addressed by non-bayesian textbooks- t-tests, analysis of variance (ANOVA) and comparisons in ANOVA, multiple regression, and chi-square (contingency table analysis).Coverage of experiment planningR and BUGS computer programming code on websiteExercises have explicit purposes and guidelines for accomplishment finding large groups of friends on Facebook. The Golden Ticket explores what we truly can and cannot achieve computationally, describing the benefits and unexpected challenges of this compelling problem.
This book is intended to serve as an invaluable reference for anyone concerned with the application of wavelets to signal processing. It has evolved from material used to teach "wavelet signal processing" courses in electrical engineering departments at Massachusetts Institute of Technology and Tel Aviv University, as well as applied mathematics departments at the Courant Institute of New York University and École Polytechnique in Paris.Provides a broad perspective on the principles and applications of transient signal processing with waveletsEmphasizes intuitive understanding, while providing the mathematical foundations and description of fast algorithmsNumerous examples of real applications to noise removal, deconvolution, audio and image compression, singularity and edge detection, multifractal analysis, and time-varying frequency measurementsAlgorithms and numerical examples are implemented in Wavelab, which is a Matlab toolbox freely available over the InternetContent is accessible on several level of complexity, depending on the individual reader's needs
New to the Second Edition
Optical flow calculation and video compression algorithmsImage models with bounded variation functionsBayes and Minimax theories for signal estimation200 pages rewritten and most illustrations redrawnMore problems and topics for a graduate course in wavelet signal processing, in engineering and applied mathematics
"Kline is a first-class teacher and an able writer. . . . This is an enlarging and a brilliant book." ― Scientific American "Dr. Morris Kline has succeeded brilliantly in explaining the nature of much that is basic in math, and how it is used in science." ― San Francisco ChronicleSince the major branches of mathematics grew and expanded in conjunction with science, the most effective way to appreciate and understand mathematics is in terms of the study of nature. Unfortunately, the relationship of mathematics to the study of nature is neglected in dry, technique-oriented textbooks, and it has remained for Professor Morris Kline to describe the simultaneous growth of mathematics and the physical sciences in this remarkable book. In a manner that reflects both erudition and enthusiasm, the author provides a stimulating account of the development of basic mathematics from arithmetic, algebra, geometry, and trigonometry, to calculus, differential equations, and the non-Euclidean geometries. At the same time, Dr. Kline shows how mathematics is used in optics, astronomy, motion under the law of gravitation, acoustics, electromagnetism, and other phenomena. Historical and biographical materials are also included, while mathematical notation has been kept to a minimum. This is an excellent presentation of mathematical ideas from the time of the Greeks to the modern era. It will be of great interest to the mathematically inclined high school and college student, as well as to any reader who wants to understand ― perhaps for the first time ― the true greatness of mathematical achievements.
This book is primarily intended for junior-level students who take the courses on 'signals and systems'. It may be useful as a reference text for practicing engineers and scientists who want to acquire some of the concepts required for signal proce- ing. The readers are assumed to know the basics about linear algebra, calculus (on complex numbers, differentiation, and integration), differential equations, Laplace R transform, and MATLAB . Some knowledge about circuit systems will be helpful. Knowledge in signals and systems is crucial to students majoring in Electrical Engineering. The main objective of this book is to make the readers prepared for studying advanced subjects on signal processing, communication, and control by covering from the basic concepts of signals and systems to manual-like introduc- R R tions of how to use the MATLAB and Simulink tools for signal analysis and lter design. The features of this book can be summarized as follows: 1. It not only introduces the four Fourier analysis tools, CTFS (continuous-time Fourier series), CTFT (continuous-time Fourier transform), DFT (discrete-time Fourier transform), and DTFS (discrete-time Fourier series), but also illuminates the relationship among them so that the readers can realize why only the DFT of the four tools is used for practical spectral analysis and why/how it differs from the other ones, and further, think about how to reduce the difference to get better information about the spectral characteristics of signals from the DFT analysis.
The main theme of this book is the homotopy principle for holomorphic mappings from Stein manifolds to the newly introduced class of Oka manifolds. The book contains the first complete account of Oka-Grauert theory and its modern extensions, initiated by Mikhail Gromov and developed in the last decade by the author and his collaborators. Included is the first systematic presentation of the theory of holomorphic automorphisms of complex Euclidean spaces, a survey on Stein neighborhoods, connections between the geometry of Stein surfaces and Seiberg-Witten theory, and a wide variety of applications ranging from classical to contemporary.
This volume offers a collection of non-trivial, unconventional problems that require deep insight and imagination to solve. They cover many topics, including number theory, algebra, combinatorics, geometry and analysis. The problems start as simple exercises and become more difficult as the reader progresses through the book to become challenging enough even for the experienced problem solver. The introductory problems focus on the basic methods and tools while the advanced problems aim to develop problem solving techniques and intuition as well as promote further research in the area. Solutions are included for each problem.
This book contains 104 of the best problems used in the training and testing of the U. S. International Mathematical Olympiad (IMO) team. It is not a collection of very dif?cult, and impenetrable questions. Rather, the book gradually builds students' number-theoretic skills and techniques. The ?rst chapter provides a comprehensive introduction to number theory and its mathematical structures. This chapter can serve as a textbook for a short course in number theory. This work aims to broaden students' view of mathematics and better prepare them for possible participation in various mathematical competitions. It provides in-depth enrichment in important areas of number theory by reorganizing and enhancing students' problem-solving tactics and strategies. The book further stimulates s- dents' interest for the future study of mathematics. In the United States of America, the selection process leading to participation in the International Mathematical Olympiad (IMO) consists of a series of national contests called the American Mathematics Contest 10 (AMC 10), the American Mathematics Contest 12 (AMC 12), the American Invitational Mathematics - amination (AIME), and the United States of America Mathematical Olympiad (USAMO). Participation in the AIME and the USAMO is by invitation only, based on performance in the preceding exams of the sequence. The Mathematical Olympiad Summer Program (MOSP) is a four-week intensive training program for approximately ?fty very promising students who have risen to the top in the American Mathematics Competitions.
This book deals with the mathematical analysis and the numerical approximation of eddy current problems in the time-harmonic case. It takes into account all the most used formulations, placing the problem in a rigorous functional framework.
The subject of this book is probabilistic number theory. In a wide sense probabilistic number theory is part of the analytic number theory, where the methods and ideas of probability theory are used to study the distribution of values of arithmetic objects. This is usually complicated, as it is difficult to say anything about their concrete values. This is why the following problem is usually investigated: given some set, how often do values of an arithmetic object get into this set? It turns out that this frequency follows strict mathematical laws. Here we discover an analogy with quantum mechanics where it is impossible to describe the chaotic behaviour of one particle, but that large numbers of particles obey statistical laws. The objects of investigation of this book are Dirichlet series, and, as the title shows, the main attention is devoted to the Riemann zeta-function. In studying the distribution of values of Dirichlet series the weak convergence of probability measures on different spaces (one of the principle asymptotic probability theory methods) is used. The application of this method was launched by H. Bohr in the third decade of this century and it was implemented in his works together with B. Jessen. Further development of this idea was made in the papers of B. Jessen and A. Wintner, V. Borchsenius and B.
Putnam and Beyond takes the reader on a journey through the world of college mathematics, focusing on some of the most important concepts and results in the theories of polynomials, linear algebra, real analysis in one and several variables, differential equations, coordinate geometry, trigonometry, elementary number theory, combinatorics, and probability. Using the W.L. Putnam Mathematical Competition for undergraduates as an inspiring symbol to build an appropriate math background for graduate studies in pure or applied mathematics, the reader is eased into transitioning from problem-solving at the high school level to the university and beyond, that is, to mathematical research.
* Each chapter systematically presents a single subject within which problems are clustered in every section according to the specific topic.
* The exposition is driven by more than 1100 problems and examples chosen from numerous sources from around the world; many original contributions come from the authors.
* Complete solutions to all problems are given at the end of the book. The source, author, and historical background are cited whenever possible.
This work may be used as a study guide for the Putnam exam, as a text for many different problem-solving courses, and as a source of problems for standard courses in undergraduate mathematics. Putnam and Beyond is organized for self-study by undergraduate and graduate students, as well as teachers and researchers in the physical sciences who wish to to expand their mathematical horizons.
Written in a cookbook style, this book offers solutions using a recipe based approach. Each recipe contains step-by-step instructions followed by an analysis of what was done in each task and other useful information. The cookbook approach means you can dive into whatever recipes you want in no particular order. The CryENGINE3 Cookbook is written to be accessible to all developers currently using the CryENGINE3. It also explores the depth and power of the CryENGINE3 and is a useful guide to follow when becoming familiar with this award winning middle-ware game engine. This book is written with the casual and professional developer in mind. Fundamental knowledge of some Digital Content Creation Tools, like Photoshop and 3d Studio Max is required. The Software Development Kit version of the CryENGINE is used for all examples, so the reader should have a version of the development kit to follow the recipes contained in this book thatIntuitive Probability and Random Processes using MATLAB® is an introduction to probability and random processes that merges theory with practice. Based on the author's belief that only "hands-on" experience with the material can promote intuitive understanding, the approach is to motivate the need for theory using MATLAB examples, followed by theory and analysis, and finally descriptions of "real-world" examples to acquaint the reader with a wide variety of applications. The latter is intended to answer the usual question "Why do we have to study this?" Other salient features are:
*heavy reliance on computer simulation for illustration and student exercises
*the incorporation of MATLAB programs and code segments
*discussion of discrete random variables followed by continuous random variables to minimize confusion
*summary sections at the beginning of each chapter
*in-line equation explanations
*warnings on common errors and pitfalls
*over 750 problems designed to help the reader assimilate and extend the concepts
Intuitive Probability and Random Processes using MATLAB® is intended for undergraduate and first-year graduate students in engineering. The practicing engineer as well as others having the appropriate mathematical background will also benefit from this book.
About the Author
Steven M. Kay is a Professor of Electrical Engineering at the University of Rhode Island and a leading expert in signal processing. He has received the Education Award "for outstanding contributions in education and in writing scholarly books and texts..." from the IEEE Signal Processing society and has been listed as among the 250 most cited researchers in the world in engineering.
This book outlines an elementary, one-semester course thatThis new edition is extensively revised and updated with a refocused layout. In addition to the inclusion of extra exercises, the quality and focus of the exercises in this book has improved, which will help motivate the reader. New features include a discussion of infinite products, and expanded sections on metric spaces, the Baire category theorem, multi-variable functions, and the Gamma function.
Reviews from the first edition:
"This is a dangerous book. Understanding Analysis is so well-written and the development of the theory so well-motivated that exposing students to it could well lead them to expect such excellence in all their textbooks. ... Understanding Analysis is perfectly titled; if your students read it that's what's going to happen. This terrific book will become the text of choice for the single-variable introductory analysis course; take a look at it next time you're preparing that class."
-Steve Kennedy, The Mathematical Association of America, 2001
"Each chapter begins with a discussion section and ends with an epilogue. The discussion serves to motivate the content of the chapter while the epilogue points tantalisingly to more advanced topics. ... I wish I had written this book! The development of the subject follows the tried-and-true path, but the presentation is engaging and challenging. Abbott focuses attention immediately on the topics which make analysis fascinating ... and makes them accessible to an inexperienced audience."
These counterexamples, arranged according to difficulty or sophistication, deal mostly with the part of analysis known as "real variables," starting at the level of calculus. The first half of the book concerns functions of a real variable; topics include the real number system, functions and limits, differentiation, Riemann integration, sequences, infinite series, uniform convergence, and sets and measure on the real axis. The second half, encompassing higher dimensions, examines functions of two variables, plane sets, area, metric and topological spaces, and function spaces. This volume contains much that will prove suitable for students who have not yet completed a first course in calculus, and ample material of interest to more advanced students of analysis as well as graduate students. 12 figures. Bibliography. Index. Errata.
A half-century ago, advanced calculus was a well-de?ned subject at the core of the undergraduate mathematics curriulum. The classic texts of Taylor [19], Buck [1], Widder [21], and Kaplan [9], for example, show some of the ways it was approached. Over time, certain aspects of the course came to be seen as more signi?cant—those seen as giving a rigorous foundation to calculus—and they - came the basis for a new course, an introduction to real analysis, that eventually supplanted advanced calculus in the core. Advanced calculus did not, in the process, become less important, but its role in the curriculum changed. In fact, a bifurcation occurred. In one direction we got c- culus on n-manifolds, a course beyond the practical reach of many undergraduates; in the other, we got calculus in two and three dimensions but still with the theorems of Stokes and Gauss as the goal. The latter course is intended for everyone who has had a year-long introduction to calculus; it often has a name like Calculus III. In my experience, though, it does not manage to accomplish what the old advancedcalculus course did. Multivariable calculusnaturallysplits intothreeparts:(1)severalfunctionsofonevariable,(2)one function of several variables, and (3) several functions of several variables. The ?rst two are well-developed in Calculus III, but the third is really too large and varied to be treated satisfactorily in the time remaining at the end of a semester. To put it another way: Green's theorem ?ts comfortably; Stokes' and Gauss' do not.
This is the first comprehensive monograph on the mathematical theory of the solitaire game "The Tower of Hanoi" which was invented in the 19th century by the French number theorist Édouard Lucas. The book comprises a survey of the historical development from the game's predecessors up to recent research in mathematics and applications in computer science and psychology. Apart from long-standing myths it contains a thorough, largely self-contained presentation of the essential mathematical facts with complete proofs, including also unpublished material. The main objects of research today are the so-called Hanoi graphs and the related Sierpiński graphs. Acknowledging the great popularity of the topic in computer science, algorithms and their correctness proofs form an essential part of the book. In view of the most important practical applications of the Tower of Hanoi and its variants, namely in physics, network theory, and cognitive (neuro)psychology, other related structures and puzzles like, e.g., the "Tower of London", are addressed.
Numerous captivating integer sequences arise along the way, but also many open questions impose themselves. Central among these is the famed Frame-Stewart conjecture. Despite many attempts to decide it and large-scale numerical experiments supporting its truth, it remains unsettled after more than 70 years and thus demonstrates the timeliness of the topic.
Enriched with elaborate illustrations, connections to other puzzles and challenges for the reader in the form of (solved) exercises as well as problems for further exploration, this book is enjoyable reading for students, educators, game enthusiasts and researchers alike. | 677.169 | 1 |
Encyclopedia of Mathematics is a useful reference providing current and accurate information on the subject for high school and college students. Comprehensive coverage includes significant discoveries in mathematics, in addition to definitions of basic terms, thought-provoking essays, and capsule biographies of notable scientists in mathematics - all presenting a wide range of valuable information compiled into a single source. Written in easy-to-understand language, the encyclopedia explains the importance of mathematics to society and includes summaries of notable events throughout history related to the subject. Featuring more than 800 cross-referenced entries, the encyclopedia includes six essays, interspersed throughout the text, that discuss the evolution of algebra and equations, calculus, functions, geometry, probability and statistics, and trigonometry. The encyclopedia also includes three helpful appendixes - bibliographies and Web resources, a chronology of notable discoveries in mathematics, and an extensive list of associations that provide information about mathematics - as well as a comprehensive index. Encyclopedia of Mathematics is an indispensable resource that will meet the specific demands of students, interested laypeople, and professionals who need accurate and straightforward information on historical or current issues in mathematics.
"Sinopsis" puede pertenecer a otra edición de este libro.
About the Author:
James Tanton is the founding director of the St. Mark's Institute of Mathematics.
From Booklist:
Researcher, author, and educator Tanton has compiled this encyclopedia to share his enthusiasm for thinking about and doing mathematics. More than 800 alphabetically arranged entries present a wide variety of mathematical definitions, theorems, historical figures, formulas, examples, charts, and pictures. Many cross-references serve to connect concepts or extend a concept further. A mathematical time line listing major accomplishments is available following the entries, along with a list of current mathematics organizations. The bibliography contains print and Web resources, and the index is helpful in locating terms and concepts.
Each entry varies in length depending on the term, concept, or person being described. Six longer essays describe the history of the branches of mathematics. The writing style is straightforward and readable and sometimes contains parenthetical notes that add background or context. If an entry contains a word or words in capital letters, that term or person is also an entry in the encyclopedia.AVS*##6603
Descripción Facts On File. Estado de conservación: BRAND NEW. BRAND NEW Hardcover - This title is now printed on demand - please allow added time for shipment! A Brand New Quality Book from a Full-Time Bookshop in business since 1992!. Nº de ref. de la librería 2052962 | 677.169 | 1 |
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Geometry in Problems (MSRI Mathematical Circles Library)
Alexander Shen
Classical Euclidean geometry, with all its triangles, circles, and inscribed angles, remains an excellent playground for high-school mathematics students, even if it looks outdated from the professional mathematician's viewpoint. It provides an excellent choice of elegant and natural problems that can be used in a course based on problem solving. The book contains more than 750 (mostly) easy but nontrivial problems in all areas of plane geometry and solutions for most of them, as well as additional problems for self-study (some with hints). Each chapter also provides concise reminders of basic notions used in the chapter, so the book is almost self-contained (although a good textbook and competent teacher are always recommended). More than 450 figures illustrate the problems and their solutions. The book can be used by motivated high-school students, as well as their teachers and parents. After solving the problems in the book the student will have mastered the main notions and methods of plane geometry and, hopefully, will have had fun in the process. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession.
Algebra
Israel M. Gelfand
This book is about algebra. This is a very old science and its gems have lost their charm for us through everyday use. We have tried in this book to refresh them for you. The main part of the book is made up of problems. The best way to deal with them is: Solve the problem by yourself - compare your solution with the solution in the book (if it exists) - go to the next problem. However, if you have difficulties solving a problem (and some of them are quite difficult), you may read the hint or start to read the solution. If there is no solution in the book for some problem, you may skip it (it is not heavily used in the sequel) and return to it later. The book is divided into sections devoted to different topics. Some of them are very short, others are rather long. Of course, you know arithmetic pretty well. However, we shall go through it once more, starting with easy things. 2 Exchange of terms in addition Let's add 3 and 5: 3+5=8. And now change the order: 5+3=8. We get the same result. Adding three apples to five apples is the same as adding five apples to three - apples do not disappear and we get eight of them in both cases. 3 Exchange of terms in multiplication Multiplication has a similar property. But let us first agree on notation.
Algorithms and Programming: Problems and Solutions (Springer Undergraduate Texts in Mathematics and Technology)
Alexander Shen
This text is structured in a problem-solution format that requires the student to think through the programming process. New to the second edition are additional chapters on suffix trees, games and strategies, and Huffman coding as well as an Appendix illustrating the ease of conversion from Pascal to C.
Mr. Alexander F. Hickethier MBA
We II: Revised 12/18/2015 Terrestrial Navigation Calculations worked-out for Master 500 GT through 2nd Mate Unlimited Upon Oceans. Volume II provides an in-depth understanding of Terrestrial Calculation found US Coast Guard Merchant Mariner Examinations through 2nd Mate Unlimited. Chapter 1, Tides and Currents. Chapter 2, Speed by RPM, SOA, Slip and Fuel Consumption Calculations. Chapter 3, Compass Deviation Table Construction, Deviation by Celestial Observation (Amplitude and Azimuth), and Deviation by Terrestrial Observation. Chapter 4, Time Zone Calculations, (Sunrise and Sunset, Time tick, and Estimated Time of Arrival). To see all Marine Navigation Publications offered by this author click on authors name above.
Mr Alexander F. Hickethier MBA
THIS PUBLICATION WAS REVISED ON 09/24/2015, We have developed this series of training manuals to assist the Merchant Mariner in passing the U.S. Coast Guard Licensing Examinations, from Master 500 GT to 2nd Mate Unlimited upon Oceans. THIS PUBLICATION WAS REVISED ON 09/24/2015, A description of the Deck and Stability Workbook follows: Volume I provides an in-depth understanding of Basic Deck and Stability Calculation found US coast Guard Merchant Mariner Examinations through 2nd Mate Unlimited, subject matter areas includes Stowage Calculations, Lumber and Dunnage Stowage, Stowage Factors, Size of lines and Block and Tackle, lifting stress, Anchoring calculations, Humidity and Dew Point Calculations and Rules for bearings, Stability Terminology, Calculating Period of Roll and Estimating GM, Freesurface, Floodable Length Curves, Loll, Final Draft, TPI, Trim, LCG, Final KG, Freeboard Draft, VCG and LCG; Deck cargo Loading D0036-37DG and Double Bottom Tankage. To see all Marine Navigation Publications offered by this author click on authors name above. Revised 9-24-2015
Mr. Alexander F. Hickethier MBA
THIS BOOK WAS REVISED ON 12/18/2015, We III: Celestial Navigation Calculations worked-out for Master 500 GT through 2nd Mate Unlimited Upon Oceans Endorsement. Volume III provides an in-depth understanding of the Celestial Calculation found on the US Coast Guard Merchant Mariner Examinations through 2nd Mate Unlimited. Chapter 1, The Sailings, Parallel, Mid-Latitude, Mercator and Great Circle. Chapter 2 Time Zone calculations, (Sunrise and Sunset, Time Tick, Time of Meridian Transit and Estimated Time of Arrival). Chapter 3, Deviation by Celestial Observation (Amplitude and Azimuth). Chapter 4, Latitude Observations (Local Apparent Noon and Latitude by Polaris). Chapter 5, Sight Reduction - Running Fixes (any body). Chapter 6, Star Identification, and Star and Planet Selection. Chapter 6, Sight Reduction - Running Fixes (any body. ) Chapter 7, Miscellaneous Problems (Ho, He and intercept). To see all Marine Navigation Publications offered by this author click on authors name above. Last Review and update 12-18-2015
Thirteen Movements to Stretch the Body and Make it More Supple, and Guiding and Harmonising Energy to Regulate the Breath: Dao Yin Yang Sheng Gong Foundation Sequences 2 (Dao Yin Yang Shen Gong)
Zhang Guangde
Dao Yin Yang Sheng Gong exercises regulate the body, breathing and the mind to achieve an improved quality of life. This book focuses on concentration, stretching the body to increase suppleness and adjusting the body's energy flow to control respiration.
Featuring clear and detailed explanations of every movement along with photographs and an accompanying DVD, this book will be equally useful to practitioners and beginners interested in taking up Qigong. This is the second volume in Professor Zhang Guangde's foundation Dao Yin Yang Sheng Gong sequences, and the movements described synchronize breathing and concentration by enhancing the body's flow of blood and energy to achieve higher levels of physical and mental agility.
This will be an essential text for Qigong practitioners and students and is accessible enough for the beginning Dao Yin Yang Sheng Gong student.
Zhang Guangde
Professor Zhang Guangde's Dao Yin Yang Sheng Gong forms part of the Chinese national health program and is the most popular form of Qigong practiced in China and across the globe. Professor Zhang has spent nearly forty years creating and developing his system of Qigong. He combines the philosophy of the Classical Chinese text The I Ching with the primary theories of Chinese traditional medicine. His methods have proven highly effective in maintaining health and well-being.
This book contains detailed step-by-step instructions and illustrations to show every step of the foundation sequences, and the accompanying DVD will enable even the least experienced of students to begin to grasp this form of Qigong. The exercises featured promote controlled breathing and concentration. They are formed of calming movements which enhance the flow of blood and energy through the body to achieve increased flexibility and higher levels of physical and mental agility.
This will be an essential text for Qigong practitioners and students and is accessible enough for the beginning Dao Yin Yang Sheng Gong student. | 677.169 | 1 |
Free Community Plan
Curriculum Upgrade Plan 1 2 Geometry 06 Math Course your Book Access login via the Pearson website.
The Home School Learning Network Curriculum Upgrade Plan is an affordable, dynamic online K-12 curriculum service that allows home educators and teachers to access thousands of unit studies, lesson plans, and worksheets.
Not only can you browse over 250 unit studies and thousands of worksheets, you can also choose a structured learning program with 36 weeks of learning for every grade!
As a new edition to the Curriculum Upgrade Plan, HLN now offers full courses in Middle School and High School Math starting August 2011, and courses in Middle School Science will be offered starting August 2012r. As always at HLN, all learning is administered by YOUR Homeschool family!
Please Note: You must be a member of the Curriculum Upgrade Plan to have access to these courses.
Learn more about these courses below!
How Courses Work
Log in to HLN with your subscription and design a program of learning for your family that matches your children's Learning Styles. You may use ALL resources on HLN at all times!
If you decide to follow a Math or Science Course as part of your learning plan, click the "Book Resource Fees" button on the main members page, and pay any fees applicable to the courses you wish your children to take. View Book Resource Fees per course.
You will be assigned a Pearson User ID and password, separate from your HLN User ID and password, and this will be emailed to you within 1 business day.
Click the Courses link on HLN to read and/or print your assignments for the week. Make sure to allot 3-4 days per week, 1-2 hours per day, to complete the work for each course you take.
When instructed on the HLN Course Pages, you will log in to Pearson to access learning resources and videos.
When instructed on the HLN Course Pages, you will download worksheets and tests to take directly from the HLN website.
HLN does NOT grade homework, assignments, or tests from any course. Parents and teachers are provided all grading and answer sheets needed for the course.
Middle School Math
6th Grade Math7th Grade Math consists of a structured approach to a variety of topics such as ratios, percents, equations, inequalities, geometry, graphing and probability.
Test Taking Strategies provide a guide to problem solving approaches
8th Grade Math: Pre-Algebra is designed for the middle school learner and provides a smooth transition from 6th and 7th Grade Math Topics covered include algebraic expressions and integers, solving one-step equations and inequalities, area and volume, and linear functions.
High School Math
Algebra 1 For many students who struggle, math shows up as a collection of rules, formulas, and properties that they learn temporarily, forget quickly, and never use again. Students find mathematics meaningless if they don't see the connections. Prentice Hall Algebra 1For many students who struggle, math shows up as a collection of rules, formulas, and properties that they learn temporarily, forget quickly, and never use again. Students find mathematics meaningless if they don't see the connections. Prentice Hall GeometryAlgebra 2 For many students who struggle, math shows up as a collection of rules, formulas, and properties that they learn temporarily, forget quickly, and never use again. Students find mathematics meaningless if they don't see the connections. Prentice Hall Algebra 2E-Book Resource Fees
The Math Courses currently offered are 36-week courses, and use Pearson Online Books and/or Book Resources. Each Course will provide you access via a special login to learning material, worksheets, and tests to give you a full Middle or High School curriculum in each subject. A subscription to HLN is required at this time for all courses taken
These courses are optional, but they do require a one-time E-Book Resources Fee, as listed below. These fees provide you with a one-year access to the course material and any required e-books and/or e-book resources. If you decide to add a course to your subscription, you will be guided to a page to buy that course's access once you are logged in via your subscription.
Once you are a member of the Curriculum Upgrade Community, you will have access to ala carte curriculum resources such as unit studies, lesson plans, and worksheets. In addition, you can also be guided, week-by-week, through curriculum, either by grade or by theme.
Below is a summary of both our thematic curricula and our structured curriculum.
Thematic Units
Use our weekly unit studies, grade-specific workplans and worksheets to learn about specific themes. Each unit study includes background information, resources, and 6-8 lessons. We provide weekly suggestions for themes, but you may choose any theme from our archive of over 300 themes at any time!
Create a Society Challenge
What type of culture can you imagine? Each week, complete a new aspect of the design of your very own culture! If you choose to submit all 12 projects, we will post it online, and at the end of the summer, HLN will award certificates and prizes to the most creative and well-developed culture! The ability to transmit artwork electronically is recommended for this project.
The projects each week are as follows:
Week 01: What, Where and When?! Making decisions about your culture...
Week 11: Create a Hero! Every culture needs a hero! This week, create your own hero story, and draw a pciture of their heroic act!
Week 12: Final Presentation This week, create your final masterpiece! Create an image that represents as many aspects of the culture as possible!
Art Challenge
Would you like to create a new work of art each week this summer? Try our Art Challenge! Each week, you will be provided with a topic to draw, paint, sculpt or illustrate. If ten submissions are received, we will award a weekly winner, and at the end of the summer, we will award certificates and prizes for the best works of art submitted!
The projects each week are as follows:
Week 01: Let's Warm Up! Copy a Master Who is your favorite artist in history? This week, create your own version of your favorite painting!
Week 02: Portraits Portraits are drawings, paintings, or sculptures of people - this week, create a portrait of yourself or someone you love!
Week 03: Landscapes What does your neighborhood or favorite vacation spot look like? Landscapes can be full of nature or full of city life - you choose!
Week 07: Create a Book Cover! This week, you become an illustrator and create a cover for your favorite book!
Week 08: Tromp L'Oeil Tromp L'Oeil is a French word that means "deceive the eye"! This week, you will learn about and create a Tromp L'Oeil piece of art!
Week 09: Clay Creations This week, you will try your hand at sculpture! You'll learn about clay, and the types of artwork you can create with it!
Week 10: Perspective Perspective is the cornerstone of realistic artwork - what is it, and how can you create a piece of artwork that includes perspective? Join us this week and find out!
Week 11: Collage Dig through magazines and old pictures... rummage through your junk drawers... use your own words, pictures, fabrics... and more!
Week 12: Fantasy Art Do you imagine a creature that does not exist, or what life is like on another planet? This week, create your own fantasy artwork to match your wildest imagination!
Poetry Challenge
Do you love poetry? Each week, learn about a new type of poem, then try your hand at writing your own! Each week, your entries will be entered into our Poetry Challenge. If ten submissions are received, we will award a weekly winner, and at the end of the summer, we will award certificates and prizes for the best poetry submitted!
Week 12: Poetry Theme: Open In our final week of summer, submit your favorite poem, or write one more!
Fiction Writing
Take a look at the photos below... can you see a story in each picture?! Each week, you will be provided with one of these pictures along with some writing ideas... then it is up to YOU to come up with a creative story to go with the picture! Each week, your entries will be entered into our Fiction Writing Challenge. If ten submissions are received, we will award a weekly winner, and at the end of the summer, we will award certificates and prizes for the best fiction submitted!
The projects each week are as follows:
Week 01
Week 05
Week 09
Week 02
Week 06
Week 10
Week 03
Week 07
Week 11
Week 04
Week 08
Week 12
Science and Nature Challenge
Does science make you smile? Summer offers the unique opportunity to have fun with all types of science activities! If ten submissions are received, we will award a weekly winner, and at the end of the summer, we will award certificates and prizes for the science project submitted!
The projects each week are as follows:
Week 01: Special Effects Photography Experiment with your imaging software to create a unique version of a favorite photo!
Week 02: Build a Solar System Model Build a Solar System diorama, mobile, or model!
Week 03: Bird Watching! How many birds can you find and identify in your neighborhood?
Week 04: Create a Terrarium Create your own terrarium from seeds and more!
Movies
These Movie Learning Guides are 6-10 pages, and provide vocabulary, lessons, activities and discussion questions. The actual DVD is not included; you may rent them or purchase them in order to use these guides.
The Sound of Music
Lord of the Rings
Beyond the Lord of the Rings
To Kill a Mockingbird
Chronicles of Narnia
James and the Giant Peach
A Little Princess
Perfect Storm
Stand and Deliver
Moby Dick
Return to Oz
Swiss Family Robinson
Arts and Culture Programs
These Movie Learning Guides are 6-10 pages, and provide vocabulary, lessons, activities and discussion questions. The actual DVD is not included; you may rent them or purchase them in order to use these guides.
Drawing the Marvel Way
Norman Rockwell
Frank Lloyd Wright
The Mind's Eye
Thomas Jefferson
The Hearst Castle
Ken Burns: The Statue of Liberty
Alaska: Spirit of the Wild
Africa
Africa: The Serengeti
Surviving Everest
Into Thin Air
Science Programs
These Movie Learning Guides are 6-10 pages, and provide vocabulary, lessons, activities and discussion questions. The actual DVD is not included; you may rent them or purchase them in order to use these guides. | 677.169 | 1 |
Graphing calculators for LHS students
Here is some information on the Texas Instruments graphing calculators in the TI-83/TI-84 series that Lexington High School strongly recommends at least through the class of 2021. (By strongly recommends they mean you will not be able to complete some classwork, homework and tests without one!) Information on this page is correct to the best of my knowledge, but please visit the LHS Math Dept calculator page for the official word or ask your teacher if you have, or would like to buy, any calculator besides the recommended models.
Acceptable calculators include (roughly in order from cheapest to most expensive; some are discontinued and only available used):
Color models. The TI-84C and TI-84CCE have a color, higher resolution screen and use rechargeable batteries. The batteries are reported to last roughly a week. Today's students are adept with rechargeable batteries, but not always reliable, so keep that in mind, especially since these calculators are often needed on (ahem) tests. The other calculators in this series use AAA batteries which can last roughly a year. The CE is slimmer and lighter, which should make it easier to carry.
