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Grade 7 Math Course Lesson Plan: 34 weeks
Transcription
1 Grade 7 Math Course Lesson Plan: 34 weeks Welcome to Thinkwell s 7th Grade Math! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson plan is meant to be a guide for you and your homeschool student. Each day, you ll tackle a different topic and all the materials associated with that topic, such as video lectures, worksheets, exercises, and interactivities. If you follow our day by day schedule, you ll complete the full curriculum for the course in 34 weeks. Feel free to modify and amend the plan as it best works for you. And, as always, please let us know what we can do to help you get you up and running with Thinkwell s 7th Grade Math! Week 1 Chapter 1: Algebraic Reasoning Week 1, Day Numbers and Patterns Week 1, Day Exponents Week 1, Day Applying Exponents: Scientific Notation Week 1, Day Order of Operations Week 1, Day Properties Week 2 Chapter 1: Algebraic Reasoning Week 2, Day 1 Subchapter 1.1 Worksheet, Part I Week 2, Day 2 Subchapter 1.1 Worksheet, Part II Week 2, Day Variables and Algebraic Expressions Week 2, Day Translate Words into Math Week 2, Day Simplifying Algebraic Expressions Week 3 Chapter 1: Algebraic Reasoning Week 3, Day Equations and Their Solutions Page 1 of 12
Variables and Expressions Problem Solving: Using a Problem-Solving Plan Use a four-step plan to solve problems. Choose an appropriate method of computation. Numbers and Expressions Use the order of operations
2012-2013 Math Content PATHWAY TO ALGEBRA I Unit Lesson Section Number and Operations in Base Ten Place Value with Whole Numbers Place Value and Rounding Addition and Subtraction Concepts Regrouping Concepts
A Correlation of Prentice Hall Mathematics Courses 1-3 Common Core Edition 2013 to the Topics & Lessons of Pearson A Correlation of Courses 1, 2 and 3, Common Core Introduction This document demonstrates
MATH Activities ver3 This content summary list is for the 2011-12 school year. Detailed Content Alignment Documents to State & Common Core Standards are posted on NOTE: Penda continues
Understanding the Progression of Math Courses in NEISD According to House Bill 1 (HB1), students in Texas are required to obtain credits for four courses in each subject area of the foundation curriculum
Algebra I COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics, with an emphasis
New York State Mathematics Content Strands, Grade 6, Correlated to Glencoe MathScape, Course 1 and The lessons that address each Performance Indicator are listed, and those in which the Performance Indicator
Academic Content Standards Grade Eight Ohio Pre-Algebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and smallStandard: Number, Number Sense and Operations Number and Number C. Develop meaning for percents including percents greater than 1. Describe what it means to find a specific percent of a number, Systems
MATH There are many activities parents can involve their children in that are math related. Children of all ages can always practice their math facts (addition, subtraction, multiplication, division) in
MATHS Step 1 To achieve Step 1 in Maths students must master the following skills and competencies: Number Add and subtract positive decimal numbers Add and subtract negative numbers in context Order decimal
SOL 8.1 exponents order of operations expression base scientific notation Represents repeated multiplication of the number. 10 4 Defines the order in which operations are performed to simplify an expression.2013 2014 Scheme of Work Subject MATHS Year 9 Course/ Year Term 1 Key Topics What will ALL students learn? What will the most able students learn? Number Written methods of calculations Decimals Rounding
Pre-Algebra IA Pre-Algebra IA introduces students to the following concepts and functions: number notation decimals operational symbols inverse operations of multiplication and division rules for solving
Nativity Catholic School Rising 7th grade IXL Language Arts and Math Summer Homework Please work on the following skills listed in the 6th Grade Math and Language Arts IXL Program for a minimum of 60 minutes
Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards
Consumer Math 15 INDEPENDENT LEAR NING S INC E 1975 Consumer Math Consumer Math ENROLLED STUDENTS ONLY This course is designed for the student who is challenged by abstract forms of higher This math. course
Terms and Definitions Absolute Value the magnitude of a number, or the distance from 0 on a real number line Additive Property of Area the process of finding an the area of a shape by totaling the areas
MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers andRAVEN S GUIDE TO BRITISH COLUMBIA MATHEMATICS GRADE 6 LINKED DIRECTLY TO NEW CURRICULUM REQUIREMENTS FROM THE WESTERN PROTOCOLS FOR 2008 AND BEYOND STUDENT GUIDE AND RESOURCE BOOK Key to Student Success | 677.169 | 1 |
GCSE Mathematics
What will I learn?
Gaining a grade C in GCSE Mathematics is a stepping stone into many types of education and employment. You should take this course if you intend to progress to a Level 3 program, apply for a university course or this qualification may be required by specific employers.
Maths is about more than numbers. It encourages you to think logically and apply the principles you learn to everyday situations.
You will start with the basics, making calculations and solving problems before moving on to number systems, formulae, graphs, functions and algebraic equations.
Geometry will be explored, and you will discover ways to calculate lengths, areas and volumes of different shapes.
GCSE Mathematics includes practical development of your skills in the five main topics: Number, Algebra, Geometry, Data Handling and Probability.
You will be assessed in two final exams; Non-Calculator and Calculator, both 105 minutes in duration and completed approximately four days apart. The scores from each are aggregated to produce your final grade.
What could I progress on to?
Achieving a grade C in Mathematics can help you achieve your personal or career goals, including further study at Level 3, university, employment or promotion.
Along with English Language, a good grade in Mathematics is an absolute requirement for any university course or career.
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All
practical
teachers
know how few
students understand and
appreciate the more difficult parts of the theory.
specially
2.
giving to the student complete familiarity
with
all
the essentials of the subject."
this book.
owing
has certain distinctive features. All parts of the theory whicJi are beyond the comprehension
of
the student or wliicli are logically
unsound are
omitted. but "cases" that are taught only on account of tradition. chief
:
among
These
which are the following
1. Until recently the tendency was to multiply as far as possible. Typical in this respect is the
treatment of factoring in
many
text-books
In this book
all
methods which are of
and which are applied in advanced work are given. etc.
and ingenuity
while the cultivation of the student's reasoning power is neglected.
manufactured for this purpose.
omissions serve not only practical but distinctly pedagogic " cases " ends. in order to make every example a
social
case of a memorized method.. not only taxes a student's memory unduly but in variably leads to mechanical modes of study. short-cuts that solve only examples
real value.
however.
Elementary Algebra.PREFACE
IN
this
book the attempt
while
still
is
made
to shorten the usual course
in algebra.
Such a large number of methods. The entire study of algebra becomes a mechanical application of memorized
rules. and conse-
. are omitted.
"
While
in
many
respects
similar to the author's
to its peculiar aim.
All unnecessary methods
and "cases" are
omitted.
as quadratic equations and
graphs.
Topics of practical importance.g. are placed early in the course.
differ
With very few
from those
exceptions
all
the exer
cises in this
book
in the
"Elementary Alge-
bra". and it is hoped that this treatment will materially diminish the difficulty of this topic for young students. the following
may
be quoted from the author's "Elementary Algebra":
which
"Particular care has been bestowed upon those chapters in the customary courses offer the greatest difficulties to
the beginner. however. e. hence either book
4. all elementary proofs theorem for fractional exponents. all proofs for the sign age
of the product of
of the binomial
3.
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. especially problems and factoring.
may
be used to supplement the other. For the more ambitious student.
The best way to introduce a beginner to a new topic is to offer Lim a large number of simple exercises. This made it necessary to introduce the theory of proportions
.
In regard
to
some other features of the book. there has been placed at the end of the book a collection of exercises which contains an abundance
of
more
difficult
work.
TJie exercises are slightly simpler than in the larger look. Moreover.vi
PREFACE
quently hardly ever emphasize the theoretical aspect of alge bra. etc.
two negative numbers. 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. a great deal of the theory offered in the avertext-book is logically unsound . " The book is designed to meet the requirements for admis-
sion to our best universities
and
colleges. in particular the
requirements of the College Entrance Examination Board.
such examples.
of the Mississippi or the height of Mt. and commercial are numerous. and
of the
hoped that some
modes of representation given
will be considered im-
provements upon the prevailing methods.
viz. Moreover. physics. based upon statistical abstracts.PREFACE
vii
and graphical methods into the first year's work. elementary way.
but they unquestionably furnish a very good antidote against 'the tendency of school algebra to degenerate into a mechanical application of
memorized
rules.
to solve a
It is
undoubtedly more interesting for a student
problem that results in the height of Mt.
and they usually involve difficult numerical calculations. The entire work in graphical methods has been so arranged that teachers who wish
a shorter course
may omit
these chapters. in
"
geometry
. an innovation which seems to mark a distinct gain from the pedagogical point
of view."
Applications taken from geometry.
McKinley
than one that gives him the number of Henry's marbles. " Graphical methods have not only a great practical value. and hence the student is more easily led to do the work by rote
than when the arrangement
braic aspect of the problem. are
frequently arranged in sets that are algebraically uniform. but the true study of algebra has not been sacrificed in order to make an impressive display of sham
life
applications.
is
based principally upon the alge-
.
nobody would find the length Etna by such a method.
while in the usual course proportions
are studied a long time after their principal application. But on the other hand very few of such applied examples are
genuine applications of algebra. the student will be able to utilize this knowledge where it is most
needed.
By studying proportions during the first year's work.'
This topic has been preit is
sented in a simple.
however. Manguse for the careful reading of the proofs and
many
valuable suggestions. is such problems involves as a rule the teaching of physics by the
teacher of algebra. desires to acknowledge his indebtedness to Mr. 1910.
April.
William P.
pupil's knowlso small that an extensive use of
The average
Hence the
field of
suitable for secondary school
tations.
NEW
YORK.
ARTHUR SCHULTZE.
.viii
PREFACE
problems relating to physics often
offer
It is true that
a field
for genuine applications of algebra.
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.
edge of physics.
if
:
a = 2.
28. 6 = 2. = 3. 6 = 6. physics.
:
6.
30.
26.
sible to state
Ex. 6 = 5.
Twice a3 diminished by 5 times the square root of the quantity a minus 6 square. 25. 33. 6 = 7. 6 = 1.
then
8 = \ V(a + 6 + c) (a 4.
24. a = 4.
a =3.
. 6 = 3. 6 = 5. a
a=3.
Read the expressions
of Exs.
12 cr6
-f-
6 a6 2
6s.
Six
2
.
w
cube plus three times the quantity a minus
plus 6 multiplied
6.
22.c) (a . Six times a plus 4 times
32.
37. 30. 10-14
The
representation of numbers by letters makes it posvery briefly and accurately some of the principles of arithmetic. of this exercise?
What kind of expressions are Exs.
6=2. a =4.
23.
a = 3.
35. 38. and the area of the
is
triangle
S
square feet (or squares of other units selected). 6.
34.12
17
&
*
ELEMENTS OF ALGEBRA
18
'
8
Find the numerical value of 8 a3
21.
The quantity a
6
2 by the quantity a
minus
36. a = 3.
6 = 4.
29. a = 4. geometry. and other sciences.
6. 6 = 6.
a.
27. a = 2.6 -f c) (6
a
+ c).
Express in algebraic symbols 31. 2-6 of the exercise.
Six times the square of a minus three times the cube of Eight x cube minus four x square plus y square.6 .
and
If the three sides of a triangle contain respectively c feet (or other units of length).
if v
:
a.) Assuming g
. if v = 30 miles per hour. the three sides of a triangle are respectively 13.seconds.
S = | V(13-hl4-fl5)(13H-14-15)(T3-14-i-15)(14-13-f-15)
= V42-12-14.16
1
= 84.
i. c. b 14. 14. How far does a body fall from a state of rest in T ^7 of a (c)
A
second ?
3. if v = 50 meters per second 5000 feet per minute. 12. 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.
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. count the resistance of the atmosphere.
2. then a 13.g. if v . b.16 centimeters per second. d. A train in 4 hours. and c
13
and
15
=
=
=
.
84 square
EXERCISE
1.
. and 15 feet.
By
using the formula
find the area of a triangle
whose
sides are respectively
(a) 3.
(c) 4.
=
(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. Find the height of the tree. and 13 inches.
4.
the area of the triangle equals
feet. and 5 feet. An electric car in 40 seconds. A carrier pigeon in 10 minutes.INTRODUCTION
E.e. 15 therefore feet.
(b) 5. 13.
If cated on the Fahrenheit scale.
5.).14
square meters. the
3.
~
7n
cubic feet.
ELEMENTS OF ALGEBRA
If
the radius of a circle
etc.
diameter of a sphere equals d
feet.14d (square units).
then the
volume
V=
(a) 10 feet.
(c)
5
F.). and the value given above is only an
surface
$=
2
approximation. This number cannot be expressed exactly.14
4. Find the area of a circle whose radius is
It
(b)
(a) 10 meters.
the area
etc.
meters.
$ = 3.) Find the surface of a sphere whose diameter equals
(a)
7. then
=p
n
*
r
%>
or
Find by means
(a)
(b)
6.
:
8000 miles.
fo
If
i
represents the simple interest of
i
p
dollars at r
in
n
years.
If the
(b) 1 inch.
square units (square inches.
2 inches. (The number 3.
32 F.
(c)
8000 miles.
If the diameter of a sphere equals d units of length.
denotes the number of degrees of temperature indi8.14 is frequently denoted by the Greek letter TT.
to Centigrade readings:
(b)
Change the following readings
(a)
122 F.
is
H
2
units of length (inches.
.
(c)
5 miles.
(c)
10
feet.
6
Find the volume of a sphere whose diameter equals:
(b)
3
feet.
on $ 500 for 2 years at 4 %. the equivalent reading C on the Centigrade scale may be found by the formula
F
C
y
= f(F-32).
of this formula
:
The The
interest on interest
$800
for 4 years at
ty%.
In algebra.
While
in arithmetic the
word sum
refers only to
the
result obtained
by adding positive numbers. or that
and
(+6) + (+4) = +
16
10.$6) + (-
$4) = (-
$10). 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.
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
(. In arithmetic we add a gain of $ 6 and a gain of $ 4.
Since similar operations with different units always produce analogous results. the fact that a loss of
loss of
+ $2. SUBTRACTION.CHAPTER
II
ADDITION. or positive and negative numbers. Thus a gain of $ 2 is considered the sum of a gain of $ 6 and a loss of $ 4. Or in the symbols of algebra
$4) =
Similarly.
. we call the aggregate value of a gain of 6 and a loss of 4 the sum of the two. however. in algebra this word includes also the results obtained by adding negative. AND PARENTHESES
ADDITION OF MONOMIALS
31.
+ -12.
is 0. 24. '. the average of 4 and 8
The average The average
of 2. c =
4.
19.
4
is
3 J.
.
(-17)
15
+ (-14). if :
a
a
= 2. 5.3.
of 2.
+ (-9). and the sum of the numbers divided by n.
5. find the numerical values of a + b
-f c-j-c?. = 5.
6
6
= 3. 4.
l-f(-2).
22. is 2. 23-26. c = = 5. the one third their sum.
33. subtract their absolute values and
.
21.
(always) prefix the sign of the greater.
23.
EXERCISE
Find the sum
of:
10
Find the values
17. add their absolute values if they have opposite signs. 12.
(_
In Exs.
d = 5. d = 0.
The average
of two
numbers
is
average of three numbers average of n numbers is the
is one half their sum.16
32.
Thus. 10.
-
0.
ELEMENTS OF ALGEBRA
These considerations lead to the following principle
:
If two numbers have the same sign.
of:
20.
18.
$3000 gain. 3.5. and 4. and 3 yards.7.
= -23. 38.
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. 74.
= 22.
29.
'
Find the average of the following
34.
are similar terms. 72.
.
4
F.
42.
:
and
1. 7 a. SUBTRACTION. and 3 a.7.. 09. 6. & = 15.
or 16
Va + b
and
2Vo"+~&.
d=
3. 1..
:
Find the average temperature of Irkutsk by taking the average of the following monthly temperatures 12. and 3 F. 66.
5 a2 &
6 ax^y and
7 ax'2 y..
35.
. .
36.13. 60.
27.
25. 12. c=14.
. $1000 loss.
and
-8
F. 10. or
and
. Find the average temperature of New York by taking the average of the following monthly averages 30. $7000 gain.
^
'
37. $500 loss. 39. = -13.
Similar or like terms are terms which have the same
literal factors. . if his yearly gain or loss during 6 years was $ 5000 gain. 55.