Some teachers may allow some other graphing calculators, but there is often keystroke-by-keystroke instruction in class so students with other calculators would be on their own as far as figuring out the keystrokes. Also this year's teacher may allow a different calculator but maybe not next year's teacher. There are a few TI-Nspire calculators that can emulate the TI-84, including the TI-Nspire with Touchpad (With available TI-84 Plus Keypad for compatibility) and the TI-Nspire with Clickpad (with TI-84 Plus Keypad for compatibility), but those don't seem to be recommended by the department. More advanced calculators, especially those with computer algebra systems (CAS), would probably be disallowed by most teachers. TI has a comparison page for its graphing calculators.
All the recommended calculators are also legal on the PSAT, SAT and SAT Math Level 1 and Level 2 subject, and the ACT Math tests. Many of the more advanced calculators with CAS, which are not recommended by LHS, are allowed on some standardized tests like the SAT, for which some of the CAS features might be useful, so you may want to consider getting one of those as well, but only if you will get enough practice with it.
You can buy these locally (e.g. Staples) or online (often roughly half price if bought used). Even better, get them from siblings or graduating friends.
If purchase is a hardship, you can probably get a loaner from the math department.
There are software tools that can do much more than graphing calculators. For graphing there's GeoGebra. WolframAlpha can probably do any problem you throw at it, numerically, algebraically and graphically. I also have made a list of open source software tools. | 677.169 | 1 |
In a few classes, all it takes to pass an test is notice getting, memorization, and remember. On the other hand, exceeding in a very math course normally takes another variety of effort and hard work. You can not just exhibit up for the lecture and check out your teacher "talk" about calculus and . You learn it by accomplishing: being attentive at school, actively researching, and solving math difficulties – even if your instructor hasn't assigned you any. Should you find yourself struggling to carry out perfectly in the math class, then visit very best site for fixing math problems to find out the way you can become an even better math college student.
Low cost math gurus on the net
Math classes abide by a purely natural progression – every one builds upon the knowledge you've acquired and mastered with the previous study course. In case you are locating it hard to follow new ideas at school, pull out your old math notes and review former product to refresh yourself. Ensure that you meet the prerequisites before signing up for a course.
Evaluate Notes The Night time In advance of Course
Despise when a trainer phone calls on you and you've neglected ways to clear up a particular dilemma? Stay away from this moment by examining your math notes. This tends to help you identify which ideas or thoughts you'd wish to go in excess of at school the next day.
The thought of doing homework just about every night may seem annoying, however, if you need to achieve , it can be essential that you consistently follow and grasp the problem-solving approaches. Use your textbook or on the web guides to operate via prime math issues with a weekly foundation – even when you might have no homework assigned.
Utilize the Health supplements That come with Your Textbook
Textbook publishers have enriched modern publications with excess substance (which include CD-ROMs or on the internet modules) which can be utilized to assistance college students gain extra follow in . Many of these materials could also contain a solution or clarification information, that may assist you to with doing work through math troubles all on your own.
Browse Ahead To remain Ahead
If you prefer to minimize your in-class workload or the time you spend on homework, use your spare time soon after faculty or about the weekends to read ahead to your chapters and concepts that can be covered the following time you are in class.
Evaluate Aged Assessments and Classroom Illustrations
The get the job done you are doing at school, for research, and on quizzes can supply clues to what your midterm or closing test will look like. Make use of your outdated assessments and classwork to make a own review information on your forthcoming examination. Glance on the way your instructor frames thoughts – this is likely how they may surface on the test.
Learn how to Operate With the Clock
It is a popular study tip for persons using timed tests; in particular standardized exams. If you only have forty minutes for your 100-point check, you'll be able to optimally spend 4 minutes on every 10-point issue. Get information and facts about how extensive the examination is going to be and which varieties of queries will likely be on it. Then approach to assault the simpler questions initially, leaving you ample time for you to commit around the a lot more hard types.
Maximize your Resources to obtain math research assistance
If you're owning a hard time comprehension principles at school, then be sure to get assist beyond class. Check with your pals to produce a research group and check out your instructor's workplace hrs to go over tough challenges one-on-one. Go to analyze and evaluation sessions whenever your instructor announces them, or use a non-public tutor if you want just one.
Converse To Yourself
Any time you are reviewing difficulties for an examination, consider to explain out loud what method and procedures you accustomed to get the options. These verbal declarations will appear in helpful all through a exam whenever you should recall the measures you ought to take to find a answer. Get more practice by hoping this tactic having a pal.
Use Analyze Guides For Additional Apply
Are your textbook or class notes not assisting you have an understanding of what you needs to be studying at school? Use review guides for standardized tests, such as the ACT, SAT, or DSST, to brush up on outdated substance, or . Examine guides ordinarily come equipped with extensive explanations of the way to clear up a sample difficulty, , so you can normally obtain wherever would be the superior invest in mathdifficulties. | 677.169 | 1 |
Download And Read Books Everywhere, Anywhere
The following are the results of "Mastering Essential Math Skills 2" books in our database. Click on the download or Read Now button to download or read "Mastering Essential Math Skills 2" ebook in pdf, epub, mobi, tuebl and audiobooks.
📝Mastering Essential Math Skills Book 2 Book Synopsis : Veteran sixth-grade teacher Richard Fisher shares his proven system of teaching that motivates students to learn and produces dramatic results. Using Fisher's method, students quickly gain confidence and excitement that leads quickly to success.
📝Mastering Essential Math Skills Book 2 Middle High School With Companion Dvd Book Synopsis : This is the extra-sturdy, non-consumable, Redesigned Library Version with a companion DVD. Through each and every lesson included in the DVD,award-winning teacher, Richard W.Fisher, carefully guides students to mastery.He fully explains each topic, captivating the student's interest as they master each math concept. The student can then easilycomplete the exercises in the book armed with full confidence. An excellent program for students who have struggled with math in the past. Students will master the necessary topics for success in algebra and beyond, and have fun while doing so. A must book/DVD set for every library.
📝The Ultimate Math Survival Guide Part 2 Book Synopsis : Geometry Problem Solving Pre-Algebra These three essentials areas of math skills are absolutely necessary for success in school, college, a career, and in everyday life. INCLUDED WITH THE BOOK IS FREE ACCESS TO ALL OF MATH ESSENTIALS VIDEOS FROM math essentials.net. SELECT FROM HUNDREDS OF LESSONS! PASSWORDS COME INCLUDED WITH THE BOOK. Award-winning teacher and author Richard W. Fisher shares his proven system of teaching that motivates students to learn and produces dramatic results. Using Mr. Fisher's method, students rapidly gain confidence and excitement that quickly lead to success. *Presented in a simple format that everyone can easily understand. *Each lesson flows smoothly and logically to the next. *Each lesson is short, concise, and straight to the point. *Each new topic is clearly explained. *Lots of examples with step-by-step solutions. *Each lesson includes valuable helpful hints. Review is built into each lesson. Students will retain what they have learned. *Each lesson includes Problem Solving. This ensures that students will learn to apply their knowledge to real-life situations. *Final tests to measure progress. *Includes solutions for each lesson and a Math Resource Center.
📝Mastering Essential Math Skills Pre algebra Concepts With Companion Dvd Book Synopsis : This is the new extra-sturdy, non-consumable Library Version with a companion DVD. This set is a must for students about to take Algebra I. It is also excellent for those struggling with algebra. Throughout the DVD, award-winning teacher, Richard W. Fisher carefully guides students through each and every topic necessary for success in algebra. His clear explanations and encouraging style quickly captivates student's interest and he makes learning these sophisticated topics fun and easy. After each of his carefully crafted DVD lessons, students are ready to complete the lessons in the book armed with full confidence. Even students who have struggled with math will find the DVD and book lessons fun and exciting. This is a must book/DVD set for every library.
📝Shaping the Future with Math Science and Technology Book Synopsis : Shaping the Future with Math, Science, and Technology examines how ingenuity, creativity, and teamwork skills are part of an intellectual toolbox associated with math, science, and technology. The book provides new ideas, proven processes, practical tools, and examples useful to educators who want to encourage students to solve problems and express themselves in imaginative ways. The development of a technological knowledge-based economy depends on the development of educational systems that allow schools, teachers, and students of diverse capabilities, backgrounds and learning preferences do better with both content and imaginative problem solving. This book makes the case that it is, indeed, possible to educate our way to a better economy and a better future. Paying attention to 21st century approaches and skills can help accomplish those goals.
📝Essentials of Assessment Report Writing Book Synopsis : Instructive guide to preparing informative and accurate assessment reports for a variety of individuals and settings Assessment reports are central to the diagnostic process and are used to inform parents, clients, and clinicians, among others, about academic problems, personality functioning, neuropsychological strengths and weaknesses, behavioral problems, and the like. Essentials of Assessment Report Writing provides handy, quick-reference information, using the popular Essentials format, for preparing effective assessment reports. This book is designed to help busy mental health professionals quickly acquire the knowledge and skills they need to write effective psychological assessment reports. Each concise chapter features numerous callout boxes highlighting key concepts, bulleted points, and extensive illustrative material, as well as test questions that help you gauge and reinforce your grasp of the information covered. This practical guide focuses on efficiently and effectively communicating referral and background information, appearance and behavioral observations, test results and interpretation, summary and diagnostic impressions, and treatment recommendations. The authors provide examples of both good and bad case report writing and highlight ethical issues and topics relevant to presenting feedback. Essentials of Assessment Report Writing is the only pocket reference illustrating how to prepare an effective assessment report.
📝Practical Problems in Mathematics for Heating and Cooling Technicians Book Synopsis : Practical Problems for Heating And Cooling Technicians, 6th Edition, provides students with the essential quantitative skills they need for success in the HVAC field. This text presents mathematical theories in concise, easy to understand segments, and reinforces each concept with multiple examples and practice problems from real-world HVAC tasks, including the latest in geothermal systems, and zone heating and cooling. Loaded with helpful visual features and study aids, Practical Problems for Heating And Cooling Technicians, 6th Edition puts key information at the students' fingertips with critical formula conversion charts, a glossary of updated HVAC-specific terms, and hands-on exercises designed to build confidence and comfort with basic mathematical skills. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
📝Woodcock Johnson IV Book Synopsis : Includes online access to new, customizable WJ IV score tables, graphs, and forms for clinicians Woodcock-Johnson IV: Reports, Recommendations, and Strategies offers psychologists, clinicians, and educators an essential resource for preparing and writing psychological and educational reports after administering the Woodcock-Johnson IV. Written by Drs. Nancy Mather and Lynne E. Jaffe, this text enhances comprehension and use of this instrument and its many interpretive features. This book offers helpful information for understanding and using the WJ IV scores, provides tips to facilitate interpretation of test results, and includes sample diagnostic reports of students with various educational needs from kindergarten to the postsecondary level. The book also provides a wide variety of recommendations for cognitive abilities; oral language; and the achievement areas of reading, written language, and mathematics. It also provides guidelines for evaluators and recommendations focused on special populations, such as sensory impairments, autism, English Language Learners, and gifted and twice exceptional students, as well as recommendations for the use of assistive technology. The final section provides descriptions of the academic and behavioral strategies mentioned in the reports and recommendations. The unique access code included with each book allows access to downloadable, easy-to-customize score tables, graphs, and forms. This essential guide Facilitates the use and interpretation of the WJ IV Tests of Cognitive Abilities, Tests of Oral Language, and Tests of Achievement Explains scores and various interpretive features Offers a variety of types of diagnostic reports Provides a wide variety of educational recommendations and evidence-based strategies
📝Essential Math Skills Skills and Activities for Proficiency in Third Grade Book Synopsis : Transform your 3rd grade math outcomes with these 95 engaging activities. Each activity supports an essential math skill. Created to support the Common Core and other national standards, this resource is a great tool for educators.
📝Unleashing the Positive Power of Differences Book Synopsis : Move from entrenched differences to common goals! All too often, education initiatives collapse because leaders fail to learn from the concerns of those charged with implementation. Acclaimed education coach Jane Kise demonstrates how polarity thinking—a powerful approach to bridging differences—can help organizations shift from conflict to collaboration. Readers will find: Ways to recognize polarities, map the positive and negative aspects, and channel energy wasted on disagreement toward a greater common purpose Tools for introducing and working with polarities Polarity mapping to help leaders improve processes for leading change and creating buy-in Ways to use polarity with students as a framework for higher-level thinking | 677.169 | 1 |
06 Jun 2013
views:1315224Curricular content
Each state sets its own curricular standards and details are usually set by each local school district. Although there are no federal standards, 45 states have agreed to base their curricula on the Common Core State Standards in mathematics beginning in 2015. The National Council of Teachers of Mathematics(NCTM) published educational recommendations in mathematics education in 1991 and 2000 which have been highly influential, describing mathematical knowledge, skills and pedagogical emphases from kindergarten through high school. The 2006 NCTM Curriculum Focal Points have also been influential for its recommendations of the most important mathematical topics for each grade level through grade 8.
What is a function? | Functions and their graphs | Algebra II | Khan Academy
What is a function? | Functions and their graphs | Algebra II | Khan AcademyAlgebra Basics: What Are Functions? - Math Antics19:35
Definition of Function in Hindi
Definition of Function in Hindi
Definition of Function in Hindi
This video helps you to understand:
1. what is function..?
2. what is domain and codomain of function..?
3. graph of a function...
42:12
Class 12 XII Maths CBSE Functions 01
Class 12 XII Maths CBSE Functions 01
Class 12 XII Maths CBSE Functions 01
6:57:18
Function By GB Sir Part 1 Of 6 | IITJEE-(Mains+Advance)
Function By GB Sir Part 1 Of 6 | IITJEE-(Mains+Advance)there is always your choice)
So Subscribe me and I will upload all video lectures soon...
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7:36
Relations & Functions
Relations & Functions
Relations & Functions
This video looks at relations and functions. It includes six examples of determining whether a relation is a function, using the vertical line test and by looking for repeated x values.
Function-1(Definition and basic concept)
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1:10:11
Function Boiler Room Berlin Live Set
Function Boiler Room Berlin Live Set4:2112:40
❤︎² How to Find the Domain of Any Function (mathbff)
❤︎² How to Find the Domain of Any Function (mathbff) functi...
published: 06 Jun 2013 easypublished: 02 Dec 2008
Definition of Function in Hindi
This video helps you to understand:
1. what is function..?
2. what is domain and codomain of function..?
3. graph of a function...
published: 28 Jul 2015
Class 12 XII Maths CBSE Functions 01
published: 30 Jan 2015published: 07 Oct 2017
Relations & Functions
This video looks at relations and functions. It includes six examples of determining whether a relation is a function, using the vertical line test and by looking for repeated x values....
What is a function? | Functions and their graphs | Algebra II | Khan Academy
Watch the next lesson"Function video Lecture of Maths for IIT-JEEMain and Advanced by GB Sir. GB Sir is known for his focused and simplified JEE teaching to bring to students an ea... high what4...published: 09 May 2015
Class 12 XII Maths CBSE Functions 01
published: 30 Jan 2015 th...
Function in TweakFM (Ostgut Ton, Sandwell District, Synewave)published: 22 Sep 2012
SPM - Form 5 - Modern Math - Graph Function
You will learn all the different pattern graph in this video. At the same time, you will be understand how to differentiate the function of the graph and their shape. Hope this video will help you better understanding on this chapter.
Python Tutorial for Beginners 8: Functions...
published: 17 May 2017
Function Operations +−×÷ [fbt] (Operations on Functions)
"Function video Lecture of Maths for IIT-JEEMain and Advanced by GB Sir. GB Sir is known for his focused and simplified JEE teaching to bring to students an ea...
"Function lear... propert......SPM - Form 5 - Modern Math - Graph Function
You will learn all the different pattern graph in this video. At the same time, you will be understand how to differentiate the function of the graph and their ...
You will learn all the different pattern graph in this video. At the same time, you will be understand how to differentiate the function of the graph and their shape. Hope this video will help you better understanding on this chapter.
You will learn all the different pattern graph in this video. At the same time, you will be understand how to differentiate the function of the graph and their shape. Hope this video will help you better understanding on this chapter.
In this PythonBeginner Tutorial, we willLearn More at mathantics.com
Visit for more Free math videos and7:36
Relations & Functions
This video looks at relations and functions. It includes six examples of determining whet
Pre We out here tryna function, bitch You're f-cking off my high, get up out my mix You're messing up my vibe, I'm trying to get some crackers Put'em in my ride, take her to the Ritz I'm tossing this sloppy, offa my broccoli, Bacardi One fifty one out my body, I'm about that green like wasabi Like... we robbin', we bouncin, back the f-ck off me Getting money my hobby, not getting money is nothing The rappers I listen to is E-40 and Pac I'm having my revenue playa having this guap I'm on my fly big n-gga shit I'm stayin laced and groomed I spray myself with sucka repellent my n-gga, not perfume You think you God, I can sell it a hustler think I can't Gift of Gab sell the White House black paint Word candy SLANG I'm thinkin bout takin a million dollar insurance policy out on my mouthpiece Pre Hey bitch, show cake bitch It's uncle Earl and the HBK bitch Mention the gang they already know that we ballin I'm coming straight out the Rich I got family down in New Orleans Where you from, you say you're lying Out here we say that you jawsin You probably thought this never would happen my n-ggas been called that Alcoholic, sippin on that liquor, oh I'm drunk as hell F-ckin witta a lil bitch over in Vallejo Got a whole pack of pre-rolled Young L's And I'm never down to uno, pockets on sumo Haters respect the pedigree, ballin heavily A phony homie, I never be for methamphetamine That means it's crack ho, young G, hotter than Tabasco I smash hoes, collect two hunnid and pass GO My flow so Lamborghini, yo shit's so Rav 4 Now you understand why everything I do I gas ho, Suzy, n-gga Pre-Chorus: Hey, hey bitch, try this! Guarantee turn a square to a bop bitch You ain't down, bye bitch I ain't got time for playin, I'm just saying man We out here tryna function, we out here tryna function We out here tryna function, we out here tryna function I ain't got time for playin, I'm just sayingOperation theaters, intensive care units and recovery wards with other super specialty services will function in the building adjacent to the Omandurar medical college. The new government hospital shall provide for the need of trauma care facility in the locality ... ....
However, the direction of associations remains unclear and we still have limited understanding of how associations might change across the life course ... His research is focused on the association between physical activity and cognitive function across the life course, determining whether physical activity and cognitive function have the same association throughout the life course, and mechanisms and moderators of this association....
A group of cane farmers created a flutter when they raised slogans against the State Government at the function to inaugurate the cane crushing period at the SalemCooperativeSugarMills in Mohanur town near here on Saturday ... They also demanded permission for speaking at the function, but were refused....
George Okoh in Makurdi... The NULGE boss said autonomy at the third tier of government would make council areas functional ... What we have now is that there is a disconnection between the people and their leaders but our agitation is focused on having functional democratic government directly elected by the people" ... ....
Q. I need to know the law for kids under 16 for night time. I need to know if it is OK for them to drive to school functions like practices and then straight home... A. Drivers under the age of 16 can only drive a half hour before sunrise and a half an hour after sunrise when driving anywhere except from a school function if they get a restricted school attendance driving permit ... and 9.00 p.m ... ....
Puducherry chief minister V Narayanasamy has appealed to Tamil Nadu ministers not to surrender their rights and authority, bestowed by the electorate, by allowing the governor to interfere in the day-to-day functioning of the government ... "We have written many letters to the Prime Minister and Unionhome minister (on the arbitrary functioning of the lt governor). He stressed that the Aare Ona Kakanfo title is not a title of a town or its functions limited to a town or city, but for the entire Yoruba land, adding that the title shares the same origin with Ilaro ... .... | 677.169 | 1 |
Representations of Functions Practice
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This practice activity requires students to represent functions in word problems, equations, tables, and graphs. It provides students with one of the representations and then asks them to come up with the other 3. Students also need to evaluate functions and write their own from a provided statement. | 677.169 | 1 |
Pages
Friday, December 7, 2012
Friday, December 7, 2012
Prepare for the Absolute Value Equations and Inequalities Assessment. There are a variety of ways to do that including, but not limited to: review the online pre-assessment; review your notebook and/or the openers and lessons posted on the blog; review the video, work some practice problems in your textbook or that you find online. You can, of course, also get help from me, another math teacher, a teacher in the Study Center, a peer tutor in the Study Center, or a parent, sibling or friend. Do whatever works best for you, but make sure you're prepared. The expectation is that you should all be able to do very well on this assessment.
Sometime in the next two weeks I would really, really, really appreciate it if you would fill out this evaluation of me. | 677.169 | 1 |
Technical Mathematics (2nd Edition)
Author:Dale Ewen - Joan S. Gary - James E. Trefzger
ISBN 13:9780130488107
ISBN 10:130488100
Edition:2
Publisher:Pearson
Publication Date:2004-06-14
Format:Hardcover
Pages:832
List Price:$255.20
 
 
This book provides readers with necessary mathematics skills. Mathematics provides the essential framework for and is the basic language of all the technologies. Mathematical, problem-solving, and critical thinking skills are crucial to understanding the changing face of technology. It presents the following major areas: fundamental concepts and measurement; fundamental algebraic concepts; exponential and logarithmic functions; right-triangle trigonometry; the trigonometric functions with formulas and identities; complex numbers; matrices; polynomial and rational functions; basic statistics for process control; and analytic geometry. An excellent learning aid and resource tool for engineers, especially computer software, hardware, and peripheral manufacturers. Its comprehensive appendices make this an excellent desktop reference. | 677.169 | 1 |
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Symmetry: An Introduction to Group Theory and its Application is an eight-chapter text that covers the fundamental bases, the development of the theoretical and experimental aspects of the group theory.
Chapter 1 deals with the elementary concepts and definitions, while Chapter 2 provides the necessary theory of vector spaces. Chapters 3 and 4 are devoted to an opportunity of actually working with groups and representations until the ideas already introduced are fully assimilated. Chapter 5 looks into the more formal theory of irreducible representations, while Chapter 6 is concerned largely with quadratic forms, illustrated by applications to crystal properties and to molecular vibrations. Chapter 7 surveys the symmetry properties of functions, with special emphasis on the eigenvalue equation in quantum mechanics. Chapter 8 covers more advanced applications, including the detailed analysis of tensor properties and tensor operators.
This book is of great value to mathematicians, and math teachers and students. | 677.169 | 1 |
Bringing the material up to date to reflect modern applications, Algebraic Number Theory, Second Edition has been completely rewritten and reorganized to incorporate a new style, methodology, and presentation. This edition focuses on integral domains, ideals, and unique factorization in the first chapter; field extensions in the second chapter; and class groups in the third chapter. Applications are now collected in chapter four and at the end of chapter five, where primality testing is highlighted as an application of the Kronecker–Weber theorem. In chapter five, the sections on ideal decomposition in number fields have been more evenly distributed. The final chapter continues to cover reciprocity laws.
This introduction to algebraic number theory discusses the classical concepts from the viewpoint of Arakelov theory. The treatment of class theory is particularly rich in illustrating complements, offering hints for further study, and providing concrete examples. | 677.169 | 1 |
You will need to be good at algebra before you start the course. Throughout the course you will learn many new mathematical techniques and apply them to a variety of practical situations. Studying Mathematics will support the study of several other su...
You will need to be good at algebra before you start the course. Throughout the course you will learn many new mathematical techniques and apply them to a variety of practical situations. Studying Mathematics will support the study of several other su | 677.169 | 1 |
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Calculus Bundle. Updated 8 - 19 - 2017
I have bundled 15 of my most popular Calculus activities and resources together to keep your students engaged, having fun, and on track, and additionally to make your lesson and assessment planning easier. With over 245 pages including over 200 Task Cards, Station Cards, Guided notes, scores of Quizzes and Homework Problems, Games, Graphic Organizers, Cheat Sheets, QR's, Answer Keys, Student Response Sheets and much more, you will have plenty of new activities, resources and assessments to help you plan many lessons throughout the year. Plus I have included a Bonus resource you will find very helpful!
These activities are designed for Calculus 1, AP Calculus AB, Honors Calculus, and first semester AP Calculus BC | 677.169 | 1 |
CALCULUS Advice
Showing 1 to 1 of 1
This course lays the foundation for calculus and has really important real-world applications. For example, the derivative of a position function is the velocity function, the derivative of which is the acceleration function. Optimization is another cool application of the skills you learn in calculus. These skills have wide-ranging applications. Calculus is certainly challenging, but it's worth it for people who are willing to work to master it.
Course highlights:
The highlights of this course was the many real-world applications of the skills we learned. The main two skills are differentiation and integration, which are reciprocal functions that "undo" each other, a relationship similar to that of multiplication and division.
Hours per week:
6-8 hours
Advice for students:
Do as many different problems as you can, even more than is assigned. The more practice, the better. The professor will teach you the skills you need to work out a type of problem, but there are so many different sub-types of problems that the professor cannot show you all the different ways a problem can show up. Your best bet is to get comfortable with as many different types of problems as possible. There are some that have certain tricks to them so don't be afraid to get help from your professor, TA, tutor, classmates, the Internet, etc. | 677.169 | 1 |
3/1 -- Beginners: Modular Arithmetic, Part II
We will continue to discuss modular arithmetic this week. More specifically, we will learn about congruency classes and how we can use modular arithmetic when discussing powers. Please redo the combinatorics quiz from last week and bring in your solutions. | 677.169 | 1 |
Math
(Pre) Algebra People who understand algebra find it easy, believe it or not. The people who struggle and suffer with it and learn to hate it are the ones trying to skip understanding and just memorize procedures. It has nothing to do with talent, and everything to do with whether you understand. Every good algebra teacher on Earth recommends understanding and tries to steer you away from the agony of memorizing meaningless procedures for exams. That's why awesome sites such as Khan Academy with it's Pre-Algebra and Algebra sections. Also, let YouTube be a help to you!
Geometry/Trigonometry
If you can figure out algebra, maybe geometry and trigonometry is a challenge for you. Luckily, Khan Academy is there to help you with both geometry and trig! Also, you can always try Math is Fun! We also can't forget Math Geo and Math Trig!
(Pre) Calculus
The word "calculus" comes from "rock", and also means a stone formed in a body. People in ancient times did arithmetic with piles of stones, so a particular method of computation in mathematics came to be known as calculus. Therefore, we all know very well that even PreCalculus is a tough subject. Khan Academy as usual is here to save the day on that! There's also the MIT Coursebook, where the greatest of the mathematicians came together to make this coursework open for you! Also, WikiBooks is a great place to find simple explanations for problem structure!
Statistics
Statistics is the study of the collection, organization, analysis, interpretation and presentation of data. While many see it as the easiest form of math, it is still difficult to get through until you get the hang of things. We have Khan Academy Stats here to save the day and bring you to that A+ you want! Math is Fun brings out great assistance for the grade, all for FREE | 677.169 | 1 |
TI-83/84: Common Error Messages
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This is a quick reference sheet I give my students so they don't have to stop class and ask me for help when they receive an ERROR message on their calculators. It shows the most common error messages that they run into during pre-calculus (WINDOW RANGE, INVALID DIM, DIM MISMATCH, SYNTAX) and how to solve it. Many of my students are intimidated by the many buttons of the TI-83/84, especially because it is so much less intuitive than their calculator apps on their phones. They've found this to be a very useful exercise and I have them keep it in the front of their binders. | 677.169 | 1 |
Welcome to Calculus! I hope that you are enjoying a relaxing and fun summer!
In Calculus, we will begin the year with a brief review of Trigonometry. Then we move quickly into working with Pre-Calculus and begin to cover lots of new material from there. We also will work a lot with derivatives, as well as antiderivatives. In addition, there will be a heavy emphasis on applying the concepts covered to real-world situations focusing on Integrals. The material in Calculus will be new and challenging, and we will have fun working through it together!
The supplies that are needed for Calculus are listed below. They should all be a part of your general supply list. Please bring your supplies in as soon as possible and have them all labeled with your first and last name, as well as your homeroom teacher.
? Spiral notebook. (3-Subject)
? Pencils
? Calculator will be provided
? Protractor / Ruler / 1/4" Graph Paper
? Organizational Binder (We will have many handouts to put in.)
I hope you enjoy your last week of summer! I look forward to meeting you soon, as we embark on an exciting year together! | 677.169 | 1 |
Mathematica includes functions for performing a variety of specific algebraic transformations. Some are algorithmically straightforward; others include highly sophisticated algorithms, many developed and refined at Wolfram Research. | 677.169 | 1 |
The Importance of Applied Mathematics
These principles in arithmetic have been esteemed in pc research to be able to offer software applications oriented toward to quickly and effective information conversion. By data transformation, we suggest any information which can be efficaciously refined in order to achieve a pre-established goal. So, the major change happened as soon as when theorems in arithmetic were properly translated into advanced program, applying pleasant and user friendly interfaces. Actually, any software request, having been implemented to do a certain rationale-based task, is an advanced representation of a sample utilized in arithmetic or in economy.
An easy means to fix raising arithmetic skills of United Claims pupils in the area of rating could be accomplished quickly if we were to combine our businessmen with this educators in a mixed work to improve our bodies of rating mymathlab the out-dated British program to metrics. To date, the United States, Liberia, and Myanmar (Burma), are the only real three nations on the planet where in actuality the English system of rating is the principal program of measurement. Every other place in the world employs the full process of measurement.
Within the United Claims everyday rating based on the old English system is typically taught and applied, while metrics is taught as another means of measurement. Consequently, available earth, technicians, architects and experts who work for organizations who cope with international places should record their proportions in two programs, the British system and the more acceptable world-wide metric system. The United States hasn't confined the usage of metrics. The issue lies in the fact that producers have not made a genuine energy to make the move to metrics, while other nations have inked so properly and with a relatively smooth transition by simply appearance and labeling goods in metrics. The reluctance to change to metrics is mind boggling. In the United States the liter and two liter container of soda seems to be the sole change that's been made. No body looked to mind that transition. The change was not so difficult for many Americans to accept. It is the sort of transformation that should be completed with all sold products. Unfortunately, further progress in making the modify hasn't been attempted.
The major problem that maybe not converting to the metrics process presents to training is that all pupils in the United Claims should learn both the British and metric system. However, since the full process isn't generally used, it's rarely mastered. For most students, and specifically for students who've learning deficiencies, metrics stays a non-mastered skill. For when pupils are shown metrics, they've little or no frequent utilization of the machine and, therefore, find it also harder to understand and internalize. The change to metrics is extended overdue. Now is the full time for teachers to insist that the full process of rating be the primary program of measurement used in the United Claims, thus enabling teachers and students to focus entirely on the more common full program of measurement.
As an example, linear development algorithms have already been effectively changed into intensive software providing profitability answers for different demands. To put it differently, methods are explored as cutting-edge answers in computer technology, as an example linear coding examples have a totally different price in pc science. As a matter of fact, these cases are optimized models, changed into superior platforms and interfaces; A advanced algorithm has exactly the same starting place as a mathematics product, yet, the variations are apparent whenever we examine effects and effectiveness parameters. In the form of a linear programming software software, consumers can lower a really demanding and painstaking method based on extended calculation.
The advantages of linear programming solutions are unquestionable. Yet the implementation a computer software software counting on linear development models has assigned to algorithmic approach a wider understanding. And by larger availability, we mean the fact that the simplex process or the changed process has been aligned to consumers who need the last result of the design, and are less interested in how a computerized program has done the rationale. The system, which could demonstrate to them the way to optimum income, is the only thing that matters. Additionally, linear development solver takes over the most difficult part of the method, creating the LP optimization an easy-to-access solution. Combined with easy-access function, a computer-based solution using the simplex strategy or the modified technique could be customized for various task domains. Even though transportation and logistics, engineering, or computer sciences take advantage of the exact same algorithm, the functioning theory is somehow modified to the specific top features of the realm, considering the truth that profit is differently approached by various people. | 677.169 | 1 |
Help for Students with the Maths Step Exam
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This course is aimed primarily at students who will be sitting the STEP Mathematics exams in June, although we welcome any student who is interested in practising STEP questions. The aims of the course are as follows
To familiarise students with the style of questions used in the STEP exams.
To teach some of the common techniques and strategies that are useful for STEP questions.
To advise on the required level of detail and clarity of presentation when answering questions.
To advise on other aspects of STEP exam technique.
The course is divided into a number of broad topic-areas (Calculus, Trigonometry, Vectors etc.) with a different focus each week. The materials used in the course follow the "Road Map" format developed by Michael Gibson and are provided free of charge – these materials may also be available via your local FMSP coordinator and we strongly recommend that you find out what provision the FMSP offers in your area before enrolling on this course. In a typical session we will look at 3 STEP questions using the Road Map format. You will be advised to study other questions in your own time in between sessions. Each session will be 1 hour 30 minutes long and the course will start in February 2017. | 677.169 | 1 |
Our 302-page guide to Number Properties & Algebra assumes that you know nothing. Everything is presented as if you haven't done math in years. To keep things simple, we discuss math in language that's easy to understand and focus on smart strategies for every level of material. No matter how difficult a topic may be, we walk you through each concept, step by step, to ensure that everything makes sense.
Need help with the advanced stuff?
At the ends of our chapters, you'll find a treatment of every rare and advanced concept tested by the GRE involving Integer Properties, Exponents, Roots, Functions, Sequences, or Algebra. You won't find these concepts discussed anywhere else. Master Key to the GRE is the only resource that covers them. We know that some of you only need help solving the most difficult questions -- the questions that determine who scores above the 90th percentile. We've made sure that our materials teach everything, so that students who need superstar scores get all the support they need.
Tricks and shortcuts you won't find anywhere else.
Time is a major concern for most test-takers, so we've included every time-saving strategy out there to help you "beat the clock". We don't care how well you think you know math. These shortcuts will save you valuable minutes, no matter what your current skill level may be.
Over 250 practice questions.
Like our lessons, our practice questions are sorted by difficulty and topic so that you can focus on material that is right for you. Nearly a quarter of these questions involve the most rare or advanced topics tested by the GRE. So if you're looking for a lot of help with diabolical fare, you'll find it here. Along the way, we let you know which topics are commonly tested and which ones are not so that you can determine for yourself which topics are worth your time.
Animated video solutions.
Every practice question comes with an animated video solution. Short, written explanations are typically insufficient for students who find math challenging. By providing video solutions, we can talk you through our practice problems, every step of the way, so that you can follow along easily and see where your solution went wrong.
Let g be a continuous real-valued function defined on with the
following properties(pre-algebra through AP calculus), computer science, AP statistics, and
..Number and Quantity, Algebra, Functions,(2) Volume of a pyramid = 1/3 * base area * heightWorking with expressions and
functions: manipulating algebraic ..
Number properties (1) Product of any two numbers = Product of their HCF and
LCM
Use the algebraic approach to solve this week's
challenge.
Srinivasa Iyengar Ramanujan FRS was an Indian mathematical genius and
autodidact who lived during the British RajIn Pearson's biography of Galton (Pearson [1924, VolPolygons are named according to the number of sides they
have...Algebra: These questions include factoring and simplifying algebraic ..22 ...
Content areas include arithmetic, algebra, geometry and data analysisII, pthe GRE tends to test the same triangles over and over, you just need to
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calculate interest rates without a calculator that has an exponents key? ...I have a master degree...
What is the volume of the solid in xyz-space bounded by
the surfacesDrill 1Number Properties.......68% and Calculus..use of a number of fundamental numerical scales, such as mass, time,
.. b2ff6ad845 | 677.169 | 1 |
Calculus by Stanley I Grossman
Stressing an intuitive instead of rigorous/formal method of calculus, this student-oriented textual content covers the entire commonplace themes. Integration of chapter-end machine routines and calculator workouts through the textual content and a graphing calculator appendix know the impression of expertise on calculus. An abundance of attention-grabbing purposes from engineering, physics, biology, chemistry, economics, astronomy, drugs and natural arithmetic indicates the far-reaching relevance of calculus. An algebra evaluation is supplied in bankruptcy 10. various examples during the textual content comprise all of the algebraic steps, with key steps highlighted in color, had to entire the answer. The examples are complimented via greater than 7,000 part and bankruptcy routines that includes drill, program, calculator, ''show/prove/disprove'', and problem difficulties. each one challenge set starts off with ''Self-quiz'' inquiries to support scholars review their realizing of uncomplicated principles within the part. the advance of calculus is printed in vast old notes. Biographical sketches impart details on popular mathematicians
Here's an summary of contemporary computational stabilization tools for linear inversion, with purposes to a number of difficulties in audio processing, clinical imaging, seismology, astronomy, and different parts. Rank-deficient difficulties contain matrices which are precisely or approximately rank poor. Such difficulties usually come up in reference to noise suppression and different difficulties the place the target is to suppress undesirable disturbances of given measurements.
This publication leads readers from a uncomplicated starting place to a complicated point figuring out of algebra, common sense and combinatorics. excellent for graduate or PhD mathematical-science scholars searching for assist in realizing the basics of the subject, it additionally explores extra particular components corresponding to invariant concept of finite teams, version thought, and enumerative combinatorics.
X - 3)2 + ( y + 2)2 = 5 ; (4, 0) 20. (x + 2)2 + ( y - 3)2 = 9; (0, 3 + y'S) 22. (a) Find all points on the line y = 0 that are twice as far from (0, 0) as from ( 12, 0). (b) Show that the set of all points in the plane that are twice as far from (0, 0) as from ( 12, 0) is a circle. *23. Suppose that x2 + y2 + Ax + By + C = 0 and x2 + y2 + ax + by + c = 0 are different circles that meet at two distinct points. Show that the line through those two points of inter section has the equation *24. (A - a)x + (B - b)y + (C - c) Suppose the point (a, b) is on the circle x2 + y2 **25. | 677.169 | 1 |
Download and read online The Humongous Book of Statistics Problems in PDF and EPUB Following the successful, 'The Humongous Books', in calculus and algebra, bestselling author Mike Kelley takes a typical statistics workbook, full of solved problems, and writes notes in the margins, adding missing steps and simplifying concepts and solutions. By learning how to interpret and solve problems as they are presented in statistics courses, students prepare to solve those difficult problems that were never discussed in class but are always on exams. - With annotated notes and explanations of missing steps throughout, like no other statistics workbook on the market - An award-winning former math teacher whose website (calculus-help.com) reaches thousands every month, providing exposure for all his books
Download and read online The Humongous Book of SAT Math Problems in PDF and EPUB The Humongous Books are typically 464 pages and contain 650 to 1,000 completed problems. They are designed to look like textbooks with problems and answers that have had handwritten notes added by a mentor, peer, or previous student who clarified the process, formula, and steps that went into solving the problem. The Humongous Book of SAT Math Problems takes a typical SAT study guide of solved math problems and provides easy-to-follow margin notes that add missing steps and simplify the solutions, thereby preparing students to solve all types of problems that appear in both levels of the SAT math exam. | 677.169 | 1 |
Algebra I includes the study of properties and operations of the real number system; evaluating rational algebraic expressions; solving and graphing first degree equations and inequalities; translating word problems into equations; operations with and factoring of polynomials; solving simple quadratic equations; and simplifying radical expressions.Jr. High students will receive High School Algebra credit with a grade of 75% or better.
Please see the links below for video help on specific Units and Targets.
I am available every morning from 7:00 - 7:55 am for students that would like extra help. Mr. Fortier is available every afternoon until 4:15 and Mrs. Smart is available every afternoon until 4:30 pm.
Announcements
WELCOME TO THE 2017-2018 SCHOOL YEAR
I am excited and looking forward to a fantastic school year! Please make sure to contact me if you have any questions or concerns at any time throughout the school year! Here are a couple of quote gems from Albert Einstein...
"A person who never made a mistake never tried anything new." Albert Einstein
"Do not worry too much about your difficulties in mathematics, I can assure you that mine are still greater." Albert Einstein | 677.169 | 1 |
manufactured for this purpose. in order to make every example a
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real value.
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omitted. The entire study of algebra becomes a mechanical application of memorized
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and ingenuity
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which
"Particular care has been bestowed upon those chapters in the customary courses offer the greatest difficulties to
the beginner. The presenwill be found to be tation of problems as given in Chapter
V
quite a departure from the customary way of treating the subject. hence either book
4.
may
be used to supplement the other.vi
PREFACE
quently hardly ever emphasize the theoretical aspect of alge bra. etc.g. in particular the
requirements of the College Entrance Examination Board.
enable students
who can devote only a minimum
This arrangement will of time to
algebra to study those subjects which are of such importance for further work. and it is hoped that this treatment will materially diminish the difficulty of this topic for young students. there has been placed at the end of the book a collection of exercises which contains an abundance
of
more
difficult
work. This made it necessary to introduce the theory of proportions
.
In regard
to
some other features of the book. e.
Topics of practical importance. as quadratic equations and
graphs. a great deal of the theory offered in the avertext-book is logically unsound .
nobody would find the length Etna by such a method.
is
based principally upon the alge-
.PREFACE
vii
and graphical methods into the first year's work. physics.
to solve a
It is
undoubtedly more interesting for a student
problem that results in the height of Mt.'
This topic has been preit is
sented in a simple. But on the other hand very few of such applied examples are
genuine applications of algebra. " Graphical methods have not only a great practical value.
and they usually involve difficult numerical calculations. based upon statistical abstracts. the student will be able to utilize this knowledge where it is most
needed."
Applications taken from geometry. are
frequently arranged in sets that are algebraically uniform. The entire work in graphical methods has been so arranged that teachers who wish
a shorter course
may omit
these chapters.
but they unquestionably furnish a very good antidote against 'the tendency of school algebra to degenerate into a mechanical application of
memorized
rules. Moreover. and commercial are numerous. in
"
geometry
. elementary way.
McKinley
than one that gives him the number of Henry's marbles. an innovation which seems to mark a distinct gain from the pedagogical point
of view.
of the Mississippi or the height of Mt.
By studying proportions during the first year's work. and hence the student is more easily led to do the work by rote
than when the arrangement
braic aspect of the problem. and
of the
hoped that some
modes of representation given
will be considered im-
provements upon the prevailing methods. but the true study of algebra has not been sacrificed in order to make an impressive display of sham
life
applications.
while in the usual course proportions
are studied a long time after their principal application.
viz. such examples.
ARTHUR SCHULTZE.
William P. desires to acknowledge his indebtedness to Mr.
. however. is such problems involves as a rule the teaching of physics by the
teacher of algebra.
NEW
YORK. Manguse for the careful reading of the proofs and
many
valuable suggestions.
genuine applications of elementary algebra work seems to have certain limi-
but within these limits the author has attempted to
give as
many
The author
for
simple applied examples as possible. 1910.
edge of physics.
April.viii
PREFACE
problems relating to physics often
offer
It is true that
a field
for genuine applications of algebra.
pupil's knowlso small that an extensive use of
The average
Hence the
field of
suitable for secondary school
tations.
6=2. 33.
w
cube plus three times the quantity a minus
plus 6 multiplied
6.
27. a
a=3.
Six
2
. 2-6 of the exercise. a = 4. 6 = 2.
Twice a3 diminished by 5 times the square root of the quantity a minus 6 square.
then
8 = \ V(a + 6 + c) (a 4.
28.
12 cr6
-f-
6 a6 2
6s.12
17
&
*
ELEMENTS OF ALGEBRA
18
'
8
Find the numerical value of 8 a3
21. 6 = 6.
35. and the area of the
is
triangle
S
square feet (or squares of other units selected). a = 4. and other sciences. 38.c) (a .
:
6. 6 = 5.
sible to state
Ex.
34. 6 = 7.
a =3.
and
If the three sides of a triangle contain respectively c feet (or other units of length). if
:
a = 2. 6 = 5. Six times a plus 4 times
32.
22.
30.
26. 6 = 3.
29.
6.6 .6 -f c) (6
a
+ c). 10-14
The
representation of numbers by letters makes it posvery briefly and accurately some of the principles of arithmetic. 6 = 6. 6. a =4.
6 = 4. of this exercise?
What kind of expressions are Exs. 25. 30. = 3.
23.
24.
a = 3.
Six times the square of a minus three times the cube of Eight x cube minus four x square plus y square.
The quantity a
6
2 by the quantity a
minus
36. geometry.
Express in algebraic symbols 31. a = 2.
. a = 3.
Read the expressions
of Exs.
a. 6 = 1.
37. physics.
count the resistance of the atmosphere. if v = 50 meters per second 5000 feet per minute. and 5 feet.
(c) 4. Find the height of the tree. and 13 inches. and c
13
and
15
=
=
=
.
2. if v
:
a. b.
By
using the formula
find the area of a triangle
whose
sides are respectively
(a) 3.
4.16
1
= 84. then a 13.
9
distance s passed over by a body moving with the uniform velocity v in the time t is represented by the formula
The
Find the distance passed over by A snail in 100 seconds.e.16 centimeters per second. 12. A carrier pigeon in 10 minutes. 15 therefore feet. and 15 feet.
=
(a)
How
far does
a body fall from a state of rest in 2
seconds ?
(b)
*
stone dropped from the top of a tree reached the ground in 2-J. b 14. if v . An electric car in 40 seconds. c.
S = | V(13-hl4-fl5)(13H-14-15)(T3-14-i-15)(14-13-f-15)
= V42-12-14. d. How far does a body fall from a state of rest in T ^7 of a (c)
A
second ?
3. A body falling from a state of rest passes in t seconds 2 over a space S (This formula does not take into ac^gt 32 feet.
the area of the triangle equals
feet. 13.INTRODUCTION
E.g.
(b) 5. the three sides of a triangle are respectively 13.) Assuming g
.
84 square
EXERCISE
1.
i.
. if v = 30 miles per hour. A train in 4 hours.seconds. 14.
Find the area of a circle whose radius is
It
(b)
(a) 10 meters. the
3. and the value given above is only an
surface
$=
2
approximation.
~
7n
cubic feet.
If the
(b) 1 inch.) Find the surface of a sphere whose diameter equals
(a)
7.
(c)
5 miles.14
4.
ELEMENTS OF ALGEBRA
If
the radius of a circle
etc.
If the diameter of a sphere equals d units of length.14
square meters.
the area
etc.
to Centigrade readings:
(b)
Change the following readings
(a)
122 F.
5.
meters. the equivalent reading C on the Centigrade scale may be found by the formula
F
C
y
= f(F-32).14d (square units).
fo
If
i
represents the simple interest of
i
p
dollars at r
in
n
years.
(c)
5
F.
then the
volume
V=
(a) 10 feet.
square units (square inches.
denotes the number of degrees of temperature indi8. then
=p
n
*
r
%>
or
Find by means
(a)
(b)
6.14 is frequently denoted by the Greek letter TT.
of this formula
:
The The
interest on interest
$800
for 4 years at
ty%.
6
Find the volume of a sphere whose diameter equals:
(b)
3
feet.
.
:
8000 miles.
32 F.
$ = 3.
(c)
8000 miles.
2 inches.
on $ 500 for 2 years at 4 %.
diameter of a sphere equals d
feet. (The number 3.). This number cannot be expressed exactly.
(c)
10
feet. If cated on the Fahrenheit scale.
is
H
2
units of length (inches.).
In arithmetic we add a gain of $ 6 and a gain of $ 4.
Since similar operations with different units always produce analogous results.
of
$6 and a gain
$4
equals a
$2 may be
represented thus
In a corresponding manner we have for a loss of $6 and a
of
loss
$4
(.$6) + (-
$4) = (-
$10).CHAPTER
II
ADDITION. In algebra. or positive and negative numbers. in algebra this word includes also the results obtained by adding negative. Thus a gain of $ 2 is considered the sum of a gain of $ 6 and a loss of $ 4. we call the aggregate value of a gain of 6 and a loss of 4 the sum of the two. however. the fact that a loss of
loss of
+ $2. or that
and
(+6) + (+4) = +
16
10. AND PARENTHESES
ADDITION OF MONOMIALS
31. Or in the symbols of algebra
$4) =
Similarly.
While
in arithmetic the
word sum
refers only to
the
result obtained
by adding positive numbers.
. but we cannot add a gain of $0 and a loss of $4. we define the sum of two numbers in such a way that these results become general. SUBTRACTION.
4
is
3 J.
(_
In Exs. 5.3. add their absolute values if they have opposite signs. c = = 5.
19.
(-17)
15
+ (-14). is 2. if :
a
a
= 2. 10. the one third their sum. find the numerical values of a + b
-f c-j-c?.
5. c =
4. 23-26. + -12.
of 2.
The average
of two
numbers
is
average of three numbers average of n numbers is the
is one half their sum.
(always) prefix the sign of the greater.
. 24.
18.
33.
d = 5. 12.
ELEMENTS OF ALGEBRA
These considerations lead to the following principle
:
If two numbers have the same sign.
6
6
= 3.
21.16
32. d = 0. 4.
+ (-9).
22. and the sum of the numbers divided by n. the average of 4 and 8
The average The average
of 2.
23.
EXERCISE
Find the sum
of:
10
Find the values
17.
l-f(-2).
Thus.
of:
20. '. = 5.
is 0. subtract their absolute values and
.
-
0.
1. 09. -4. }/ Add 2 a.13.
42. -11 (Centigrade).
= 22. 43. SUBTRACTION. 72.. & = 15. and 4. and 3 yards. 7 a. 3.ADDITION.
2.
-'
1?
a
26.7. 39. 12. $1000 loss.
:
48.
4
F.
36. . = -13. if his yearly gain or loss during 6 years was $ 5000 gain.3.
2. 6.
3 and 25.
6. 60.
&
28. and 3 F.
'
Find the average of the following
34.
.
:
and
1.. 55. 66.
30. .
and
-8
F. $500 loss. 5 and
12. c = 0. 10.
13. 7 yards.
:
34. 0.
5 a2 &
6 ax^y and
7 ax'2 y.
which are not
similar.
33. $3000 gain. 10.
29.
. Find the average gain per year of a merchant.
.
Dissimilar or unlike terms are terms
4 a2 6c and o
4 a2 6c2 are dissimilar terms. or
and
.
35. 38. 7 a. \\ Add 2 a. Find the average temperature of New York by taking the average of the following monthly averages 30. 32.
sets of
numbers:
13.
AND PARENTHESES
d = l. c=14.
27.
or 16
Va + b
and
2Vo"+~&.7. .
^
'
37.
40. 10.4. 34.5.
= -23. $7000 gain.
25.
31.
d=
3. and 3 a. 41..
are similar terms. and 3 a.
:
Find the average temperature of Irkutsk by taking the average of the following monthly temperatures 12.
. 32. 74. 6. affected
by the same exponents.
Similar or like terms are terms which have the same
literal factors.
. 37.
Find the average of the following temperatures 27 F.
What number must be added to 9 to give 12? What number must be added to 12 to give 9 ? What number must be added to 3 to give 6 ? C* What number must be added to 3 to give 6? **j Add 2 yards. and $4500 gain.5.
7 rap2. b wider sense than in arithmetic.
12(a-f b)
12.
2(a-f &).
In algebra the word sum is used in a 36. or
a
6.
11
-2 a +3a -4o
2.
EXERCISE
Add:
1. and 4 ac2
is
a
2 a&
-|-
4 ac2. The indicated by connecting and a 2 and
a
is
is
-f-
a2
.
11.
+ 6 af
.13 rap
25 rap 2.
b
a
-f (
6). in algebra it may be considered b.
9(a-f-6).
14
.
12
2 wp2 .
5 a2
.
-f
4 a2.
sum of two such terms can only be them with the -f. While in arithmetic a denotes a difference only.
The sum
x 2 and
f
x2 .
ab
7
c
2
dn
6.
2 a&.
5Vm + w.
The sum The sum
of a of a
Dissimilar terms cannot be united into a single term.
ELEMENTS OF ALGEBRA
The sum
of 3
of
two similar terms
x2
is
is
another similar term. Algebraic sum.
12
13
b sx
xY xY 7 #y
7.
:
2 a2.
.18
35.
12Vm-f-n.
Vm
-f.
1
\
-f-
7 a 2 frc
Find the sum of
9.ii.
13.
10.
-3a
. either the difference of a and b or the sum of a and
The sum
of
a.sign.
5l
3(a-f-6).
2
.
3.
2.
1. may be stated number added to 3 will give 5? To subtract from a the number b means to find the number which added to b gives a.
State the other practical examples which show that the number is equal to the addition of a
40.
AND PARENTHESES
23
subtraction of a negative
positive number.
7. from What 3. The student should perform mentally the operation of chang8 2 6 from 6 a 2 fc. Therefore any example in subtraction
different
. SUBTRACTION. the given number the subtrahend.
From
5 subtract
+ 3.
From
5 subtract
to
.
Ex.
. the algebraic sum and one of the two numbers is
The algebraic sum is given. and their algebraic sum is required. ing the sign of the subtrahend thus to subtract 6 a 2 6 and 8 a 2 6 and find the sum of change mentally the sign of
. a-b =
x.
41.
NOTE. In subtraction.
+b
3.
6
-(-3) = 8.
3 gives
3)
The number which added
Hence. change the sign
of the subtrahend and
add. Subtraction is the inverse of addition.2.g.
To
subtract. and the
required number the difference.
3.
if
x
Ex. Or in symbols. called the minvend.
This gives by the same method.
From
5 subtract
to
The number which added
Hence. the other number is required.
Ex.
5
is
2. In addition. two numbers are given.
3 gives 5
is
evidently 8.
a.
The
results of the preceding examples could be obtained
by the following
Principle.ADDITION.
may
be stated in a
:
5 take form e.
(-
6)
-(-
= .
6 b -f (. we may begin either at the innermost or outermost.a~^~6)]} = 4 a -{7 a 6 b -[.
A
moved
w
may be resign of aggregation preceded by the sign inserted provided the sign of evei'y term inclosed is
E.
& -f
c.
(b
c)
a
=a
6 4-
c.g. Simplify 4 a f
+ 5&)-[-6& +(-25.
Hence
the
it is
sign
may
obvious that parentheses preceded by the -f or be removed or inserted according to the fol:
lowing principles
44. SUBTRACTION. one occurring within the other.
.
II.
45.
4a-{(7a + 6&)-[-6&-f(-2&.
If there is no sign before the first term within a paren*
-f-
thesis.& c
additions
and sub-
+ d) = a + b
c
+ d. The beginner will find it most convenient
at every step to
remove only those parentheses which contain
(7 a
no others. I. may be written as follows:
a
-f ( 4.
tractions
By using the signs of aggregation.
66
2&-a + 6
4a
Answer.
changed. If we wish to remove several signs of aggregation. the sign
is
understood.b c = a a
&
-f-
-f.
46.
a+(b-c) = a +b .ADDITION.c.a^6)]
-
}
.
AND PARENTHESES
27
SIGNS OF AGGREGATION
43. A sign of aggregation preceded by the sign -f may be removed or inserted without changing the sign of any term.c. Ex.2 b .a
-f-
= 4a
sss
7a
12
06
6.
6
o+(
a
+ c) = a =a 6 c) ( 4-.
7. The product of the sum and the difference
of
m and n.
4.7-fa.
3.
and the subtrahend the second.
p + q + r-s.2 tf .
The product The product
m
and
n. )X
6.
8.
3.
terms
5. 9.
2m-n + 2q-3t.
m
x
2
4.
12. The minuend is always the of the two numbers mentioned. SUBTRACTION.
of the cubes of
m and
n.
EXERCISE
AND PARENTHESES
16
29
In each of the following expressions inclose the last three terms in a parenthesis
:
1.
z
+ d.
2.
In each of the following expressions inclose the last three in a parenthesis preceded by the minus sign
:
-27i2 -3^ 2 + 4r/.
6.
7.
The sum
of the fourth powers of a of
and
6.
EXERCISES
IN"
ALGEBRAIC EXPRESSION
17
:
EXERCISE
Write the following expressions
I.4 y* .
The sum^)f
m
and
n.
5. .1.
10.
5^2
_ r .
The sum
of tKe squares of a
and
b.
The
difference of a
and
6.
6 diminished
.
The The
difference of the cubes of
m
and
n.
II.
The square of the difference of a and b.
Nine times the square of the sum of a and by the product of a and b.
4 xy
7 x* 4-9 x + 2.
5 a2
2.ADDITION.
'
NOTE.
Three times the product of the squares of The cube of the product of m and n.
first.
13.
m and n.
a-\-l>
>
c
+
d.
y
-f-
8
.
difference of the cubes of n and m.
)
.
b. 16.
d.30
14.
The sum
The
of a
and
b multiplied
b is equal to the difference of
by the difference of a and a 2 and b 2
.
dif-
of the squares of
a and
b increased
by the
square root of
15.
x cube minus quantity 2 x2 minus 6 x plus
The sum
of the cubes of a. (Let a and b represent the numbers.
6 is equal to the square of
b.
difference of the cubes of a
and
b divided
by the
difference of
a and
6.
ELEMENTS OF ALGEBRA
The sum
x.
and
c
divided by the
ference of a and
Write algebraically the following statements:
V 17. 6.
a plus the prod-
uct of a and
s
plus the square of
-19. The difference of the squares of two numbers divided by the difference of the numbers is equal to the sum of the two numbers.
18.
4. applied at let us indicate a downward pull at by a positive sign.
two loads balance.CHAPTER
III
MULTIPLICATION
MULTIPLICATION OF ALGEBRAIC NUMBERS
EXERCISE
18
In the annexed diagram of a balance.
A
A
A
1.
force is produced
therefore.
3. therefore. and forces produced at by 3 Ib.
5. let us consider the and JB. If the two loads balance. what force is produced by the Ib. 2. what force
31
is
produced by tak(
ing away 5 weights from
B ? What therefore is
5)
x(
3) ?
.
By what sign is an upward pull at A represented ? What is the sign of a 3 Ib. what force is produced by the addition of 5 weights at B ? What. is 5 x ( 3) ?
7. weight at B ?
If the
addition of five 3
plication example. weight at A ? What is the sign of a 3 Ib. weights at A ? Express this as a multibalance. weights.
If the two loads
what
What.
If the
two loads balance. is
by
taking away 5 weights from
A?
5
X 3?
6.
This definition has the additional advantage of leading to algenumbers which are identical with those for positive numbers. or
4x3 =
=
(_4) X
The preceding
3=(-4)+(-4)+(-4)=-12. thus. becomes meaningless
if
definition. or plied by 3.
examples were generally
method of the preceding what would be the values of
(
5x4. and we may choose any definition that does not lead to contradictions. times is just as meaningless as to fire a gun
tion
7
Consequently we have to define the meaning of a multiplicaif the multiplier is negative.4)-(-4) = +
12.
(
(.
4x(-3)=-12. 4 multi44-44-4 12. Multiplication by a positive integer is a repeated addition.
9
9. Practical examples^
it
however.
ELEMENTS OF ALGEBRA
If
the signs obtained by the
true.
48. 9 x
(-
11). 5x(-4). To take a number 7 times.
(-
9)
x (-
11) ?
State a rule by which the sign of the product of
two
fac-
tors can be obtained.4) x
braic laws for negative
~ 3> = -(.
4
x(-8) = ~(4)-(4)-(4)=:-12.
x
11.9) x
11. (-5)X4.
In multiplying integers we have therefore four cases
trated
illus-
by the following examples
:
4x3 = 4-12.32
8.
however.
Multiplication
by a negative
integer is a repeated
sub-
traction.
the multiplier is a negative number.
NOTE. such as given in the preceding exercise.4)-(. a result that would not be obtained by other assumptions. 4 multiplied by 3.
(.
. make venient to accept the following definition
:
con-
49.
Thus.
3 a 2 + a8
a
a = =-
I
1
=2
-f
2
a
4. To multiply two polynomials.
The most convenient way of adding the partial products is to place similar terms in columns. are far more likely to occur in the coefficients than anywhere else. as illustrated in the following example
:
Ex.2 a2 6 a8
2 a*
*
-
2"
a2
-7
60. Since errors. this method
tests only the values of the coefficients
and not the values of
the exponents.
59.3 a
3
2
by 2 a
:
a2 + l. Multiply 2 a .3
b
by a
5
b.1.a6 4 a 8 + 5 a* .a
. 1 being the most convenient value to be substituted for all letters.
If
Arranging according to ascending powers
2
a
.M UL TIP LICA TION
37
58.
2.
. Since all powers of 1 are 1.a6
=2
by numerical
Examples
in multiplication can be checked
substitution.3 ab
2
2 a2
10 ab
-
13 ab
+ 15 6 2 + 15 6 2
Product.4. the work becomes simpler and more symmetrical by arranging these expressions according to either ascending or descending powers.
Multiply 2
+ a -a.
Check.
a2
+ a8 + 3 . If the polynomials to be multiplied contain several powers of the same letter.3 a 2 + a8 .
Ex. however. multiply each term of one by each term of the other and add the partial products thus formed.
2a-3b a-66 2 a . the student should
apply this test to every example.
4.
(2a-3)(a + 2). plus the
last terms.
. the product of two binomials whose corresponding terms are similar is equal to the product of the first two
terms.
sum of the
cross products.
11.
5.
7%e square of a polynomial is equal to the sum of the squares of each term increased by twice the product of each term with each that follows it.
(5a6-4)(5a&-3). (3m + 2)(m-l). ) (2
of a polynomial.
that the square of each term is while the product of the terms may have plus always positive.
The middle term
or
Wxy-12xy
Hence in general.
8. and are represented as
2 y and 4y 3 x. (100 + 3)(100 + 4).
:
25
2.
or
The student should note
minus
signs.
2
10.
2
(2m-3)(3m + 2). (4s + y)(3-2y).
7.
6.
65.-f
2 a&
-f
2 ac
+ 2 &c. 14.42
ELEMENTS OF ALGEBRA
of the result is obtained
product of 5 x
follows:
by adding the These products are frequently called the cross products.
2 2 2 2 (2a 6 -7)(a & +
5).
The square
2
(a 4.
3. 9.&
+ c) = a + tf + c
.
(5a-4)(4a-l).
2
(2x y
(6
2
2
+ z )(ary + 2z ).
(x
i-
5
2
ft
x 2 -3 6 s).
((5a?
(10
12.
2
2
+ 2) (10 4-3). 13. plus the product of the
EXERCISE
Multiply by inspection
1.
is the process of finding one of two factors and the other factor are given. The dividend is the product of the two factors, the divisor the given factor, and the quotient is the required factor.
67.
Division
if
their product
is
Thus
by
-f
to divide
12.
12
by
+
3,
we must find
is
the
;
number which
3 gives
But
this
number
4
hence
_
multiplied
12 r +3
=4.
68.
Since
-f
a
-
-f b
-fa
_a
and
it
-f-
a
= -f ab = ab b = ab b = ab,
b
-f-
follows that
4-a
=+b
ab
a
ab
a
69.
Hence the law
:
of signs
is
the same in division as in
multiplication
70.
Like signs produce plus, unlike signs minus.
Law
of
,
a8 -5- a5
=a
3
for a 3
It follows from the definition that Exponents. X a5 a8
=
.
Or
in general, if
greater than
m n, a
-f-
and n are positive integers, and m ~ n an = a m a" = a'"-", for a
<
m
m
is
45
46
ELEMENTS OF ALGEBRA
71. TJie exponent of a quotient of two powers with equal bases equals the exponent of the dividend diminished by the exponent
of the divisor.
DIVISION OF MONOMIALS
7 3 72. To divide 10x y z by number which multiplied by number is evidently
2x y
6
2
,
we have
z
to
find
the
2x*y
gives 10 x^ifz.
This
Therefore,
the quotient
*
,
= - 5 a*yz.
is
Hence,
sign,
of two monomials of their
part
coefficients,
is the
a monomial whose
coefficient is the quotient
preceded by the proper
literal
and whose
literal
found
in accordance with the
quotient of their law of exponents.
parts
73. In dividing a product of several factors by a number, only one of these factors is divided by that number. Thus (8 12 20)-?-4 equals 2 12 20, or 8 3 20 or 8 12 5.
-
-
.
-
.
-
.
EXERCISE
Perform the divisions indicated
'
:
28
'
2
.
76-H-15.
-39-*- 3.
2
15
3"
7
7'
3.
-4*
'
4.
5.
-j-2
12
.
4
2
9
5 11
68
3 19 -j-3
5
10.
(3
38
-
-2 4 )^(3 4 .2 2).
56
'
11.
3
(2
.3*.5 7 )-f-(
2
'
12
'
2V
14
36 a
'
13
''
y-ffl-g
35
-5.25
-12 a
2abc
15
-42^
'
-56aW
'
UafiV
DIVISION
lg
47
-^1^. 16 w
7
20>
7i
9
_Z^L4L.
22.
10 iy.
132 a V* 14 1
*
01
-240m
120m-
40
6c
fl
/5i.
3J)
c
23.
2 (15- 25. a ) -=- 5.
25. 26.
(18
(
.
5
.
2a )-f-9a.
2
24.
(7- 26 a
2
)
-f-
13.
DIVISION OF POLYNOMIALS BY MONOMIALS
To divide ax-}- fr.e-f ex by x we must find an expression which multiplied by x gives the product ax + bx -J- ex.
74.
But
TT
x(a
aa?
Hence
+ b e) ax + bx + ex. + bx -f ex = a 4- b +
-\.
,
.
c.
a?
To divide a polynomial by a monomial, cfc'wde each term of the dividend by the monomial and add the partial quotients thus
formed.
3 xyz
EXERCISE
Perform the operations indicated
1.
:
29
2.
5.
fl
o.
(5*
_5* + 52)
-5.