2.
Dissimilar or unlike terms are terms
4 a2 6c and o
4 a2 6c2 are dissimilar terms.
:
48.
&
28. c = 0. 5 and
12.4.
which are not
similar. 32. affected
by the same exponents. 7 yards. \\ Add 2 a.
2. 41. 43. -4. 37. 10.
33. and 3 a. 32.
6.
Find the average of the following temperatures 27 F.
:
34. 10.
. }/ Add 2 a.
3 and 25.
40.
31. 34.
30. 6. 7 a. . -11 (Centigrade).ADDITION. and $4500 gain.
AND PARENTHESES
d = l. Find the average gain per year of a merchant.5. 0.
13.
.3.
sets of
numbers:
13.
-'
1?
a
26.
ELEMENTS OF ALGEBRA
The sum
of 3
of
two similar terms
x2
is
is
another similar term.13 rap
25 rap 2. or
a
6. b wider sense than in arithmetic.
11. in algebra it may be considered b.
:
2 a2.
+ 6 af
. and 4 ac2
is
a
2 a&
-|-
4 ac2.
5Vm + w.
ab
7
c
2
dn
6. While in arithmetic a denotes a difference only.
12(a-f b)
12.
12Vm-f-n.
2(a-f &).
.
12
13
b sx
xY xY 7 #y
7.
2 a&.
13. Algebraic sum.
Vm
-f.
10.
The sum The sum
of a of a
Dissimilar terms cannot be united into a single term.
b
a
-f (
6).
5 a2
.
-f
4 a2.
-3a
.
5l
3(a-f-6). The indicated by connecting and a 2 and
a
is
is
-f-
a2
.
1
\
-f-
7 a 2 frc
Find the sum of
9.
2
.
EXERCISE
Add:
1.
11
-2 a +3a -4o
2.
12
2 wp2 .
The sum
x 2 and
f
x2 . either the difference of a and b or the sum of a and
The sum
of
a.
sum of two such terms can only be them with the -f.
In algebra the word sum is used in a 36.18
35.sign.ii. 7 rap2.
14
.
9(a-f-6).
To
subtract.
3.
6
-(-3) = 8. 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. Subtraction is the inverse of addition. SUBTRACTION. from What 3.g.3. the algebraic sum and one of the two numbers is
The algebraic sum is given. 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
.
NOTE.2. the given number the subtrahend.
From
5 subtract
to
.
This gives by the same method.
Ex. change the sign
of the subtrahend and
add. The student should perform mentally the operation of chang8 2 6 from 6 a 2 fc.
AND PARENTHESES
23
subtraction of a negative
positive number. the other number is required.
5
is
2.
(-
6)
-(-
= . In subtraction.
Ex. Therefore any example in subtraction
different
. called the minvend.ADDITION. In addition.
3 gives
3)
The number which added
Hence. a-b =
x. and their algebraic sum is required.
41.
2.
The
results of the preceding examples could be obtained
by the following
Principle.
1.
a.
From
5 subtract
+ 3. Or in symbols.
.
may
be stated in a
:
5 take form e.
State the other practical examples which show that the number is equal to the addition of a
40. and the
required number the difference.
if
x
Ex.
7. two numbers are given.
From
5 subtract
to
The number which added
Hence.
+b
3.
3 gives 5
is
evidently 8.
& -f
c.a~^~6)]} = 4 a -{7 a 6 b -[. SUBTRACTION.c.c.
AND PARENTHESES
27
SIGNS OF AGGREGATION
43.
4a-{(7a + 6&)-[-6&-f(-2&.
6
o+(
a
+ c) = a =a 6 c) ( 4-.
changed. I.
II.a^6)]
-
}
.
45.
(b
c)
a
=a
6 4-
c. we may begin either at the innermost or outermost.a
-f-
= 4a
sss
7a
12
06
6. Simplify 4 a f
+ 5&)-[-6& +(-25.ADDITION.
a+(b-c) = a +b .
.2 b . Ex.
If there is no sign before the first term within a paren*
-f-
thesis. one occurring within the other.b c = a a
&
-f-
-f.
46.& c
additions
and sub-
+ d) = a + b
c
+ d.6 b -f (. The beginner will find it most convenient
at every step to
remove only those parentheses which contain
(7 a
no others. may be written as follows:
a
-f ( 4. If we wish to remove several signs of aggregation.
tractions
By using the signs of aggregation.g.
66
2&-a + 6
4a
Answer. A sign of aggregation preceded by the sign -f may be removed or inserted without changing the sign of any term.
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. the sign
is
understood.
A
moved
w
may be resign of aggregation preceded by the sign inserted provided the sign of evei'y term inclosed is
E.
and the subtrahend the second. 9.
13.
3. The minuend is always the of the two numbers mentioned.
m and n.4 y* .
a-\-l>
>
c
+
d.
6 diminished
.
y
-f-
8
.
2. SUBTRACTION.
EXERCISES
IN"
ALGEBRAIC EXPRESSION
17
:
EXERCISE
Write the following expressions
I.
12.
The sum
of the fourth powers of a of
and
6.
In each of the following expressions inclose the last three in a parenthesis preceded by the minus sign
:
-27i2 -3^ 2 + 4r/.
The
difference of a
and
6.7-fa.
first.
The square of the difference of a and b.
z
+ d. The product of the sum and the difference
of
m and n.
5 a2
2.
4.
5^2
_ r .
II.
6.
The sum
of tKe squares of a
and
b.2 tf .ADDITION.
10.
terms
5.1.
p + q + r-s.
difference of the cubes of n and m.
of the cubes of
m and
n.
7.
5.
4 xy
7 x* 4-9 x + 2.
The The
difference of the cubes of
m
and
n.
The product The product
m
and
n.
Three times the product of the squares of The cube of the product of m and n. )X
6.
7.
'
NOTE.
m
x
2
4.
3.
Nine times the square of the sum of a and by the product of a and b.
2m-n + 2q-3t. .
8.
EXERCISE
AND PARENTHESES
16
29
In each of the following expressions inclose the last three terms in a parenthesis
:
1.
The sum^)f
m
and
n.
6 is equal to the square of
b.
d.
x cube minus quantity 2 x2 minus 6 x plus
The sum
of the cubes of a. 6.
dif-
of the squares of
a and
b increased
by the
square root of
15. 16.
ELEMENTS OF ALGEBRA
The sum
x.
difference of the cubes of a
and
b divided
by the
difference of
a and
6.30
14.)
. (Let a and b represent the numbers.
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
.
b.
18.
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.
and
c
divided by the
ference of a and
Write algebraically the following statements:
V 17.
A
A
A
1. what force
31
is
produced by tak(
ing away 5 weights from
B ? What therefore is
5)
x(
3) ?
. If the two loads balance. let us consider the and JB. weight at B ?
If the
addition of five 3
plication example.
5. what force is produced by the Ib.
If the two loads
what
What.
If the
two loads balance. applied at let us indicate a downward pull at by a positive sign. therefore. 2.
two loads balance. and forces produced at by 3 Ib.
force is produced
therefore.CHAPTER
III
MULTIPLICATION
MULTIPLICATION OF ALGEBRAIC NUMBERS
EXERCISE
18
In the annexed diagram of a balance.
By what sign is an upward pull at A represented ? What is the sign of a 3 Ib.
3. weight at A ? What is the sign of a 3 Ib.
4. is
by
taking away 5 weights from
A?
5
X 3?
6. what force is produced by the addition of 5 weights at B ? What. weights at A ? Express this as a multibalance. is 5 x ( 3) ?
7. weights.
or
4x3 =
=
(_4) X
The preceding
3=(-4)+(-4)+(-4)=-12. and we may choose any definition that does not lead to contradictions. 9 x
(-
11). 4 multiplied by 3. such as given in the preceding exercise. 5x(-4). make venient to accept the following definition
:
con-
49.
4x(-3)=-12. Multiplication by a positive integer is a repeated addition.
48.
4
x(-8) = ~(4)-(4)-(4)=:-12.
NOTE.4)-(-4) = +
12.32
8.
In multiplying integers we have therefore four cases
trated
illus-
by the following examples
:
4x3 = 4-12.
(-
9)
x (-
11) ?
State a rule by which the sign of the product of
two
fac-
tors can be obtained.
Thus.
ELEMENTS OF ALGEBRA
If
the signs obtained by the
true. This definition has the additional advantage of leading to algenumbers which are identical with those for positive numbers.9) x
11.
Multiplication
by a negative
integer is a repeated
sub-
traction. or plied by 3.
however. To take a number 7 times. a result that would not be obtained by other assumptions.
.
x
11.
the multiplier is a negative number. 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.
(
(.
examples were generally
method of the preceding what would be the values of
(
5x4.
(.4) x
braic laws for negative
~ 3> = -(.4)-(. 4 multi44-44-4 12. thus. (-5)X4. becomes meaningless
if
definition.
9
9. Practical examples^
it
however.
as illustrated in the following example
:
Ex.3 a 2 + a8
a
a = =-
I
1
=2
-f
2
a
4.3 a 2 + a8 .
Check.1.3 ab
2
2 a2
10 ab
-
13 ab
+ 15 6 2 + 15 6 2
Product.
2.3 a
3
2
by 2 a
:
a2 + l. If the polynomials to be multiplied contain several powers of the same letter.
59. are far more likely to occur in the coefficients than anywhere else.
2a-3b a-66 2 a . this method
tests only the values of the coefficients
and not the values of
the exponents. Multiply 2 a .
If
Arranging according to ascending powers
2
a
. 1 being the most convenient value to be substituted for all letters.M UL TIP LICA TION
37
58.
.3
b
by a
5
b.
Multiply 2
+ a -a. Since all powers of 1 are 1. multiply each term of one by each term of the other and add the partial products thus formed.
a2
+ a8 + 3 . however.a
. To multiply two polynomials. the work becomes simpler and more symmetrical by arranging these expressions according to either ascending or descending powers.
Ex.4. the student should
apply this test to every example.a6
=2
by numerical
Examples
in multiplication can be checked
substitution. Since errors.
The most convenient way of adding the partial products is to place similar terms in columns.a6 4 a 8 + 5 a* .2 a2 6 a8
2 a*
*
-
2"
a2
-7
60.
14.
2
10. 9.
sum of the
cross products.
8.
((5a?
(10
12. and are represented as
2 y and 4y 3 x.
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.
or
The student should note
minus
signs. the product of two binomials whose corresponding terms are similar is equal to the product of the first two
terms.
:
25
2.
2
2
+ 2) (10 4-3).
11.
5.
.
(2a-3)(a + 2). (3m + 2)(m-l).
4.
(5a6-4)(5a&-3).
2 2 2 2 (2a 6 -7)(a & +
5).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. (100 + 3)(100 + 4).
(x
i-
5
2
ft
x 2 -3 6 s).
2
(2m-3)(3m + 2).
The middle term
or
Wxy-12xy
Hence in general. 13.&
+ c) = a + tf + c
.
The square
2
(a 4.
65.
3.-f
2 a&
-f
2 ac
+ 2 &c.
that the square of each term is while the product of the terms may have plus always positive. (4s + y)(3-2y).
2
(2x y
(6
2
2
+ z )(ary + 2z ). plus the product of the
EXERCISE
Multiply by inspection
1. plus the
last terms.
7.
(5a-4)(4a-l).
6. ) (2
of a polynomial.
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. in the equation 2 x 0.
82.
83.
y
=
7 satisfy
the equation x
y
=
13.
in
Thus x
12 satisfies the equation x
+
1
13.
.
hence
it
is
an equation
of
condition.
The
first
member
or left side of an equation
is
that part
The secof the equation which precedes the sign of equality.
which
is
true for all values
a2 6 2 no matter what values we assign to a Thus. (a + ft) (a b) and b. The sign of identity sometimes used is = thus we may write
.r
-f9
= 20
is
true only
when
a.
ber
equation is employed to discover an unknown num(frequently denoted by x.
Thus.
(rt+6)(a-ft)
=
2
-
b'
2
.
=11.
.
the
first
member
is
2 x
+
4.
81. ond member or right side is that part which follows the sign of
equality.
A set of numbers which when substituted for the letters an equation produce equal values of the two members.
.CHAPTER V
LINEAR EQUATIONS AND PROBLEMS
79. An identity is an equation of the letters involved. y y or z) from its relation to
63
An
known numbers.
the
80.
x
20. is said to satisfy an equation. An equation of condition is usually called an equation. second member is x
+
4
x
9.
E.
.
89.
.
= bx
expressed by a letter or a combination of
c.
NOTE. 87.
The process
of solving equations depends upon the
:
lowing principles.2. called axioms
1.g.
86.54
84.
one member to another by changing
x + a=. Consider the equation b Subtracting a from both members.
If equals be added
to equals.
3. the quotients are equal.
Transposition of terms.
(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.
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.
If equals be multiplied by equals.
To
solve
an equation
to find its roots. x
I.
2. the products are equal. an^ unknown quantity which satisfies the equation is
a root of the equation.
If equals be divided by equals.
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.
Like powers or
like roots
of equals are equal.
85.
the divisor equals zero.
90.
If equals be subtracted from equals.
the
known quan
x) (x -f 4)
tities are
=
.
5. the remainders are
equal. A
2
a.
2
= 6#-f7. the
sums are
equal.
a.b.
A numerical
equation is one in
which
all
.
ELEMENTS OF ALGEBRA
If
value of the
an equation contains only one unknown quantity.
9
is
a root of the equation 2 y
+2=
is
20.
fol-
A
linear equation is also called a simple equation.e.
A
term may be transposed from
its sign.
4.
but 4 does not equal
5.
Divide 100 into two
12.
x
-f-
y yards cost $ 100
.
9.
3.
6. or 12 7. greater one is g. The difference between two numbers Find the smaller one.
17. one yard will cost
100
-dollars.
Find the greater one.
is
a?
2
is
c?.
7. so that one part Divide a into two parts.
smaller one
16.
13. Hence 6 a must be added
to a to give
5.
What number divided by 3 will give the quotient a? ? What is the dividend if the divisor is 7 and the quotient
?
.58
Ex.
two numbers
and the and the
2
Find the greater one. 11.
one part equals
is 10. 6.
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.
Ex. so that one part
The
difference between
is s.
a. 15. and the smaller one
parts.
1.
$> 100 yards cost one hundred dollars.
5.
10.
EXERCISE
1.
14.
find the cost of one yard. one yard will cost -
Hence
if
x
-f
y yards cost $ 100.
33
2. is b. so that
of c ?
is
p.
If 7
2.
Divide a into two parts.
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. is d.
4.
amount each will then have. 33. Find the sum of their ages
5 years ago.
How many years
A
older than
is
B?
old.
Find
35. A room is x feet long and y feet wide.
and 4
floor of a room that is 3 feet shorter wider than the one mentioned in Ex.
and
c cents.
How many
cents had he left ?
28.
19.
is
A A
is
# years
old. find the of their ages 6 years hence.
59
What must
The
be subtracted from 2 b to give a?
is a.
A
feet wide.
24.
A man
had a
dollars.
34.
numbers
is x. find the
has ra dollars.
26.
How many
cents
has he ?
27. and B's age is y years.
22.
A
dollars.
and spent
5 cents.
How many
cents are in d dollars ? in x dimes ?
A has
a
dollars. rectangular field is x feet long and the length of a fence surrounding the field. The greatest of three consecutive the other two.
20. square feet are there in the area of the floor ?
How many
2 feet longer
29.
smallest of three consecutive numbers
Find
the other two.
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
.
?/
31.
feet wider than the one
mentioned in Ex.
and
B
is
y years old. b dimes. 28.
Find
21.
32.
y years
How
old was he 5 years ago ?
How
old will he be 10 years hence ?
23. and B has n dollars.
sum
If A's age is x years.LINEAR EQUATIONS AND PROBLEMS
18.
Find the area of the Find the area of the
feet
floor of
a room that
is
and 3
30. 28.
If
B
gave
A
6
25.
50. What fraction of the cistern will be second by the two pipes together ?
44. and the second pipe alone fills it in
filled
y minutes.60
ELEMENTS OF ALGEBRA
wil\
36.
-46.
a. how many miles he walk in n hours ?
37.
-.
m is the
denominator.
.