52
.
3.
97
.
(2
(G^-G^-G^-i-G
(11- 2
4.
(8- 3
+
11 -3
+ 11
-5)-*- 11.
18 aft- 27 oc
Q y.
9a
4
-25 -2 )^-2
<?
2
.
+8- 5 + 8-
7) -*-8.
5a5 +4as -2a
2
-a
-14gV+21gy
Itf
15 a*b
-
12
aW + 9 a
2
2
3a
48
,
ELEMENTS OF ALGEBRA
22
4,
m n - 33 m n
4
s
2
-f
55
mV
- 39 afyV + 26 arVz 3
- 49 aW + 28 a -W - 14 g 6 c
4 4
15. 16.
2 (115 afy -f 161 afy
- 69
4
2
a;
4
?/
3
- 23 ofy
3
4
)
-5-
23 x2y.
(52
afyV - 39
4
?/
oryz
- 65 zyz - 26 tf#z)
-5-
13 xyz.
-f-
,
17.
(85 tf
- 68 x + 51 afy - 34 xy* -f 1 7
a;/)
- 17
as.
DIVISION OF A POLYNOMIAL BY A POLYNOMIAL
75.
Let
it
be required to divide 25 a
- 12 -f 6 a - 20 a
3
2
by
2 a 2 -f 3 a, divide
4
a, or, arranging according to
2
descending powers of
6a3 -20a
-f
25a-12
2 by 2a -
The term containing the highest power of a in the dividend (i.e. a 8 ) is evidently the product of the terms containing respectively the highest power of a in the divisor and in the quotient.
Hence the term containing the highest power
of a in the quotient is
If
the product of 3 a and 2
2
4 a
+
3, i.e.
6 a3
12 a 2
-f
9 a, be sub-
8 a 2 -f 16 a tracted from the dividend, the remainder is 12. This remainder obviously must be the product of the divisor and the rest of the quotient. To obtain the other terms of the quotient we have
therefore to divide the remainder,
8 a2
-f-
16 a
12,
2 by 2 a
4 a
+
3.
consequently repeat the process. By dividing the highest term in the new dividend 8 a 2 by the highest term in the divisor 2 a 2 we obtain
,
We
4,
the next highest term in the quotient. 4 by the divisor 2 a2 4 a Multiplying
-I-
+ 3, we
obtain the product
8 a2
16 a
12,
which subtracted from the preceding dividend leaves
the required quotient.
no remainder. Hence 3 a
4
is
DIVISION
The work
is
49
:
usually arranged as follows
- 20 * 2 + 3 0a-- 12 a 2 +
a3
25 a
{)
-
12
I
2 a2 8 a
-
4 a 4
a
_
12
+3
I
-
8 a? 4- 16
a-
76. The method which was applied in the preceding example may be stated as follows 1. Arrange dividend and divisor according to ascending or
:
descending powers of a common letter. 2. Divide the first term of the dividend by the first term of the divisor, and write the result for the first term of the quotient.
3.
Multiply this term of the quotient by the whole divisor, and
subtract the result
4.
from
it
the dividend.
the same order as the given new dividend, and proceed as before.
Arrange
the
remainder in
as a
expression, consider
5.
until the highest poiver
Continue the process until a remainder zero is obtained, or of the letter according to which the dividend
is less
was arranged
the divisor.
than the highest poiver of the same
letter in
77.
Checks.
Numerical substitution constitutes a very con-
venient, but not absolutely reliable check. An absolute check consists in multiplying quotient and divisor. The result must equal the dividend if the division
was
exact, or the dividend diminished by the remainder division was not exact.
An equation of condition is an equation which is true only for certain values of the letters involved.
=11. is said to satisfy an equation.
A set of numbers which when substituted for the letters an equation produce equal values of the two members.
.
. second member is x
+
4
x
9.
The
first
member
or left side of an equation
is
that part
The secof the equation which precedes the sign of equality.CHAPTER V
LINEAR EQUATIONS AND PROBLEMS
79.
the
first
member
is
2 x
+
4.
the
80.
x
20. in the equation 2 x 0. (a + ft) (a b) and b. y y or z) from its relation to
63
An
known numbers.
in
Thus x
12 satisfies the equation x
+
1
13.
83.r
-f9
= 20
is
true only
when
a.
y
=
7 satisfy
the equation x
y
=
13. ond member or right side is that part which follows the sign of
equality.
. An equation of condition is usually called an equation.
hence
it
is
an equation
of
condition.
(rt+6)(a-ft)
=
2
-
b'
2
.
ber
equation is employed to discover an unknown num(frequently denoted by x. The sign of identity sometimes used is = thus we may write
.
Thus.
which
is
true for all values
a2 6 2 no matter what values we assign to a Thus. An identity is an equation of the letters involved.
82.
81.
an^ unknown quantity which satisfies the equation is
a root of the equation. the remainders are
equal.
ELEMENTS OF ALGEBRA
If
value of the
an equation contains only one unknown quantity.
The process
of solving equations depends upon the
:
lowing principles.
a.
one member to another by changing
x + a=. the quotients are equal.
A
linear equation or
which when reduced
first
to its simplest
an equation of the first degree is one form contains only the
as
9ie
power of the unknown quantity.
the divisor equals zero.
90.
(Axiom
2)
the term a has been transposed from the left to thQ
right
member by changing
its
sign.
Axiom
4
is
not true
if
0x4
= 0x5.
9
is
a root of the equation 2 y
+2=
is
20. the
sums are
equal.
but 4 does not equal
5. 87.
2
= 6#-f7.
85.
= bx
expressed by a letter or a combination of
c.
86. Consider the equation b Subtracting a from both members.
4.54
84.
E.
A
term may be transposed from
its sign.
Like powers or
like roots
of equals are equal.
If equals be added
to equals.
If equals be multiplied by equals.
NOTE.
2.
.
If equals be subtracted from equals.
If equals be divided by equals.e.
.
A numerical
equation is one in
which
all
.b.
Transposition of terms. the products are equal.
3. called axioms
1.g.
the
known quan
x) (x -f 4)
tities are
=
.
5.
fol-
A
linear equation is also called a simple equation.2. x
I. A
2
a.
89.
To
solve
an equation
to find its roots.
expressed in arithmetical numbers
literal
is
as (7
equation
is
one in which at least one of the
known
quantities as x -f a letters
88.
x
-f-
y yards cost $ 100
.
By how much does a exceed 10 ? By how much does 9 exceed x ? What number exceeds a by 4 ? What number exceeds m by n ? What is the 5th part of n ? What is the nth part of x ? By how much does 10 exceed the third part of a? By how much does the fourth part of x exceed b ? By how much does the double of b exceed one half Two numbers differ by 7.
Ex.
17.
13. greater one is g.
ELEMENTS OF ALGEBRA
What must
be added to a to produce a sum b ?
:
Consider the arithmetical question duce the sum of 12 ?
What must
be added to 7 to pro-
The answer is 5.
Find the greater one.
$> 100 yards cost one hundred dollars. is b.
smaller one
16.
1.
5.
3. 11.
Divide a into two parts.58
Ex.
14. 15. so that one part Divide a into two parts. Divide 100 into two
12. Hence 6 a must be added
to a to give
5. so that one part
The
difference between
is s.
two numbers
and the and the
2
Find the greater one. so that
of c ?
is
p.
What number divided by 3 will give the quotient a? ? What is the dividend if the divisor is 7 and the quotient
?
.
one part equals
is 10. and the smaller one
parts.
is
a?
2
is
c?.
If 7
2. or 12 7.
4.
6. The difference between two numbers Find the smaller one.
9.
EXERCISE
1. is d.
33
2.
7.
find the cost of one yard.
10. 6.
a. one yard will cost -
Hence
if
x
-f
y yards cost $ 100. one yard will cost
100
-dollars.
33.
and
B
is
y years old. amount each will then have.
How many
cents are in d dollars ? in x dimes ?
A has
a
dollars.
How many
cents had he left ?
28.
Find
21.
A
feet wide.
and spent
5 cents.
A man
had a
dollars.
A
dollars.
feet wider than the one
mentioned in Ex.LINEAR EQUATIONS AND PROBLEMS
18. 28. and B's age is y years.
19. 28.
20.
24. and B has n dollars.
numbers
is x.
22.
59
What must
The
be subtracted from 2 b to give a?
is a.
26.
What What What What
is
the cost of 10 apples at x cents each ?
is
is is
x apples cost 20 cents ? the price of 12 apples if x apples cost 20 cents ? the price of 3 apples if x apples cost n cents ?
the cost of 1 apple
if
.
Find the area of the Find the area of the
feet
floor of
a room that
is
and 3
30.
smallest of three consecutive numbers
Find
the other two.
and
c cents.
If
B
gave
A
6
25. Find the sum of their ages
5 years ago.
sum
If A's age is x years. b dimes.
Find
35. square feet are there in the area of the floor ?
How many
2 feet longer
29.
How many years
A
older than
is
B?
old.
How many
cents
has he ?
27.
34. rectangular field is x feet long and the length of a fence surrounding the field.
is
A A
is
# years
old.
32.
y years
How
old was he 5 years ago ?
How
old will he be 10 years hence ?
23. The greatest of three consecutive the other two. find the of their ages 6 years hence. find the
has ra dollars.
and 4
floor of a room that is 3 feet shorter wider than the one mentioned in Ex. A room is x feet long and y feet wide.
?/
31.
-.
How many
x years ago
miles does a train
move
in
t
hours at the
rate of x miles per hour ?
41.
48.60
ELEMENTS OF ALGEBRA
wil\
36.
miles does
will
If a man walks r miles per hour. The first pipe x minutes.
how many
how many
miles will
he walk in n hours
38.
Find x
% %
of 1000.
Find
a.
as
a exceeds
b
by as much
as c exceeds 9. If a man walks n miles in 4 hours.
A
cistern
is
filled
43. If a man walks 3 miles per hour." we have to consider that in this by statement "exceeds" means minus ( ). how many miles he walk in n hours ?
37.
of 4. What fraction of the cistern will be filled by one pipe in one minute ?
42.
of m.
.
If a
man walks
?
r miles per hour.
49.
Find the
number. he walk each hour ?
39.
% % %
of 100
of
x. Find a
47.50.
A
was 20 years
old. What fraction of the cistern will be second by the two pipes together ?
44.
A
cistern can be filled
in
alone
fills it
by two pipes. and the second pipe alone fills it in
filled
y minutes. in how many hours he walk n miles ?
40.
-46.
The numerator
If
of a fraction exceeds the denominator
by
3. find the fraction.
The two
digits of a
number
are x and
y. and "by as much as" Hence we have means equals (=)
95.
m is the
denominator.
b
To express in algebraic symbols the sentence: " a exceeds much as b exceeds 9.
per
Find 5 Find 6
45.
c
a
b
=
-
9.
How
old
is
he
now ?
by a pipe in x minutes.
a.
by one third of b equals 100.
a exceeds b by c.
In
many
word
There are usually several different ways of expressing a symbolical statement in words.
double of a
is
10.
of x increased by 10 equals
x.
third of x equals
difference of x
The
and y increased by 7 equals
a.
-80. 2.
80.
c.
8
-b ) + 80 = a
.
of a and 10 equals 2
c.
3.
4. the difference of the squares of a
61
and
b increased
-}-
a2
i<5
-
b'
2
'
by 80 equals the excess of a over
80
Or.
same
result as 7
subtracted from
.
=
2
2
a3
(a
-
80.
c.
The product
of the
is
diminished by 90 b divided by 7.
6. Four times the difference of a and b exceeds c by as
d exceeds
9.
5.
of a increased
much
8.
equal to the
sum and the difference of a and b sum of the squares of a and
gives the
Twenty subtracted from 2 a
a.
9.
The double
as
7.
cases it is possible to translate a sentence word by in algebraic symbols in other cases the sentence has to be changed to obtain the symbols. thus:
a
b
= c may
be expressed as follows
difference between a
:
The
and
b is
c. a is greater than b by b is smaller than a by
c.LINEAR EQUATIONS AND PROBLEMS
Similarly. etc.
The
excess of a over b
is c.
EXERCISE
The The double The sum
One
34
:
Express the following sentences as equations
1.
A
If
and
B
B together have $ 200 less than C. In 3 years A will be twice as old as B.
(c)
If each
man
gains $500.
symbols
B.
and C's age
4
a. B's.
a. express in algebraic
3x
:
10. and C have respectively 2 a.
12.
x
is
100
x%
is
of 700. the
sum
and C's
money
(d)
(e)
will be $ 12.
ELEMENTS OF ALGEBRA
Nine
is
as
much below a
13. A is 4 years older than
Five years ago A was x years old. a second sum.
(a)
(b)
(c)
A is twice as old as B.
amounts.000. the first sum exceeds b % of the second sum by
first
(e)
%
of the first plus 5
%
of the second plus 6
%
of the
third
sum
equals $8000.
A
gains
$20 and B
loses
$40. they have equal amounts.
first
00
x % of the
equals one tenth of the third sum.
#is5%of450. the first sum equals 6 % of the third sura. B's age
20. B's. they have equal
of A's. express in algebraic symbols
:
-700.
is
If A's age is 2 x.
5x
A sum of money consists of x dollars.*(/)
(g)
(Ji)
Three years ago the sum of A's and B's ages was 50.
a. and
(a)
(6)
A
If
has $ 5 more than B.
6
%
of m. and C's ages will be 100.
18.
..
11. a third sum of 2 x + 1 dollars. sum equals $20. of 30 dollars.
->.
17. (d) In 10 years A will be n years old. In 10 years the sum of A's. pays to C $100.
x
4-
If A.
as 17
is
is
above
a. 14. 16. (e) In 3 years A will be as old as B is now.
m is x %
of n.
50
is
x % of
15. Express as
:
equations of the (a) 5
(b)
(c)
% a%
of the second
(d)
x c of / a % of
4
sum equals $ 90. 3 1200 dollars. B.62
10.
4 x = 80. Find A's present age. The student should note that x stands for the number of and similarly in other examples for number of dollars. denote the unknown
96.
Three times a certain number exceeds 40 by as Find the number.
Transposing. be three times as old as he was 5 years ago.
A
will
Check. verbal statement (1)
(1) In 15 years
A
will
may be expressed in symbols (2).
Ex. In 15 years A will be three times as old as he was 5 years ago. the
. the required
. x = 20.
much
as 40 exceeds the number. x + 15 = 3 x
3x 16
15.
number. 2.
number by x (or another letter) and express the yiven sentence as an equation.
3
x
+
16
=
x
x
(x
-
p)
Or.
Check.
NOTE.
Uniting. Three times a certain no.
1.
.
-23 =-30. The solution of the equation (jives the value of the unknown number.
Dividing.
3z-40:r:40-z.
Let x
The
(2)
= A's present age.
x+16 = 3(3-5).
be 30
. etc.
6 years ago he was 10
. exceeds 40 by as much as 40 exceeds the no.
Uniting. Write the sentence in algebraic symbols. x= 15.
3 x or 60 exceeds 40
+ x = 40 + 40.
In 15 years
10.
but
30
=3
x
years. The equation can frequently be written by translating the sentence word by word into algebraic symbols in fact. = x x
3x
-40
3x
40-
Or. number of
yards.
equation is the sentence written in alyebraic shorthand.
by 20
40 exceeds 20 by 20.
Transposing.
Ex.
Let x = the number. 15.LINEAR EQUATIONS AND PROBLEMS
63
PROBLEMS LEADING TO SIMPLE EQUATIONS
The simplest kind of problems contain only one unknown number.
Simplifying. In order to solve them.
Four times the length of the Suez Canal exceeds 180 miles by twice the length of the canal.
.
What number
7
%
of
350?
Ten times the width of the Brooklyn Bridge exceeds 800 ft.
13. exceeds the width of the bridge.
EXERCISE
1. How many miles per hour does it run ?
. A train moving at uniform rate runs in 5 hours 90 miles more than in 2 hours.
Let x
3.
300
56. then the
problem expressed in symbols
W
or. % of
120.
How
old
is
man will be he now ?
twice as old as he was
9.
Uldbe
66
| x x
5(5 is
=
-*-.2.
5.
3.
Find the number whose double increased by 14 equals Find the number whose double exceeds 40 by 10.
47 diminished by three times a certain number equals 2.
twice the number plus
7.64
Ex. 14.
35
What number added
to twice itself gives a
sum
of
39?
44. Find the number. by as much as 135 ft.
Six years hence a
12 years ago.
Find
8. How long is the Suez
Canal?
10. 11.
Find the number.
to
42 gives a
sum
equal to 7 times the
original
6.
A will
be three times as old as to-da3r
.
Hence
40
= 46f.
Find the number whose double exceeds 30 by as much
as 24 exceeds the number.
ELEMENTS OF ALGEBRA
56
is
what per cent
of 120 ?
= number
of per cent.
120.
14 50
is
is
4
what per cent of 500 ? % of what number?
is
12.
Forty years hence
his present age.
A
number added
number.
Find the width of the Brooklyn Bridge.
4.
Dividing.
then dollars has each ? many
have equal amounts of money.
How many
dol-
A has
A
to
$40. B will have lars has A now?
17. If A gains A have three times as much
16. while in the more complex probWe denote one of the unknown
x. The other verbal statement. written in algebraic
symbols.
How many dollars
must
?
B
give to
18. Vermont's population increased by 180.
five
If
A
gives
B
$200.
Ex. If a problem contains two unknown quantities. Find
the population of Maine in 1800.
The problem consists of two statements I. In 1800 the population of Maine equaled that of Vermont.
A
and
B
have equal amounts of money.
97.000.000. and
B
has $00.
times as
much
as A. Maine's population increased by 510.
how many
acres did he wish to
buy
?
19.
One number exceeds another by
:
and their sum
is
Find the numbers. One number exceeds the other one by II. two verbal statements must be given. which gives the value of
8. and Maine had then twice as many inhabitants as Vermont.
make A's money
equal to 4 times B's
money
wishes to purchase a farm containing a certain He found one farm which contained 30 acres too many. If the first farm contained twice as many acres as
A man
number
of acres.
14.
statements are given directly.
x.LINEAR EQUATIONS AND PROBLEMS
15. The sum of the two numbers is 14. Ill the simpler examples these two
lems they are only implied. and
as
15. and another which lacked 25 acres of the required number.
F
8. During the following 90 years.
the second one.
65
A
and
B
$200.
B
How
will loses $100.
.
numbers (usually the smaller one) by
and use one of the
given verbal statements to express the other unknown number in terms of x.
is
the equation.
1.
= 14.= The second statement written
the equation ^
smaller number.
Statement
x
in
=
the larger number.
Uniting.
The two statements
I. unknown quantity
in
Then. 2.
o\
(o?-f 8)
Simplifying.
Let
x
14
I
the smaller number.
x
x =14
8.
/
.
+
a-
-f
-f
8
= 14.
26
= B's number of marbles after the exchange.
expressed symbols is (14 x) course to the same answer as the first method. 25 marbles to B.
If
A gives
are
:
A
If
II.
has three times as many marbles as B. the greater number. Let
x
3x
express one
many as A.
. I. the sum of the two numbers is 14. B will have twice as
viz. = 3. 26 = A's number of marbles after the exchange.
To
express statement II in algebraic symbols. 8 the greater number.
x
3x
4-
and
B
will gain.
If
we
select the first one.
and
Let x
= the
Then x -+. to
Use the simpler statement.
Another method for solving this problem is to express one unknown quantity in terms of the other by means of statement II viz. although in general the simpler one should be selected.
in algebraic
-i
symbols produces
#4a. the smaller number.66
ELEMENTS OF ALGEBRA
Either statement may be used to express one unknown number in terms of the other.
terms of the other.
2x
a?
x
-j-
= 6.
Dividing.
.
A will lose.
.
x
= 8.
which leads
ot
Ex. 8 = 11. A has three times as many marbles as B.
= B's number of marbles. consider that by the
exchange
Hence. B will have twice as many as A. A gives B 25 marbles. = A's number of marbles.
Then.
<
Transposing.
Selecting the cent as the denomination (in order to avoid fractions). then.10.
60.
2. 3 x = 45.
Never add the number number of yards to their
Ex.5 x .
1. Check.LINEAR EQUATIONS AND PROBLEMS
Therefore. x = 15. 45 ..
(Statement II)
Qx
. their sum
+ +
10 x 10 x
is
EXERCISE
36
is five
v
v. B's number of marbles. of dollars to the number of cents. cents.
67
x
-f
25 25
Transposing. etc.
*
'
.240.
. How many are there of each ?
The two statements are I.
we
express the statement II in algebraic symbols..75.
differ
differ
and the greater and their sum
times
Two numbers
by
60.
Find
the numbers. 40 x . 15 + 25 = 40. x = the number of half dollars. 11 x = 5. Find the numbers. Eleven coins. The number of coins II.25 = 20. consisting of half dollars and dimes. 3.
50.10. the number of dimes.
Uniting.
Uniting.
by 44.$3.
is 70.550 -f 310.
greater
is
.10.
Dividing.
6 times the smaller.
*
98. the
price..
50(11 660 50 x
-)+ 10 x = 310.
x x
+
= 2(3 x = 6x
25
25).
dollars
and dimes
is
$3.
w'3.
Simplifying. 6 half dollars = 260 cents.
The numbers which appear
in the equation should
always
be expressed in the same denomination.
50 x
Transposing. the number of half dollars.
Two numbers
the smaller.
6 dimes
= 60
= 310. A's number of marbles. but 40 = 2 x 20. Dividing.
Let
11
= the number of dimes.
Check. have a value of $3.
Simplifying.
The sum of two numbers is 42. The value of the half
:
is 11.
x
from
I. and the Find the numbers. . x = 6.
and in Mexico
?
A
cubic foot of aluminum.
How many
hours does the day last ?
. one of which increased by
9.
it
If the smaller
one contained 11 pints more.68
4. and twice the altitude of Mt.
On December
21.
7. Find their ages. Mount Everest is 9000 feet higher than Mt. How many volcanoes are
in the
8. the number.
cubic foot of iron weighs three times as much as a If 4 cubic feet of aluminum and
Ibs. and four times the former equals five times the latter.
tnree times the smaller by 65.
Find
Find two consecutive numbers whose sum equals 157.. and in 5 years A's age will be three times B's.
What
is
the altitude of each
mountain
12.
?
Two
vessels contain together 9 pints.
2 cubic feet of iron weigh 1600 foot of each substance.
find the
weight of a cubic
Divide 20 into two parts. the larger part exceeds five times the smaller part by 15 inches.
5.
of volcanoes in
Mexico exceeds the number
of volcanoes in the United States by 2.
ELEMENTS OF ALGEBRA
One number
is
six
times another number. Everest by 11.
11. Twice 14.
United
States.
9.
How many
14 years older than B.
as the larger one. and the
greater increased by five times the smaller equals 22.
3 shall be equal to the other increased by
10. McKinley.
A's age is four times B's.000
feet.
and twice the greater exceeds Find the numbers.
would contain three times as
pints does each contain ?
much
13. What are their ages ?
is
A A
much
line 60 inches long is divided into two parts. and B's age is as below 30 as A's age is above 40. McKinley exceeds the altitude of
Mt. the night in
Copenhagen
lasts 10 hours
longer than the day. How many inches are in each part ?
15.
Two numbers
The number
differ
by
39.
6.
then three times the sum of A's and B's money would exceed C's money by as much as A had originally.
has.
times as
much
as
A.LINEAR EQUATIONS AND PROBLEMS
99. III.
number
had. x = 8.
number
of dollars of dollars
B
C
had.
5
5
Expressing in symbols Three times the sum of A's and B's money exceeds C's money by A's 3 x ( x _5 + 3z-5) (90-4z) = x. = 48. and C together have $80.
A
and B each gave $ 5
respectively. original amount. bers is denoted by x. they would have 3. II. try to obtain
it
by a
series of successive steps. 19.
8(8
+ 19)
to C. and the other
of x
problem contains three unknown quantities.
I. 4 x = number of dollars C had after receiving $10.
the
the
number
of dollars of dollars of dollars
A
B
C
has.
first
According to
3 x
number
number
and according
to
80
4
x
=
the
express statement III by algebraical symbols.
are
:
C's
The three statements
A. and B has three as A. and C together have $80. Tf it should be difficult to express the selected verbal state-
ment
directly in algebraical symbols. three One of the unknown num-
two are expressed in terms by means of two of the verbal statements.
sum of A's and B's money would exceed much as A had originally."
To
x
8x
90
= number of dollars A had after giving $5.
69
If a
verbal statements must be given. B. = number of dollars B had after giving $5.
If
4x
= 24. The solution gives
:
3x
80
Check. B has three times as much as A. then
three times the
money by
I.
.
has. B.
Let
x
II.
Ex. and 68.
1. let us consider the words ** if A and B each gave $ 5 to C. If A and B each gave $5 to C. The third
verbal statement produces the equation. or 66 exceeds 58 by 8. number of dollars A had. If A and B each gave $5 to C.
number of horses.
x 35
-f
+
=
+
EXERCISE
1. according to III. each cow $ 35. + 35 x 4.
Uniting.
37
Find three numbers such that the second is twice the first. The number of cows exceeds the number of horses by 4.
three statements are
:
IT. 90
may
be written.
first
the third exceeds the second by and third is 20. sheep.
+
8
90 x
and. 90 x -f 35 x + GO x = 140 20 + 1185. 28 2 (9 5).
+ 35 (x +-4) -f 15(4z-f 8) = 1185. each horse costing $ 90. the third five times the first. number of sheep.
1 1
Check. and the difference between the third and the second is 15
2. 185 a = 925. 9 cows. and Ex.
Find three numbers such that the second is twice the 2. 2.140 + (50 x x 120 = 185. number of cows. x = 5.70
ELEMENTS OF ALGEBRA
man spent $1185 in buying horses.
28 x 15 or 450
5 horses. 4 x -f 8 = 28. cows.
and. according to II.
The total cost equals $1185. The number of cows exceeded the number of horses by
4. 2 (2 x -f 4) or 4 x
Therefore. and 28 sheep would cost 6 x 90 -f 9 + 316 420 = 1185. and each sheep $ 15.
The
I.
A
and the number of sheep was twice as large as the number How many animals of each kind did he buy ?
of horses and cows together.
x
-j-
= the
number of horses. x -f 4 = 9. = the number of dollars spent for sheep
Hence statement
90 x
Simplifying.
85 (x 15 (4 x
I
+ 4)
+
8)
= the number of sheep.
Dividing. and the sum of the
.
x
Transposing. 9 -5 = 4 . number of cows.
III. The number of sheep is equal to twice tho number of horses and
x 4
the
cows together.
first. Let
then. = the number of dollars spent for horses. = the number of dollars spent for cows.
is five
numbers such that the sum of the first two times the first.000.000 more inhabitants than Philaand Berlin has 1.
twice as old as B. and children together was 37.
-
4. If the population of New York is twice that of Berlin.
7.LINEAR EQUATIONS AND PROBLEMS
3. and the sum of the first and third is 36. increased by three times the second side.
71
the
Find three numbers such that the second is 4 less than the third is three times the second.
first.
"Find three
is 4.
A
12. men. women. the second one is one inch longer than the first.
v
-
Divide 25 into three parts such that the second part first. and the third part exceeds the second by 10.
9.000. equals 49 inches.
A
is
Five years ago the What are their ages ?
C. If twice
The sum
the third side.
New York
delphia. what is the population of each city ?
8. the copper. what is the length of each?
has 3. In a room there were three times as many children as If the number of women.
what are the three angles ?
10.
v
. and the third exceeds the
is
second by
5.000 more than Philadelphia (Census 1905).
the third
2.
Find three consecutive numbers whose sum equals
63. and 2 more men than women.
13.
and
of the three sides of a triangle is 28 inches. how many children
were present ?
x
11. The gold.
first.
twice the
6.
the
first
Find three consecutive numbers such that the sum of and twice the last equals 22.
If the second angle of a triangle is 20 larger than the and the third is 20 more than the sum of the second and
first. and the pig iron produced in one year (1906) in the United States represented together a value
. and is 5 years younger than sum of B's and C's ages was 25 years.
The
three angles of any triangle are together equal to
180.
000. statement "A and B walk from two towns 27 miles apart until they meet " means the sum of the distances walked by A and B equals 27 miles. width.
California has twice as
many
electoral votes as Colorado.72
of
ELEMENTS OF ALGEBRA
$ 750.
Dividing.
14. but stops 2 hours on the way.
A and B
apart.
The copper had twice
the value of the gold.
= 35.000.
and Massachusetts has one more than California and Colorado If the three states together have 31 electoral votes.
7
Uniting. Let x = number of hours A walks.
how many
100. speed. such as length. we obtain 3 a. 8 x = 15. After how many hours will they meet and how
E.000 more than that
the copper.
. or time. First fill in all the numbers given directly.
has each state
?
If the example contains Arrangement of Problems. 3 and 4. of 3 or 4 different kinds.
Find the value of each.
3z + 4a:-8 = 27.
start at the same hour from two towns 27 miles walks at the rate of 4 miles per hour. and 4 (x But the 2) for the last column.e. then x 2 = number of hours B walks.
of
arid the value of the iron
was $300.
B
many
miles does
A
walk
?
Explanation.000.g. number of hours.
3x
+
4 (x
2)
=
27. = 5. together. and quantities
area. and A walks at the rate of 3 miles per hour without stopping. number of miles A
x
x
walks. and distance.
Hence
Simplifying. Since in uniform motion the distance is always the product of
rate
and time. i. it is frequently advantageous to arrange the quantities in a systematic manner.
After how many hours will B overtake A.
A
sets out later
two hours
B
. The second is 5 yards longer than the first. and the sum Find the length of their areas is equal to 390 square yards. and its width decreased
by 2 yards. each of the others had to pay
$ 100 more. What are the
two sums
5. twice as large. and follows on horseback traveling at the rate of 5 miles per hour.
3. but four men failed to pay their shares.74
ELEMENTS OF ALGEBRA
EXERCISE
38
rectangular field is 10 yards and another 12 yards wide.
invested at 5 %.
as a
4.
A
If its length
rectangular field is 2 yards longer than it is wide.
2. the area would remain the same. and in order to raise the required sum each of the remaining men had to pay one dollar more.
1.
sum $ 50
larger invested at 4
brings the same interest Find the first sum.
Find the share of each. A man bought 6 Ibs. of coffee for $ 1.
Six persons bought an automobile.
A
of each.55.
If the silk cost three times as
For a part he 7.
A sum
?
invested at 4 %. and the cost
of silk
of the auto-
and 30 yards of cloth cost together much per yard as the cloth.
mobile.
Ten yards
$
42.
Find the dimen-
A
certain
sum
invested at 5
%
%. How many pounds of each kind did he buy ?
8.
sions of the field. paid 24 ^ per pound and for the rest he paid 35 ^ per pound.
Twenty men subscribed equal amounts
of
to raise a certain
money. but as two of them were unable to pay their share. together bring $ 78 interest. and a second sum. and how far will each then have traveled ?
9. How much did each man subscribe ?
sum
walking at the rate of 3 miles per hour. how much did each cost per yard ?
6. were increased by 3 yards.
After how many hours. traveling by coach in the opposite direction at the rate of 6 miles per hour.LINEAR EQUATIONS AND PROBLEMS
v
75
10.
walking at the same time in the same If A walks at the rate
of 2
far
miles per hour.will they be 36 miles apart ?
11.
A
sets out
two hours
later
B
starts
New York to Albany is 142 miles. and from the same point.
The
distance from
If a train starts at
. and B at the rate of 3 miles per hour. but
A has
a start of 2 miles. how many miles from New York will they meet?
X
12.
A
and
B
set out
direction. how must B walk before he overtakes A ?
walking at the rate of 3 miles per hour. and another train starts at the same time from New York traveling at the rate of 41 miles an hour. Albany and travels toward New York at the rate of 30 miles per hour without stopping.
stage of the work.
vV
.
we
shall not.
The prime
factors of 10 a*b are 2.
104.
\-
V&
is
a
rational with respect to
and
irrational with respect
102. if it is integral to all letters contained in it.
it is
composite. if this letter does not occur in any denominator. as.
The
factors of
an algebraic expression are the quantities
will give the expression. at this
6
2
. An expression is integral with respect to a letter.
this letter.
a.
a2
to 6. 6.
An
after simplifying. consider
105. it contains no indicated root of this letter
.CHAPTER
VI
FACTORING
101.
a factor of a 2
A
factor is said to be prime. a.
irrational.
J Although Va'
In the present chapter only integral and rational expressions
b~
X
V
<2
Ir
a2
b'
2
2
?>
.
expression is rational with respect to a letter.
-f-
db
6
to b.
a-
+
2 ab
+ 4 c2
. if it contains
no other
factors (except itself
and unity)
otherwise
. 5.
+ 62
is
integral with respect to a. if.
which multiplied together
are considered factors. but fractional with respect
103. if it does contain
some indicated root of
.
76
.
An
expression
is integral
and rational with respect
and rational.
Ex. 2.g.
it fol-
lows that every method of multiplication will produce a method
of factoring.
109.
It (a.
Divide
6
a% .62
can be
&).
01.
Since factoring
the inverse of multiplication.3 6a + 1).
.9 x2^ + 12 sy* = 3 Z2/2 (2 #2 . 2. .9 x if + 12 xy\
2
The
greatest factor
common
2
to all terms
flcy*
is
8
2
xy'
.
An
the process of separating an expression expression is factored if written in the
form of a product.
y.
107.
POLYNOMIALS ALL OF WHOSE TERMS CONTAIN A COMMON FACTOR
(
mx + my+ mz~m(x+y + z).9 x2 y 8 + 12
3 xy
-f
by
3
xy\
and the quotient
But.
in the
form
4)
+3.
1. for this result is a sum.
it
follows
that a 2
.3 sy + 4 y8).
The factors
of a
monomial can be obtained by inspection
2
The prime
108.
Factor
14 a*
W-
21 a 2 6 4 c2
+ 7 a2 6
2
c2
7
a2 6 2 c 2 (2 a 2
. since (a + 6) (a 2 IP factored.
x.
2 4 x + 3) is factored if written (x' would not be factored if written x(x and not a product.
2.
factors of 12
&V
is
are 3.
55. x.
Factor G ofy 2
.
TYPE
I. dividend
is
2 x2
4
2
1/
.
?/.)
Ex.
110. or that a
=
6)
(a
= a .
or
Factoring examples may be checked by multiplication by numerical substitution.
Hence
6 aty 2
= divisor x quotient.
E. 8) (s-1).FACTORING
106.62 + &)(a 2
.
77
Factoring
is
into its factors.
11 a
2
. Factor x? .
.
5.
2
6.
determine whether
In solving any factoring example.
Ex. but only in a limited number
of ways as a product of two numbers. however.6 = 20. of this type.
11 a2 and whose sum The numbers whose product is and a. 2 11 a?=(x + 11 a) (a.4 x .
2.5) (a 6).
.11 a + 30.
We may consider
1..
4.
Therefore
Check.
3.
tfa2 -
3.
or
77
l. and (a . + 30 = 20. the two numbers
have opposite
signs.11) (a
+
7).77 =
(a.
EXERCISE
Besolve into prime factors :
40
4.
but of these only
a:
Hence
2
. If q is negative. or 11 and 7 have a sum equal to 4. the student should first all terms contain a common monomial factor.
77 as the product of 1 77.
and the greater one has the same sign
Not every trinomial
Ex. a 2 .
Ex.
11
7.5) (a .G) = . If q is positive. the two numbers have both the same sign as p.
Factor
+ 10 ax . Hence z6 -? oty+12 if= (x -3 y)(x*-4 y ). can be factored.1 1 a
tf
a 4.
If
30 and whose
sum
is
11 are
5
a2
11
a = 1.1 afy 8 The two numbers whose product is equal to 12 yp and whose sum equals 3 8 7 y are -4 y* and -3 y*. Hence fc -f 10 ax
is
10 a are 11 a
-
12 /.30 = (a .
79
Factor a2
-4 x . or 7 11.
m -5m + 6.
as p.
is
The two numbers whose product and -6.
+
112.
Factor a2
. it is advisable to consider the factors of q first.4 .
Since a number can be represented in an infinite number of ways as the sum of two numbers.FACTORING
Ex.11.
.a).
Factor 3 x 2
.
The work may be shortened by the
:
follow-
ing considerations
1.83 x
-f-
54.
and that they must be negative. X x 18.13 x + 5 = (3 x .
11 x
2x.17 x
2o?-l
V A
5
-
13 a
combination
the correct one.
. Hence only 1 x 54 and 2 x 27 need
be considered. If py? -\-qx-\-r does not contain any monomial factor.
The
and
factors of the first term consist of one pair only.
sible
13 x
negative. or
G
114. none of the binomial factors can contain a monomial factor.
a.FACTORING
If
81
we consider that the
factors of -f 5
as
must have
is
:
like signs. the second terms of the factors have same sign as q. which has the same absolute value as the term qx. 54 x 1.e-5
V A
x-1
3xl \/ /\
is
3
a. the signs of the second terms are minus. Since the first term of the first factor (3 x) contains a 3. but the opposite sign. viz. 18 x 3. all pos-
combinations are contained in the following
6x-l
x-5 . 27 x 2.
If
the factors
a combination should give a sum of cross products.
Ex. 64 may be considered the
:
product of the following combinations of numbers 1 x 54.5 . 3 x and x.31 x
Evidently the
last
2
V A
6. and after a little practice the student possible should be able to find the proper factors of simple trinomials
In actual work
at the first trial. then the second terms of
have opposite signs. 6 x 9.1). 2.
3. 2 x 27. we have to reject every combination of factors of 54 whose first factor contains a 3.5) (2 x . exchange the
signs of the second terms of the factors. and r is negative. If p is poxiliw.
. 9 x 6.
all
it is not always necessary to write down combinations.
the
If p and r are positive.
of a 4 and a 2 b is a2
The H. of 6 sfyz. C.
24
s
. The H. and prefix it as a coefficient to H.
8
.
3.
C. of
:
48
4.
122.
121.) of
two or more
.
15
aW. The H. of
aW.
the algebraic factor of highest degree common expressions to these expressions thus a 6 is the II. F. of a 7 and a e b 7
.
6.
-
23 3
.
5
2
3
. F. If the expressions have numerical coefficients.
The
highest
is
common
factor (IT.
The student should note
H.
25
W. aW.
F. C.
54
-
32
.
expressions which have no are prime to one another.CHAPTER
VII
HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE
HIGHEST COMMON FACTOR
120. F.
+
8
ft)
and
cfiW is
2
a 2 /) 2
ft)
.
13 aty
39 afyV.
5
7
34 2s
. of
two or more monomials whose factors
.
C. C.
33
2
7
3
22 3 2
.
Two
common
factor except unity
The H. of the algebraic expressions.
89
.
are prime can be found by inspection. C.
II
2
. find by arithmetic the greatest common factor of the coefficients.
3
. Thus the H.
. of (a
and (a
+
fc)
(a
4
is
(a
+ 6)
2
.
C. F. F. is the lowest
that the power of each factor in the power in which that factor occurs in any
of the given expressions.
EXERCISE
Find the H. F. F. and GO aty 8 is 6 aty.
5.
5
s
7
2
5. F. F.
2
2
. C. C.
2. 12 tfifz. C.
two lowest common multiples.
2. M. is equal to the highest power in which it occurs in any of the
given expressions. C.6 3 ).
&)
2
M. C.
M of the algebraic expressions.
300 z 2 y.
M. M.
1. which
also
signs.
etc.
.
4 a 2 &2
_
Hence.C. but opposite
. C. of the
general. C.
Find the L. Obviously the power of each factor in the L.) of
two or more
expressions is the common multiple of lowest degree. 60
x^y'
2
. To find the L.
Hence the L. of tfy and xy*.
127. C.
C.
=4 a2 62 (a2 . L. M.
Ex.M.
M.
. of several expressions which are not completely factored.
NOTE.
The
lowest
common
multiple (L.
Find the L. C. thus.
= (a -f
last
2
&)'
is
(a
-
6) .
of 4 a 2 6 2 and 4 a 4
-4 a 68
2
.
2
The The
L.
2 multiples of 3 x
and 6 y are 30 xz y. resolve each expression into prime factors and apply the method for monomials. L.M. find by arithmetic their least common multiple and prefix it as a coefficient to the L.
Ex.
Common
125. ory is the L.
The
L.
128. C.
126.
M. of as -&2 a2 + 2a&-f b\ and 6-a.
a^c8
3
.
6
c6 is
C a*b*c*.
of 3
aW.
A
common
remainder.
of 12(a
+
ft)
and (a
+ &)*( -
is
12(a
+ &)( . each set of expressions has
In example ft).(a + &) 2 (a
have the same absolute value.LOWEST COMMON MULTIPLE
91
LOWEST COMMON MULTIPLE
multiple of two or more expressions is an which can be divided by each of them without a expression
124. If the expressions have a numerical coefficient. M.C. C. M.6)2. C.
A
-f-
fraction is
b. Thus.
the product of two fractions is the product of their numerators divided by the product of their denominators.
Reduce
~-
to its lowest terms. F.
successively all
2
j/' .
Ex. however.
All operations with fractions in algebra are identical
with the corresponding operations in arithmetic.
131.
and
i
x mx = my y
terms
A
1.
Remove
tor.
a b
= ma
mb
. C. but we
In arithmetic. thus -
is identical
with a
divisor b the denominator.
TT
Hence
24
2 z = --
3x
. as 8.
common
6
2
divisors of
numerator and denomina-
and z 8
(or divide the terms
.CHAPTER
VIII
FRACTIONS
REDUCTION OF FRACTIONS
129.
a?.
rni
Thus
132.ry ^
by
their H. If both terms of a fraction are multiplied or divided by the same number) the value of the fraction is not altered. etc. only positive integral numerators shall assume that the
all
arithmetic principles are generally true for
algebraic numbers.
fraction
is
in its lowest
when
its
numerator
and
its
denominator have no
common
factors.
an indicated quotient. the value of a fraction is not altered by multiplying or dividing both its numerator and its denominator by the same number.
and denominators are considered.
130.
The dividend a is called the numerator and the The numerator and the denominator
are the terms of the fraction.
Reduce -^-. C. C.
and the terms of
***.96
134.
ELEMENTS OF 'ALGEBRA
Reduction of fractions to equal fractions of lowest common Since the terms of a fraction may be multiplied
denominator.
+
3).
1).
M.
Ex
-
Reduce
to their lowest
common
denominator. we may use the same process as in arithmetic for reducing fractions to the lowest
common
denominator.3) (-!)'
=
.
and
(a-
8).
by any quantity without altering the value of the fraction. multiplying the terms of
22
.
^
to their lowest
com-
The
L.
TheL.
we have
the quotients (x
1).
-
by 4
6' .
and
Tb reduce fractions to their lowest common denominator.C. and
135.
we may extend this method
to integral expressions. Divide the L.
Multiplying these quotients by the corresponding numerators and writing the results over the common denominator.
Since a
(z
-6 + 3)(s-3)O-l)'
6a.~16
(a
+ 3) (x.
mon
T denominator.
2>
.
3 a\ and 4
aW
is
12 afo 2 x2 .
.-1^22
' . =(z
(x
+ 3)(z.
To reduce
to a fraction with the
denominator 12 a3 6 2 x2 numerator
^lA^L O r 2 a 3
'
and denominator must be multiplied by
Similarly.r
2
2
.by 3 ^
A
2
' . by the denominator of each fraction.
we have
-M^.C. of the denominators for the common denominator. take the L.
.
1.
Ex. we have
(a
+ 3) (a -8) (-!)'
NOTE.M. and 6rar 3 a? kalr
.
-
of
//-*
2
.
.3)O -
Dividing this by each denominator.
multiply each quotient by the corresponding numerator.M.D.
g. (In
order to cancel
common
factors.
2.
fractions to integral numbers.
we may extend any
e. each
numerator and denomi-
nator has to be factored.
expressed in symbols:
c
a
_ac b'd~bd'
principle proved for
b
141.
integer.
or. multiply the
142. and the product of the denominators for the
denominator.
Since -
= a.
!.
Simplify 1 J
The
expreeaion
=8
6
. Fractions are multiplied by taking the product of tht numerators for the numerator. Common factors in the numerators and the denominators should be canceled before performing the multiplication.
F J Simplify
.102
ELEMENTS OF ALGEBRA
MULTIPLICATION OF FRACTIONS
140.
2
a
Ex.
-x
b
c
=
numerator by
To multiply a fraction by an
that integer.)
Ex.
.
The
reciprocal of a
number
is
the quotient obtained by
dividing 1
by that number. and the principle of division follows
may
be expressed as
145.y3
+
xy*
x*y~ -f y
8
y
-f
3
2/
x3
EXERCISE 56*
Simplify the following expressions
2
x*
'""*'-*'
:
om
2 a2 6 2
r -
3
i_L#_-i-17
ar
J
13 a& 2
5
ft2
'
u2
+a
.104
ELEMENTS OF ALGEBRA
DIVISION OF FRACTIONS
143.
* x* -f xy 2
by
x*y
+y
x'
2
3
s^jf\ =
x'
2
x*
.
1. x a + b
obtained by inverting
reciprocal of a fraction
is
the fraction. expression by the reciprocal of the fraction.
Divide X-n?/
. invert the divisor and multiply it by the dividend. Integral or mixed divisors should be expressed in fractional form before dividing. To divide an expression by a fraction.
144.
The reciprocal of ?
Hence the
:
+*
x
is
1
+ + * = _*_. :
a 4-1
a-b
* See page 272.
8
multiply
the
Ex. To divide an expression by a fraction.
The The
reciprocal of a
is
a
1
-f-
reciprocal of J
is
|
|.
A can do a piece of work in 3 days and B in 2 days. = the number of minute spaces the minute hand moves
over.
Multiplying by
Dividing. When between 3 and 4 o'clock are the hands of
a clock together
?
is
At
3 o'clock the hour hand
15 minute spaces ahead of the minute
:
hand.114
35.
A would do
each day ^ and
B
j.
ELEMENTS OF ALGEBRA
(a) Find a formula expressing degrees of Fahrenheit terms of degrees of centigrade (<7) by solving the equation
(F)
in
(ft)
Express in degrees Fahrenheit 40
If
C.
12. Ex.20
C.
and
12
= the number
over.
days by x and the piece of work while in x days they would do
respectively
ff
~ and and hence the sentence written in algebraic symbols ^. 2.
x
Or
Uniting. 1..
Find
R in terms of C and
TT.
100
C.
C
is
the circumference of a circle whose radius
R.
is
36.
then
= 2 TT#.
PROBLEMS LEADING TO FRACTIONAL AND LITERAL EQUATIONS 152. In how many days can both do it working together ?
If
we denote
then
/-
the required
number
by
1.
.
..
Ex.
of minute spaces the
hour hand moves
Therefore x
~ = the number of minute spaces the minute hand moves
more than the hour hand.minutes after x=
^
of
3 o'clock.
~^ = 15
11 x
'
!i^=15. 2 3
. hence the question would be formulated After how many minutes has the minute hand moved 15 spaces more than the hour hand ?
Let then
x x
= the required number of minutes after 3 o'clock. = 16^.180.
the required
number
of days.
But
in
uniform motion Time
=
Distance
.
= 100 + 4 x. hours more than the express train to travel 180 miles.
32
x
= |. then
Ox
j
5
a
Rate Hence the rates can be expressed.
180
Transposing. u The accommodation train needs 4 hours more than the express train." gives the equation /I).FRACTIONAL AND LITERAL EQUATIONS
A
in symbols the following sentence
115
more symmetrical but very similar equation is obtained by writing ** The work done by A in one day plus the work done by B in one day equals the work done by both in one day.
fx
xx*
=
152
+4
(1)
Hence
=
36
= rate
of express train.
or 1J."
:
Let
x -
= the
required
number
of days. 4x = 80.
Explanation
:
If
x
is
the rate of the accommodation train.
Ex. and the statement.
the rate of the express train. what is
the rate of the express train
?
180
Therefore.
in
Then
Therefore.
Solving.
= the
x
part of the
work both do
one day.
Clearing. 3. The speed of an express train is $ of the speed of an If the accommodation train needs 4 accommodation train.
length in the ground.116
ELEMENTS OF ALGEBRA
EXERCISE
60
1.
and 9
feet above water. and of the father's age. and one half the greater Find the numbers.
ceeds the smaller by
4.
Find a number whose third and fourth parts added
together
2. How
did the
much money
man
leave ?
11.
is oO.
make
21. and found that he had \ of his original fortune left.
3.
by 6. to his son.
Find A's
8.
of his present age.
9
its
A
post
is
a fifth of
its
length in water. a man had How much money had he
at first?
.
ex-
What
5. which was $4000.
Find two consecutive numbers such that
9.
A man left ^ of his property to his wife.
J-
of the greater
increased by ^ of the smaller equals
6.
Twenty years ago A's age was |
age.
is
equal
7.
fifth
Two numbers
differ
2.
The sum
10 years hence the son's age will be
of the ages of a father and his son is 50.
its
Find the number whose fourth part exceeds part by 3. How much money had he at first?
12
left
After spending ^ of his
^ of his money and $15. -|
Find their present ages.
Two numbers
differ
l to s of the smaller.
by 3.
money and $10. to his daughand the remainder.
are the
The sum of two numbers numbers ?
and one
is
^ of the other. and J of the greater Find the numbers. A man lost f of his fortune and $500. one half of What is the length
of the post ?
10
ter.
A man
has invested
J-
of his
money
at
the remainder at
6%.
after
rate of the latter ?
15. what is the
14. In how many days can both do it working together ? ( 152. and an ounce of silver -fa of an ounce. A can do a piece of work in 4 clays. If the accommodation train needs 1 hour more than the express train to travel 120 miles. 2.
air. Ex.
At what time between 7 and
8 o'clock are the hands of
?
a clock in a straight line and opposite
18.
at 4J % and P> has invested $ 5000 They both derive the same income from their How much money has each invested ?
20. 3. ounces of gold and silver are there in a mixed mass weighing
20 ounces in
21.
152. and B In how many days can both do it working together
in
?
12 days. ^ at 5%. Ex.
117
The speed
of an accommodation train
is
f of the speed
of an express train.
How much money
$500?
4%.
investments.)
22. and
it
B in 6 days. and B in 4 days. 1.
A can
A
can do a piece of work in 2 days.)
At what time between 7 and 8
o'clock are the
hands of
a clock together ?
17.
.
A has invested capital
at
more
4%.
At what time between 4 and
(
5 o'clock are the hands of
a clock together?
16. Ex. and has he invested if
his animal interest therefrom is
19. An ounce of gold when weighed in water loses -fa of an How many ounce. and after traveling 150 miles overtakes the accommodation train. If the rate of the express train is -f of the rate of the accommodation train.) (
An express train starts from a certain station two hours an accommodation train. what is the rate of the express train? 152.
?
In
how many days can both do
working together
23.FRACTIONAL AND LITERAL EQUATIONS
13.
and losing
1-*-
ounces when weighed in water?
do a piece of work in 3 days.
Find three consecutive numbers whose sum
Find three consecutive numbers whose sum
last
:
The
two examples are
special cases of the following
problem 27. In how
in the numerical values of the
:
many days
If
can both do
we
let
x
= the
it working together ? required number of days. A in 4.
make
it
m
6
A can do this work in 6 days Q = 2. 3. therefore. The problem to be solved.g.
. n x
Solving. is A can do a piece of work in m days and B in n days.
B in 5. A in 6. 2.414. if
B
in 3 days.118
153. . and n = 3. it is possible to solve all examples of this type by one example. Ex. is 57.
To
and
find the numerical answer.
is 42.
they can both do
in 2 days.
6
I
3
Solve the following problems
24. B in 30. Find three consecutive numbers whose sum equals m.
ELEMENTS OF ALGEBRA
The
last three questions
and their solutions differ only two given numbers. by taking for these numerical values two general algebraic numbers. and apply the
method of
170.
we
obtain the equation
m m
-. Then
ft
i.e. Answers to numerical questions of this kind may then be found by numerical substitution. A in 6.
25. e.
:
In
how many days
if
can
A
and
it
B
working together do a
piece of
work
each alone can do
(a)
(6)
(c)
in the following
number
ofdavs:
(d)
A in 5. m and n. B in 12.
.
26. Find the numbers if m = 24 30.= -. B in 16. Hence.009 918.=
m
-f-
n
it
Therefore both working together can do
in
mn
-f-
n
days.
(c) 16.
is ?n
. 2 miles per hour. (d) 1. Find the side of the square. solve the following ones Find two consecutive numbers the difference of whose squares
:
find the smaller number. respectively (a) 60 miles.
34. 2 miles per hour. the rate of the
first.
119
Find two consecutive numbers the difference of whose
is 11.
:
(c)
64 miles.
4J-
miles per hour.721.
The
one:
31. two pipes together ? Find the numerical answer. If each side of a square were increased by 1 foot.
. and how many miles does each travel ?
32.
Find two consecutive numbers -the difference of whose
is 21. 3 miles per hour. 88 one traveling 3 miles per hour. 3J miles per hour. and how many miles does each travel ? Solve the problem if the distance.
squares
30. and the second 5 miles per hour. the second at the apart. the area would be increased by 19 square feet.
and
the rate of the second are.
by two pipes in m and n minutes In how many minutes can it be filled by the respectively.FRACTIONAL AND LITERAL EQUATIONS
28. (b) 149.
last three
examples are special cases of the following
The
difference of the squares of
two consecutive numbers
By using the result of this problem.
Two men
start at the
first
miles
apart.001. if m and n are.
is (a)
51. (a) 20 and 5 minutes.
same hour from two towns. After how many hours do they meet. 5 miles per hour.000.
squares
29. After how many hours do they rate of n miles per hour. d miles the first traveling at the rate of m. respectively. the
Two men start at the same time from two towns. (b) 8 and 56 minutes.
A cistern can
be
filled
(c)
6 and 3 hours.
33. (b) 35 miles.
meet.
.
the symbol
being a sign of division.g.
term of a ratio
a
the
is
is
the antecedent.CHAPTER X
RATIO AND PROPORTION
11ATTO
154. the denominator
The
the
157. the antecedent.
antecedent.
E.
The
first
156.
is
numerator of any fraction
consequent.5.) The ratio of 12 3 equals 4.
b.
b. 158. the second
term the consequent.
1.
:
:
155.
" a Thus."
we may
write
a
:
b
= 6.
terms are multiplied or divided by the same number.
The
ratio of
first
dividing the
two numbers number by the
and
:
is
the quotient obtained by
second.
b
is
a
Since a ratio
a
fraction.
The
ratio -
is
the inverse of the ratio -.
:
A somewhat shorter way
would be to multiply each term by
120
6.
In the ratio a
:
ft. b is the consequent. instead of writing
6 times as large as
?>. etc.
Ex.
A
ratio
is
used to compare the magnitude of two
is
numbers. all principles
relating
to
fractions
if its
may
be af)plied to ratios. 6 12 = . a ratio
is
not changed
etc.
Thus the
written a
:
ratio of a
b
is
.or a *
b
The
ratio is also frequently
(In most European countries this symbol is employed as the usual sign of division.
Simplify the ratio 21 3|.
The last
first three.
extremes. The last term d is the fourth proportional to a.
27 06: 18 a6. the second
and fourth terms of a proportion are the and third terms are the means.
terms.
11.
159.
J:l.
and
c
is
the third proportional to a and
.
16a2 :24a&.
b is the
mean
b.
3:1}.
17.
7|:4 T T
4
.
$24: $8.
= |or:6=c:(Z are
The
first
160.
3:4.
In the proportion a b
:
=
b
:
c.
12. and c.
two
|
ratios.
7f:6J.
Transform the following
unity
15.
6. b and c the means.
16.
:
ratios so that the antecedents equal
16:64.
16 x*y
64 x*y
:
24 48
xif. b.
:
1.
61
:
ratios
72:18.
proportional between a
and
c. and the last term the third proportional to the first and second
161. a and d are the extremes.
Simplify the following ratios
7.
4|-:5f
:
5.
10.
:
a-y
.
equal
2.
3
8.
9.
5 f hours
:
2.
62:16.
18.
4.
term
is
the fourth proportional to the
:
In the proportion a b = c c?.
:
is
If the means of a proportion are equal.RATIO
Ex.
AND PROPORTION
ratio 5
5
:
121
first
Transform the
3J so that the
term will
33
:
*~5
~
3
'4*
5
EXERCISE
Find the value of the following
1. either mean the mean proportional between the first and the last terms.
8^-
hours.
3.
A
proportion
is
a statement expressing the equality of
proportions.
1.
The mean proportional
of their product. and we
divide both
members by
we have
?^~ E.e. of iron weigh .)
mn = pq. and the
other pair the extremes. or 8 equals the inverse ratio of 4 3. of iron weigh 45 grams. Hence the number of men required to do some work.
Clearing of fractions.
briefly.
164. In any proportion product of the extremes. i. then G ccm. if the ratio of any two of the first kind.
ad =
be.
If 6 men can do a piece of work in 4 days. a b
:
bettveen two
numbers
is
equal to
the square root
Let the proportion be
Then Hence
6
=b = ac. of a proportion.
163.
:
:
directly proportional
may say.'*
Quantities of one kind are said to be inversely proportional to quantities of another kind. then 8 men can do it in 3 days.
:
c.
q~~ n
. " we " NOTE.
If
(Converse of
nq.
ccm. 6 ccm. If the product of two numbers is equal to the product of two other numbers^ either pair may be made the means. 3 4. = 30 grams 45 grams. is equal to the ratio of the corresponding two
of the other kind. Hence the weight of a mass of iron is proportional to its volume. are
: : :
inversely proportional. Instead of u
If 4
or 4 ccm.
ELEMENTS OF ALGEBRA
Quantities of one kind are said to be directly proper
tional to quantities of another kind.__(163.
2
165.
pro-
portional. if the ratio of any two of the first kind is equal \o the inverse ratio of the corresponding two of
the other kind.
163.122
162.
!-.
t/ie
product of the means
b
is
equal
to the
Let
a
:
=c
:
d.30 grams.) b = Vac. and the time necessary to do it.
the squares of their radii
(e)
55. the volume of a
The temperature remaining
body of gas inversely proportional to the pressure. The number of men (m) is inversely proportional to the number of days (d) required to do a certain piece of work.
What
will be the
volume
if
the pressure
is
12 pounds per square inch ?
.
and the speed
of the train.
what
58.
57. under a pressure of 15 pounds per square inch has a volume of
gas
is
A
16 cubic
feet.
areas of circles are proportional to the squares of If the radii of two circles are to each other as
circle is
4
:
7. and the area
of the rectangle.
(d)
The sum
of
money producing $60
interest at
5%.
the area of the larger? the same.
and the area of the smaller
is
8 square inches.
(b)
The time a
The length
train needs to travel 10 miles.
56.
A
line 7^. (e) The distance traveled by a train moving at a uniform rate. and the time. othei
(a) Triangles
as their basis (b
and
b'). and
the time necessary for it.
1
(6) The circumferences (C and C ) of two other as their radii (R and A"). and the
:
total cost.inches long represents
map corresponds to how many miles ?
The
their radii.
ELEMENTS OF ALGEBEA
State the following propositions as proportions : T (7 and T) of equal altitudes are to each.
A
line 11 inches long
on a certain
22 miles.126
54.
(c)
of a rectangle of constant width. State whether the quantities mentioned below are directly or inversely proportional (a) The number of yards of a certain kind of silk. (c) The volume of a body of gas (V) is
circles are to
each
inversely propor-
tional to the pressure (P).
(d)
The
areas
(A and
A') of two circles are to each other as
(R and R').
11 x = 66 is the first number. so that
Find^K7and BO.
Let
A
B
AC=1x. 11 x -f 7 x = 108.
Then
Hence
BG = 5 x.
is
A line AB. What is the greatest distance a person can see from an elevation of 5 miles ? From h miles
the
Metropolitan
Tower (700
feet high) ?
feet
high) ?
From Mount
McKinley (20. When a problem requires the finding of two numbers which are to each other as m n.
4
'
r
i
1
(AC): (BO) =7: 5.
x=2. 2. = the second number.
:
Ex. AB = 2 x. it is advisable to represent these unknown numbers by mx and nx.
as 11
Let
then
:
1.
Divide 108 into two parts which are to each other
7. 18 x = 108.
2 x
Or
=
4. produced to a point C.
Hence
or
Therefore
Hence
and
= the first number.RATIO AND PROPORTION
69.
Therefore
7
=
14
= AC.
127
The number
is
of miles one can see from an elevation of
very nearly the mean proportional between h and the diameter of the earth (8000 miles). x = 6.
. 7 x = 42 is the second number.
11
x
x
7
Ex. 4 inches long.000
168.
128
ELEMENTS OF ALGEBRA
EXERCISE
63
1.
m
in the ratio x:
y
%
three sides of a triangle are 11.
consists of 9 parts of copper and one part of ounces of each are there in 22 ounces of gun-
metal ?
Air is a mixture composed mainly of oxygen and nitrowhose volumes are to each other as 21 79.
3. Brass is an alloy consisting of two parts of copper and one part of zinc.
of water?
Divide 10 in the ratio a
b.
:
Divide 39 in the ratio 1
:
5.)
.
11.
Gunmetal
tin.
7.
Divide 20 in the ratio 1 m.
Divide 44 in the ratio 2
Divide 45 in the ratio 3
:
9. How
The
long are the parts ? 15.
What
are the parts ?
5. and 15 inches.000 square miles.000. and c inches. cubic feet of oxygen are there in a room whose volume is 4500
:
cubic feet?
8.
How many
7. 12.
The
total area of land is to the total area of
is
water as
7 18.
12. and the longest is divided in the ratio of the other two. How many ounces of copper and zinc are in 10 ounces of brass ?
6. The three sides of a triangle are respectively a.
:
197.
14.
:
Divide a in the ratio 3
Divide
:
7.
:
4. 6. what are
its
parts ?
(For additional examples see page 279. How many gen. 9. find the number of square miles of land and of water.
13. If c is divided in the ratio of the other two.
A line 24 inches
long
is
divided in the ratio 3
5.
How many
grams of hydrogen are contained in 100
:
grams
10. Water consists of one part of hydrogen and 8 parts of
If the total surface of the earth
oxygen. 2.
there is only one solution. From (3) it follows y 10 x and since
by the same values of x and
to be satisfied
y. y =
5
/0 \ (2)
of values.
a?
(1)
then
I. expressing a y. is x = 7. x = 1. etc.
the equations have the two values of
y must be equal. which substituted in (2) gives y both equations are to be satisfied by the same Therefore.CHAPTER XI
SIMULTANEOUS LINEAR EQUATIONS
169.
However.-.
The
root of (4)
if
K
129
. values of x and y.
Hence
2s -5
o
= 10 _ ^
(4)
= 3.
if
there
is
different relation
between x and
*
given another equation. y = 1.
2 y = . the equation is satisfied by an infinite number of sets Such an equation is called indeterminate.y=--|.
Hence. if
.
y
(3)
these
unknown numbers can be found.
If
satisfied
degree containing two or more by any number of values of
2oj-3y =
6. such as
+ = 10.
An
equation of the
first
unknown numbers can be the unknown quantities.e. =.-L
x
If
If
= 0.
3 y = 80. and 3 x + 3 y =. y = 2.
A
system of two simultaneous equations containing two
quantities is solved by combining them so as to obtain
unknown
one equation containing only one
173.
Substitution. 26 y = 60.
172.
6x
.
are simultaneous equations. Independent equations are equations representing different relations between the unknown quantities such equations
.26. 6 and 4 x y not simultaneous.
21 y
.
174.
E. 30 can be reduced to the same form -f 5 y Hence they are not independent.24.
~ 50. for they cannot be satisfied by any value of x and y.
The process of combining several equations so as make one unknown quantity disappear is called elimination.
Therefore.
ELIMINATION BY ADDITION OR SUBTRACTION
175. The first set of equations is also called consistent.
to
The two methods
I.
unknown
quantity. y
I
171.
x
-H
2y
satisfied
6 and 7 x 3y = by the values x = I. for they express the x -f y 10. for they are 2 y = 6 are But 2 x 2.
of elimination
most frequently used
II.
Solve
-y=6x
6x
-f
Multiply (1) by
2.
ELEMENTS OF ALGEBRA
A
system
of simultaneous equations is
tions that can be satisfied
a group of equa by the same values of the unknown
numbers.
(3)
(4)
Multiply (2) by
-
Subtract (4) from (3).
cannot be reduced to the same form. Any set of values satisfying 5 x + 6 y = 60 will also satisfy the equation 3 x -f. the last set inconsistent.X. 3. same relation.
viz.
= .
4y
.
By By
Addition or Subtraction.130
170.
# 4.)
it is advisable to represent a different letter.
to express
it is difficult
two of the required
digits in
terms
hence we employ 3
letters for the three
unknown
quantities.
(1)
100s
+ lOy + z + 396 = 100* + 10y + x.
.
=
l. 1 digit in the tens place. the first and the last digits
will be interchanged. however.
. either directly or implied.
y
*
z
30. The sum of three digits of a number is 8.
Check.
symbols:
x
+
y
+z-
8.
x
:
z
=1
:
2.