% % %
of 100
of
x. Find a
47.
as
a exceeds
b
by as much
as c exceeds 9.
of m.
how many
how many
miles will
he walk in n hours
38. The first pipe x minutes." we have to consider that in this by statement "exceeds" means minus ( ).
c
a
b
=
-
9. in how many hours he walk n miles ?
40. and "by as much as" Hence we have means equals (=)
95.
miles does
will
If a man walks r miles per hour.
per
Find 5 Find 6
45. What fraction of the cistern will be filled by one pipe in one minute ?
42. he walk each hour ?
39.
A
was 20 years
old.
A
cistern
is
filled
43.
49.
of 4.
b
To express in algebraic symbols the sentence: " a exceeds much as b exceeds 9.
The numerator
If
of a fraction exceeds the denominator
by
3. find the fraction.
Find the
number.
How
old
is
he
now ?
by a pipe in x minutes.
Find x
% %
of 1000.
48. If a man walks n miles in 4 hours.
How many
x years ago
miles does a train
move
in
t
hours at the
rate of x miles per hour ?
41.
The two
digits of a
number
are x and
y. If a man walks 3 miles per hour.
If a
man walks
?
r miles per hour.
A
cistern can be filled
in
alone
fills it
by two pipes.
Find
a.
a exceeds b by c.
The
excess of a over b
is c. 2. a is greater than b by b is smaller than a by
c.
The product
of the
is
diminished by 90 b divided by 7.
3.LINEAR EQUATIONS AND PROBLEMS
Similarly.
by one third of b equals 100.
of a increased
much
8.
6.
-80.
The double
as
7. 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.
4. Four times the difference of a and b exceeds c by as
d exceeds
9.
9.
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.
c.
of x increased by 10 equals
x.
double of a
is
10. thus:
a
b
= c may
be expressed as follows
difference between a
:
The
and
b is
c.
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.
c.
5.
In
many
word
There are usually several different ways of expressing a symbolical statement in words. etc.
8
-b ) + 80 = a
.
80.
EXERCISE
The The double The sum
One
34
:
Express the following sentences as equations
1.
same
result as 7
subtracted from
.
=
2
2
a3
(a
-
80.
third of x equals
difference of x
The
and y increased by 7 equals
a.
of a and 10 equals 2
c.
000.
first
00
x % of the
equals one tenth of the third sum. B.
5x
A sum of money consists of x dollars. 14.
50
is
x % of
15.62
10.
. and
(a)
(6)
A
If
has $ 5 more than B. the
sum
and C's
money
(d)
(e)
will be $ 12.
18. B's. a third sum of 2 x + 1 dollars. B's. A is 4 years older than
Five years ago A was x years old.
->.
and C's age
4
a. of 30 dollars. a second sum.
(a)
(b)
(c)
A is twice as old as B. they have equal
of A's.
as 17
is
is
above
a. express in algebraic symbols
:
-700.
17. (d) In 10 years A will be n years old.
symbols
B. they have equal amounts. 16. In 3 years A will be twice as old as B.
11. In 10 years the sum of A's.
a.
A
gains
$20 and B
loses
$40..
x
is
100
x%
is
of 700. 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.
6
%
of m. and C's ages will be 100. B's age
20.*(/)
(g)
(Ji)
Three years ago the sum of A's and B's ages was 50.
#is5%of450.
m is x %
of n. (e) In 3 years A will be as old as B is now.
(c)
If each
man
gains $500.
A
If
and
B
B together have $ 200 less than C.
ELEMENTS OF ALGEBRA
Nine
is
as
much below a
13. and C have respectively 2 a. 3 1200 dollars.
is
If A's age is 2 x. sum equals $20.
a.
x
4-
If A. pays to C $100. Express as
:
equations of the (a) 5
(b)
(c)
% a%
of the second
(d)
x c of / a % of
4
sum equals $ 90.
amounts. the first sum equals 6 % of the third sura. express in algebraic
3x
:
10.
12.
3
x
+
16
=
x
x
(x
-
p)
Or. number of
yards. The equation can frequently be written by translating the sentence word by word into algebraic symbols in fact. 4 x = 80. The student should note that x stands for the number of and similarly in other examples for number of dollars.
Uniting.
number by x (or another letter) and express the yiven sentence as an equation. x + 15 = 3 x
3x 16
15. the required
.
Uniting. Find A's present age.
Let x
The
(2)
= A's present age. In 15 years A will be three times as old as he was 5 years ago.
Let x = the number. verbal statement (1)
(1) In 15 years
A
will
may be expressed in symbols (2). x= 15.
Dividing. exceeds 40 by as much as 40 exceeds the no.
Simplifying.LINEAR EQUATIONS AND PROBLEMS
63
PROBLEMS LEADING TO SIMPLE EQUATIONS
The simplest kind of problems contain only one unknown number.
3 x or 60 exceeds 40
+ x = 40 + 40.
3z-40:r:40-z.
x+16 = 3(3-5). The solution of the equation (jives the value of the unknown number.
Transposing. x = 20. etc. Three times a certain no.
1. denote the unknown
96.
Transposing.
by 20
40 exceeds 20 by 20. 15. = x x
3x
-40
3x
40-
Or.
-23 =-30.
but
30
=3
x
years.
NOTE. In order to solve them.
Ex.
.
be 30
.
equation is the sentence written in alyebraic shorthand.
In 15 years
10.
Check.
Three times a certain number exceeds 40 by as Find the number. Write the sentence in algebraic symbols.
A
will
Check. the
.
6 years ago he was 10
.
much
as 40 exceeds the number. 2.
number.
Ex. be three times as old as he was 5 years ago.
What number
7
%
of
350?
Ten times the width of the Brooklyn Bridge exceeds 800 ft.
Find the number whose double exceeds 30 by as much
as 24 exceeds the number.
A
number added
number.
35
What number added
to twice itself gives a
sum
of
39?
44. % of
120.
A will
be three times as old as to-da3r
.
.
Find
8.
twice the number plus
7. How many miles per hour does it run ?
.
Find the number. then the
problem expressed in symbols
W
or.
13.
120.
EXERCISE
1.
Find the number whose double increased by 14 equals Find the number whose double exceeds 40 by 10.
Uldbe
66
| x x
5(5 is
=
-*-.
Forty years hence
his present age.
47 diminished by three times a certain number equals 2.
Six years hence a
12 years ago. 11.
How
old
is
man will be he now ?
twice as old as he was
9.64
Ex. by as much as 135 ft.
Let x
3. 14.
3.
to
42 gives a
sum
equal to 7 times the
original
6.
4. How long is the Suez
Canal?
10. Find the number.
5. Four times the length of the Suez Canal exceeds 180 miles by twice the length of the canal.
14 50
is
is
4
what per cent of 500 ? % of what number?
is
12.
ELEMENTS OF ALGEBRA
56
is
what per cent
of 120 ?
= number
of per cent.
Find the width of the Brooklyn Bridge.2.
Hence
40
= 46f.
300
56.
Dividing. A train moving at uniform rate runs in 5 hours 90 miles more than in 2 hours. exceeds the width of the bridge.
000. written in algebraic
symbols. and another which lacked 25 acres of the required number.
The problem consists of two statements I. Vermont's population increased by 180. If a problem contains two unknown quantities.LINEAR EQUATIONS AND PROBLEMS
15.
65
A
and
B
$200.
the second one.
B
How
will loses $100.
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. The sum of the two numbers is 14. One number exceeds the other one by II. Maine's population increased by 510.
numbers (usually the smaller one) by
and use one of the
given verbal statements to express the other unknown number in terms of x. and
B
has $00. B will have lars has A now?
17.
How many
dol-
A has
A
to
$40.
times as
much
as A.
97. During the following 90 years.
F
8.
x.
is
the equation.
One number exceeds another by
:
and their sum
is
Find the numbers. Find
the population of Maine in 1800.
statements are given directly.
. and Maine had then twice as many inhabitants as Vermont. Ill the simpler examples these two
lems they are only implied.
14. which gives the value of
8.000. then dollars has each ? many
have equal amounts of money. If the first farm contained twice as many acres as
A man
number
of acres.
How many dollars
must
?
B
give to
18. while in the more complex probWe denote one of the unknown
x.
1. In 1800 the population of Maine equaled that of Vermont. If A gains A have three times as much
16.
five
If
A
gives
B
$200. The other verbal statement.
how many
acres did he wish to
buy
?
19. and
as
15.
Ex.
A
and
B
have equal amounts of money. two verbal statements must be given.
To
express statement II in algebraic symbols. 8 = 11. A gives B 25 marbles.
= B's number of marbles.
The two statements
I. unknown quantity
in
Then.
/
. A has three times as many marbles as B. 8 the greater number.= The second statement written
the equation ^
smaller number. 26 = A's number of marbles after the exchange. the smaller number. = 3.
Let
x
14
I
the smaller number. I.
26
= B's number of marbles after the exchange.
. Let
x
3x
express one
many as A.
x
3x
4-
and
B
will gain.
terms of the other.
expressed symbols is (14 x) course to the same answer as the first method.
2x
a?
x
-j-
= 6. the greater number. consider that by the
exchange
Hence. = A's number of marbles.
in algebraic
-i
symbols produces
#4a. = 14.
Dividing.
x
x =14
8.
A will lose.
o\
(o?-f 8)
Simplifying. although in general the simpler one should be selected.
.
+
a-
-f
-f
8
= 14. to
Use the simpler statement.
which leads
ot
Ex.66
ELEMENTS OF ALGEBRA
Either statement may be used to express one unknown number in terms of the other.
Another method for solving this problem is to express one unknown quantity in terms of the other by means of statement II viz.
<
Transposing.
x
= 8.
has three times as many marbles as B.
If
A gives
are
:
A
If
II.
and
Let x
= the
Then x -+.
. B will have twice as
viz. the sum of the two numbers is 14. 2.
If
we
select the first one. B will have twice as many as A.
Then.
Statement
x
in
=
the larger number.
Uniting. 25 marbles to B.
15 + 25 = 40.. the number of dimes.
Two numbers
the smaller..
x
from
I. How many are there of each ?
The two statements are I.
by 44. The value of the half
:
is 11.10.550 -f 310. their sum
+ +
10 x 10 x
is
EXERCISE
36
is five
v
v. the number of half dollars. x = 6. have a value of $3. Dividing.
*
'
.25 = 20.
6 times the smaller. B's number of marbles.
(Statement II)
Qx
. 3 x = 45.
*
98.
Simplifying.
x x
+
= 2(3 x = 6x
25
25).240. 11 x = 5. etc.
6 dimes
= 60
= 310.
2.
Uniting.10.
50(11 660 50 x
-)+ 10 x = 310.
Find
the numbers.75. x = 15.
Simplifying.
The sum of two numbers is 42.$3. the
price.LINEAR EQUATIONS AND PROBLEMS
Therefore. Check.5 x .
we
express the statement II in algebraic symbols.
50.
w'3.
differ
differ
and the greater and their sum
times
Two numbers
by
60. 6 half dollars = 260 cents. and the Find the numbers.
greater
is
.
Selecting the cent as the denomination (in order to avoid fractions).
50 x
Transposing. 45 . Eleven coins.
Let
11
= the number of dimes. The number of coins II.
Dividing. A's number of marbles. consisting of half dollars and dimes. 40 x . Find the numbers. x = the number of half dollars.10.
1.. 3. cents.
60.
67
x
-f
25 25
Transposing.
dollars
and dimes
is
$3.
Check.
is 70. .
Uniting.
The numbers which appear
in the equation should
always
be expressed in the same denomination. then.
.
Never add the number number of yards to their
Ex. but 40 = 2 x 20. of dollars to the number of cents.
and four times the former equals five times the latter.
find the
weight of a cubic
Divide 20 into two parts.
and twice the greater exceeds Find the numbers. and twice the altitude of Mt.
How many
hours does the day last ?
. the larger part exceeds five times the smaller part by 15 inches.
of volcanoes in
Mexico exceeds the number
of volcanoes in the United States by 2.
9. McKinley. Everest by 11. and in 5 years A's age will be three times B's.
What
is
the altitude of each
mountain
12.
it
If the smaller
one contained 11 pints more. Mount Everest is 9000 feet higher than Mt.
6. McKinley exceeds the altitude of
Mt.. and B's age is as below 30 as A's age is above 40.
2 cubic feet of iron weigh 1600 foot of each substance.
How many
14 years older than B.
A's age is four times B's.
ELEMENTS OF ALGEBRA
One number
is
six
times another number. Find their ages.
tnree times the smaller by 65.
11.
5. and the
greater increased by five times the smaller equals 22.
3 shall be equal to the other increased by
10.
Two numbers
The number
differ
by
39.68
4.
7. the number.000
feet. How many inches are in each part ?
15. How many volcanoes are
in the
8.
On December
21.
as the larger one. the night in
Copenhagen
lasts 10 hours
longer than the day. Twice 14.
cubic foot of iron weighs three times as much as a If 4 cubic feet of aluminum and
Ibs.
and in Mexico
?
A
cubic foot of aluminum.
would contain three times as
pints does each contain ?
much
13.
Find
Find two consecutive numbers whose sum equals 157.
?
Two
vessels contain together 9 pints.
United
States. What are their ages ?
is
A A
much
line 60 inches long is divided into two parts. one of which increased by
9.
69
If a
verbal statements must be given."
To
x
8x
90
= number of dollars A had after giving $5.
the
the
number
of dollars of dollars of dollars
A
B
C
has. 4 x = number of dollars C had after receiving $10. then three times the sum of A's and B's money would exceed C's money by as much as A had originally. = 48. If A and B each gave $5 to C.
Ex. Tf it should be difficult to express the selected verbal state-
ment
directly in algebraical symbols. bers is denoted by x. III. and C together have $80. II. and the other
of x
problem contains three unknown quantities. B.
If
4x
= 24. they would have 3. The solution gives
:
3x
80
Check.
Let
x
II.
has.
are
:
C's
The three statements
A. 19.
times as
much
as
A.LINEAR EQUATIONS AND PROBLEMS
99. and C together have $80. x = 8. number of dollars A had. = number of dollars B had after giving $5.
I. three One of the unknown num-
two are expressed in terms by means of two of the verbal statements. let us consider the words ** if A and B each gave $ 5 to C.
sum of A's and B's money would exceed much as A had originally.
first
According to
3 x
number
number
and according
to
80
4
x
=
the
express statement III by algebraical symbols. The third
verbal statement produces the equation. B.
1. and 68. and B has three as A. If A and B each gave $5 to C.
8(8
+ 19)
to C. or 66 exceeds 58 by 8.
A
and B each gave $ 5
respectively. original amount.
number
of dollars of dollars
B
C
had. then
three times the
money by
I. try to obtain
it
by a
series of successive steps.
number
had. B has three times as much as A.
.
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.
has.
2 (2 x -f 4) or 4 x
Therefore. The number of cows exceeded the number of horses by
4.140 + (50 x x 120 = 185. 9 -5 = 4 . number of cows. each horse costing $ 90.
28 x 15 or 450
5 horses. 185 a = 925. The number of sheep is equal to twice tho number of horses and
x 4
the
cows together. = the number of dollars spent for horses.
+ 35 (x +-4) -f 15(4z-f 8) = 1185.
1 1
Check.
+
8
90 x
and. and 28 sheep would cost 6 x 90 -f 9 + 316 420 = 1185.70
ELEMENTS OF ALGEBRA
man spent $1185 in buying horses.
and. + 35 x 4. = the number of dollars spent for sheep
Hence statement
90 x
Simplifying.
Uniting.
37
Find three numbers such that the second is twice the first.
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.
The total cost equals $1185. and Ex.
x
-j-
= the
number of horses. according to II. and each sheep $ 15. x -f 4 = 9.
x 35
-f
+
=
+
EXERCISE
1. and the sum of the
. each cow $ 35. and the difference between the third and the second is 15
2. the third five times the first. = the number of dollars spent for cows.
Dividing. according to III.
x
Transposing.
85 (x 15 (4 x
I
+ 4)
+
8)
= the number of sheep. 9 cows. 4 x -f 8 = 28. 28 2 (9 5).
Find three numbers such that the second is twice the 2.
three statements are
:
IT. 2. number of horses. number of sheep.
first. number of cows.
first
the third exceeds the second by and third is 20. Let
then. 90
may
be written.
The
I.
III. cows. The number of cows exceeds the number of horses by 4. 90 x -f 35 x + GO x = 140 20 + 1185. sheep. x = 5.
and the pig iron produced in one year (1906) in the United States represented together a value
. and the sum of the first and third is 36.000 more than Philadelphia (Census 1905). and the third exceeds the
is
second by
5.
twice the
6.
twice as old as B.