= 2 m.2/
2/
PROBLEMS LEADING TO SIMULTANEOUS EQUATIONS
183.SIMULTANEOUS LINEAR EQUATIONS
143
x
29. z + x = 2 n. 1. Problems involving several unknown quantities must contain.y
125
(3)
The solution of these equations gives x Hence the required number is 125. and to express
In complex examples.
Ex.
2
= 6. The digit in the tens' place is | of the sum of the other two digits.
and Then
100
+
10 y
+z-
the digit in the units' place.
(
99.
1
=
2. + z = 2p.
The
three statements of the problem can
now be
readily expressed in
.
M=i.
+2+
6
= 8.
unknown quantity by
every verbal statement as an equation.
2
= 1(1+6). y
31.
Obviously
of the other
. the number. Simple examples of this
kind can usually be solved by equations involving only one
unknown
every
quantity.
+
396
= 521.
Let
x
y z
= the
the digit in the hundreds' place. and if 396 be added to the number. as many verbal statements as there are unknown quantities.
Find the number.
By
expressing the two statements in symbols.
= the
fraction.
3. the fraction
Let and
then y
is
reduced to
nurn orator. x 3x-4y = 12. and C travel from the same place in the same B starts 2 hours after A and travels one mile per hour faster than A.
.144
Ex.
3+1 5+1
4_2.
increased by one.
x 3
= 24.
(1) (2)
12. the fraction is reduced to | and if both numerator and denominator of the reciprocal of the fraction be dimin-
ished by one.
direction.
Or
(4)-2x(3).
2.
x
y
= the = the
x
denominator
.
we
obtain.
6
x 4
= 24. 5_
_4_
A.
4
x
= 24. who travels 2 miles an hour faster than B. = 8.
=
Hence the
fraction
is
f. y = 3. the distance traveled by A.
Find the
fraction. B. starts 2 hours after B and overtakes A at the same How many miles has A then traveled? instant as B.
xy
a:
2y 4y
2.
ELEMENTS OF ALGE13KA
If both numerator and denominator of a fraction be
.
2.
Ex.
8
= xy + x xy = xy -f 3 x 2 y = 2. C.
3
xand y
I
1
(2)
5.
(3) C4)
=
24 miles.
+
I
2
(1)
and
These equations give x
Check. From (3)
Hence xy
Check.
Since the three
men
traveled the
same
distance.
)
added to a number of two digits. both terms. and the fourth 3. Find the fraction. to the number the digits will be interchanged. and twice the numerator What is the fracincreased by the denominator equals 15.
the
number
(See Ex.
Half the sum of two numbers equals 4.
Find the numbers.
5. the Find the fraction.
?
What
9.SIMULTANEOUS LINEAR EQUATIONS
EXERCISE
70
145
1. the value of the fraction is fa.
. the last two digits are interchanged. its value added to the denominator. Find the number.
183. If the denominator be doubled. A fraction is reduced to J.
number by
the
first
3. fraction is reduced to \-.
added to the numerator of a fraction.
The sum
18
is is
and
if
added
of the digits of a number of two figures is 6. If
9 be added to the number.
If 4 be
Tf 3 be
is J. and four times the first digit exceeds the second digit by 3.
to
L
<>
Find the
If the
numerator and the denominator of a fraction be If 1 be subtracted from increased by 3. and the two digits exceeds the third digit by 3. Four times a certain number increased by three times another number equals 33.
Five times a certain number exceeds three times another 11. and the numerator increased by 4.
2.
tion ?
8.
part of their difference equals
4. and the second increased by 2 equals three times the first.
If 27 is
10. if its numerator and its denominator are increased by 1. the fraction equals . and
its
denomi-
nator diminished by one. Find the numbers.}.
1.
6. Find the numbers.
7.
If the
numerator of a fraction be trebled.
The sum
of the first
sum
of the three digits of a number is 9. the fraction is reduced
fraction. Find the number. it is reduced to J. and the second one increased by 5 equals twice
number. the digits will be interchanged.
much money
is
invested at
A sum
of
money
at simple interest
amounted
in 6 years
to $8000. Twice A's age exceeds the sum of B's and C's ages by 30.
19.
A
sum
of $10. and B's age is \ the sum of A's and C's ages.
14. Two cubic centimeters of gold and three cubic centimeters of silver weigh
together 69 J.000
is
partly invested at
6%. the rate of interest?
18. If the sum of
how
old
is
each
now ?
at
invested $ 5000.
. partly at 5% and partly at 4%. and in 5 years to $1125. Find
the rates of interest. A man invested $750.
and
money and
17. the annual interest would be $ 195. now.
13. and The 6 investment brings $ 70 more interest than the 5
%
%
4%
investments together. Find the weight of one cubic centimeter of gold and one cubic centimeter of silver.
in 8 years to $8500. and the 5% investment brings $15 more interest than the 4 % investment. the rate of interest ?
What was
the
sum
of
A sum
of
money
at simple interest
amounted
in 2 years
to $090. respectively ?
16. What was the amount of each investment ?
A man
%
5%.
Ten years ago A was B was as
as old as
B
is
old as
will be 5 years hence . a part at 6 and the remainder bringing a total yearly interest of $260. Ten years ago the sum of their ages was 90. Three cubic centimeters of gold and two cubic centimeters of silver weigh together 78 grains. Find their present ages.146
ELEMENTS OF ALGEBRA
11.
partly at
5 %. How 6 %.grams. 12. What was the amount of each investment ?
15. and partly at 4 %.
and 5 years ago
their ages is 55.
What was
the
sum and
rates
est
The sums of $1500 and $2000 are invested at different and their annual interest is $ 190. and 4 %. bringing a total yearly interest of $530. If the rates of interwere exchanged. 5 %.
and F. for $ 740.
. and AC = 5 inches.
points. If angle ABC = GO angle BAG = 50. The number of sheep was twice the number of horses and cows together. and sheep. and F.
triangle
Tf
AD.SIMULTANEOUS LINEAR EQUATIONS
147
20. and angle e angle/.
Find their
rates of walking.
It takes
A two hours
longer
24 miles. and CF?
is
a circle
inscribed in the
7<7.
An C touch ing the sides in D. BD = HE. BE.
E. and GE = CF. then AD = AF. and $15 for each sheep.
the three sides of a triangle E.
A
r
^
A
circle is inscribed in triangle
sides in D. The sum of the 3 angles of a triangle is 180. three
AD = AF. $ 50 for each cow. ED = BE. How many did he sell
of each if the total
number
of animals
was 24?
21. he would walk it in two hours less than
than
to travel
B
B. what are the angles of the triangle ?
22. BC = 7 inches.
24. B find angles a.
Find the parts of the
ABC touching the three sides if AB = 9.
1
NOTE. what is
that
=
OF.
.
is
the center of the circum-
scribed circle.
25. andCL4 = 8. and F '(see diagram). and CE If AB = G inches. A farmer sold a number of horses. receiving $ 100 for each horse. and e.
BC=7.
respectively. are taken so
ABC. If one angle exceeds the sum of the other two by 20. but if A would double his pace. cows. and their difference by GO . c. angle c = angle d.
In the annexed diagram angle a = angle b.
23. and angle BCA = 70.
On
/).
the length of
NOTE.
jr. then
the position of point is determined if the lengths
of
P
P3f and
185..
is
The point whose abscissa is a.
* This chapter
may
be omitted on a
148
reading.
the ordinate of point P. and ordinates abore the x-axis are considered positive .
(2. Abscissas measured to the riyht of the origin. and whose ordinate is usually denoted by (X ?/).
?/. Thus the points A.
two fixed straight lines XX' and YY' meet in at right angles. and
respectively represented
Dare
and
by
(3 7 4). and point the origin.
It'
Location of a point.
.
or its equal
OM. and PJ/_L XX'.
The
abscissa
is
usually denoted by
line XX' is called the jr-axis.
2).
PN.
-3).
lines
PM
the
and P^V are
coordinates
called
point P. (7. and PN _L YY'.
first
3).
The
of
Coordinates.
186.
(3. YY' they-axis. PM. and r or its equal OA is
.CHAPTER
XII*
GRAPHIC REPRESENTATION OF FUNCTIONS AND
EQUATIONS
184.
(2. B. hence
The
coordinates lying in opposite directions are negative.
is the abscissa.
PN are given. the ordinate by ?/.
.
Plot the points
:
(0.1). (4.and(l. -2).
(-1. (4. which of its coordinates
known ?
13. 11. 0).
(4.
4).
6.GRAPHIC REPRESENTATION OF FUNCTIONS
The
is
149
process of locating a point called plotting the point.
2.
Graphs.
4)
and
(4.e. (-2.
Plot the points: (-4.
and measure
their
distance. i. 0).
(-3. 0).
(-5.
Plot the points
(6. -3).
3. 3).(!. (-4.
0).3).
all all
points
points
lie
lie
whose abscissas equal zero ?
whose ordinates equal zero?
y) if y
10.
.
What
Draw
is
the distance of the point
(3. (4.
-2). -!).
4.4).
Draw
the triangle whose vertices are respectively
(-l. -4).
Plot the points:
(4.
2J-).
71
2).
8. paper ruled with two sets of equidistant and parallel linos intersecting at right angles. (0.
6. the mutual dependence of the two quantities may be represented
either by a table or
by a diagram.2). Graphic constructions are greatly facilitated by the use of cross-section paper.
two variable quantities are so related that
changes of the one bring about definite changes of the other. 1).
1).)
EXERCISE
1.
whose coordinates are given
NOTE. (-4.
the quadrilateral whose vertices are respectively
(4.
What
are the coordinates of the origin ? If
187.
What
is
the locus of
(a?. (0. 3).
=3?
is
If a point lies in the avaxis.
4)
from the
origin ?
7.
12. (See diagram on page 151.
Where do Where do
Where do
all
points
lie
whose ordinates
tfqual
4?
9.
10
. may be represented graphby making each number in one column the abscissa.
we meas1
.. B.
Thus the average temperature on May
on April 20. A.
1.
By representing
of points.
representation does not allow the same accuracy of results as a numerical table.
.
may be found
on Jan.
from January 1 to December 1. or the curved line the temperature.
A graphic
and
it
impresses upon the eye
all
the peculiarities of
the changes better and quicker than any numerical compilations.
188. however.
ure the ordinate of F.
15. C. Thus the first table produces 12 points.150
ELEMENTS OF ALGEBRA
tables represent the average temperature
Thus the following
of
New
volumes
1
Y'ork City of a certain
to 8 pounds. and the corresponding number in the adjacent column the ordinate of a point.
ically
each representing a temperature at a certain date. but it indicates in a given space a great many more
facts than a table. and the amount of gas subjected to pressures from
pound
The same data. in like manner the average temperatures for every value of the time. we obtain an uninterrupted sequence
etc.
ABCN
y
the so-called graph of
To
15
find
from the diagram the temperature on June
to be 15
. D.
. The engineer. concise representation of a
number
of numerical data
is
required.GRAPHIC REPRESENTATION OF FUNCTIONS
151
i55$5St5SS 3{utt|s33<0za3
Graphs are possibly the most widely used devices of applied matheThe scientist uses them to compile the data found from experiments. (c) January 15. uses them. Whenever a clear. the graph
is
applied. the
matics.
:
72
find approximate answers to the following
Determine the average temperature of New York City on (a) May 1.
EXERCISE
From the diagram
questions
1. Daily papers represent ecpnoniical facts graphically. and to deduce general laws therefrom.
physician. the merchant. (d) November 20. (b) July 15. etc. as the prices and production of commodities. the rise and fall of wages.
. 1?
11
0.?
is
is
the average temperature of
New York
6.
How much warmer
1 ?
on the average
is it
on July 1 than
on
May
17.
Which month
is
is
the coldest of the year?
Which month
the hottest of the year?
16. ?
-
3. At what date is the average temperature lowest? the lowest average temperature ?
5.
During what month does the temperature decrease most
rapidly ?
13. 1 to Oct. At what date is the average temperature highest the highest average temperature?
?
What What
is
4.
When
the average temperature below
C.
is
10.
How
much.
on
1 to
the
average.
From what
date to what date does the temperature
increase (on the average)?
8..152
2. (c)
the average temperature oi 1 C. (1)
10
C.
When
What
is
the temperature equal to the yearly average of
the average temperature from Sept.
During what months
above 18 C..
1 ?
does
the
temperature
increase from
11. ?
9.
15. (freezing
point) ?
7. from what date to what date would it extend ?
If
.
ELEMENTS OF ALGEKRA
At what date
(a) G
or dates
is
New York
is
C.
June
July
During what month does the temperature increase most
?
rapidly
12.
During what month does the temperature change least?
14.
is
ture
we would denote the time during which the temperaabove the yearly average of 11 as the warm season. (d) 9 0.
19.
Draw
.
20. From the table on page 150 draw a graph representing the volumes of a certain body of gas under varying pressures.
NOTE.
a temperature chart of a patient.
153
1?
When is the average temperature the same as on April
Use the graphs of the following examples for the solution of concrete numerical examples.
Represent graphically the populations
:
(in
hundred thou-
sands) of the following states
22.09 yards.
Hour
Temperature
. Construct a diagram containing the graphs of the mean temperatures of the following three cities (in degrees Fahren-
heit)
:
21.GRAPHIC REPRESENTATION OF FUNCTIONS
18. transformation of meters into yards.
Draw
a
graph for the
23. One meter equals 1. in a similar manner as the temperature graph was applied in examples 1-18.
books from
for printing.
A
10 wheels a day.
+7
If
will
respec-
assume the values 7. binding.g.
3.50. the value of a of this quantity will change. if each copy sells for $1.inch.
function
If the value of a quantity changes. 2 x -f 7 gradually from 1 to 2. e.
to 20 Represent graphically the weight of iron from cubic centimeters.5
grams. etc. if 1 cubic centimeter of iron weighs 7. 2 ..
2
is
called
x
2 xy
+ 7 is a function of x.
x*
x
19. 9. x
7 to 9.50.
190.154
24.
26..
to
27. 1 to 1200 copies.
.
ELEMENTS OF ALGEBRA
If
C
2
is
the circumference of a circle whose radius
is J2. 3. gas.
4. represent his daily gain (or loss).
2.
The
initial cost of
cost of manufacturing a certain book consists of the $800 for making the plates.)
T
circumferences of
25.
then
C
irJl.
x increases will change gradually from
13. if he sells 0. and $.
28.
If
dealer in bicycles gains $2 on every wheel he sells.
Show
graphically the cost of the
REPRESENTATION OF FUNCTIONS OF ONE VARIABLE
189.50 per copy
(Let 100 copies = about \.
from
R
Represent graphically the = to R = 8 inches. if x assumes
successively the
tively
values
1. amount to $8.
(Assume ir~
all circles
>2
2
. 2 8 y' + 3 y is a function of x and
y. etc.
29. An expression involving one or several letters a function of these letters..) On the same diagram represent the selling price of the books. the daily average expenses for rent. Represent graphically the cost of butter from 5 pounds if 1 pound cost $. Represent graphically the distances traveled by a train in 3 hours at a rate of 20 miles per hour.
however. E.
values of x2
nates are the corresponding i.
.2 x
may
4 from x
=
4. The values of func192.
2). x a variable. 3 50.
it is
In the example of the preceding article.
may
. plot points which
lie
between those constructed above.
-J).
(-
2.1).
Draw the graph of x2 -f.
9).0). Thus the table on page 1G4 gives the values of the functions x 2 x3 and Vsr. for x=l. as
1.g. while 7 is a constant.1).
9). be also represented by a graph. 2.
(1^. (2. construct
'. to con struct the graph x of x 2 construct a series of -3 points whose abscissas rep2 resent X) and whose ordi1
tions
.
is
A
constant
a quantity whose value does not change in the
same discussion.
3
(0.e. If a more exact diagram
is
required. (1. hence
various values of x
The values of a function for the be given in the form of a numerical table. 4).
Graph
of a function. 4).
etc. to
x = 4.
Ex.
Q-. and (3.
155
-A
variable is a quantity
whose value changes in the
same
discussion.
To
obtain the values of the functions for the various values of
the
following arrangement
be found convenient
:
.
a*.
1
the points (-3.
2
(-1.GRAPHIC REPRESENTATION OF FUNCTIONS
191.
may.
is
supposed to change.
and join
the
points in order.
r
*/
+*
01
.2 x
.
2. and join(0.
(-3.)
For brevity.
(To avoid
very large ordinatcs. Thus 4x + 7.156
ELEMENTS OF ALGEBRA
Locating the points(
4. (-2.
rf
71
. Thus in the above example.
Draw
y
z x
the graph of
= 2x-3.
A
Y'
function of the
first
degree is an integral
rational function
involving only
the
power of the variable.
7
. or ax + b -f c are funclirst
tions of the first degree. the function
is
frequently represented
by a single letter... (4.-. j/=-3.
if
/*
4
>
1i >
>
?/
=
193. as y. 4). 2 4 and if y = x -f. 5). hence two points are sufficient for the construction
of these graphs. etc.
Ex.4).
4J.
.
= 0. straight line produces the required graph. -1)..
194.20). the scale unit of the ordinatcs is taken smaller than that of the x. and joining in order produces
the graph
ABC.
If
If
Locating
ing
by a
3) and (4. = 4. y = 6..
It can be
proved that the
graph is a straight
of a function of the first degree
line.
ELEMENTS OF ALGEBRA
Degrees of the Fahrenheit
(F.
. the abscissas of 3.24. then
y = . 14 F.. 9 F... then
cXj
where
c is a constant.
y=
formula graphically.
that
graph with the o>axis.) scale by the formula
(a)
Draw
the graph of
C = f (F-32)
from
to
(b)
4 F F=l.
C. If two variables x and y are directly proportional. i.
GRAPHIC SOLUTION OF EQUATIONS INVOLVING ONE
UNKNOWN QUANTITY
Since we can graphically determine the values of x make a function of x equal to zero.24 or x =
P and
Q.
If
two variables x and y are inversely proportional..
From
grade equal to
(c)
the diagram find the number of degrees of centi-1 F.158
24.
that the graph of two variables that are directly proportional is a straight line passing through the origin (assume
for c
27.
to Fahrenheit readings
:
Change
10
C. it is evidently possible Thus to find to find graphically the real roots of an equation. we have to measure the abscissas of the intersection of the
195. Represent 26.
A body
moving with a uniform
t
velocity of 3 yards per
second moves in
this
seconds a distance d
=3
1.where x
c is
a constant.
1
C. 32 F.. Therefore x = 1.) scale are
expressed in
degrees of the Centigrade (C.
Show
any convenient number).e.
25.
if c
Draw
the locus of this equation
= 12. what values of x make the function x2 + 2x 4 = (see 192).
Hence.
y=
A
and construct
x
(
-
graphically.
199.
4)
and
(2.
fc
= 3.
.
and joining by a straight
line.
If the given equation is of the we can usually locate two
y.
Ex.
If
x
=
0.
y y
2.1.
Thus
If
in
points without solving the equation for the preceding example:
3x
s
. = 0.
3x
_
4
.e.
== 2.
?/
=4
AB.
represent graphically equations of the form y function of x ( 1D2).
unknown
quantities. 0). Represent graphically
Solving for
y ='-"JJ y.
NOTE.
Hence we may
join (0.
Graph
of
equations involving two
unknown
quantities.
Ex. locate points
(0. Draw the locus of 4 x + 3 y = 12.
i. we can construct the graph or locus of any
Since
we can
=
equation involving two
to the above form.
4) and
them by
straight line
AB
(3.
first
degree.
produces the
7*
required locus. because their graphs are straight lines.
1)
and
0).
Hence
if
if
x
x
-
2. solve for
?/. 2). Equations of the first degree are called linear equations.
?/.2.
X'-2
Locating the points
(2.
(f
.
T
. that can be reduced
Thus
to represent
x
-
-
-L^-
\
x
=2
-
graphically.
if
y
=
is
0. y = -l.160
ELEMENTS OF ALGEBRA
GRAPHIC SOLUTION OF EQUATIONS INVOLVING TWO UNKNOWN QUANTITIES
198.2 y ~ 2. and join the required graph.
and CD. The roots of two simultaneous equations are represented by the coordinates of the point (or points) at which their
graphs intersect.
parallel have only one point of intersection.
(2)
.
202. P.
The
every
coordinates
of
point in satisfy the equation
(1). the point of intersection of the coordinate of P.
203.
AB
but only one point
in
AB
also satisfies
(2). we obtain the roots.1=0. The coordinates of every point of the graph satisfy the given equation.
AB
y
= .15.
To
find the roots of
the system.
201. linear equations have only one pair of roots.GRAPHIC REPRESENTATION OF FUNCTIONS
161
200. and every set of real values of x and y satisfying the given equation is represented by a point in
the locus. viz.
3.
Solve graphically the equations
:
(1)
\x-y-\.57.
Graphical solution of a linear system.
Since two straight lines which are not coincident nor simultaneous
Ex.
equation
x=
By measuring
3.
By
the
method
of
the preceding article construct the graphs
AB
and
and
CD
of
(1)
(2) respectively.
3.
and
+ 3). 3).
Solve graphically the
:
fol-
lowing system
= =
25. = 0.
2.
4.
5. and
.9. there are two pairs of By measuring the coordinates of
:
P and Q we find
204.
x2
. 0) and (0.
4.
4.162
ELEMENTS OF ALGEBRA
graph.
4.
P
graphs meet in two and $. 3. etc.
In general. construct CD the locus of (2)
of intersection.
-
4. which consist of a
pair of parallel lines. Since the two
-
we obtain DE. if x equals
respectively
0. 5. e.
V25
5.
Using the method of the preceding para. and joining by a
straight line.
intersection. (-4.5. This is clearly shown by the graphs of (1) arid (2)..
(1)
(2)
cannot be satisfied by the same values of x and y.
Measuring the coordinates
of P. the graph
of
points
roots.
(1)
(2)
-C. we of the
+
y*
= 25.
3x
2 y = -6.
Locating two points of equation (2).y~ Therefore. (4.
2 equation x
3). 2. AB the locus of (1).
Inconsistent equations. 0.
obtain the graph (a circle)
AB C
joining.
1.
4.
4.g. 0.
Locating the
points
(5. (-2. parallel graphs indicate inconsistent equations. they are inconsistent.e.
Solving (1) for y. 4. i.
.
y equals
3.0).
3.0.
4. the point
we
obtain
Ex.
The equations
2
4
= 0.5. 1.
There can be no point of
and hence no
roots.
etc.
V9 = +
3. or y
~
3.
27
=y
means
r'
=
27. for distinction. and
all
other numbers are.
numbers.
\/a
=
x means x n
=
y
?>
a.
Every odd root of a quantity has
same sign as
and
2
the
quantity. called real
numbers.
\/"^27=-3.
a)
4
= a4
.
1. (_3) = -27. for (+ a) = a \/32 = 2.
or
-3
for
(usually written
3)
.
109
. which can be simplified no further.
Evolution
it is
is the operation of finding a root of a quan the inverse of involution. it is evidently impossible to express an even root of a negative quantity by Such roots are called imaginary the usual system of numbers. Since even powers can never be negative.CHAPTER XIV
EVOLUTION
213.
2.
V
\/P
214.
tity
. and ( v/o* = a. quantity
may
the
be either 2wsitive
or negative.
It follows
from the law of signs
in evolution that
:
Any
even root of a positive.
215.
Thus
V^I is an imaginary number.
or x
&4 .
= x means
= 6-.
for (-f 3) 2
(
3)
equal
0.
4
4
.
a2
+ & + c + 2 a& .
15. it is not known whether the given
expression is a perfect square.
2
49a 8 16 a 4
9.e.72 aW + 81 &
4
.2 &c.
12.
mV-14m??2)-f 49. let us consider the relation of a -f. and b (2 a -f b).
+ 6 + 4a&.
.
ELEMENTS OF ALGEBEA
4a2 -44a?> + 121V2 4a
s
.
the given expression is a perfect square.
10.>
13.
term a of the root
is
the square root of the
first
The second term
of the root can be obtained
a.
The
term
a'
first
2
. the that 2 ab -f b 2
=
we have then to consider sum of trial divisor 2 a.
14.b 2 2 to its square.
2 2
218.172
7.
multiplied by b must give the last two terms of the
as follows
square.
a-\-b
is
the root
if
In most cases.
8
.
2
.2 ac .
and
b. however.2 ab + b
. 11.
#2
a2
-
16.
second term 2ab by the double of
by dividing the the so-called trial divisor. a -f. i.
The work may be arranged
2
:
a 2 + 2 ab
+ W \a + b
. In order to find a general method for extracting the square root of a polynomial.
2ab
.
EVOLUTION
Ex. and consider Hence the their sum one term.
of x.
First trial divisor. We find the first two terms of the root by the method used in Ex.
219. 8 a 2
2. 8 a 2
-
12 a
+4
a
-f 2.
The square
.
double of this term
find the next
is
the
new
trial divisor.
.24 a + 4 -12 a + 25 a8
s
. the required root
(4
a'2
8a
+
2}.
.
. 2 Subtracting the square of 4x' from the trinomial gives the remainder '24 x'2 + y. Second trial divisor.
.
First complete divisor.
Explanation. As there is no remainder. 8 a 2 Second complete divisor. 8 a 2 .
is
As
there
is
no remainder.24 afy* -f 9 tf.
2. The process of the preceding article can be extended to polynomials of more than three terms.
Extract the square root of
16 a 4
. we obtain the next term of the root 3 y 3 which has to be added to 2 the trial divisor.
Ex.
10 a 4
8
a. Multiply the complete divisor Sx' 3y 3 by Sy 8 and subtract the product from the remainder.
. 6 a.
173
x*
Extract the square root of 1G
16x4
10 x*
__
.
and so
forth.
by division we
term of the
root. 24# 2 y 3 by the trial divisor Dividing the first term of the remainder.
*/''
. 1.
-
24 a
3
+
25 a 2
-
12 a
+4
Square of 4 a First remainder.
1. the first term of the answer. By doubling 4x'2 we obtain 8x2 the trial divisor. 8 /-.
\
24 a 3
4-f
a2
10 a 2
Second remainder.
4 x2
3
?/
8 is
the required square foot.
Arranging according to descending powers of
10 a
4
a.
Arrange the expression according to descending powers root of 10 x 4 is 4 # 2 the lirst term of the root.
the first of which is 4.
From
A
will
show the
comparison of the algebraical and arithmetical method given below identity of the methods.
Ex. and we may apply the method used in algebraic process. then the number of groups is equal to the number of digits in the square root.
a 2 = 6400. Therefore 6 = 8.EVOLUTION
220.
2.
1.000 is 100.176.
a
f>2'41 '70
6
c
[700
+ 20 + 4 = 724
2 a
a2 = +6=
41)
00 00
1400
+ 20 = 1420
4
341 76
28400
=
1444
57 76
6776
. the square root of 7744 equals 88.000 is 1000.
The
is
trial divisor
=
160.
Find the square root of 7744. Thus the square root of 96'04' two digits. etc.
7744 80 6400
1
+8
160
+ 8 = 168
1344
1344
Since a
2 a
Explanation. and the square root of the greatest square in
units. Hence if we divide the digits of the number into groups. of 10.
the preceding explanation it follows that the root has two digits.000. and the complete divisor
168. of a number between 100 and 10.
As
8
x 168
=
1344. Hence the root is 80 plus an unknown number. beginning at the
and each group contains two digits (except the last. the integral part of the square root of a number less than 100 has one figure.
175
The
by a method very similar
expressions.
first
. two figures. and the first remainder is..
the
consists of
group is the first digit in the root. etc.1344. of 1. the first of which is 8. the first of which is 9 the square root of 21'06'81 has three digits.000. which may contain one or two).
square root of arithmetical numbers can be found to the one used for algebraic
Since the square root of 100 is 10.
Ex.
= 80.
Find the square root of 524.
and if the righthand group contains only one digit.
we must
Thus the groups
1'67'24.688
4
45 2 70
2 25
508
4064
6168 41)600
41344
2256
222.
ELEMENTS OF ALGEKRA
In marking
off groups in a number which has decimal begin at the decimal point.0961
are
'. or by transforming the common fraction into a decimal.GO'61.
places.1T6
221.
EXERCISE
Extract the square roots of
:
82
.
The groups
of 16724.10.
12.7 to three decimal places.
3.
Roots of common fractions are extracted either by divid-
ing the root of the numerator by the root of the denominator.1 are
Ex.70
6.
Find the square root of
6/.
in . annex a cipher.
2
:
3.
9
&
-{-
c#
a
x
+a
and
c. 24.
.
2
.
If a 2 4.
Find the side
of each field.
The
two numbers
(See
is
2
:
3.
.
26.b 2 If s
If
=c
.
=
a
2
2
(' 2
solve for solve for
= Trr
.
If 22
= ~^-.
A
number multiplied by
ratio of
its fifth
part equals 45. is 5(5.
22
a.
29. Find the side of each field.
find a in terms of 6
.
Find the numbers.180
on
__!_:L
ELEMENTS OF ALGEBRA
a.
27.
'
4.
solve for v.
If 2
-f 2 b*
= 4w
2
-f c
sol ve for
m.
108. If the hypotenuse
whose angles
a
units of length.
and the sum
The
sides of
two square
fields are as
3
:
5.
EXERCISE
1. The sides of two square fields are as 7 2.
solve for d.
3. 25.
and they con-
tain together 30G square feet. opposite the right angle is called the hypotenuse (c in the diagram).
2a
-f-
1
23.
A
right triangle is a triangle.
28.
:
6.
2.
is
one of
_____
b
The side right angle.
and their product
:
150. and the first exceeds the second by 405 square yards.
Find
is
the number.
228.
If
G=m m
g
.
2
.
may
be considered one half of a
rec-
square units.
Three numbers are to each other as 1 Find the numbers. and the two other sides respectively
c
2
contains
c
a and b units. then
Since such a triangle
tangle.
r.)
of their squares
5.
84
is
Find a positive number which
equal to
its
reciprocal
(
144).
solve for
r. its area contains
=a
2
-f-
b2
.
If s
= 4 Trr
'
2
.
4.
The area
:
sides are as 3
4.)
COMPLETE QUADRATIC EQUATIONS
229.
add
(|)
Hence
2
.
Find the
sides. and the third side is 15 inches.
The following
ex-
ample
illustrates the
method
or
of solving a complete quadratic
equation by completing the square. Find the unknown sides and the area.
radii are as 3
14.
8 = 4 wr2 Find 440 square yards.)
13. and the
other two sides are as 3
4.
of a right triangle Find these sides. passes in t seconds 2 over a space s yt Assuming g 32 feet.
and the two smaller
11.
24.
181
The hypotenuse
of a right triangle
:
is
35 inches.
2m.
Find the
radii. The hypotenuse of a right triangle is to one side as 13:12. the radius of a sphere whose surface equals
If the radius of a sphere is r.
member can be made a complete square by adding 7 x with another term.
is
and the other
two
sides are equal.
The hypotenuse
of a right triangle is 2. we have
of
or
m = |.
The area $
/S
of a circle
2
.
. (b) 100 feet?
=
.
8.
9.
Solve
Transposing.7 x -f 10 = 0.
-J-
=
12. x* 7 x=
10.
4.
.
. (b) 44 square feet.QUADRATIC EQUATIONS
7. its surface
(Assume
ir
=
2 .
Method
of completing the
square.
Find these
10.
the formula
= Trr
whose radius equals r is found by Find the radius of circle whose area S
equals (a) 154 square inches.2
7
.
sides.
Two
circles together contain
:
3850 square
feet. let us compare x 2
The
left
the perfect square x2
2
mx -f m
to
2
. To find this term.
make x2
Evidently 7 takes the place 7x a complete square
to
to
which corresponds
m
2
. A body falling from a state of rest.
7r
(Assume
and their
=
2 7
2
. in how many seconds will a body fall (a) G4 feet.
231.
o^
or
-}-
3 ax == 4 a9
7 wr
.
49.
.
ao.
x
la
48.
Solution
by formula.184
ELEMENTS OF ALGEBRA
45
46.
any quadratic equation may be obtained by 6. -\-bx-\.c
= 0.
= 12.
Solving this equation we obtain
by the method of the preceding
2a
The
roots of
substituting the values of a.
=8
r/io?.
2x
3
4. and c in the general answer.
=0.
article.
2
Every quadratic equation can be
reduced to the general form.
and equals 190 square inches.
8.
52.
58.3.
number by 10.
54. -5.
The sum
of the squares of
two consecutive numbers
85. Find the number.
Problems involving quadratics have
lems of this type have only one solution.
2.
3.
Find
the numbers.QUADRATIC EQUATIONS
Form
51.
55.
-2.
5.0.3.
Find the number.
What
are the
numbers
of
?
is
The product
two consecutive numbers
210. The
11.1.
is
Find two numbers whose product
288.
0.
:
3.
2. Find the sides. and consequently many prob-
235.