7. women.000.
A
is
Five years ago the What are their ages ?
C.LINEAR EQUATIONS AND PROBLEMS
3. The gold.
"Find three
is 4.
The
three angles of any triangle are together equal to
180. the copper.
New York
delphia.
-
4.
13. and children together was 37. If the population of New York is twice that of Berlin.
A
12. In a room there were three times as many children as If the number of women. and 2 more men than women. what is the length of each?
has 3.
v
. and is 5 years younger than sum of B's and C's ages was 25 years.
is five
numbers such that the sum of the first two times the first.
first.000 more inhabitants than Philaand Berlin has 1. and the third part exceeds the second by 10.
the
first
Find three consecutive numbers such that the sum of and twice the last equals 22.
71
the
Find three numbers such that the second is 4 less than the third is three times the second. what is the population of each city ?
8.
v
-
Divide 25 into three parts such that the second part first.
and
of the three sides of a triangle is 28 inches.
first. men.
the third
2.
what are the three angles ?
10.
Find three consecutive numbers whose sum equals
63. how many children
were present ?
x
11.
9. equals 49 inches.
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. the second one is one inch longer than the first. If twice
The sum
the third side. increased by three times the second side.000.
After how many hours will they meet and how
E.
B
many
miles does
A
walk
?
Explanation.000. and quantities
area.e. or time. and A walks at the rate of 3 miles per hour without stopping. of 3 or 4 different kinds. such as length.
Hence
Simplifying. Let x = number of hours A walks.000.
3x
+
4 (x
2)
=
27. 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. we obtain 3 a.
start at the same hour from two towns 27 miles walks at the rate of 4 miles per hour.000 more than that
the copper.
how many
100. width. and distance. number of miles A
x
x
walks.
14. = 5.g. 8 x = 15.
The copper had twice
the value of the gold.
A and B
apart.
of
arid the value of the iron
was $300. number of hours.000.
. and 4 (x But the 2) for the last column.
7
Uniting. 3 and 4. but stops 2 hours on the way. Since in uniform motion the distance is always the product of
rate
and time.
3z + 4a:-8 = 27.
Find the value of each. it is frequently advantageous to arrange the quantities in a systematic manner. speed.
= 35.72
of
ELEMENTS OF ALGEBRA
$ 750.
California has twice as
many
electoral votes as Colorado.
Dividing. i. then x 2 = number of hours B walks. First fill in all the numbers given directly. together.
has each state
?
If the example contains Arrangement of Problems.
and Massachusetts has one more than California and Colorado If the three states together have 31 electoral votes.
and in order to raise the required sum each of the remaining men had to pay one dollar more. A man bought 6 Ibs. but four men failed to pay their shares. How many pounds of each kind did he buy ?
8. and the sum Find the length of their areas is equal to 390 square yards. and its width decreased
by 2 yards.55.
Six persons bought an automobile. and how far will each then have traveled ?
9. and follows on horseback traveling at the rate of 5 miles per hour. and the cost
of silk
of the auto-
and 30 yards of cloth cost together much per yard as the cloth.
Find the dimen-
A
certain
sum
invested at 5
%
%.
Twenty men subscribed equal amounts
of
to raise a certain
money.
If the silk cost three times as
For a part he 7. twice as large.
Find the share of each. but as two of them were unable to pay their share. were increased by 3 yards.
invested at 5 %. What are the
two sums
5. each of the others had to pay
$ 100 more.
A
of each.
A
sets out later
two hours
B
.
1. together bring $ 78 interest.
3. paid 24 ^ per pound and for the rest he paid 35 ^ per pound.
as a
4. the area would remain the same.
Ten yards
$
42. After how many hours will B overtake A. How much did each man subscribe ?
sum
walking at the rate of 3 miles per hour.
mobile.
A sum
?
invested at 4 %.
sum $ 50
larger invested at 4
brings the same interest Find the first sum.
A
If its length
rectangular field is 2 yards longer than it is wide. and a second sum. of coffee for $ 1. how much did each cost per yard ?
6.
sions of the field.
2.74
ELEMENTS OF ALGEBRA
EXERCISE
38
rectangular field is 10 yards and another 12 yards wide. The second is 5 yards longer than the first.
and another train starts at the same time from New York traveling at the rate of 41 miles an hour. and B at the rate of 3 miles per hour.
The
distance from
If a train starts at
. but
A has
a start of 2 miles. 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. how must B walk before he overtakes A ?
walking at the rate of 3 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.
A
and
B
set out
direction. Albany and travels toward New York at the rate of 30 miles per hour without stopping.
walking at the same time in the same If A walks at the rate
of 2
far
miles per hour. how many miles from New York will they meet?
X
12. and from the same point.
An
expression
is integral
and rational with respect
and rational.
a2
to 6.
J Although Va'
In the present chapter only integral and rational expressions
b~
X
V
<2
Ir
a2
b'
2
2
?>
. a. 6.
\-
V&
is
a
rational with respect to
and
irrational with respect
102. 5.
we
shall not. if it does contain
some indicated root of
. if this letter does not occur in any denominator. if.
which multiplied together
are considered factors. if it is integral to all letters contained in it.
a factor of a 2
A
factor is said to be prime. but fractional with respect
103. if it contains
no other
factors (except itself
and unity)
otherwise
.
-f-
db
6
to b.CHAPTER
VI
FACTORING
101. at this
6
2
.
it is
composite. as.
vV
.
irrational.
An
after simplifying.
a-
+
2 ab
+ 4 c2
.
a.
The
factors of
an algebraic expression are the quantities
will give the expression.
expression is rational with respect to a letter. An expression is integral with respect to a letter. it contains no indicated root of this letter
.
this letter.
stage of the work.
The prime
factors of 10 a*b are 2.
+ 62
is
integral with respect to a. consider
105.
76
.
104.
.
POLYNOMIALS ALL OF WHOSE TERMS CONTAIN A COMMON FACTOR
(
mx + my+ mz~m(x+y + z).
Divide
6
a% .
77
Factoring
is
into its factors.
Ex.
An
the process of separating an expression expression is factored if written in the
form of a product. 8) (s-1).3 sy + 4 y8).
Factor
14 a*
W-
21 a 2 6 4 c2
+ 7 a2 6
2
c2
7
a2 6 2 c 2 (2 a 2
.9 x2^ + 12 sy* = 3 Z2/2 (2 #2 .
it
follows
that a 2
.
E.
Factor G ofy 2
.
factors of 12
&V
is
are 3.
The factors
of a
monomial can be obtained by inspection
2
The prime
108.
55.
107.62 + &)(a 2
. 2.3 6a + 1).
2.62
can be
&). x.)
Ex.
Hence
6 aty 2
= divisor x quotient.
TYPE
I.
y.
109.
?/.
01.
it fol-
lows that every method of multiplication will produce a method
of factoring. for this result is a sum. since (a + 6) (a 2 IP factored.9 x if + 12 xy\
2
The
greatest factor
common
2
to all terms
flcy*
is
8
2
xy'
. .
110.
2 4 x + 3) is factored if written (x' would not be factored if written x(x and not a product.
x. dividend
is
2 x2
4
2
1/
.
in the
form
4)
+3. or that a
=
6)
(a
= a .g.
It (a.
1.
Since factoring
the inverse of multiplication.9 x2 y 8 + 12
3 xy
-f
by
3
xy\
and the quotient
But.
or
Factoring examples may be checked by multiplication by numerical substitution. 2.FACTORING
106.
4 x .30 = (a .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*.
m -5m + 6.
is
The two numbers whose product and -6. but only in a limited number
of ways as a product of two numbers.
2.
.
tfa2 -
3..
Ex.
79
Factor a2
-4 x . If q is positive.
If
30 and whose
sum
is
11 are
5
a2
11
a = 1.4 . the student should first all terms contain a common monomial factor.6 = 20. the two numbers
have opposite
signs.a). however.
Ex. and (a .
.11) (a
+
7).5) (a 6).
EXERCISE
Besolve into prime factors :
40
4.
or
77
l.11 a + 30.
. of this type.1 1 a
tf
a 4.G) = . 2 11 a?=(x + 11 a) (a. + 30 = 20.
5. Factor x? .11 a
2
.77 =
(a.
3. Hence z6 -? oty+12 if= (x -3 y)(x*-4 y ).11. If q is negative.
11
7.
as p.FACTORING
Ex.
determine whether
In solving any factoring example.5) (a . it is advisable to consider the factors of q first.
and the greater one has the same sign
Not every trinomial
Ex. or 11 and 7 have a sum equal to 4.
4. a 2 . the two numbers have both the same sign as p. or 7 11.
Factor
+ 10 ax .
Therefore
Check.
2
6. can be factored.
Factor a2
.
but of these only
a:
Hence
2
.
+
112.
Since a number can be represented in an infinite number of ways as the sum of two numbers.
We may consider
1.
11 a2 and whose sum The numbers whose product is and a. Hence fc -f 10 ax
is
10 a are 11 a
-
12 /.
77 as the product of 1 77.
If py? -\-qx-\-r does not contain any monomial factor.
and that they must be negative.
sible
13 x
negative.
11 x
2x. 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. 64 may be considered the
:
product of the following combinations of numbers 1 x 54.
The work may be shortened by the
:
follow-
ing considerations
1. 9 x 6. or
G
114. the signs of the second terms are minus.17 x
2o?-l
V A
5
-
13 a
combination
the correct one. 2 x 27. but the opposite sign. 6 x 9.
If
the factors
a combination should give a sum of cross products.1). Since the first term of the first factor (3 x) contains a 3.e-5
V A
x-1
3xl \/ /\
is
3
a.
3.FACTORING
If
81
we consider that the
factors of -f 5
as
must have
is
:
like signs. we have to reject every combination of factors of 54 whose first factor contains a 3. exchange the
signs of the second terms of the factors. 18 x 3.
The
and
factors of the first term consist of one pair only.
the
If p and r are positive. 2.
Factor 3 x 2
. 54 x 1.
Ex. all pos-
combinations are contained in the following
6x-l
x-5 . If p is poxiliw. 3 x and x. viz. X x 18.5 . then the second terms of
have opposite signs.
.
all
it is not always necessary to write down combinations.13 x + 5 = (3 x . and r is negative. which has the same absolute value as the term qx. 27 x 2.
.31 x
Evidently the
last
2
V A
6.83 x
-f-
54.5) (2 x .
a. Hence only 1 x 54 and 2 x 27 need
be considered. the second terms of the factors have same sign as q. none of the binomial factors can contain a monomial factor.
33
2
7
3
22 3 2
.
121. and GO aty 8 is 6 aty. of
aW. C.
54
-
32
. The H. aW. C. The H. Thus the H. F. C.
3. of 6 sfyz. F.
13 aty
39 afyV. F. F. find by arithmetic the greatest common factor of the coefficients.
The
highest
is
common
factor (IT. F.
Two
common
factor except unity
The H.
are prime can be found by inspection.
3
.) of
two or more
.
5
2
3
.
5
7
34 2s
. C. 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. of
:
48
4.
2. of the algebraic expressions. F. of (a
and (a
+
fc)
(a
4
is
(a
+ 6)
2
.
2
2
. If the expressions have numerical coefficients. F. F. and prefix it as a coefficient to H.CHAPTER
VII
HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE
HIGHEST COMMON FACTOR
120. of a 4 and a 2 b is a2
The H.
C.
.
C.
F.
the algebraic factor of highest degree common expressions to these expressions thus a 6 is the II.
5. of a 7 and a e b 7
.
The student should note
H.
+
8
ft)
and
cfiW is
2
a 2 /) 2
ft)
.
5
s
7
2
5.
8
. C.
122. F. C.
II
2
.
6.
expressions which have no are prime to one another.
24
s
.
15
aW. of
two or more monomials whose factors
.
C.
89
.
25
W. C. 12 tfifz.
-
23 3
.
each set of expressions has
In example ft). C. 60
x^y'
2
.
2.
A
common
remainder. M.
The
lowest
common
multiple (L.
M. If the expressions have a numerical coefficient. M.
The
L.
= (a -f
last
2
&)'
is
(a
-
6) .
4 a 2 &2
_
Hence.
etc. C. M. find by arithmetic their least common multiple and prefix it as a coefficient to the L. C.
C. ory is the L.
6
c6 is
C a*b*c*.
of 4 a 2 6 2 and 4 a 4
-4 a 68
2
. two lowest common multiples.C.
Ex.6)2. but opposite
.
300 z 2 y.
1. C. M.
128. L.
.
126.C.
Hence the L.
Find the L.
2
The The
L.
of 3
aW.
a^c8
3
.
M. C. of as -&2 a2 + 2a&-f b\ and 6-a.
Common
125.
=4 a2 62 (a2 .
2 multiples of 3 x
and 6 y are 30 xz y.M.
M of the algebraic expressions.
Ex. of several expressions which are not completely factored. which
also
signs. C.
&)
2
M.
127.
. of the
general.
M. C.M.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
124.6 3 ). Obviously the power of each factor in the L.) of
two or more
expressions is the common multiple of lowest degree. thus.
of 12(a
+
ft)
and (a
+ &)*( -
is
12(a
+ &)( . To find the L. L. is equal to the highest power in which it occurs in any of the
given expressions. M. of tfy and xy*. C. C. resolve each expression into prime factors and apply the method for monomials.
Find the L.(a + &) 2 (a
have the same absolute value.
NOTE.
and
i
x mx = my y
terms
A
1.
Remove
tor.
rni
Thus
132. If both terms of a fraction are multiplied or divided by the same number) the value of the fraction is not altered. the value of a fraction is not altered by multiplying or dividing both its numerator and its denominator by the same number.
Reduce
~-
to its lowest terms.
the product of two fractions is the product of their numerators divided by the product of their denominators. C.
A
-f-
fraction is
b.
common
6
2
divisors of
numerator and denomina-
and z 8
(or divide the terms
.
The dividend a is called the numerator and the The numerator and the denominator
are the terms of the fraction.
131.
TT
Hence
24
2 z = --
3x
. 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.CHAPTER
VIII
FRACTIONS
REDUCTION OF FRACTIONS
129.
an indicated quotient.
a?.ry ^
by
their H.
and denominators are considered. as 8. but we
In arithmetic.
a b
= ma
mb
.
successively all
2
j/' . F. thus -
is identical
with a
divisor b the denominator. however. etc.
All operations with fractions in algebra are identical
with the corresponding operations in arithmetic.
130.
Ex. Thus.
3) (-!)'
=
.
1. multiplying the terms of
22
.
and the terms of
***.
3 a\ and 4
aW
is
12 afo 2 x2 . by the denominator of each fraction. and
135. Divide the L.
Ex
-
Reduce
to their lowest
common
denominator. take the L.
we have
-M^. of the denominators for the common denominator.
-
of
//-*
2
.
.
and
(a-
8). we have
(a
+ 3) (a -8) (-!)'
NOTE.-1^22
' .
TheL.C. we may use the same process as in arithmetic for reducing fractions to the lowest
common
denominator.
.
Ex.96
134.C.3)O -
Dividing this by each denominator.
we may extend this method
to integral expressions.
Reduce -^-.
-
by 4
6' .by 3 ^
A
2
' .
1).
M.
Since a
(z
-6 + 3)(s-3)O-l)'
6a. and 6rar 3 a? kalr
.
Multiplying these quotients by the corresponding numerators and writing the results over the common denominator.M.D.
2>
.
ELEMENTS OF 'ALGEBRA
Reduction of fractions to equal fractions of lowest common Since the terms of a fraction may be multiplied
denominator.
mon
T denominator.
.
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.
^
to their lowest
com-
The
L.
and
Tb reduce fractions to their lowest common denominator.
+
3).r
2
2
.
multiply each quotient by the corresponding numerator.~16
(a
+ 3) (x. C.
we have
the quotients (x
1).
by any quantity without altering the value of the fraction. =(z
(x
+ 3)(z.M. C.
2.
fractions to integral numbers. and the product of the denominators for the
denominator.
!.
we may extend any
e.
F J Simplify
.
or.
2
a
Ex.)
Ex. each
numerator and denomi-
nator has to be factored. Fractions are multiplied by taking the product of tht numerators for the numerator.
Simplify 1 J
The
expreeaion
=8
6
. (In
order to cancel
common
factors.
expressed in symbols:
c
a
_ac b'd~bd'
principle proved for
b
141.
integer.