.
57.
Divide CO into two parts whose product
is 875.2.
Twenty-nine times a number exceeds the square of the 190.
of their reciprocals is
4.
G.
189
the equations whose roots are
53.
two numbers is 4.
88
its reciprocal
A
number increased by three times
equals
6J.
3. but frequently the conditions of the problem exclude negative or fractional answers.
EXERCISE
1.
its
sides of a rectangle differ by 9 inches.
Find two numbers whose difference
is 40. feet.
and whose
product
9.
Find a number which exceeds
its
square by
is
-|.
56.3.
PROBLEMS INVOLVING QUADRATICS
in general two answers. and the difference Find the numbers.
7.
The
difference of
|.9. -2.
-2.
-4.
1.
area
A
a perimeter of 380
rectangular field has an area of 8400 square feet and Find the dimensions of the field.
1.
and whose sum
is
is 36.0.
6.
-2.
and Find the sides of the rectangle.
A man
A man
sold a
as the watch cost dollars.
watch cost
sold a watch for $ 21.
ply between the same two ports. ABCD.
watch for $ 24. and the slower reaches its destination one day
before the other.
A man
cent as the horse cost dollars.
The diagonal
:
tangle as 5 4.
If he
each horse ?
. he would have received two horses more for the same money.
vessel sail ?
How many
miles per hour did the faster
If 20.
other. one of which sails two miles per hour faster than the other. dollars.10.
15. and lost as many per cent Find the cost of the watch.
Two steamers
and
is
of 420 miles. Find the rate
of the train.
.
and gained as many per Find the cost of the horse. What did he pay for each
apple ?
A man bought a certain number of horses for $1200. it would have needed two hours less to travel 120 miles. a distance One steamer travels half a mile faster than the two hours less on the journey. he had paid 2 ^ more for each apple.
as the
16. What did he pay for
21. At what rates do
the steamers travel ?
18.
17.190
12.
14.
19. and the line BD joining
two opposite
vertices (called "diagonal")
feet. If a train had traveled 10 miles an hour faster. he would have received 12 apples less for the same money.
of a rectangle is to the length of the recthe area of the figure is 96 square inches.
c equals 221
Find
AB and AD. and lost as many per cent Find the cost of the watch. start together on voyages of 1152 and 720 miles respectively. had paid $ 20 less for each horse. A man bought a certain number of apples for $ 2. exceeds its widtK AD by 119 feet. Two vessels.
13.
ELEMENTS OF ALGEBRA
The length
1
B
AB of a rectangle. sold a horse for $144.
Find
TT r (Area of a circle .QUADRATIC EQUATIONS
22. A needs 8 days more than B to do a certain piece of work.
Ex.
237. 23 inches long. 30 feet long and 20 feet wide.
EQUATIONS IN THE QUADRATIC FORM An equation is said to be in the quadratic form
if it
contains only two unknown terms. the two men can do it in 3 days. so that the rectangle. and the unknown factor of one of these terms is the square of the unknown factor of the
other.
B
AB
AB
-2
191
grass plot.
24.
. The number of eggs which can be bought for $ 1 is equal to the number of cents which 4 eggs cost.
or x
= \/l = 1.
=9
Therefore
x
=
\/8
= 2.
Solve
^-9^ + 8 =
**
0.
By formula.
Find the side of an equilateral triangle whose altitude
equals 3 inches.
and the area of the path
the radius of the basin. How many eggs can be bought for $ 1 ?
236.
(tf. Find and CB. contains B 78 square inches. how wide is the walk ?
23. constructed with and CB as sides.I) -4(aj*-l)
2
= 9. is surrounded by a walk of uniform width. a point taken.) 25.
27. and working together.
A rectangular
A
circular basin is surrounded
is
-
by a path 5
feet wide. If the area of the walk is equal to the area of the plot.
1. Equations in the quadratic form can be solved by the methods used for quadratics. In how many days can B do the work ?
=
26.
is
On the prolongation of a line AC. as
0.
^-3^ = 7.
of the area of the basin.
while the second of the first.a" = a m n
mn .CHAPTER XVI
THE THEORY OF EXPONENTS
242.
We assume.
we may choose
for such
symbols any definition that
is
con-
venient for other work. we let these quantities be what they must be if the exponent law of multiplication is generally true.
no
Fractional and negative exponents.
(ab)
.
for all values
1
of
m and n. The following four fundamental laws for positive integral exponents have been developed in preceding chapters
:
I.
the direct consequence of the defiand third are consequences
FRACTIONAL AND NEGATIVE EXPONENTS
243. 4~ 3 have meaning according to the original definition of power. (a ) s=a m = aw bm
a
. such as 2*. that a
an
= a m+n
.
provided
w > n.
must be
*The symbol
smaller than.
II.
244. however. instead of giving a formal definition of fractional and negative exponents.*
III. ~ a m -f."
means "is greater than"
195
similarly
means "is
. hence. (a m ) w
.
It is. very important that all exponents should be governed by the same laws.
>
m therefore.
m
IV.
Then the law
of involution.
a m a" = a m+t1 . and
.
The
first
of these laws
is
nition of power.
= a""
<
.
(bed)*.
= a.
29.
0?=-^. 25.
28.
n 2 a.
Hence
Or
Therefore
Similarly.
Let
x
is
The operation which makes the fractional exponent disappear evidently the raising of both members to the third power.
'&M
A
27.
laws.
a\
26.g.
31.
To
find the
meaning
of
a fractional exponent.196
ELEMENTS OF ALGEBRA
true for positive integral values of n.
-
we
find
a?
Hence we
define a* to be the qth root of of. as. fractional.
245.
a*.
30.
23. a .
etc. 3*. (xy$. at.
Assuming these two
8*.
Write the following expressions as radicals :
22.
disappear.
a?*. since the raising to a positive integral power is only a repeated multiplication.
24.
we
try to discover the
let the
meaning of
In every case we
unknown quantity
and apply to both members of the equation that operation which makes the negative.
e.
.
m$. 4~ .
^=(a^)
3*
3
. ml. or zero exponent
equal
x.
248.
a
a
a
= =
a a a
a1
1
a.
Multiplying both members by
a".
by changing the sign of
NOTE.
etc.
each
is
The
fact that a
if
=
we
It loses its singularity
1 sometimes appears peculiar to beginners.
ELEMENTS OF ALGEBRA
To
find the
meaning
of a negative exponent. e.
vice versa.
.198
247.g. in which
obtained from the preceding one by dividing both
members by
a.
cr n. consider the following equations.
Or
a"#
= l. or
the exponent.2
=
a2
.
a8 a
2
=
1
1
.
Let
x=
or".
an x = a. Factors
may
be transferred
from
the
numerator
to
the
denominator of a fraction.
V ra
4/
3
-\/m
33.
1.
Divide
by
^
2a
3 qfo
4.
2.
we wish to arrange terms according to descending we have to remember that.
1.2 d
.
lix
=
2x-l
=+1
Ex. the term which does not contain x may be considered as a term containing #.
6
35.
34.
If
powers of
a?.
1 Multiply 3 or
+x
5 by 2 x
x.
powers of x arranged are
:
Ex.202
ELEMENTS OF ALGEBRA
32. The
252.
40.
Arrange in descending powers of
Check.
V3 .
E.y.
Ex.
it
more convenient to multiply dividend and divisor by a factor which makes the divisor rational.
.
Va
-v/a. the quotient of the surds
is
If. a
VS
-f-
a?Vy
= -\/ -
x*y
this
Since surds of different orders can be reduced to surds of
the same order.
52.
(5V2+V10)(2V5-1).
a fraction.
49.
(2
45.
-v/a
-
DIVISION OF RADICALS
267.
all
monomial surds may be divided by
method.
is
1
2.
(5V7-2V2)(2VT-7V2).
47.
51.
43.
48.
Ex. Monomial surdn of the same order may be divided by multiplying the quotient of the coefficients by the quotient of the
surd factors.
(3V5-2V3)(2V3-V3).
44.
60.
(3V3-2Vo)(2V3+V5).V5) ( V3 + 2 VS).
(V50-f 3Vl2)-4-V2==
however.
268.214
42.
53.
46.
ELEMENTS OF ALGEHRA
(3V5-5V3)
S
.
g.
+ 4\/5 _ 12v 3 + 4\/5 V8 V8
V2 V2
269. we have
V3
But
if
1. called rationalizing the
the following examples
:
215
divisor. To show that expressions with rational denominators are simpler than those with irrational denominators.
VTL_Vll '
~~"
\/7_V77
.
Divide
VII by v7. the rationalizing factor
x
' g
\/2. arithTo find..
we have
to multiply
In order to make the divisor (V?) rational.
. is illustrated
by
Ex.
1.
Divide 12 V5
+ 4V5 by V.73205. Evidently.by the usual arithmetical method. metical problems afford the best illustrations.
. the by 3 is much easier to perform than the division by
1.
by V7.
is
Since \/8
12 Vil
=
2 V*2.73205
we
simplify
JL-V^l
V3
*>
^>
division
Either quotient equals .
.
Hence
in arithmetical
work
it
is
always best to
rationalize the denominators before dividing.
3.
/~
}
Ex.57735. e.
The
2.
Divide 4 v^a by
is
rationalizing factor
evidently \/Tb
hence. however.
4\/3~a'
36
Ex.RADICALS
This method.
Factor
consider
m
m
6
n9
.
it
follows from the Factoi
xn y n is always divisible by x y.
2.xy +/).
286. If n is a Theorem that
1.
Two
special cases of the preceding propositions are of
viz. if w is odd.
ar
+p=
z6
e.
Factor 27 a* -f
27 a 6
8. xn y n y n y n = 0.
Ex.
2
Ex.
-
y
5
=
(x
-
can readily be seen that #n -f either x + y or x y.230
285.g. if n For ( y) n -f y n = 0.
x* -f-/
= (x +/)O .
2.
1.
:
importance."
.
It
y is
not divisible by
287.
and have
for
any positive integral value of
If n
is
odd.
The
difference of
two even powers should always be
considered as a difference of two squares.
2 8 (3 a )
+8=
+
288.
ELEMENTS OF ALGEBRA
positive integer. For substituting y for x.
By
we obtain the other
factors.y n is divisible by x -f ?/. if n is even.
We may
6
n 6 either a difference of two squares or a
dif-
* The symbol
means " and so forth to.
is
odd.
xn -f.
actual division
n.
By making x
any * assigned
zero.
(1)
= 0.
Or.
The
~~f
fraction .
306. while the
remaining terms do not
cancelj the root is infinity.
customary to represent this result
by the equation ~
The symbol
304.
1.e.decreases
X
if
called infinity.
= 10.
the
If in an equation
terms containing
unknown quantity
cancel.g.
x
-f 2.
i.
be the numbers.
Hence such an equation
identity.e.242
303.
of the second exceeds the product of the first
Find three consecutive numbers such that the square and third by 1.
and becomes infinitely small.x'2 2 x =
1. without exception. cancel.
Interpretation of
QO
The
fraction
if
x
x
inis
infinitely large.increases
if
x
de-
x
creases.
i. however
x approaches the value
be-
comes
infinitely large.
(a:
Then
Simplifying.
and
.
.
(1). or that x may equal any finite number. the answer is indeterminate.
The
solution
x
=-
indicates that the problem
is indeter-
If all terms of an minate.
TO^UU"
sufficiently small.
ELEMENTS OF ALGEBRA
Interpretation of ?
e. or infinitesimal) This result is usually written
:
305.
is satisfied
by any number.
great.
as
+ l.
ToU"
^-100 a.
creases.
I. it
is
an
Ex.
.
+
I)
2
x2
'
-f
2x
+
1
-x(x + 2)= .
1.
Let
2.i
solving
a problem
the result
or oo indicates that the
all
problem has no solution.000
a.can be
If
It is
made
larger than
number.
Hence any number will satisfy equation the given problem is indeterminate. equation.
oo is
= QQ.
(1)
is
an
identity.
6.)
53 yards. The sum of the areas of two squares is 208 square feet. Find two numbers whose product whose squares is 514.
Find the other two
sides. and the side of one increased by the side of the other e.quals 20 feet. To inclose a rectangular field 1225 square feet in area.
the
The mean proportional between two numbers sum of their squares is 328.
ELEMENTS OF ALGEBRA
The
difference between
is
of their squares
325.
and the diago(Ex. and
its
The diagonal
is
is
perimeter
11. Find the edge of each cube. Find the edges.
255 and the sum of
5. But if the length is increased by 10 inches and
12.
14. the area becomes
-f%
of
the original area. 12.
is 6.
9.
of a right triangle is 73.
The area of a
nal 41 feet. increased by the edge of the other. Find the numbers.)
The area
of a right triangle is 210 square feet.
103.
Find the
sides. and the edge of one exceeds the edge of the other by 2 centimeters.
is
is
17 and the
sum
4.
146 yards.
of a rectangular field
feet.
and the sum of
(
228. equals 4 inches.
Find the
sides of the rectangle.244
3.
The hypotenuse
is
the other two sides
7. 190.
and the
hypotenuse
is 37. Find these sides. Find the side of each square.
rectangle is 360 square Find the lengths of the sides. Two cubes together contain 30| cubic inches.
10.
. 148 feet of fence are required.
two numbers Find the numbers. Find the dimensions of the
field. and the edge of one.
8.
and
is
The area of a rectangle remains unaltered if its length increased by 20 inches while its breadth is diminished by 10 inches. The volumes of two cubes differ by 98 cubic centimeters.
13.
is
the breadth
diminished by 20 inches. p.
and the equal to the surface of a sphere Find the radii.
the quotient
is 2.
by the product of 27 be added to the number.)
17.
The
radii of
two spheres
is
difference of their surfaces
whose radius = 47T#2. their areas are together equal to the area of a circle whose radius is 37 inches. Find the number.
.
differ by 8 inches.SIMULTANEOUS QUADRATIC EQUATIONS
15.
is
20 inches.) (Area of circle
and
=
1
16.
245
The sum of the radii of two circles is equal to 47 inches. irR *. Find the radii.
and
if
the digits will be interchanged. (Surface of sphere
If a
number
of
two
digits be divided
its digits.
11.
-4. to produce the 4th term. a -f d. An arithmetic progression (A.1) d. progression.
To
find the
nth term
/
of an A. except the first. The first is an ascending..7. the first
term a and
the
common difference d being given..
The common
Thus each
difference is the
number which added
an A.
a. and d..
of the following series is
3.
series 9.
:
7..) is a series.. 3 d must be added to a.
. 3. a + 2 d..CHAPTER XX
PROGRESSIONS
307.11 246
(I)
Thus the 12th term of the
3
or 42. the second a descending. P. is derived from the preceding by the addition of a constant number. 2 d must be added to a..
of a series are its successive numbers.
. each term of which. The progression is a. a
11.
to
A series
is
a succession of numbers formed according
some
fixed law. 16. to produce the nth term.
Hence
/
= a + (n . (n 1) d must be added to a. a
3d.
17.
added to each term to obtain the next one.
10. 12.
to each
term produces the next term.
... 19.
a
+
d.
The common differences are respectively 4.
The terms
ARITHMETIC PROGRESSION
308. to produce the 3d term.. .
+
2 d. P. P.
Since d
is
a
-f
3
d.
309. 15 is 9 -f.
-f
.
= -2.
8.
1.
9.
Find the 7th term of the Find the 21st term
series
..
first
2
Write down the
(a)
(6)
(c)
6 terms of an A.
Or
Hence
Thus
from
(I)
= (+/).
Which
(6)
(c)
of the following series are in A. 7.16. -7.
Find the nth term of the
series 2..
. 21..-.
-|. 3.
2
EXERCISE
1. 19..
series
.. 6.
-3.4.
series 2.
1.
3.
of the series 10. the
term
a.
6
we have
Hence
.PROGRESSIONS
310.. a = 2.. 5.
2.-... 2J. -24.
7.
if
a = 5. 2
sum
of the first 60
I
(II)
to find the
' '
odd numbers. 3.
2*=(a + Z) + (a + l) + (a + l)
2s = n
*
. ?
(a) 1.
. .. -4^. 99) = 2600.
the last term
and the common difference d being given. 4. 6. 8.
115.3 a = -l..
. 5. 3. P.
247
first
To
find the
sum s
19
of the first
n terms of an A.. 5.
= I + 49 = *({ +
...' cZ == .
= 99.
(d) 1J.
.
5. 8. d
..
Find the 12th term of the
-4..-
(a
+ + (a +
l)
l).
5.
Find the 101th term of the
series 1.
Find the 10th term of the
series 17. d = 3. P.
= a + (a
Reversing the order. -10.
Adding.
.
Find the 5th term of the
4. 2.
6..
1-J.8.
9. P.
21. 11.
8.
to 20 terms.
.248
Find the
10.
22. 1|. P.
12.
6.
3.
2.
:
3.
23. 11.
to 8 terms. 31. In most problems relating to A.
>
2-f
2.
ELEMENTS OF ALGEBRA
last
term and the sum of the following series :
. 2J.
15.1 -f 3.
33. 16.5
H + i-f
-f-
to 10 terms. 15. and a yearly increase of $ 120..
to 16 terms. Jive quantities are involved.
17.
19. the other two may be found by the solution of the simultaneous equations
.
\-n.
Q^) How many times
in 12 hours ?
(&fi)
does a clock.
1+2+3+4H
Find the sum of the
first
n odd numbers.
18. strike
for the first yard.
to 7 terms. 12.
1.
11.
to 15 terms. and for each than for the preceding one.7 -f
to 12 terms. 7.
.
-.
.
+ 2-f-3 + 4 H
hlOO. 7.
.
'.
rf.
20.
7. hence if any three of them are given.
to 10 terms.
.
(x +"l) 4.(#
1
2) -f (x -f 3) H
to
a terms.
(i)
(ii)
.
4.
1. How much does he receive (a) in the 21st year (6) during the first 21 years ?
j
311.
.
to 20 terms.
to 20 terms. striking hours only. 29.
1J.
16.
.
$1
For boring a well 60 yards deep a contractor receives yard thereafter 10^ more How much does he receive all
together ?
^S5 A bookkeeper accepts a position at a yearly salary of $ 1000.
13.
15.
Sum
the following series
14. 11.
+ 3.
n = 4. Find d and Given a = 1700. Find a Given a = 7. n = 20. n = 13. n = 17.
.
a+
and
b
a
b
5. n.
m
and
n
2.
17. s == 440.
10.
A
$300
is
divided
among 6 persons
in such a
way
that each
person receives $ 10 did each receive ?
more than the preceding
one. and all his savings in 5 years amounted to $ 6540. Find?. 74.
f
J 1 1
/
. = 17.
6?
9.
a x
-f-
b
and a
b.
produced.
Between 10 and 6
insert 7 arithmetic
means
. of 5 terms
6.
has the series 82. d = 5.
7.
I.
= 16.
I
Find
I
in terms of a. ceding one.
16.
15. = 45.
man saved each month $2 more than in the pre 18. = 52.
y and #-f-5y.
How much
.
Between 4 and 8
insert 3 terms (arithmetic
is
means)
so
that an A. Given a = |.3. Find d. = 1870. = 83.
T?
^. Find n. 78. n = 16.
How many terms How many terms
Given d = 3.
3. Find d.
and
s.
14. Given a = . n
has the series
^
j
. Find a and Given s = 44.
f?
.
4. Given a = 1.
11. Given a = 4. How much did he save the first month?
19.
8. P. Find w.
12. = ^ 3 = 1. s = 70.250
ELEMENTS OF ALGEBRA
EXERCISE
116
:
Find the arithmetic means between
1.
13.
.
Therefore
Thus the sum
= ^ZlD.PROGRESSIONS
251
GEOMETRIC PROGRESSION
313. 24. P. 108.
4. 36. To find the sum s of the first n terms term a and the ratio r being given.
fl
lg[(i)
-l]
==
32(W -
1)
= 332 J. the first term a and
the ratios r being given.arn ~ l . the first
= a + ar -for ar -f ar Multiplying by r.
4-
(1)
.
(I)
of the series 16.
.
2 a. 24..g..
s(r
1)
8
= ar"
7*
JL
a.
..
-2. 12.
E.
the following form 8
nf +
q(l-r")
1
r
.
Hence
Thus the 6th term
l
= ar
n~l
.
or 81
315.
. P. -I.
NOTE. <zr .
|..)
is
a series each term of
which..
A geometric progression
first. 36. called the ratio.
2
arn
(2)
Subtracting (1) from
(2).
The progression is a.
is
16(f)
4
. ar. rs =
s
2
-.
.
ar8
r.. except the
multiplying
derived from the preceding one by by a constant number. +1. or.
g==
it is
convenient to write formula' (II) in
*.
4. is
it
(G.
and
To
find the
nth term
/ of
a G.
.
r
n~ l
..
The
314. P.
(II)
of the
8 =s
first
6 terms of the series 16.
ratios are respectively 3.
If
n
is less
:
than unity.
of a G. a?*2 To obtain the nth term a must evidently be multiplied by
. 36...
9.
series 5. f.
7.
.
. 144.
a
=
I. if any three of them are given..
first
term
4.
(d) 5.
. 4.
.
..
(b) 1. the other two be found by the solution of the simultaneous equations :
may
(I)
/=<!/-'..
..252
ELEMENTS OF ALGEBRA
316.
36.
0.
volved .
Write down the first 5 terms of a G.
-fa.
Hence n
=
7. whose and whose common ratio is 4.
.l.. 25. whose
. P.
or
7.
2
term
3. 18. In most problems relating to G.72. hence. 72.288. 3..
series 6.
\
t
series
.
.
series
Find the llth term of the Find the 7th term of the
ratio is
^.
Evidently the total
number
of terms is 5
+ 2. 36.
117
Which
(a)
of the following series are in G.
72.
6.
And the
required
means are
18.
is 3.
Ex.
first
term
is
125 and
whose common
. 144...
Find the 5th term of a G. P.
Find the 6th term of the
series J.54.
(it.18.
576.
r^2.
EXERCISE
1. P. 9.
Write down the first 6 terms of a G.
..4.
|. 80.
10. 36.
+-f%9 %
. 20.
l. Jive quantities are in. P.5...
676
t
Substituting in
= r6 = 64.. 1. P._!=!>..
Find the 7th term of the Find the 6th term of the
Find the 9th term of the
^.
4.
144. 676.
.
Hence the
or
series is
0.
-fa.
series
.6. + 5. I
= 670.-. whose and whose second term is 8.5.
f.
8. ?
(c)
2.
To
insert 5 geometric
means between 9 and 576.18.
288.
first
5.
is 16.
. 9.
.
i 288.
.*.
187. Four years ago a father was three times as old as his son is now.
power one of the two Find the power of each.
.
The length
is
of a floor exceeds its width
by 2
feet. Find the age
5 years older than his sister
183. 189. 10x 2 192.
number divided by
3. and 5 h. + 11 ~ 6.
and | as old as his Find the age of the
Resolve into prime factors
:
184. and the father's present age is twice what the son will be 8 years
hence. 15 m.
186. z 2 + x .
dimension
182.
+x-
2.
respectively.
180.
younger than his Find the age of
the father.
181.
if
each
increased 2 feet.
What
is
the distance?
if
square grass plot would contain 73 square feet more Find the side of the plot.
2
2
+
a
_ no.
two boys is twice that of the younger.
13 a + 3.
The age
of the elder of
it
three years ago of each. the ana of the floor will be increased 48 square feet.
same
result as the
number
diminished by
175.
is
What are their ages ? Two engines are together
more than the
of 80 horse
16 horse power
other.266
173.
179. 6 in each row the lowest row has 2 panes of glass in each window more than the middle row. How many are there in each window ?
. A house has 3 rows of windows.
7/
191.-36.
3 gives the
same
result as the
numbet
multiplied by
Find the number.
A
each
177.
190.
178. was three times that of the younger.
A
boy
is
father. z 2
-92.
Find the number. x*
185. side were one foot longer. 12 m. Find the dimensions of the floor.
A
the
boy
is
as old as his father
and
3 years
sum
of the ages of the three is 57 years. aW + llab-2&.
An
The two
express train runs 7 miles an hour faster than an ordinary trains run a certain distance in 4 h.
+
a. the
sum
of the ages of all three is 51.
train.
.
-ll?/-102. 4 a 2
y-y
-42.56.
176.
188.
father.
ELEMENTS OF ALGEBRA
A A
number increased by
3.
. and the middle row has 4 panes in each window more than the upper row there are in all 168 panes of glass.
3 gives the
174.
sister
.
a
x
)
~
a
2 b
2
ar
a
IJ a. the order of the digits will be inverted. and was out 5 hours.
down again
How
person walks up a hill at the rate of 2 miles an hour. 411.
A man
drives to a certain place at the rate of 8 miles an
Returning by a road 3 miles longer at the rate of 9 miles an hour.c) .278
410. and at the rate of 3^ miles an hour.
-f
a
x
-f
x
-f c
1
1
a-b
b
x
415.(c rt
a)(x
-
b)
=
0.
mx ~
nx
(a
~
mx
nx
c
d
d
c)(:r
lfi:r
a
b)(x
.
a
x
a
x
b
b
x
c
b
_a
b
-f
x
414.
420. 18 be subtracted from the number.
418 ~j-o.(5 I2x
~r
l
a)
.
4x
a
a
2 c
6
Qx
3 x
c
419.
hour.a)(x b
b)
(x
b
~
)
412. Find the number. Find the number of miles an hour that A and B each walk. In a
if
and
422.
-
a)
-2
6 2a.
2 a
x
c
x
6
-f c
a
+
a
+
a
+
6
-f
walks 2 miles more than B walks in 7 hours more than A walks in 5 hours.
(x
. far did he walk all together ?
A
. he takes 7 minutes longer than in going. x
1
a
x
x1
ab
1
1
a
x
a
c
+
b
c
x
a
b
b
~
c
x
b
416
417.
A
in 9 hours
B walks
11 miles
number of two digits the first digit is twice the second. How long is each road ?
423.
(x
-f
ELEMENTS OF ALGEBRA
a)(z
-
b)
=
a
2 alb
=
a
(x
-f
b)(x
2
.
421. Tn 6 hours
.
and the other number least. Find two numbers such that twice the greater exceeds the by 30. If 31 years were added to the age of a father it would be also if one year were taken from the son's age
. There are two numbers the half of the greater of which exceeds the less by 2.
487.
Find
the number. Find the numbers. and a fifth part of one brother's age that of the other. had each at first?
B
B
then has
J
as
much
spends } of his money and as A. A sum of money at simple interest amounts in 8 months to $260. and in 18 months to $2180. How much money
less
484. Find the sum and the rate of
interest. age.282
ELEMENTS OF ALGEBRA
476.
half the
The greatest exceeds the sum of the greatest and
480. the Find their ages. Find the
fraction.
481.
latter
would then be twice the
son's
A
and B together have $6000.
to
.
if
the
sum of
the digits be multiplied by
the digits will be inverted.
by 4. A sum of money at simple interest amounted in 10 months to $2100.
A
number
consists of
two
digits
4. also a third of the greater exceeds half the less by 2. fraction becomes equal to |. What is that fraction which becomes f when its numerator is doubled and its denominator is increased by 1.
If 1
be added to the numerator of a fraction
it
if 1
be added to the denominator
it becomes equal becomes equal to ^. Find the principal and the rate of
interest.
486. and in 20 months to $275.
477. and 5 times the less exceeds the greater by 3. and becomes when its denominator is doubled and its numerator increased by 4 ?
j|
478. Find their ages.
least
The sum
of three
numbers
is
is
21. In a certain proper fraction the difference between the nu merator and the denominator is 12.
thrice that of his son
and added to the father's.
whose difference
is
4.
years.
. Of the ages of two brothers one exceeds half the other by 4 is equal to an eighth of 482. Find the numbers.
A
spends \ of his. and if each be increased by 5 the Find the fraction.
485.
483.
479.
it is filled in 35 minutes. if the number be increased by Find the number. If they had walked toward each other. 90. BC = 5.
touches
and
F respectively.
527. and BE. L. N. Find the numbers. CD. Throe numbers are such that the
A
the
first
and second equals
. When weighed in water. How long will B and C take to do
.
530. if and L.
.
and CA=7.REVIEW EXERCISE
285
525.
and B together can do a piece of work in 2 days. What are their rates of
travel?
.
AC
in /). 37 pounds of tin lose 5 pounds.
532. and one overtakes the other in 6 hours. A can do a piece of work in 12 days B and C together can do the same piece of work in 4 days A and C can do it in half the time in which B alone can do it. B and C and C and A in 4 days.
it
separately
?
531. (a) How many pounds of tin and lead are in a mixture weighing 120 pounds in air. Tf and run together.
sum of the reciprocals of of the reciprocals of the first of the reciprocals of the second and
the
sum
528. In
circle
A ABC. Two persons start to travel from two stations 24 miles apart. if L and Af in 20 minutes. his father is half as old again as his mother was c years ago. Tu what time will it be filled if all run
M
N
N
t
together?
529.
and third equals \\ the sum third equals \. A vessel can be filled by three pipes. in 28 minutes. A boy is a years old his mother was I years old when he was born. In how many days can each alone do the same work?
526. they would have met in 2 hours. A number of three digits whose first and last digits are the same has 7 for the sum of its digits. M. the first and second digits will change places. and 23 pounds of lead lose 2 pounds. Find the present ages of his father and mother.
E
533. and losing 14 pounds when weighed in water? (b) How many pounds of tin and lead are in an alloy weighing 220 pounds in air and 201 pounds in water ?
in 3 days.
. An (escribed) and the prolongations of BA and BC in Find AD. AB=6.
How
is
t /
long will
I
take 11
men
2
t' .
c. 2|. x*
-
2
x.
-
3 x. The roots of the equation 2 + 2 x x z = 1.
547. The value of x that produces the greatest value of y. AND BRITISH ISLES
535. i.
to
do the work? pendulum. GERMANY. x 8
549.
of
Draw
a graph for the trans-
The number
in
of
workmen Draw
required to finish a certain piece
the graph
work
D
days
it
is
from
D
1 to
D=
12. x
2
+
x. the function.
545.
548.
The values of y.
2. One dollar equals 4. x 2 544. FRANCE. formation of dollars into marks.e.3
Draw
down
the time of swing for a
pendulum
of length
8 feet. 2 541.
542. if x = f 1.
b. 2 x
+
5.
.
.
-
7.
543.10 marks.
x*. The greatest value of the function.
546.
Draw
the graphs of the following functions
:
538. 536.
550. Draw the graph of y 2 and from the diagram determine
:
+
2 x
x*.
from x
=
2 to x
= 4. then / = 3 and write
=
3. Represent the following table graphically
TABLE OF POPULATION (IN MILLIONS) OF UNITED STATES.
d.
+
3. the time of whose swing a graph for the formula from / =0
537.
-
3 x.
e. If
to
feet is the length of a
seconds.286
ELEMENTS OF ALGEBRA
:
534. The values of x if y = 2.
a. z 2
-
x x
-
5. 2
-
x
-
x2
. x *-x
+
x
+
1.
540. 3 x
539.
717. If a pound of tea cost 30 J* more than a pound of coffee. **-13a: 2
710. Find the altitude of an equilateral triangle whose side equals a. and working together they can build it in 18 days.
sum is a and whose product equals J. A man bought a certain number of shares in a company for
$375. paying $ 12 for the tea and $9 for the coffee.292
709. Find two consecutive numbers whose product equals 600.25 might have bought five more for the same money.44#2 + 121 = 0.
723.40 a 2* 2 + 9 a 4 = 0. In how many days can A build the wall?
718.
What two numbers
are those whose
sum
is
47 and product
A man
bought a certain number of pounds of tea and
10 pounds more of coffee. What number exceeds its reciprocal by {$. 12
-4*+
-
8. 16 x* . needs 15 days longer to build a wall than B.
A
equals CO feet. what is the
price of the coffee per
pound ?
:
Find the numerical value of
728.
in value.
729.
a:
713.
ELEMENTS OF ALGEBRA
+36 = 0.
.
The area
the price of 100 apples by $1.
The
difference of the cubes of
two consecutive numbers
is
find them.
217
.
722.
___ _ 2* -5 3*2-7
715.
716. Find the price of an apple. 727. 721. he
many
312?
he had waited a few days until each share had fallen $6.l
+
8
-8
+
ft)'
(J)-*
(3|)*
+
(a
+
64-
+ i. if 1 more for 30/ would diminish
720. How shares did he buy ?
if
726. 3or
i
-16 .
714
2
*2
'
+
25
4
16
|
25 a2
711. 724. Find two numbers whose 719. 2n n 2 2 -f-2aar + a -5 = 0. 725.
of a rectangle is 221 square feet and its perimeter Find the dimensions of the rectangle. Find four consecutive integers whose product is 7920.