Since -
= a.
-x
b
c
=
numerator by
To multiply a fraction by an
that integer. Common factors in the numerators and the denominators should be canceled before performing the multiplication.g. multiply the
142.102
ELEMENTS OF ALGEBRA
MULTIPLICATION OF FRACTIONS
140.
x a + b
obtained by inverting
reciprocal of a fraction
is
the fraction.
* x* -f xy 2
by
x*y
+y
x'
2
3
s^jf\ =
x'
2
x*
. and the principle of division follows
may
be expressed as
145. To divide an expression by a fraction.
1.
144.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
. :
a 4-1
a-b
* See page 272.
.
The reciprocal of ?
Hence the
:
+*
x
is
1
+ + * = _*_.
Divide X-n?/
. Integral or mixed divisors should be expressed in fractional form before dividing. To divide an expression by a fraction. expression by the reciprocal of the fraction.
8
multiply
the
Ex.
The The
reciprocal of a
is
a
1
-f-
reciprocal of J
is
|
|.104
ELEMENTS OF ALGEBRA
DIVISION OF FRACTIONS
143.
The
reciprocal of a
number
is
the quotient obtained by
dividing 1
by that number. invert the divisor and multiply it by the dividend.
~^ = 15
11 x
'
!i^=15. 2 3
..114
35. = the number of minute spaces the minute hand moves
over..
is
36.
and
12
= the number
over.
PROBLEMS LEADING TO FRACTIONAL AND LITERAL EQUATIONS 152. 2.
.
then
= 2 TT#.minutes after x=
^
of
3 o'clock. 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. In how many days can both do it working together ?
If
we denote
then
/-
the required
number
by
1.
12.
Multiplying by
Dividing. = 16^. A can do a piece of work in 3 days and B in 2 days.
100
C. 1.
Find
R in terms of C and
TT. 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.180.20
C.
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.
A would do
each day ^ and
B
j. Ex.
.
x
Or
Uniting.
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 ^.
of minute spaces the
hour hand moves
Therefore x
~ = the number of minute spaces the minute hand moves
more than the hour hand.
Ex.
C
is
the circumference of a circle whose radius
R.
" gives the equation /I).
Clearing. what is
the rate of the express train
?
180
Therefore.
in
Then
Therefore.
or 1J.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.
180
Transposing.
But
in
uniform motion Time
=
Distance
. 3.
the rate of the express train.
Solving.
= the
x
part of the
work both do
one day. and the statement. 4x = 80. The speed of an express train is $ of the speed of an If the accommodation train needs 4 accommodation train. u The accommodation train needs 4 hours more than the express train. hours more than the express train to travel 180 miles.
Ex. the required
number
of days.
32
x
= |."
:
Let
x -
= the
required
number
of days.
= 100 + 4 x.
Explanation
:
If
x
is
the rate of the accommodation train. then
Ox
j
5
a
Rate Hence the rates can be expressed.
fx
xx*
=
152
+4
(1)
Hence
=
36
= rate
of express train.
-|
Find their present ages. How
did the
much money
man
leave ?
11.
The sum
10 years hence the son's age will be
of the ages of a father and his son is 50. How much money had he at first?
12
left
After spending ^ of his
^ of his money and $15.
of his present age. a man had How much money had he
at first?
.
Find two consecutive numbers such that
9.
Find A's
8.
its
Find the number whose fourth part exceeds part by 3.
3.
fifth
Two numbers
differ
2.
Twenty years ago A's age was |
age.
ceeds the smaller by
4. which was $4000.
money and $10. and one half the greater Find the numbers.
are the
The sum of two numbers numbers ?
and one
is
^ of the other.
length in the ground.
9
its
A
post
is
a fifth of
its
length in water. one half of What is the length
of the post ?
10
ter. to his daughand the remainder.
Two numbers
differ
l to s of the smaller. to his son.
make
21. and found that he had \ of his original fortune left. and J of the greater Find the numbers.
is
equal
7. A man lost f of his fortune and $500.
is oO.
A man left ^ of his property to his wife.
ex-
What
5.
by 3.
J-
of the greater
increased by ^ of the smaller equals
6.
by 6.
and 9
feet above water.
Find a number whose third and fourth parts added
together
2. and of the father's age.116
ELEMENTS OF ALGEBRA
EXERCISE
60
1.
At what time between 7 and
8 o'clock are the hands of
?
a clock in a straight line and opposite
18.
A can
A
can do a piece of work in 2 days. Ex. If the accommodation train needs 1 hour more than the express train to travel 120 miles.
A has invested capital
at
more
4%. ounces of gold and silver are there in a mixed mass weighing
20 ounces in
21.FRACTIONAL AND LITERAL EQUATIONS
13. Ex.)
At what time between 7 and 8
o'clock are the
hands of
a clock together ?
17.
air.) (
An express train starts from a certain station two hours an accommodation train.
?
In
how many days can both do
working together
23. A can do a piece of work in 4 clays. and B in 4 days. and B In how many days can both do it working together
in
?
12 days. and
it
B in 6 days.
How much money
$500?
4%.
At what time between 4 and
(
5 o'clock are the hands of
a clock together?
16.
and losing
1-*-
ounces when weighed in water?
do a piece of work in 3 days.
after
rate of the latter ?
15.
investments. what is the
14. In how many days can both do it working together ? ( 152.
A man
has invested
J-
of his
money
at
the remainder at
6%. Ex. 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 ounce of gold when weighed in water loses -fa of an How many ounce. and an ounce of silver -fa of an ounce.
. 2.)
22. what is the rate of the express train? 152. 3.
152. 1.
at 4J % and P> has invested $ 5000 They both derive the same income from their How much money has each invested ?
20.
117
The speed
of an accommodation train
is
f of the speed
of an express train. and has he invested if
his animal interest therefrom is
19. ^ at 5%.
Find three consecutive numbers whose sum
Find three consecutive numbers whose sum
last
:
The
two examples are
special cases of the following
problem 27.
we
obtain the equation
m m
-. B in 30. Hence.
B in 5.
. B in 16. A in 6.g.
ELEMENTS OF ALGEBRA
The
last three questions
and their solutions differ only two given numbers. it is possible to solve all examples of this type by one example. n x
Solving. Find three consecutive numbers whose sum equals m. therefore. 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.
26. Ex.e.
is 42. if
B
in 3 days.
25. is 57. by taking for these numerical values two general algebraic numbers. Find the numbers if m = 24 30.009 918. and n = 3.
. A in 6.414. B in 12. and apply the
method of
170. . e. Then
ft
i. m and n. 2. is A can do a piece of work in m days and B in n days.
6
I
3
Solve the following problems
24.118
153.= -. Answers to numerical questions of this kind may then be found by numerical substitution.
make
it
m
6
A can do this work in 6 days Q = 2.
:
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. A in 4. The problem to be solved.
To
and
find the numerical answer. 3.=
m
-f-
n
it
Therefore both working together can do
in
mn
-f-
n
days.
they can both do
in 2 days.
The
one:
31.
is ?n
. and how many miles does each travel ?
32.001.000. 2 miles per hour.
33. (c) 16. 2 miles per hour. and the second 5 miles per hour. 3J miles per hour.
squares
30. (b) 35 miles. d miles the first traveling at the rate of m. two pipes together ? Find the numerical answer.
by two pipes in m and n minutes In how many minutes can it be filled by the respectively.FRACTIONAL AND LITERAL EQUATIONS
28.
same hour from two towns. the rate of the
first. Find the side of the square. 5 miles per hour. If each side of a square were increased by 1 foot. respectively.
119
Find two consecutive numbers the difference of whose
is 11.
squares
29. the area would be increased by 19 square feet.
4J-
miles per hour. the
Two men start at the same time from two towns. 3 miles per hour.
. (b) 8 and 56 minutes. respectively (a) 60 miles.
Find two consecutive numbers -the difference of whose
is 21.
is (a)
51. 88 one traveling 3 miles per hour. After how many hours do they rate of n miles per hour. if m and n are. and how many miles does each travel ? Solve the problem if the distance. (b) 149.
:
(c)
64 miles.
meet.
34.
A cistern can
be
filled
(c)
6 and 3 hours. After how many hours do they meet. solve the following ones Find two consecutive numbers the difference of whose squares
:
find the smaller number. (a) 20 and 5 minutes.
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.
and
the rate of the second are.721. the second at the apart. (d) 1.
Two men
start at the
first
miles
apart.
g.
The
ratio of
first
dividing the
two numbers number by the
and
:
is
the quotient obtained by
second.5.
antecedent.) The ratio of 12 3 equals 4. the antecedent."
we may
write
a
:
b
= 6.CHAPTER X
RATIO AND PROPORTION
11ATTO
154.
the symbol
being a sign of division.
Simplify the ratio 21 3|.
Ex.
term of a ratio
a
the
is
is
the antecedent. 6 12 = . etc. b is the consequent.
b
is
a
Since a ratio
a
fraction. all principles
relating
to
fractions
if its
may
be af)plied to ratios.
" a Thus.
:
A somewhat shorter way
would be to multiply each term by
120
6. the second
term the consequent.
b.
The
first
156.
.
1. a ratio
is
not changed
etc.
A
ratio
is
used to compare the magnitude of two
is
numbers.
is
numerator of any fraction
consequent.
The
ratio -
is
the inverse of the ratio -.
b. the denominator
The
the
157.
Thus the
written a
:
ratio of a
b
is
.
E.
:
:
155. instead of writing
6 times as large as
?>.
In the ratio a
:
ft. 158.or a *
b
The
ratio is also frequently
(In most European countries this symbol is employed as the usual sign of division.
terms are multiplied or divided by the same number.
equal
2.
proportional between a
and
c.
27 06: 18 a6. either mean the mean proportional between the first and the last terms.
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.
61
:
ratios
72:18.
$24: $8.
3:1}.
Simplify the following ratios
7.
J:l.
:
a-y
.
Transform the following
unity
15.
62:16.
4|-:5f
:
5.
11.
= |or:6=c:(Z are
The
first
160.
:
is
If the means of a proportion are equal.
7|:4 T T
4
.
17. and the last term the third proportional to the first and second
161. the second
and fourth terms of a proportion are the and third terms are the means.
16 x*y
64 x*y
:
24 48
xif.
terms.
9.
A
proportion
is
a statement expressing the equality of
proportions.
3.
In the proportion a b
:
=
b
:
c.
159.
b is the
mean
b. b and c the means.
4. b.
1. a and d are the extremes.
3:4.
16a2 :24a&. The last
first three.
5 f hours
:
2.
6.RATIO
Ex.
7f:6J.
term
is
the fourth proportional to the
:
In the proportion a b = c c?.
and
c
is
the third proportional to a and
. The last term d is the fourth proportional to a.
3
8.
extremes.
16. and c.
8^-
hours.
:
1.
18.
12.
10.
:
ratios so that the antecedents equal
16:64.
two
|
ratios.
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.
q~~ n
.
ccm. a b
:
bettveen two
numbers
is
equal to
the square root
Let the proportion be
Then Hence
6
=b = ac.
ELEMENTS OF ALGEBRA
Quantities of one kind are said to be directly proper
tional to quantities of another kind.30 grams.'*
Quantities of one kind are said to be inversely proportional to quantities of another kind.
:
:
directly proportional
may say.
The mean proportional
of their product.
pro-
portional. i.
163. and we
divide both
members by
we have
?^~ E. then 8 men can do it in 3 days.
2
165. If the product of two numbers is equal to the product of two other numbers^ either pair may be made the means. or 8 equals the inverse ratio of 4 3.
!-.
briefly.
ad =
be.
If
(Converse of
nq.e.
Clearing of fractions. of a proportion. Hence the weight of a mass of iron is proportional to its volume.122
162.
:
c.
164. of iron weigh . then G ccm. are
: : :
inversely proportional. is equal to the ratio of the corresponding two
of the other kind. = 30 grams 45 grams. Instead of u
If 4
or 4 ccm.) b = Vac. Hence the number of men required to do some work.
t/ie
product of the means
b
is
equal
to the
Let
a
:
=c
:
d. if the ratio of any two of the first kind. and the
other pair the extremes.
If 6 men can do a piece of work in 4 days.)
mn = pq. of iron weigh 45 grams. 3 4. In any proportion product of the extremes. " we " NOTE.__(163. 6 ccm. and the time necessary to do it.
(e) The distance traveled by a train moving at a uniform rate.inches long represents
map corresponds to how many miles ?
The
their radii.
areas of circles are proportional to the squares of If the radii of two circles are to each other as
circle is
4
:
7.
1
(6) The circumferences (C and C ) of two other as their radii (R and A").126
54.
what
58. (c) The volume of a body of gas (V) is
circles are to
each
inversely propor-
tional to the pressure (P).
and the area of the smaller
is
8 square inches. the volume of a
The temperature remaining
body of gas inversely proportional to the pressure. and the
:
total cost.
57.
ELEMENTS OF ALGEBEA
State the following propositions as proportions : T (7 and T) of equal altitudes are to each. and the area
of the rectangle.
(d)
The sum
of
money producing $60
interest at
5%.
the area of the larger? the same.
(b)
The time a
The length
train needs to travel 10 miles. The number of men (m) is inversely proportional to the number of days (d) required to do a certain piece of work. under a pressure of 15 pounds per square inch has a volume of
gas
is
A
16 cubic
feet. and the time.
(c)
of a rectangle of constant width. othei
(a) Triangles
as their basis (b
and
b').
the squares of their radii
(e)
55.
A
line 11 inches long
on a certain
22 miles.
(d)
The
areas
(A and
A') of two circles are to each other as
(R and R').
and the speed
of the train.
56.
A
line 7^. and
the time necessary for it. State whether the quantities mentioned below are directly or inversely proportional (a) The number of yards of a certain kind of silk.
What
will be the
volume
if
the pressure
is
12 pounds per square inch ?
.
18 x = 108.
is
A line AB. AB = 2 x.
Hence
or
Therefore
Hence
and
= the first number. 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.
x=2. so that
Find^K7and BO. = the second number. When a problem requires the finding of two numbers which are to each other as m n. 2.000
168. produced to a point C.
Therefore
7
=
14
= AC. 11 x = 66 is the first number.
as 11
Let
then
:
1.
11
x
x
7
Ex. 4 inches long. it is advisable to represent these unknown numbers by mx and nx. 11 x -f 7 x = 108. x = 6.
4
'
r
i
1
(AC): (BO) =7: 5.
2 x
Or
=
4.
Let
A
B
AC=1x. 7 x = 42 is the second number.
Divide 108 into two parts which are to each other
7.
.
Then
Hence
BG = 5 x.
:
Ex.RATIO AND PROPORTION
69.
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).
How many gen. Brass is an alloy consisting of two parts of copper and one part of zinc.
:
Divide 39 in the ratio 1
:
5.
3.
7.
What
are the parts ?
5.000.)
. and 15 inches.
How many
grams of hydrogen are contained in 100
:
grams
10.
How many
7.
:
197. 12.
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 79.128
ELEMENTS OF ALGEBRA
EXERCISE
63
1. 6.
11. 2. How
The
long are the parts ? 15. find the number of square miles of land and of water. If c is divided in the ratio of the other two.
Gunmetal
tin.
Divide 20 in the ratio 1 m. and the longest is divided in the ratio of the other two.
12. The three sides of a triangle are respectively a.
14. cubic feet of oxygen are there in a room whose volume is 4500
:
cubic feet?
8. what are
its
parts ?
(For additional examples see page 279.
m
in the ratio x:
y
%
three sides of a triangle are 11.
:
Divide a in the ratio 3
Divide
:
7. and c inches.
A line 24 inches
long
is
divided in the ratio 3
5.
Divide 44 in the ratio 2
Divide 45 in the ratio 3
:
9.
The
total area of land is to the total area of
is
water as
7 18. 9.
:
4.
13.000 square miles. Water consists of one part of hydrogen and 8 parts of
If the total surface of the earth
oxygen. How many ounces of copper and zinc are in 10 ounces of brass ?
6.
of water?
Divide 10 in the ratio a
b.
Hence
2s -5
o
= 10 _ ^
(4)
= 3. which substituted in (2) gives y both equations are to be satisfied by the same Therefore.
An
equation of the
first
unknown numbers can be the unknown quantities. expressing a y.
y
(3)
these
unknown numbers can be found. From (3) it follows y 10 x and since
by the same values of x and
to be satisfied
y. such as
+ = 10. there is only one solution. y = 1.e.CHAPTER XI
SIMULTANEOUS LINEAR EQUATIONS
169.
However. is x = 7.y=--|. the equation is satisfied by an infinite number of sets Such an equation is called indeterminate.
Hence.
If
satisfied
degree containing two or more by any number of values of
2oj-3y =
6.-L
x
If
If
= 0.
the equations have the two values of
y must be equal.
The
root of (4)
if
K
129
. values of x and y.-.
a?
(1)
then
I.
2 y = . if
. x = 1. y =
5
/0 \ (2)
of values. =.
if
there
is
different relation
between x and
*
given another equation. etc.
Solve
-y=6x
6x
-f
Multiply (1) by
2. 30 can be reduced to the same form -f 5 y Hence they are not independent. the last set inconsistent. and 3 x + 3 y =. Independent equations are equations representing different relations between the unknown quantities such equations
. Any set of values satisfying 5 x + 6 y = 60 will also satisfy the equation 3 x -f.
are simultaneous equations.
By By
Addition or Subtraction.
x
-H
2y
satisfied
6 and 7 x 3y = by the values x = I. 3. y
I
171.
ELIMINATION BY ADDITION OR SUBTRACTION
175.
to
The two methods
I.
cannot be reduced to the same form.
(3)
(4)
Multiply (2) by
-
Subtract (4) from (3).
~ 50. same relation.X.
4y
.130
170. for they express the x -f y 10.
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.26.
21 y
.
The process of combining several equations so as make one unknown quantity disappear is called elimination.
= .
6x
. 26 y = 60.
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.
unknown
quantity.
viz.
Therefore. The first set of equations is also called consistent.
Substitution. y = 2.
172. 6 and 4 x y not simultaneous.
E. for they cannot be satisfied by any value of x and y.
174. for they are 2 y = 6 are But 2 x 2.
of elimination
most frequently used
II.24.3 y = 80.
y
*
z
30.
+
396
= 521.2/
2/
PROBLEMS LEADING TO SIMULTANEOUS EQUATIONS
183.
. and if 396 be added to the number.y
125
(3)
The solution of these equations gives x Hence the required number is 125.
x
:
z
=1
:
2. z + x = 2 n.
(
99. the first and the last digits
will be interchanged.)
it is advisable to represent a different letter.
M=i.SIMULTANEOUS LINEAR EQUATIONS
143
x
29.
1
=
2.
Let
x
y z
= the
the digit in the hundreds' place.
.
unknown quantity by
every verbal statement as an equation.
to express
it is difficult
two of the required
digits in
terms
hence we employ 3
letters for the three
unknown
quantities. Problems involving several unknown quantities must contain.
2
= 6. however. the number. either directly or implied.
(1)
100s
+ lOy + z + 396 = 100* + 10y + x.
Check. The digit in the tens' place is | of the sum of the other two digits. The sum of three digits of a number is 8.
= 2 m. y
31.
# 4. Simple examples of this
kind can usually be solved by equations involving only one
unknown
every
quantity.
=
l.
and Then
100
+
10 y
+z-
the digit in the units' place.
2
= 1(1+6).
Ex. 1 digit in the tens place. and to express
In complex examples.
The
three statements of the problem can
now be
readily expressed in
.
+2+
6
= 8.
symbols:
x
+
y
+z-
8.
Find the number. 1.
Obviously
of the other
. + z = 2p. as many verbal statements as there are unknown quantities.
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.
ELEMENTS OF ALGE13KA
If both numerator and denominator of a fraction be
.
2.
we
obtain.
Find the
fraction.
x 3
= 24.
(1) (2)
12.
(3) C4)
=
24 miles. the fraction is reduced to | and if both numerator and denominator of the reciprocal of the fraction be dimin-
ished by one. who travels 2 miles an hour faster than B.
direction.
3. x 3x-4y = 12. the fraction
Let and
then y
is
reduced to
nurn orator.
2. starts 2 hours after B and overtakes A at the same How many miles has A then traveled? instant as B.
8
= xy + x xy = xy -f 3 x 2 y = 2.
6
x 4
= 24.
=
Hence the
fraction
is
f.
increased by one.
Or
(4)-2x(3).
= the
fraction.
Since the three
men
traveled the
same
distance.
3+1 5+1
4_2. the distance traveled by A.
Ex.144
Ex. From (3)
Hence xy
Check.
.
x
y
= the = the
x
denominator
.
By
expressing the two statements in symbols. y = 3. = 8. C.
4
x
= 24. B.
xy
a:
2y 4y
2. 5_
_4_
A.
+
I
2
(1)
and
These equations give x
Check.
3
xand y
I
1
(2)
5.
The sum
of the first
sum
of the three digits of a number is 9. the fraction equals . Find the number. and the numerator increased by 4.
to
L
<>
Find the
If the
numerator and the denominator of a fraction be If 1 be subtracted from increased by 3.
part of their difference equals
4.
added to the numerator of a fraction. and twice the numerator What is the fracincreased by the denominator equals 15.
If 27 is
10. A fraction is reduced to J.
5.
the
number
(See Ex.
The sum
18
is is
and
if
added
of the digits of a number of two figures is 6. and the fourth 3.
1. the Find the fraction. fraction is reduced to \-. and four times the first digit exceeds the second digit by 3.
If the
numerator of a fraction be trebled. Find the fraction.
If 4 be
Tf 3 be
is J. the fraction is reduced
fraction.)
added to a number of two digits. Find the number.
2.
Find the numbers. the digits will be interchanged.
Five times a certain number exceeds three times another 11.
number by
the
first
3.
tion ?
8. If
9 be added to the number.
?
What
9.
6. Find the numbers. the last two digits are interchanged.
183.
Half the sum of two numbers equals 4. and the second one increased by 5 equals twice
number. Find the numbers. and the second increased by 2 equals three times the first. it is reduced to J.}. its value added to the denominator.SIMULTANEOUS LINEAR EQUATIONS
EXERCISE
70
145
1. If the denominator be doubled. if its numerator and its denominator are increased by 1. and the two digits exceeds the third digit by 3. to the number the digits will be interchanged. the value of the fraction is fa. Four times a certain number increased by three times another number equals 33. and
its
denomi-
nator diminished by one.
.
7. both terms.
the annual interest would be $ 195.
What was
the
sum and
rates
est
The sums of $1500 and $2000 are invested at different and their annual interest is $ 190.146
ELEMENTS OF ALGEBRA
11. partly at 5% and partly at 4%. and partly at 4 %. What was the amount of each investment ?
A man
%
5%.
A
sum
of $10.
14. Twice A's age exceeds the sum of B's and C's ages by 30.
partly at
5 %. 5 %.grams. and B's age is \ the sum of A's and C's ages. and the 5% investment brings $15 more interest than the 4 % investment.
13. If the sum of
how
old
is
each
now ?
at
invested $ 5000. a part at 6 and the remainder bringing a total yearly interest of $260.
and 5 years ago
their ages is 55.
much money
is
invested at
A sum
of
money
at simple interest
amounted
in 6 years
to $8000. If the rates of interwere exchanged. and in 5 years to $1125.
and
money and
17. How 6 %. respectively ?
16. Three cubic centimeters of gold and two cubic centimeters of silver weigh together 78 grains. and The 6 investment brings $ 70 more interest than the 5
%
%
4%
investments together.
19. the rate of interest ?
What was
the
sum
of
A sum
of
money
at simple interest
amounted
in 2 years
to $090. bringing a total yearly interest of $530.000
is
partly invested at
6%. 12. now.
in 8 years to $8500.
Ten years ago A was B was as
as old as
B
is
old as
will be 5 years hence .
. Two cubic centimeters of gold and three cubic centimeters of silver weigh
together 69 J. What was the amount of each investment ?
15. Ten years ago the sum of their ages was 90. Find their present ages. and 4 %. Find
the rates of interest. Find the weight of one cubic centimeter of gold and one cubic centimeter of silver. A man invested $750. the rate of interest?
18.
. $ 50 for each cow. 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. The number of sheep was twice the number of horses and cows together. BD = HE. If one angle exceeds the sum of the other two by 20. and $15 for each sheep.
An C touch ing the sides in D. what are the angles of the triangle ?
22.
.SIMULTANEOUS LINEAR EQUATIONS
147
20. and CF?
is
a circle
inscribed in the
7<7.
A
r
^
A
circle is inscribed in triangle
sides in D.
Find the parts of the
ABC touching the three sides if AB = 9. and GE = CF.
In the annexed diagram angle a = angle b. angle c = angle d. and angle e angle/.
the length of
NOTE. but if A would double his pace.
the three sides of a triangle E. then AD = AF. If angle ABC = GO angle BAG = 50. and AC = 5 inches. and e.
respectively.
is
the center of the circum-
scribed circle. three
AD = AF. and F.
E. and sheep. and their difference by GO .
Find their
rates of walking. c.
25. The sum of the 3 angles of a triangle is 180. ED = BE.
triangle
Tf
AD.
On
/). and F. A farmer sold a number of horses.
points. B find angles a.
It takes
A two hours
longer
24 miles. BE.
24.
1
NOTE. and CE If AB = G inches. cows. andCL4 = 8. and angle BCA = 70. and F '(see diagram).
BC=7. what is
that
=
OF. are taken so
ABC. for $ 740. receiving $ 100 for each horse. BC = 7 inches.
23.
The
abscissa
is
usually denoted by
line XX' is called the jr-axis.CHAPTER
XII*
GRAPHIC REPRESENTATION OF FUNCTIONS AND
EQUATIONS
184.
(3. YY' they-axis. jr.
The
of
Coordinates.
(2.
?/. B.
186.
. and
respectively represented
Dare
and
by
(3 7 4). PM.
PN.
-3). then
the position of point is determined if the lengths
of
P
P3f and
185. and whose ordinate is usually denoted by (X ?/). Thus the points A. and PN _L YY'.
is
The point whose abscissa is a. hence
The
coordinates lying in opposite directions are negative. and r or its equal OA is
. Abscissas measured to the riyht of the origin.
(2.
or its equal
OM. the ordinate by ?/.
lines
PM
the
and P^V are
coordinates
called
point P.
two fixed straight lines XX' and YY' meet in at right angles. and ordinates abore the x-axis are considered positive .
is the abscissa.
* This chapter
may
be omitted on a
148
reading.
It'
Location of a point.
the ordinate of point P. and point the origin.
2).
first
3).
PN are given. and PJ/_L XX'. (7..
=3?
is
If a point lies in the avaxis. (4. the mutual dependence of the two quantities may be represented
either by a table or
by a diagram.4).
12.
(4.GRAPHIC REPRESENTATION OF FUNCTIONS
The
is
149
process of locating a point called plotting the point.
.and(l.
3. (4.
Plot the points:
(4.
2. 3). -!).
and measure
their
distance. 11.3).
6.
two variable quantities are so related that
changes of the one bring about definite changes of the other.
(-1.
(-3.
6. (See diagram on page 151. 0). i.
0). Graphic constructions are greatly facilitated by the use of cross-section paper.
Graphs.
71
2). (0.
8.
2J-).
1). paper ruled with two sets of equidistant and parallel linos intersecting at right angles. 1). -2).e.
4).
What
is
the locus of
(a?.
Draw
the triangle whose vertices are respectively
(-l.
-2). (4.
Plot the points
(6.
Plot the points: (-4.
.
What
Draw
is
the distance of the point
(3.
the quadrilateral whose vertices are respectively
(4. (-2.2). -3). 0).
What
are the coordinates of the origin ? If
187.
(-5.
4.
Plot the points
:
(0. -4). (-4. (0.
4)
and
(4.
Where do Where do
Where do
all
points
lie
whose ordinates
tfqual
4?
9. (-4. 3).
whose coordinates are given
NOTE.)
EXERCISE
1.
4)
from the
origin ?
7.1).(!. which of its coordinates
known ?
13.
all all
points
points
lie
lie
whose abscissas equal zero ?
whose ordinates equal zero?
y) if y
10. 0).
from January 1 to December 1. B. but it indicates in a given space a great many more
facts than a table.
ically
each representing a temperature at a certain date. D. A.
15.
188.
ABCN
y
the so-called graph of
To
15
find
from the diagram the temperature on June
to be 15
. may be represented graphby making each number in one column the abscissa.
. we obtain an uninterrupted sequence
etc. or the curved line the temperature.
we meas1
. and the corresponding number in the adjacent column the ordinate of a point.
may be found
on Jan. and the amount of gas subjected to pressures from
pound
The same data. however. in like manner the average temperatures for every value of the time.
A graphic
and
it
impresses upon the eye
all
the peculiarities of
the changes better and quicker than any numerical compilations.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. C.
ure the ordinate of F.
By representing
of points. Thus the first table produces 12 points.
1.
Thus the average temperature on May
on April 20..
representation does not allow the same accuracy of results as a numerical table. 10
.
concise representation of a
number
of numerical data
is
required.
physician. and to deduce general laws therefrom. as the prices and production of commodities. uses them. (c) January 15. Daily papers represent ecpnoniical facts graphically.
. the
matics.
EXERCISE
From the diagram
questions
1. the merchant. The engineer. etc. (b) July 15. (d) November 20.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. the graph
is
applied.
:
72
find approximate answers to the following
Determine the average temperature of New York City on (a) May 1. Whenever a clear. the rise and fall of wages.
(c)
the average temperature oi 1 C.?
is
is
the average temperature of
New York
6.. At what date is the average temperature lowest? the lowest average temperature ?
5. 1 to Oct.
During what months
above 18 C. 1?
11
0.
Which month
is
is
the coldest of the year?
Which month
the hottest of the year?
16.
on
1 to
the
average.
15.
How
much.
When
What
is
the temperature equal to the yearly average of
the average temperature from Sept. (freezing
point) ?
7.
When
the average temperature below
C.. ?
-
3.
1 ?
does
the
temperature
increase from
11.
How much warmer
1 ?
on the average
is it
on July 1 than
on
May
17.
ELEMENTS OF ALGEKRA
At what date
(a) G
or dates
is
New York
is
C.
is
10. (1)
10
C.. At what date is the average temperature highest the highest average temperature?
?
What What
is
4. ?
9.152
2. from what date to what date would it extend ?
If
. (d) 9 0.
is
ture
we would denote the time during which the temperaabove the yearly average of 11 as the warm season.
During what month does the temperature decrease most
rapidly ?
13.
June
July
During what month does the temperature increase most
?
rapidly
12.
During what month does the temperature change least?
14.
From what
date to what date does the temperature
increase (on the average)?
8.
transformation of meters into yards.GRAPHIC REPRESENTATION OF FUNCTIONS
18.09 yards.
Hour
Temperature
.
NOTE.
a temperature chart of a patient.
Represent graphically the populations
:
(in
hundred thou-
sands) of the following states
22.
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. From the table on page 150 draw a graph representing the volumes of a certain body of gas under varying pressures.
Draw
a
graph for the
23.
19. in a similar manner as the temperature graph was applied in examples 1-18.
Draw
.
20. One meter equals 1. Construct a diagram containing the graphs of the mean temperatures of the following three cities (in degrees Fahren-
heit)
:
21.
26. represent his daily gain (or loss). 2 . if each copy sells for $1.
3. and $.
to 20 Represent graphically the weight of iron from cubic centimeters.
then
C
irJl.
A
10 wheels a day.
(Assume ir~
all circles
>2
2
. the value of a of this quantity will change.
28.. e.50. amount to $8.g.
190.
from
R
Represent graphically the = to R = 8 inches. if 1 cubic centimeter of iron weighs 7. if he sells 0.
If
dealer in bicycles gains $2 on every wheel he sells.
.50.
29. 9.
2. An expression involving one or several letters a function of these letters. Represent graphically the cost of butter from 5 pounds if 1 pound cost $.
4. 1 to 1200 copies. 3.
The
initial cost of
cost of manufacturing a certain book consists of the $800 for making the plates.
function
If the value of a quantity changes.
+7
If
will
respec-
assume the values 7. 2 8 y' + 3 y is a function of x and
y. x
7 to 9.
books from
for printing.
x increases will change gradually from
13.5
grams.. etc. Represent graphically the distances traveled by a train in 3 hours at a rate of 20 miles per hour. 2 x -f 7 gradually from 1 to 2.154
24. etc. gas. binding.) On the same diagram represent the selling price of the books.