= ar(a? -f y + 2) + a)(* + y
933.300
930.
two squares equals 140 feet.
is
3
.
y(
934. y(x + y + 2) = 133. + z) =108.
and the sum of
their areas 78$.
931.
942. A plantation in rows consists of 10. If each side was increased by 2 feet. The diagonal of a rectangle equals 17 feet.
.square inches. the area of the new rectangle would equal 170 square feet.
find
the radii of the two circles.
The sum
of the perimeters of
sum
of the areas of the squares is 16^f feet. and the Find the sides of the and
its
is
squares.
feet.
and the sum of their cubes
is
tangle
certain rectangle contains 300 square feet.
937.
the
The sum
of the perimeters of
sum
of their areas equals 617 square feet.
Assuming
= -y. and 10 feet broader. The
sum
of the circumferences of
44 inches.
is 3. and B diminishes his as arrives at the winning post 2 minutes before B. Find the length and breadth of the first rectangle. and also contains 300 square feet.
diagonal
940. (y
(* + y)(y +*)= 50. Find the numbers.
and the
difference of
936.
A
is
938.
two squares is 23 feet. (y + *) = . 152. Find the side of each two
circles is
IT
square. (3 + *)(ar + y + z) = 96.
935. a second rec8 feet shorter.
rate each
man
ran in the
first
heat. In the second
heat
A
. z(* + y + 2) = 76.
is 20.
ELEMENTS OF ALGEBRA
(*+s)(* + y)=10. there would have been 25 more trees in a row. The perimeter of a rectangle is 92 Find the area of the rectangle. How many rows are there?
941. Tf there had been 20 less rows.
943.
two numbers Find the numbers.000 trees. A and B run a race round a two-mile course.
The sum of two numbers Find the numbers.
much and A then
Find at what
increases his speed 2 miles per hour.
34
939. s(y
932.
the difference of their
The
is
difference of
their cubes
270.
2240.
944.
+ z)=18. In the first heat B reaches the winning post 2 minutes before A.102. Find the sides of the rectangle. The difference of two numbers cubes is 513. *(* + #) =24. feet.
A number consists of three digits whose sum is 14. A certain number exceeds the product of its two digits by 52 and exceeds twice the sum of its digits by 53. Find in what time both will do it. its area will be increased 100 square feet. and
its
perim-
948. What is its area?
field is
182 yards.
overtook
miles. that
B
A
955. The square described on the hypotenuse of a right triangle is 180 square inches. distance between P and Q. triangle is 6. Two men can perform a piece of work in a certain time one takes 4 days longer.
Find the number. and if 594 be added to the number. Find its length and breadth.
. Find the width of the path if its area is 216 square yards.
Two
starts
travelers. if its length is decreased 10 feet and its breadth increased 10 feet. each block. When
from
P
A
was found that they had together traveled 80 had passed through Q 4 hours before.
whose
946. and the other 9 days longer to perform the work than if both worked together.
A and
B. the area lengths of the sides of the rectangle. Find two numbers each of which
is
the square of the other. A rectangular lawn whose length is 30 yards and breadth 20 yards is surrounded by a path of uniform width. The area of a certain rectangle is equal to the area of a square side is 3 inches longer than one of the sides of the rectangle.
sum
Find an edge of
954. was 9 hours' journey distant from P.
.
952. and that B. The area of a certain rectangle is 2400 square feet. set out from two places.
P and
Q.
950. at
the same time
A
it
starts
and
B
from
Q
with the design to pass through Q. and travels in the same direction as A. The diagonal of a rectangular is 476 yards.REVIEW EXERCISE
301
945. the digits are reversed.
is
407 cubic feet.
949.
953. If the breadth of the rectangle be decreased by 1 inch and its
is
length increased by 2 inches. the difference in the lengths of the legs of the Find the legs of the triangle.
unaltered. the square of the middle digit is equal to the product of the extreme digits. Find the number. The sum of the contents of two cubic blocks
the
of the heights of the blocks is 11 feet.
Find the
eter
947.
. at Find the his rate of traveling.
951.
such that the product of the and fourth may be 55.1
+
2. Insert 8 arithmetic means between
1
and
-.)
the last term
the series
a perfect number.
985. Find the value of the infinite product 4
v'i
v7-!
v^5
.
0.
to 105?
981..
.
What
2 a
value must a have so that the
sum
of
+
av/2
+
a
+
V2
+
.REVIEW EXERCISE
978.
of n terms of an A...001
4.
303
979.001
+
.. The 21st term of an A. doubling the number for each successive square on the board.
v/2
1
+ +
+
1
4
+
+
3>/2
to oo
+
+
. The Arabian Araphad reports that chess was invented by amusement of an Indian rajah.. Insert 22 arithmetic means between 8 and 54.04
+
.
How many
sum
terms of 18
+
17
+
10
+
amount
. Find the
first
term.
all
A
perfect number
is
a number which equals the sum
divisible. 4 grains on the 3d.
Find
n.
If
of
2
of
integers + 2 1 + 2'2
by which
is
it
is
the
sum
of
the series
2 n is prime. P. and of the second and third 03.
of n terms of 7
+
9
+ 11+
is
is
40.
980.
987. and the
common
difference. P.2
.3 '
Find the 8th
983. The
term.
"(.
to infinity
may
be 8?
.-. Find the sum of the series
988. named Sheran.
1.
Find four perfect
numbers.+ lY L V.
first
984.
989.
992.
990. 5
11.---
:
+
9
-
-
V2
+
.
986. Find four numbers in A.-. and so on.
to oo. and the sum of the first nine terms is equal to the square of the sum of the first two.
to
n terms. then this
sum multiplied by
(Euclid.. is 225.01
3. who rewarded the inventor by promising to place 1 grain of wheat on
Sessa for the
the 1st square of a chess-board. 2 grains on the 2d.
Find the number of grains which Sessa should have received. P. The sum
982.
Find (a) the sum of all circumferences.
1000. in this square a circle.
and
if
so forth
What
is
the
sum
of the areas of all circles. The sum and product of three numbers in G. areas of all triangles. One of them travels uniformly 10 miles a day. Two travelers start on the same road. are 28 and find the numbers.
and the
fifth
term
is
8 times
the second . many days will the latter overtake the former?
.
third circle touches the second circle and the
to infinity. P.
999. Insert 4 geometric means between 243 and 32. inches. the sides
of a third triangle equal the altitudes of the second.
AB =
1004. in this circle a square.
ABC
A A
n
same
sides.
997. c. after how strokes would the density of the air be xJn ^ ^ ne original
density ?
a circle is inscribed. The side of an equilateral triangle equals 2. and so forth to Find (a) the sum of all perimeters. (6) the sum of the infinity. Each stroke of the piston of an air
air contained in the receiver. P. are unequal.
998. and G. find the series.304
ELEMENTS OF ALGEBRA
993. at the same time. (a) after 5 strokes. The sides of a second equilateral triangle equal the altitudes of the first.
.
The sum and sum
. The
fifth
term of a G.
1001. 995. Under the conditions of the preceding example. P. are
45 and 765
find the
numbers.
of squares of four
numbers
in
G.
pump removes
J
of the
of air
is
fractions of the original amount contained in the receiver. Insert 3 geometric means between 2 and 162. 1003. In an equilateral triangle second circle touches the first circle and the sides AB and AC.
is 4. and so forth to infinity. (6) after n
What
strokes?
many
1002. The other travels 8 miles the first day and After how increases this pace by \ mile a day each succeeding day. prove that they cannot be in A.
512
996. In a circle whose radius is 1 a square is inscribed. ft.
994. P. If a. P. (I) the sum of the perimeters of
all
squares.
$1. so that the Logarithms.
not
The Advanced Algebra is an amplification of the Elementary. Ph. 64-66 FIFTH AVBNTC. save Inequalities. than by the
. given.
HEW TOSS
. but the work in the latter subject
has been so arranged that teachers
who
wish a shorter course
may omit
it
ADVANCED ALGEBRA
By ARTHUR SCHULTZE. without the sacrifice of scientific accuracy and thoroughness. The author
has emphasized Graphical Methods more than is usual in text-books of this grade.10
The treatment of elementary algebra here is simple and practical.
which have been omitted from the body of the work Indeterminate Equahave been relegated to the Appendix. comparatively few methods are heretofore. physics. which has been retained to serve as a basis for higher work. Particular care has been bestowed upon those chapters which in the customary courses offer the greatest difficulties to the beginner. and the Summation of Series is here presented in a novel form.
xiv+563
pages.D. proportions and graphical methods are introduced into the first year's course. great many
work.ELEMENTARY ALGEBRA
By ARTHUR SCHULTZE. but these few are treated so thoroughly and are illustrated by so many varied examples that the student will be much better prepared for further
The Exercises are superficial study of a great many cases. very numerous and well graded there is a sufficient number of easy examples of each kind to enable the weakest students to do some work.25
lamo. The introsimpler and more natural than the
methods given
In Factoring. and commercial life.
xi 4-
373 pages. To meet the requirements of the College Entrance Examination Board. book is a thoroughly practical and comprehensive text-book.
A
examples are taken from geometry. especially
duction into Problem
Work
is
very
much
Problems and Factoring.
Half leather.
$1. The more important subjects
tions.
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PUBLISHERS. etc.
i2mo. All subjects now required for admission by the College Entrance Examination Board
have been omitted from the present volume. but none of the introduced illustrations is so complex as to require the expenditure of
time for the teaching of physics or geometry.
Half
leather.
The author
grade.
not
The Advanced Algebra is an amplification of the Elementary. but the work in the latter subject
has been so arranged that teachers
who
wish a shorter course
may omit
it
ADVANCED ALGEBRA
By ARTHUR SCHULTZE. without
Particular care has been the sacrifice of scientific accuracy and thoroughness. and commercial life. especially
duction into Problem
Work
is
very
much
Problems and Factoring.ELEMENTARY ALGEBRA
By ARTHUR Sen ULTZE.
xiv+56a
pages.25
i2mo. but none of the introduced illustrations is so complex as to require the expenditure of
time for the teaching of physics or geometry. The more important subjects
which have been omitted from the body of the work Indeterminate Equahave been relegated to the Appendix. To meet the requirements of the College Entrance Examination Board. book is a thoroughly practical and comprehensive text-book.
HEW YOKE
.
In Factoring. Logarithms. great many
A
examples are taken from geometry. bestowed upon those chapters which in the customary courses offer the greatest difficulties to the beginner. etc.
xi
-f-
373 pages. 64-66
7HTH
AVENUE. The Exercises are very numerous and well graded. than by the superficial study of a great many cases. save Inequalities. All subjects now required for admission by the College Entrance Examination Board
have been omitted from the present volume. Ph.
HatF leather.
$1. which has been retained to serve as a basis for higher work. but these few are treated so thoroughly and are illustrated by so many varied examples that the student will be much better prepared for further
work.
THE MACMILLAN COMPANY
PUBLISHBSS.10
The treatment
of elementary algebra here
is
simple and practical.
has emphasized Graphical Methods more than is usual in text-books of this and the Summation of Series is here presented in a novel form. there is a sufficient number of easy examples of each kind to enable the weakest students to do some work.
12010. physics. so that the tions. proportions and graphical methods are introduced into the first year's course.
$1.D.
Half leather. comparatively few methods are
given. The introsimpler and more natural than the
methods given heretofore.
Ph. These are introduced from the beginning 3. 6.
Cloth.
i2mo.
Algebraic Solution of Geometrical Exercises is treated in the Appendix to the Plane Geometry .
.
and no attempt has been made
to present these solutions in such form
that they can be used as models for class-room work.
$1. iamo. Attention is invited to the following important features I. The Schultze and Sevenoak Geometry is in use in a large number of the leading schools of the country. 9. The Analysis of Problems and of Theorems is more concrete and practical than in any other
distinct pedagogical value.
of Propositions has a
Propositions easily understood are given first and more difficult ones follow .
Half
leather.
ments from which General Principles may be obtained are inserted in the " Exercises. more than 1200 in number in 2..
NEW YORK
.10
L. SCHULTZE. State: . at the
It
same
provides a course which stimulates him to do original time. Preliminary Propositions are presented in a simple manner . 10.10
By ARTHUR
This key will be helpful to teachers who cannot give sufficient time to the Most solutions are merely outsolution of the exercises in the text-book.
KEY TO THE EXERCISES
in
Schultze and Sevenoak's Plane and Solid Geometry. 7 he
.
80 cents
This Geometry introduces the student systematically to the solution of geometrical exercises. Cloth. under the heading Remarks". $1.
text-book in Geometry
more
direct
ositions
7.
PLANE AND SOLID GEOMETRY
F. Difficult Propare made somewhat? easier by applying simple Notation . The numerous and well-graded Exercises the complete book.D.
SEVENOAK. 64-66 FIFTH AVENUE. Pains have been taken to give Excellent Figures
throughout the book. guides him in putting forth his efforts to the best
advantage.
izmo.
xii
+ 233 pages.
By ARTHUR SCHULTZE and
370 pages.
xtt-t
PLANE GEOMETRY
Separate. Proofs that are special cases of general principles obtained from the Exercises are not given in detail.
THE MACMILLAN COMPANY
PUBLISHERS.r and. aoo pages. Hints as to the manner of completing the work are inserted The Order 5.
wor. Many proofs are presented in a simpler and manner than in most text-books in Geometry 8.
lines. 4.
. enable him to
" The chief object of the speak with unusual authority. " is to contribute towards book/ he says in the preface.25
The
author's long
and successful experience as a teacher
of mathematics in secondary schools and his careful study of the subject from the pedagogical point of view. and not from the information that it imparts.The Teaching
of
Mathematics
in
Secondary Schools
ARTHUR SCHULTZE
Formerly Head of the Department of Mathematics in the High School Commerce.
.
Students
to
still
learn
demon-
strations instead of learning
how
demonstrate. and Assistant Professor of Mathematics in New York University
of
Cloth. Most teachers admit that mathematical instruction derives its importance from the mental training that it But in affords. 370 pages.
.
.
.
12mo. a great deal of mathematical spite
teaching
is
still
informational.
. $1. making mathematical teaching less informational and more disciplinary. Typical topics the value and the aims of mathematical teach-
ing
.
New York
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.
methods of teaching mathematics the first propositions in geometry the original exercise parallel lines methods of the circle attacking problems impossible constructions applied problems typical parts of algebra.
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causes of the inefficiency of mathematical teaching.
. New York City. of these theoretical views."
The treatment
treated are
:
is concrete and practical.
An exhaustive system of marginal references. and a full index are provided.
but in being fully illustrated with
many excellent maps. All
smaller movements and single events are clearly grouped under these general movements. Maps.
diagrams. The author's aim is to keep constantly before the
This book
pupil's mind the general movements in American history and their relative value in the development of our nation.
New York
SAN FRANCISCO
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narrative on which to base the general statements and other classifications made in the text.
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fa
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Illustrated. which put the main stress upon national development rather than upon military campaigns. Studies and Questions at the end of each chapter take the place of the individual teacher's lesson plans. which have been selected with great care and can be found in the average high school library. diagrams. The book deserves the attention
of history teachers/'
Journal of Pedagogy.
$1. This book is up-to-date not only in its matter and method.40
is distinguished from a large number of American text-books in that its main theme is the development of history the nation.
i2mo.
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etc.
Cloth. | 677.169 | 1 |
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EDM 220 - Algebraic Thinking in the Early Grades.
Algebraic thinking and its application to the K-8 classroom. Topics include the teaching of number and operation; proportional reasoning; variables and unknowns; the concept of function; modeling of real world situations using algebraic language; linear functions. | 677.169 | 1 |
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Solid Shape gives engineers and applied scientists access to the extensive mathematical literature on three dimensional shapes. Drawing on the author's deep and personal understanding of three-dimensional space, it adopts an intuitive visual approach designed to develop heuristic tools of real use in applied contexts. Increasing activity in such areas as computer aided design and robotics calls for sophisticated methods to characterize solid objects. A wealth of mathematical research exists that can greatly facilitate this work yet engineers have continued to "reinvent the wheel" as they grapple with problems in three dimensional geometry. Solid Shape bridges the gap that now exists between technical and modern geometry and shape theory or computer vision, offering engineers a new way to develop the intuitive feel for behavior of a system under varying situations without learning the mathematicians' formal proofs. Reliance on descriptive geometry rather than analysis and on representations most easily implemented on microcomputers reinforces this emphasis on transforming the theoretical to the practical. Chapters cover shape and space, Euclidean space, curved submanifolds, curves, local patches, global patches, applications in ecological optics, morphogenesis, shape in flux, and flux models. A final chapter on literature research and an appendix on how to draw and use diagrams invite readers to follow their own pursuits in threedimensional shape.Jan J. Koenderinck is Professor in the Department of Physics and Astronomy at Utrecht University. Solid Shape is included in the Artificial Intelligence series, edited by Patrick Winston, Michael Brady, and Daniel Bobrow
ISBN:
9780262111393
Category:
Artificial intelligence
Format:
Hardback
Publication Date:
21-03-1990
Language:
English
Series:
Artificial Intelligence Series
Publisher:
MIT Press Ltd
Country of origin:
United States
Pages:
720
Dimensions (mm):
234x160x46mm
Weight:
1.27Solid Shape | 677.169 | 1 |
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The Elements of Universal Mathematics, or Algebra; To Which Is Added, a Specimen of a Commentary on Sir Isaac Newton's Universal Arithmetic. Containin
Description
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1728 edition. Excerpt: ...before about Problems in general, the Solution of Problems of two Dimenfions will not be difficult. Problem XVI. 230. A andB together ewe 208; A pays each day 9, and B pays the firfi day 1, the fecond day 2, the third 3, &c. Qu. How many days will the-whole Debt be paid in, and how much does each pay? Let A's Debt be x, Bsj, and the Number of Days z.. x--y--108 oz. = x y is equivalent to the Sum of an Arithmetic Progreffion, whofe firftTerm is 1, laft z., and Number of Terms z.; which Sum is equal to 1 + z x fz. (126), therefore 4-+ f---J Which three Equations, by adding the two laft together, and then comparing it with die firft, are reduced to this: x 4-y--iz.z, + Jt = 208 z.z, + 19 = 416 7A + 1pZ, + 90-J--f JO5 z, = 13 Problem XVII. 240. A Perfon huys a Horfe, which he fells again for a, and gains as much per Cent, as the Horfe cofi him. Qu. What did the Horfe cofi him? Let the Price of the Horfe be x. 100, 100--x:: x, a Problem XVIII. '241. There are two Numbers fought, -whofe Pro duB is 12, and the Difference of their Squares 7. Let the Numbers be x and y. 12 144 xy = 12, y =--tyy=-22 X XX 144 xx--yy = 7, xx--= 7 xx x4--144 = 7 This Equation is folv'd as an Equation of two Dimenfions (217), if we feek the Value of the Square of xx. x4--7xx = 144 X4 7XX+ 12: ? = 15tf xx--37 37-xxs 121 XX = ' 1--9.X = 4, J = 3 I neglect the fecond Value xx----9, becaufe it is impoffible (218). I neceffarily difcover the Value of the Square xx in the Solution, becaufe not only x =4, y=2, but alfo x =--4, y =--3, folve the Problem. I alfo difcover two other Values of the Square xx, becaufe the Problem has befides two impoffible Solutions, which may be algebraically exexpreffed: Thefe are x =-/--9, y=--/--i6t and x----/--9, y----V--16. Problem XIX. 242show more | 677.169 | 1 |
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Cbse Class 9 Guide Of Maths .pdf
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Cbse class 8, 9, 10, 11 course - android apps on Jun 04, 2015 This guide will help you ace your schools exams and have fun In this app you will get solutions for maths problems for Class 9 CBSE syllabus. Free
Download ncert solutions for class- 9 maths | a NCERT text book solutions for class-9 Mathematics are now available to download in PDF file format in myCBSEguide.com. Each and every questions from NCERT book has | 677.169 | 1 |
Homework help algebra 2
Covering pre-algebra through algebra 3 with a variety of introductory and advanced lessons.These articles can help you understand more advanced Algebra concepts.
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Describe an event that has a probability of 0 and an event that has a probability of 1, algebra homework help.Homework for 110.201 Linear Algebra - Spring 2010 - Professor Consani Solutions to the Homework provided by TAs.Each section has solvers (calculators), lessons, and a place where.
Describe a situation in which you can use the Fundamental Counting Principle, algebra homework help.
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Webmath is a math-help web site that generates answers to specific math questions and problems, as entered by a user, at any particular moment. | 677.169 | 1 |
Sage-Related Stuff
What is Sage?
Sage is the free, open-source competitor to Maple, Mathematica, Magma, and
Matlab. It is a computer-algebra system ideally suited to students of mathematics.
This page is woefully incomplete. The Sage community involves hundreds of developers
and thousands of contributors world wide.
Interacts (sometimes called interactive webpages, applets, apps, or interactive
figures) are a really fun way to demonstrate a complicated math topic. A large repository
of them (made by the Sage community) can be found by
clicking here,
and I've made a few myself, which you can find by
clicking here. (For using these,
NO KNOWLEDGE of Sage whatsoever is required!)
The online electronic appendix covers plotting in color, complex functions, and
3D graphics. Those subjects are not suited to a black-and-white book, and therefore cannot
be printed inside the book itself. [Rough Draft]Click here.
Chapter 6 of the book teaches the reader how to make their own interactive webpages
or applets. To save readers from having to retype my
code into their computers,
I promised a zip-file with
some source code of the examples used.
Here are some videos that I've made
to introduce you to the basics of using Sage with its most
simple interface, the SageMathCell Server. Both are less than five minutes.
Part
Two covers factoring, 3D plotting, gradients, and symbolic solving.
After watching both videos (or even without them) you'd find it very easy to just dive
on into Chapter 1 of my book, linked above.
At the bottom of this page, I have some other videos about matrices and about linear
programming.
Want to give Sage a try?
For short and medium-sized problems (especially in 100-level and 200-level courses, but
even in higher-level courses too), the best way to use Sage is the SageMathCell Server.
(That's the competitor to WolframAlpha, and until recently it was called Sage-Aleph.)
The first bullet on this page talks about interacts (sometimes called applets or interactive
webpages) and those can be used by anyone, with no knowledge of Sage or computing required at all.
For longer problems (especially those that will require collaboration, writing your
own programs, or using data sets) the
CoCalc.com
server is the way to go. (This service was called SageMathCloud until the Summer of 2017. It is the
same service, just renamed, because the tool is being used by computer scientists, physicists,
engineers, and even geologists, not just math people.)
The beauty of Sage is that it works through the internet. There is almost never any
reason to do a local install of Sage on your laptop or home computer. This is good news,
because it saves a lot of headaches and hassles (especially for students), that you would
have to suffer if you were using Mathematica, Maple, Matlab, or Magma. The exception is if you
have limited or no internet access, such as in rural areas. In any case, the directions
for a local install can be found by
clicking here.
This is an excellent tutorial
for faculty, PhD-students in mathematics, and senior math majors about using
Sage for all sorts of problems.
Here is a large collection of quick-reference
cards for Sage, by various people, for various branches of mathematics, in many languages.
Personally, I think having a printed quick-reference card out next to the laptop while using
Sage is really handy. :-) | 677.169 | 1 |
This ebook is available for the following devices:
iPad
Windows
Mac
Sony Reader
Cool-er Reader
Nook
Kobo Reader
iRiver Story
This lucid and balanced introduction for first year engineers and applied mathematicians conveys the clear understanding of the fundamentals and applications of calculus, as a prelude to studying more advanced functions. Short and fundamental diagnostic exercises at the end of each chapter test comprehension before moving to new material.
Provides a clear understanding of the fundamentals and applications of calculus, as a prelude to studying more advanced functions
Includes short, useful diagnostic exercises at the end of each chapter
In the press
I have decided to use this book as a core text for a basic module in the first year of BSc and Higher National Diploma engineering courses. This text is suited in pace, content, range of well-chosen examples, etc., for students needing a high level of support., Dr John Baylos, Nottingham Trent University, UK | 677.169 | 1 |
Introduction to Differential Equations: Mathematics Association
The laws of science and engineering are typically expressed in differential equations, which are equations with derivatives in them. Understanding of differential equations and their solutions is important in the sciences and engineering. This course deal
Associations
The Mathematical Association of America is the largest professional society that focuses on mathematics accessible at the undergraduate level. Our members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists; statisticians; and many others in academia, government, business, and industry. We welcome all who are interested in the mathematical sciences.
Welcome to SIAM!Applied mathematics, in partnership with computational science, is essential in solving many real-world problems. Our mission is to build cooperation between mathematics and the worlds of science and technology through our publications, research, and community. | 677.169 | 1 |
MA213: Numerical Analysis
Unit 1: Computer ArithmeticWe see how real numbers are represented in computers for scientific
computation. We cannot represent all real numbers, so we must choose
which finite subset of the real numbers we will use. Most modern
scientific computing uses a set of floating point numbers. The
properties of floating point numbers affect arithmetic; in fact, we do
not even expect the computer to add two numbers correctly. We will
follow these errors through simple computations and learn some basic
rules and techniques for tracking errors. Finally, we will write a
simple program that pays careful attention to these considerations.
Unit1 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Be able to represent numbers using a normalized floating point
representation.
- Understand the implications of this representation on arithmetic and
the ideas of swamping and cancellation.
- Analyze such errors and understand the ideas of forward and backward
error analysis.
- Understand how conditioning analysis and backward stability combine
to allow an estimate of overall error in a computation.
Instructions: Click on the link above, then select the appropriate
link to download a pdf of the reading. The "Comparing Numbers"
reading is here to get you thinking about how we measure errors in
computation. Our primary tool is the absolute difference relative
to the true answer, or the relative error. When is the relative
error equal to the absolute error? The Floating Point
Representation Theorem and the Fundamental Axiom of Floating Point
Arithmetic are the two basic rules we will use to estimate rounding
errors.
This resource should take approximately 2 hours to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
Lecture: YouTube: University of South Florida: Autar Kaw's
"Floating Point Representation"
Link: YouTube: University of South Florida: Autar Kaw's "Floating
Point Representation"
(YouTube)
Instructions: Click on the link above, then watch the video
lectures in the chapter listed above. In this case there are 4
lectures that have been split into 8 videos. These lectures will be
useful for all of units 1.1 and 1.2.
This resource should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
1.1.2 Floating-Point Representation Theorem
- Reading: University of Arkansas: Mark Arnold's "Pictures of a Toy
Floating Point System"
Reading: University of Arkansas: Mark Arnold's "Pictures of a Toy
Floating Point System"
Instructions: Click on the link above, then select the appropriate
link to download a pdf of the reading. This is a plot of a 1-byte (8
bits) floating point system. The floats are too close to each other
on the top plot, so the subsequent plots are zoom-ins. We want to
relate the machine epsilon from the previous reading to points on
this plot. On the plot(s) identify the floating point range, and
the machine epsilon.
This resource should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
Instructions:. Click on the link above, then select the appropriate
link to download a pdf of the reading. Pretend you are a (base-10)
computer and add and multiply some 4-decimal digit numbers. Prove
the Fundamental Axiom of Floating Point Arithmetic using the
additional hypothesis that "an arithmetic operation on two such
floating point numbers returns the floating point number closest to
the true value". This
reading exhibits both a forward rounding error analysis and a
backward rounding error analysis. Make sure you clearly understand
the distinction. For the vast majority of computations a forward
error analysis does not provide useful error bounds (they are too
pessimistic). Do a forward error analysis for the product of two
real numbers; you should be able to mimic the forward analysis for
a+b that is in the reading. We will use this reading for the next
section.
This resource should take approximately 2 hours to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
1.3.2 Backward Error Analysis
- Reading: University of Arkansas: Mark Arnold's "What is a
Solution?" and "Conditioning and Stability"
Link: University of Arkansas: Mark Arnold's "What is a
Solution?" (PDF) and
"Conditioning and
Stability"(PDF)
Instructions: Click on the link above, then select the appropriate
link to download a pdf of the reading. In the "Error Analysis"
reading we show that if a, b and a+b are real numbers in the
floating point range, then fl(a+b) is the exact sum of two numbers
relatively close to a and b. Do the same for fl(ab). At this point
you have seen that (i) while an addition (or subtraction) may be
computed with large relative error, (ii) it is also the exact sum of
two numbers very close to a and b. Thus (ii) does not guarantee a
small error. You have shown that multiplication (which doesn't
over/underflow) will always be computed with small relative error.
We perform error analysis both to gain insight into a method (where
are its weak points?) and to predict how good our computed solution
is. This reading shows how the relationship between backward error
analysis and problem conditioning can give us an error estimate Rounding errors occur for
nearly every arithmetic operation, but sometimes circumstances
converge to set us up for a particularly bad result: cancellation.
You have seen cancellation in simple examples and in your forward
error analysis. Show that there is a risk of cancellation any time
we additively compute a result that is small compared to its
addendsWrite a program in Octave that computes the roots of ax2+bx+c given
the real numbers a, b and c. Make sure that your program does not
divide by 0, that it does not suffer from cancellation (except
possibly in b2-4ac), and all input variables (a,b,c) and all output
variables (r1, r2 maybe others?) are described carefully. There is
a good tutorial on Octave at
Since
you already have some experience programming, Octave should be
relatively easy to learn (it was born as a Matlab clone, and is very
much like fortran90. Matlab tutorials are plentiful on the web, you
might find them useful as well). As with learning most new
languages, the learning curve is steep with gross syntax; but never
fear, Octave is quite simple.
This resource should take approximately 10 hours to complete
(including the Octave download/install process).
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
Instructions: Click on the link above, and download the reading
using the "Download Links" menu in the upper right corner. Scan the
paper so that you know what is covered. This paper is a very good
resource for the IEEE 754 standard. You will want to have it as a
reference as you work through the material in this course.
This resource should take approximately 4 hours to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above essay.</span> This assessment
should take about 40 minutes to complete. | 677.169 | 1 |
MATH 354 Advice
Showing 1 to 2 of 2
This class is very important if you want to better understand efficient programing.
Course highlights:
I learn that for you to be more efficient in solving problems you have to keep practicing.
Hours per week:
9-11 hours
Advice for students:
The class is not hard but you have to study hard if you want to do well in any classes.
Course Term:Summer 2017
Professor:Justin Semonsen
Course Required?Yes
Course Tags:Great Intro to the SubjectGo to Office HoursAlways Do the Reading
Apr 18, 2016
| Would recommend.
This class was tough.
Course Overview:
Lyons is the best math professor I've ever taken a class with. He focuses on concepts more than computation, which makes the material challenging but practical, considering that the course itself is based on algorithms that are rarely executed by hand these days and easily found in software packages like Microsoft Excel. The subject itself is an essential component of applied mathematics, engineering, and computer science, so you will definitely pick up some good intuition on formulating complex everyday problems into ones in the realm of linear optimization.
Course highlights:
The highlights of the course include the intuition behind the duality of linear programming problems as well as some common applications of the cumulative theory behind linear optimization, such as the Transportation Problem and the Scheduling Problem.
Hours per week:
6-8 hours
Advice for students:
Make sure you're comfortable with the basics of linear algebra, especially linear independence, row operations, dot products, and proofs involving matrices!
And go to lecture! I can't stress this enough. The textbook becomes pretty hard to follow, because there are a lot of algorithms done by hand. Lyons stops to clarify a lot of concepts and intuition throughout lecture, not to mention he's low-key hilarious. He also never fails to understand a student's question on the first try. He's as good of a listener as he is a lecturer, which makes his lectures a real joy. | 677.169 | 1 |
Math Center
The Math Center is a non-credit, Community Education class which provides assistance
in mathematics as a completely free service. Current Allan Hancock College students
as well as other individuals who are 18 years or older may register for the Math Center
each semester and attend as frequently as they want. Registration is for one semester
only and may be done online or at Community Education in Building S.
The goal of the Math Center (sometimes called the Mathematics Lab) is to assist students
in the successful completion of any Allan Hancock College mathematics class by providing
additional instructional resources. The Math Center offers many resources, including
one-on-one, drop-in tutoring by our staff of instructors and student tutors. Please
see the full list of resources below:
Handouts on math topics, including content from various math courses as well as information
on overcoming math anxiety and preparing for and taking math tests
Two private study rooms
Make-up testing
Register to use the Math Center
The Math Center is a free service available to all students, howeveryou must registerfor the noncredit class BASK to use the services. Students must register online through
themyHancockportal for BASK 7014.
STAFF
Achieve Success at the Math Center
SPOTLIGHT
Heather Deniz Administration of Justice Student
"Allan Hancock College has given me the highest quality of instructors and greatest variety of classes, especially considering how affordable tuition is. The campus has a wonderful community feel with lots of resources to aid in reaching my goals of achieving my administration of justice degree." Submit a Testimonial » | 677.169 | 1 |
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