Show
graphically the cost of the
REPRESENTATION OF FUNCTIONS OF ONE VARIABLE
189.50 per copy
(Let 100 copies = about \. if x assumes
successively the
tively
values
1.
ELEMENTS OF ALGEBRA
If
C
2
is
the circumference of a circle whose radius
is J2..
2
is
called
x
2 xy
+ 7 is a function of x.
to
27.
x*
x
19. the daily average expenses for rent.inch.)
T
circumferences of
25.
2. 4).1). construct
'. plot points which
lie
between those constructed above.
1
the points (-3.
may. 3 50.2 x
may
4 from x
=
4.
etc. to con struct the graph x of x 2 construct a series of -3 points whose abscissas rep2 resent X) and whose ordi1
tions
. 4).
is
supposed to change.
155
-A
variable is a quantity
whose value changes in the
same
discussion. however.1). and (3.
it is
In the example of the preceding article.
(-
2. (2. hence
various values of x
The values of a function for the be given in the form of a numerical table.
may
.
To
obtain the values of the functions for the various values of
the
following arrangement
be found convenient
:
.0).
-J). If a more exact diagram
is
required. while 7 is a constant.
Q-.
a*. E.GRAPHIC REPRESENTATION OF FUNCTIONS
191.g.
2). x a variable.
Draw the graph of x2 -f.
Graph
of a function. The values of func192.
. be also represented by a graph.
2
(-1. Thus the table on page 1G4 gives the values of the functions x 2 x3 and Vsr.
(1^.
9). as
1.
3
(0.
Ex. (1.
values of x2
nates are the corresponding i.
9).
is
A
constant
a quantity whose value does not change in the
same discussion.
and join
the
points in order. to
x = 4.e. for x=l.
4). Thus in the above example.
if
/*
4
>
1i >
>
?/
=
193.
. Thus 4x + 7. etc.
7
. (-2.
r
*/
+*
01
. j/=-3. or ax + b -f c are funclirst
tions of the first degree.156
ELEMENTS OF ALGEBRA
Locating the points(
4.. 5). (4.
(To avoid
very large ordinatcs. 2 4 and if y = x -f.
If
If
Locating
ing
by a
3) and (4. and join(0. and joining in order produces
the graph
ABC. -1). the function
is
frequently represented
by a single letter. the scale unit of the ordinatcs is taken smaller than that of the x.
Ex.)
For brevity.
194.
(-3.20).
Draw
y
z x
the graph of
= 2x-3.
2.-.
4J. hence two points are sufficient for the construction
of these graphs.
It can be
proved that the
graph is a straight
of a function of the first degree
line. = 4.2 x
.
A
Y'
function of the
first
degree is an integral
rational function
involving only
the
power of the variable.. straight line produces the required graph. y = 6.
= 0... 4).
rf
71
. as y.
Represent 26.
If
two variables x and y are inversely proportional.
Show
any convenient number). 9 F.
to Fahrenheit readings
:
Change
10
C.
if c
Draw
the locus of this equation
= 12.) scale are
expressed in
degrees of the Centigrade (C.
ELEMENTS OF ALGEBRA
Degrees of the Fahrenheit
(F. then
y = .24. 14 F.
A body
moving with a uniform
t
velocity of 3 yards per
second moves in
this
seconds a distance d
=3
1.
that the graph of two variables that are directly proportional is a straight line passing through the origin (assume
for c
27.
C.. the abscissas of 3. it is evidently possible Thus to find to find graphically the real roots of an equation.158
24.e..
...) scale by the formula
(a)
Draw
the graph of
C = f (F-32)
from
to
(b)
4 F F=l.
25. If two variables x and y are directly proportional.
1
C. 32 F.
that
graph with the o>axis..where x
c is
a constant. i.
From
grade equal to
(c)
the diagram find the number of degrees of centi-1 F. then
cXj
where
c is a constant.
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. Therefore x = 1.24 or x =
P and
Q. we have to measure the abscissas of the intersection of the
195. what values of x make the function x2 + 2x 4 = (see 192).
y=
formula graphically.
160
ELEMENTS OF ALGEBRA
GRAPHIC SOLUTION OF EQUATIONS INVOLVING TWO UNKNOWN QUANTITIES
198.
Ex.
(f
.
Ex. Represent graphically
Solving for
y ='-"JJ y.
T
.2 y ~ 2.2. we can construct the graph or locus of any
Since
we can
=
equation involving two
to the above form.
== 2.
y=
A
and construct
x
(
-
graphically.
and joining by a straight
line.
?/
=4
AB. that can be reduced
Thus
to represent
x
-
-
-L^-
\
x
=2
-
graphically.
?/.
4) and
them by
straight line
AB
(3.
Thus
If
in
points without solving the equation for the preceding example:
3x
s
.
1)
and
0). 0).
i.
4)
and
(2.
fc
= 3.e.
represent graphically equations of the form y function of x ( 1D2). 2).
X'-2
Locating the points
(2.
NOTE.
if
y
=
is
0. Equations of the first degree are called linear equations. solve for
?/.
first
degree.
Graph
of
equations involving two
unknown
quantities.
Hence.
. because their graphs are straight lines. Draw the locus of 4 x + 3 y = 12. locate points
(0.
y y
2.
Hence we may
join (0.
If the given equation is of the we can usually locate two
y.
3x
_
4
.
produces the
7*
required locus. and join the required graph. = 0.
199.
unknown
quantities.1.
If
x
=
0. y = -l.
Hence
if
if
x
x
-
2.
202. the point of intersection of the coordinate of P.
and CD.
3.
201. The coordinates of every point of the graph satisfy the given equation.GRAPHIC REPRESENTATION OF FUNCTIONS
161
200. P.
AB
y
= .15.
The
every
coordinates
of
point in satisfy the equation
(1).
equation
x=
By measuring
3. we obtain the roots.
Graphical solution of a linear system. and every set of real values of x and y satisfying the given equation is represented by a point in
the locus.
parallel have only one point of intersection.
(2)
.1=0.57.
Since two straight lines which are not coincident nor simultaneous
Ex.
To
find the roots of
the system.
Solve graphically the equations
:
(1)
\x-y-\.
By
the
method
of
the preceding article construct the graphs
AB
and
and
CD
of
(1)
(2) respectively. The roots of two simultaneous equations are represented by the coordinates of the point (or points) at which their
graphs intersect. linear equations have only one pair of roots.
203. viz.
AB
but only one point
in
AB
also satisfies
(2).
and joining by a
straight line.
4.
(1)
(2)
-C.
4. and
. 0.g. 3). construct CD the locus of (2)
of intersection.
P
graphs meet in two and $. 2.0).
4.
Solving (1) for y. e. (-2.5.
1.
4.
3x
2 y = -6.e.9.
and
+ 3). (-4.
V25
5.
obtain the graph (a circle)
AB C
joining.
In general. = 0.
4.
Locating the
points
(5. (4.
5. 0. which consist of a
pair of parallel lines.0.
Locating two points of equation (2).
Solve graphically the
:
fol-
lowing system
= =
25.
3. the point
we
obtain
Ex. the graph
of
points
roots.
. we of the
+
y*
= 25.162
ELEMENTS OF ALGEBRA
graph. i.
The equations
2
4
= 0. etc. 5.
y equals
3. if x equals
respectively
0.
2.
2 equation x
3). there are two pairs of By measuring the coordinates of
:
P and Q we find
204..5.
3. 3.
(1)
(2)
cannot be satisfied by the same values of x and y.
Inconsistent equations.
There can be no point of
and hence no
roots.
Measuring the coordinates
of P. parallel graphs indicate inconsistent equations.y~ Therefore. 1.
4.
Using the method of the preceding para. This is clearly shown by the graphs of (1) arid (2). they are inconsistent. 4.
intersection.
x2
.
4. 0) and (0.
-
4. AB the locus of (1). Since the two
-
we obtain DE.
Since even powers can never be negative.
a)
4
= a4
.CHAPTER XIV
EVOLUTION
213. and
all
other numbers are. etc.
\/a
=
x means x n
=
y
?>
a. or y
~
3.
\/"^27=-3.
215.
Thus
V^I is an imaginary number.
27
=y
means
r'
=
27. it is evidently impossible to express an even root of a negative quantity by Such roots are called imaginary the usual system of numbers.
Evolution
it is
is the operation of finding a root of a quan the inverse of involution.
for (-f 3) 2
(
3)
equal
0.
numbers.
It follows
from the law of signs
in evolution that
:
Any
even root of a positive.
or
-3
for
(usually written
3)
. which can be simplified no further. for (+ a) = a \/32 = 2.
V9 = +
3.
1.
or x
&4 .
4
4
. called real
numbers.
tity
.
= x means
= 6-. quantity
may
the
be either 2wsitive
or negative. (_3) = -27. for distinction.
Every odd root of a quantity has
same sign as
and
2
the
quantity.
109
.
2. and ( v/o* = a.
V
\/P
214.
14.
multiplied by b must give the last two terms of the
as follows
square.>
13.
a2
+ & + c + 2 a& .
2ab
.2 &c.2 ab + b
.
mV-14m??2)-f 49. a -f. the that 2 ab -f b 2
=
we have then to consider sum of trial divisor 2 a.
#2
a2
-
16. In order to find a general method for extracting the square root of a polynomial.
term a of the root
is
the square root of the
first
The second term
of the root can be obtained
a.
2
.e.
+ 6 + 4a&.
The work may be arranged
2
:
a 2 + 2 ab
+ W \a + b
.
12.
the given expression is a perfect square.
2
49a 8 16 a 4
9. i. let us consider the relation of a -f. it is not known whether the given
expression is a perfect square.2 ac .
10.
8
.72 aW + 81 &
4
.
second term 2ab by the double of
by dividing the the so-called trial divisor.172
7.b 2 2 to its square. 11.
and
b.
15.
2 2
218.
.
a-\-b
is
the root
if
In most cases. however.
The
term
a'
first
2
.
ELEMENTS OF ALGEBEA
4a2 -44a?> + 121V2 4a
s
. and b (2 a -f b).
8 a 2 .
2.EVOLUTION
Ex. we obtain the next term of the root 3 y 3 which has to be added to 2 the trial divisor.
First trial divisor.
219.24 a + 4 -12 a + 25 a8
s
.
and so
forth. 8 a 2
2.
of x. Second trial divisor. the first term of the answer.
First complete divisor.
. and consider Hence the their sum one term.
is
As
there
is
no remainder. 8 a 2 Second complete divisor. We find the first two terms of the root by the method used in Ex.
.
double of this term
find the next
is
the
new
trial divisor. By doubling 4x'2 we obtain 8x2 the trial divisor. The process of the preceding article can be extended to polynomials of more than three terms.
*/''
.
4 x2
3
?/
8 is
the required square foot.
.
173
x*
Extract the square root of 1G
16x4
10 x*
__
. 1. 24# 2 y 3 by the trial divisor Dividing the first term of the remainder.
The square
.
. 8 /-.
. 2 Subtracting the square of 4x' from the trinomial gives the remainder '24 x'2 + y.
\
24 a 3
4-f
a2
10 a 2
Second remainder.
1. Multiply the complete divisor Sx' 3y 3 by Sy 8 and subtract the product from the remainder.24 afy* -f 9 tf.
Ex. 6 a.
by division we
term of the
root. the required root
(4
a'2
8a
+
2}. 8 a 2
-
12 a
+4
a
-f 2.
Arrange the expression according to descending powers root of 10 x 4 is 4 # 2 the lirst term of the root.
10 a 4
8
a.
Arranging according to descending powers of
10 a
4
a.
-
24 a
3
+
25 a 2
-
12 a
+4
Square of 4 a First remainder.
Extract the square root of
16 a 4
.
Explanation. As there is no remainder.
etc. Hence if we divide the digits of the number into groups.000.
As
8
x 168
=
1344. etc.176.. and the first remainder is. Therefore 6 = 8.
The
is
trial divisor
=
160. then the number of groups is equal to the number of digits in the square root.
From
A
will
show the
comparison of the algebraical and arithmetical method given below identity of the methods.
the preceding explanation it follows that the root has two digits.000 is 100.
1.1344.
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
.
7744 80 6400
1
+8
160
+ 8 = 168
1344
1344
Since a
2 a
Explanation. of 10. the first of which is 4. beginning at the
and each group contains two digits (except the last. Thus the square root of 96'04' two digits. the square root of 7744 equals 88.000 is 1000.
= 80.
square root of arithmetical numbers can be found to the one used for algebraic
Since the square root of 100 is 10.
Find the square root of 524.
Ex. and the square root of the greatest square in
units. the first of which is 9 the square root of 21'06'81 has three digits. and we may apply the method used in algebraic process.
Ex. of 1. Hence the root is 80 plus an unknown number.
first
.
175
The
by a method very similar
expressions. and the complete divisor
168. the integral part of the square root of a number less than 100 has one figure.
Find the square root of 7744.
a 2 = 6400.000. which may contain one or two). two figures.
2. of a number between 100 and 10. the first of which is 8.EVOLUTION
220.
the
consists of
group is the first digit in the root.
The groups
of 16724.
places.70
6. annex a cipher.
3.
Roots of common fractions are extracted either by divid-
ing the root of the numerator by the root of the denominator.10.
we must
Thus the groups
1'67'24.
Find the square root of
6/. and if the righthand group contains only one digit.
EXERCISE
Extract the square roots of
:
82
.0961
are
'.GO'61.
in .1 are
Ex. or by transforming the common fraction into a decimal.1T6
221.688
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.
12.7 to three decimal places.
is
one of
_____
b
The side right angle.
find a in terms of 6
.
29.
The
two numbers
(See
is
2
:
3.b 2 If s
If
=c
.
solve for v.
108.
4. 25.
228.
22
a.
.
2
. If the hypotenuse
whose angles
a
units of length.
r.)
of their squares
5.
and the sum
The
sides of
two square
fields are as
3
:
5. opposite the right angle is called the hypotenuse (c in the diagram).
If 2
-f 2 b*
= 4w
2
-f c
sol ve for
m.180
on
__!_:L
ELEMENTS OF ALGEBRA
a.
If 22
= ~^-.
27.
Find
is
the number.
28. 24.
may
be considered one half of a
rec-
square units.
If
G=m m
g
.
A
number multiplied by
ratio of
its fifth
part equals 45. Find the side of each field.
3.
and their product
:
150.
If s
= 4 Trr
'
2
.
2
:
3. then
Since such a triangle
tangle.
EXERCISE
1.
Find the numbers.
26.
solve for d. is 5(5.
Find the side
of each field.
A
right triangle is a triangle.
9
&
-{-
c#
a
x
+a
and
c.
84
is
Find a positive number which
equal to
its
reciprocal
(
144). The sides of two square fields are as 7 2.
solve for
r.
Three numbers are to each other as 1 Find the numbers.
:
6.
.
2a
-f-
1
23. and the first exceeds the second by 405 square yards.
=
a
2
2
(' 2
solve for solve for
= Trr
.
'
4. its area contains
=a
2
-f-
b2
.
If a 2 4.
2
.
and they con-
tain together 30G square feet. and the two other sides respectively
c
2
contains
c
a and b units.
2.
.
24.QUADRATIC EQUATIONS
7.
Find the
sides.
4.)
COMPLETE QUADRATIC EQUATIONS
229.
sides. in how many seconds will a body fall (a) G4 feet.
.
member can be made a complete square by adding 7 x with another term. passes in t seconds 2 over a space s yt Assuming g 32 feet.
181
The hypotenuse
of a right triangle
:
is
35 inches.
The area $
/S
of a circle
2
.
of a right triangle Find these sides.)
13. Find the unknown sides and the area.
The hypotenuse
of a right triangle is 2.
Solve
Transposing.
add
(|)
Hence
2
.
8.
Two
circles together contain
:
3850 square
feet.
9.
and the two smaller
11.
is
and the other
two
sides are equal. and the third side is 15 inches.
8 = 4 wr2 Find 440 square yards.
Find these
10.
7r
(Assume
and their
=
2 7
2
.2
7
.
-J-
=
12. its surface
(Assume
ir
=
2 .
2m.
.
radii are as 3
14. let us compare x 2
The
left
the perfect square x2
2
mx -f m
to
2
.
The following
ex-
ample
illustrates the
method
or
of solving a complete quadratic
equation by completing the square.
The area
:
sides are as 3
4. and the
other two sides are as 3
4. To find this term.7 x -f 10 = 0. the radius of a sphere whose surface equals
If the radius of a sphere is r.
the formula
= Trr
whose radius equals r is found by Find the radius of circle whose area S
equals (a) 154 square inches.
make x2
Evidently 7 takes the place 7x a complete square
to
to
which corresponds
m
2
. x* 7 x=
10.
Method
of completing the
square. (b) 100 feet?
=
. A body falling from a state of rest.
Find the
radii. we have
of
or
m = |. (b) 44 square feet. The hypotenuse of a right triangle is to one side as 13:12.
Solution
by formula. -\-bx-\.
= 12.
o^
or
-}-
3 ax == 4 a9
7 wr
.
x
la
48.
.
=0.
any quadratic equation may be obtained by 6.c
= 0.
2x
3
4.
2
Every quadratic equation can be
reduced to the general form.
=8
r/io?.
Solving this equation we obtain
by the method of the preceding
2a
The
roots of
substituting the values of a.
49. and c in the general answer.
ao.184
ELEMENTS OF ALGEBRA
45
46.
article.
231.
6.
and whose sum
is
is 36.
88
its reciprocal
A
number increased by three times
equals
6J.
-2.
Find two numbers whose difference
is 40.
-2.
and whose
product
9.
Twenty-nine times a number exceeds the square of the 190.
What
are the
numbers
of
?
is
The product
two consecutive numbers
210.
2.
8.
2.0.
its
sides of a rectangle differ by 9 inches.
7.
is
Find two numbers whose product
288.
57.
3.
Problems involving quadratics have
lems of this type have only one solution.
3.
number by 10.3.
56. Find the number.
-4.
The sum
of the squares of
two consecutive numbers
85. but frequently the conditions of the problem exclude negative or fractional answers. and the difference Find the numbers.
5. and consequently many prob-
235.
54. Find the sides. -2.QUADRATIC EQUATIONS
Form
51.9.
G.
52.
.
Find the number.
The
difference of
|. The
11.
1.
Find
the numbers.
area
A
a perimeter of 380
rectangular field has an area of 8400 square feet and Find the dimensions of the field.
1.
EXERCISE
1.3.3. -5.2.
189
the equations whose roots are
53.
Find a number which exceeds
its
square by
is
-|.
-2.
PROBLEMS INVOLVING QUADRATICS
in general two answers.
of their reciprocals is
4.
two numbers is 4.
:
3. feet.0. and equals 190 square inches.
0.
58.
55.1.
Divide CO into two parts whose product
is 875.
What did he pay for
21.
other.
14.
watch cost
sold a watch for $ 21. and lost as many per cent Find the cost of the watch.
vessel sail ?
How many
miles per hour did the faster
If 20. had paid $ 20 less for each horse.
19.
15.
watch for $ 24.
ELEMENTS OF ALGEBRA
The length
1
B
AB of a rectangle. and the slower reaches its destination one day
before the other. Two vessels. start together on voyages of 1152 and 720 miles respectively.
Two steamers
and
is
of 420 miles.190
12. he would have received 12 apples less for the same money.
as the
16. Find the rate
of the train. exceeds its widtK AD by 119 feet. A man bought a certain number of apples for $ 2. sold a horse for $144. dollars.
A man
A man
sold a
as the watch cost dollars. and the line BD joining
two opposite
vertices (called "diagonal")
feet.
and gained as many per Find the cost of the horse.
17.
A man
cent as the horse cost dollars.10.
of a rectangle is to the length of the recthe area of the figure is 96 square inches.
ply between the same two ports. and Find the sides of the rectangle.
If he
each horse ?
.
13.
c equals 221
Find
AB and AD.
The diagonal
:
tangle as 5 4. and lost as many per cent Find the cost of the watch. he would have received two horses more for the same money. What did he pay for each
apple ?
A man bought a certain number of horses for $1200. At what rates do
the steamers travel ?
18. If a train had traveled 10 miles an hour faster. one of which sails two miles per hour faster than the other. it would have needed two hours less to travel 120 miles.
. ABCD. 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.
Solve
^-9^ + 8 =
**
0. contains B 78 square inches.QUADRATIC EQUATIONS
22.
^-3^ = 7. Find and CB. a point taken.
Find
TT r (Area of a circle .
B
AB
AB
-2
191
grass plot.
A rectangular
A
circular basin is surrounded
is
-
by a path 5
feet wide.
24. A needs 8 days more than B to do a certain piece of work. as
0. Equations in the quadratic form can be solved by the methods used for quadratics.
27. In how many days can B do the work ?
=
26.
Ex. how wide is the walk ?
23.
of the area of the basin. How many eggs can be bought for $ 1 ?
236. is surrounded by a walk of uniform width. constructed with and CB as sides.
(tf.
By formula. The number of eggs which can be bought for $ 1 is equal to the number of cents which 4 eggs cost. the two men can do it in 3 days.) 25.
=9
Therefore
x
=
\/8
= 2.
or x
= \/l = 1. so that the rectangle.I) -4(aj*-l)
2
= 9.
237.
and the area of the path
the radius of the basin.
1.
EQUATIONS IN THE QUADRATIC FORM An equation is said to be in the quadratic form
if it
contains only two unknown terms. 23 inches long. and working together.
.
is
On the prolongation of a line AC. If the area of the walk is equal to the area of the plot. 30 feet long and 20 feet wide.
Find the side of an equilateral triangle whose altitude
equals 3 inches. and the unknown factor of one of these terms is the square of the unknown factor of the
other.
"
means "is greater than"
195
similarly
means "is
.CHAPTER XVI
THE THEORY OF EXPONENTS
242.
the direct consequence of the defiand third are consequences
FRACTIONAL AND NEGATIVE EXPONENTS
243.
(ab)
.
must be
*The symbol
smaller than. hence.
we may choose
for such
symbols any definition that
is
con-
venient for other work. very important that all exponents should be governed by the same laws. and
.
m
IV. that a
an
= a m+n
. ~ a m -f. (a ) s=a m = aw bm
a
. we let these quantities be what they must be if the exponent law of multiplication is generally true. such as 2*.
It is.
244. however. while the second of the first.
II. The following four fundamental laws for positive integral exponents have been developed in preceding chapters
:
I.
We assume.
no
Fractional and negative exponents.
Then the law
of involution.
for all values
1
of
m and n. instead of giving a formal definition of fractional and negative exponents.
= a""
<
.*
III.
a m a" = a m+t1 .
>
m therefore.
provided
w > n. 4~ 3 have meaning according to the original definition of power. (a m ) w
.a" = a m n
mn .
The
first
of these laws
is
nition of power.
since the raising to a positive integral power is only a repeated multiplication. fractional.
0?=-^. 4~ .
e.
Let
x
is
The operation which makes the fractional exponent disappear evidently the raising of both members to the third power.
245.
28.
To
find the
meaning
of
a fractional exponent.
m$.
31.
'&M
A
27.
24. a .
etc.
Hence
Or
Therefore
Similarly.
n 2 a.
Write the following expressions as radicals :
22. (xy$.
.196
ELEMENTS OF ALGEBRA
true for positive integral values of n.
29.
-
we
find
a?
Hence we
define a* to be the qth root of of.
a?*.
^=(a^)
3*
3
.
30.
23. ml.g. as. at. 25.
(bed)*. or zero exponent
equal
x.
a\
26.
disappear.
Assuming these two
8*.
a*.
= a.
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. 3*.
laws.
cr n.g.
Or
a"#
= l. in which
obtained from the preceding one by dividing both
members by
a.
.
etc.
ELEMENTS OF ALGEBRA
To
find the
meaning
of a negative exponent.
a
a
a
= =
a a a
a1
1
a.198
247.
vice versa.
248.
by changing the sign of
NOTE.
Let
x=
or".2
=
a2
.
a8 a
2
=
1
1
. Factors
may
be transferred
from
the
numerator
to
the
denominator of a fraction.
Multiplying both members by
a".
an x = a.
each
is
The
fact that a
if
=
we
It loses its singularity
1 sometimes appears peculiar to beginners. consider the following equations. or
the exponent. e.
Arrange in descending powers of
Check.
If
powers of
a?.
34.
1.
40.
1 Multiply 3 or
+x
5 by 2 x
x.
lix
=
2x-l
=+1
Ex.
we wish to arrange terms according to descending we have to remember that.
Divide
by
^
2a
3 qfo
4. The
252.
powers of x arranged are
:
Ex.
1.2 d
.202
ELEMENTS OF ALGEBRA
32.
V ra
4/
3
-\/m
33.
2. the term which does not contain x may be considered as a term containing #.
6
35.
all
monomial surds may be divided by
method.
60.
(3V3-2Vo)(2V3+V5). 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.
(5V2+V10)(2V5-1).
49.
is
1
2.
43.
Ex.
48.
268.
52.
51. Monomial surdn of the same order may be divided by multiplying the quotient of the coefficients by the quotient of the
surd factors.
it
more convenient to multiply dividend and divisor by a factor which makes the divisor rational.V5) ( V3 + 2 VS).
ELEMENTS OF ALGEHRA
(3V5-5V3)
S
.
E.
.
(2
45.
V3 .
53.
(5V7-2V2)(2VT-7V2).
Va
-v/a.
Ex.
a fraction.
(3V5-2V3)(2V3-V3).y.214
42.
46.
(V50-f 3Vl2)-4-V2==
however.
47.
-v/a
-
DIVISION OF RADICALS
267.
44.
Divide 12 V5
+ 4V5 by V.RADICALS
This method.57735.
1.g. we have
V3
But
if
1. the by 3 is much easier to perform than the division by
1. metical problems afford the best illustrations. however.
3.
.by the usual arithmetical method. e. is illustrated
by
Ex.
we have
to multiply
In order to make the divisor (V?) rational.
+ 4\/5 _ 12v 3 + 4\/5 V8 V8
V2 V2
269.
by V7.
The
2.
Hence
in arithmetical
work
it
is
always best to
rationalize the denominators before dividing.
VTL_Vll '
~~"
\/7_V77
.73205
we
simplify
JL-V^l
V3
*>
^>
division
Either quotient equals ..
/~
}
Ex.
4\/3~a'
36
Ex.
Divide
VII by v7. Evidently. called rationalizing the
the following examples
:
215
divisor.
.73205. arithTo find.
Divide 4 v^a by
is
rationalizing factor
evidently \/Tb
hence. the rationalizing factor
x
' g
\/2.
is
Since \/8
12 Vil
=
2 V*2.
. To show that expressions with rational denominators are simpler than those with irrational denominators.
and have
for
any positive integral value of
If n
is
odd."
.y n is divisible by x -f ?/.
Factor
consider
m
m
6
n9
.
2.
ELEMENTS OF ALGEBRA
positive integer.
xn -f.
Two
special cases of the preceding propositions are of
viz.
actual division
n. if w is odd.
The
difference of
two even powers should always be
considered as a difference of two squares.
x* -f-/
= (x +/)O .xy +/).
it
follows from the Factoi
xn y n is always divisible by x y.
Ex.
By
we obtain the other
factors. if n For ( y) n -f y n = 0.
2.
-
y
5
=
(x
-
can readily be seen that #n -f either x + y or x y.
ar
+p=
z6
e.
is
odd.
We may
6
n 6 either a difference of two squares or a
dif-
* The symbol
means " and so forth to.230
285.
It
y is
not divisible by
287.
2
Ex. xn y n y n y n = 0. If n is a Theorem that
1.
:
importance.
1. For substituting y for x. if n is even.
286.
Factor 27 a* -f
27 a 6
8.
2 8 (3 a )
+8=
+
288.g.
can be
If
It is
made
larger than
number.
oo is
= QQ.
be the numbers. cancel.
is satisfied
by any number.
TO^UU"
sufficiently small.
1.i
solving
a problem
the result
or oo indicates that the
all
problem has no solution.decreases
X
if
called infinity. or that x may equal any finite number.
and becomes infinitely small.
of the second exceeds the product of the first
Find three consecutive numbers such that the square and third by 1.
as
+ l.
= 10.
(1)
is
an
identity.
the
If in an equation
terms containing
unknown quantity
cancel.
(1)
= 0.
+
I)
2
x2
'
-f
2x
+
1
-x(x + 2)= .
The
solution
x
=-
indicates that the problem
is indeter-
If all terms of an minate.
i. while the
remaining terms do not
cancelj the root is infinity.
Or.
creases.
great.242
303.
ELEMENTS OF ALGEBRA
Interpretation of ?
e.
.
Hence any number will satisfy equation the given problem is indeterminate.e.
.
i. or infinitesimal) This result is usually written
:
305.increases
if
x
de-
x
creases.
By making x
any * assigned
zero. however
x approaches the value
be-
comes
infinitely large.
I.
ToU"
^-100 a.
(1).
306. equation. it
is
an
Ex. the answer is indeterminate.
Hence such an equation
identity. without exception.x'2 2 x =
1.
(a:
Then
Simplifying.
Interpretation of
QO
The
fraction
if
x
x
inis
infinitely large.
The
~~f
fraction .000
a.
Let
2.e.
and
.
x
-f 2.
1.
customary to represent this result
by the equation ~
The symbol
304.g.
)
53 yards. equals 4 inches. The sum of the areas of two squares is 208 square feet.
is
is
17 and the
sum
4. The volumes of two cubes differ by 98 cubic centimeters. and the edge of one.
is
the breadth
diminished by 20 inches.
Find the other two
sides.
255 and the sum of
5. Find these sides.
the
The mean proportional between two numbers sum of their squares is 328. Find the edge of each cube. p.
146 yards. and the edge of one exceeds the edge of the other by 2 centimeters. To inclose a rectangular field 1225 square feet in area. Find two numbers whose product whose squares is 514.
and the
hypotenuse
is 37. 148 feet of fence are required. and the side of one increased by the side of the other e. Find the dimensions of the
field.quals 20 feet.
The hypotenuse
is
the other two sides
7.
Find the
sides of the rectangle.
and the diago(Ex.)
The area
of a right triangle is 210 square feet.
two numbers Find the numbers. But if the length is increased by 10 inches and
12.
10.
of a rectangular field
feet.
Find the
sides. increased by the edge of the other.
103.
of a right triangle is 73. Find the edges. Find the side of each square. Two cubes together contain 30| cubic inches.
6. and
its
The diagonal
is
is
perimeter
11. 190.
14. the area becomes
-f%
of
the original area.
and the sum of
(
228.
is 6. 12. Find the numbers.
13.
9.
and
is
The area of a rectangle remains unaltered if its length increased by 20 inches while its breadth is diminished by 10 inches.
rectangle is 360 square Find the lengths of the sides.
ELEMENTS OF ALGEBRA
The
difference between
is
of their squares
325.
.
The area of a
nal 41 feet.
8.244
3.
their areas are together equal to the area of a circle whose radius is 37 inches.
the quotient
is 2.
and
if
the digits will be interchanged. Find the number.
245
The sum of the radii of two circles is equal to 47 inches. and the equal to the surface of a sphere Find the radii.SIMULTANEOUS QUADRATIC EQUATIONS
15.
is
20 inches. Find the radii. (Surface of sphere
If a
number
of
two
digits be divided
its digits. irR *.
.) (Area of circle
and
=
1
16.)
17.
differ by 8 inches.
by the product of 27 be added to the number.
The
radii of
two spheres
is
difference of their surfaces
whose radius = 47T#2.
10.
The common differences are respectively 4. the first
term a and
the
common difference d being given. .
a
+
d. P.. 12.. to produce the 3d term.
of the following series is
3. 2 d must be added to a.
309.
to
A series
is
a succession of numbers formed according
some
fixed law.
Since d
is
a
-f
3
d..11 246
(I)
Thus the 12th term of the
3
or 42.
series 9.. to produce the 4th term. An arithmetic progression (A. and d. a
3d.. 16.. The progression is a.
Hence
/
= a + (n . The first is an ascending. 15 is 9 -f. each term of which.
.
:
7. 3 d must be added to a.
.. a + 2 d.
-4. a -f d. (n 1) d must be added to a. progression..
The common
Thus each
difference is the
number which added
an A.
of a series are its successive numbers.CHAPTER XX
PROGRESSIONS
307. P.7.
11.) is a series. is derived from the preceding by the addition of a constant number.1) d..
.. 19.
a.
The terms
ARITHMETIC PROGRESSION
308.
added to each term to obtain the next one. a
11. to produce the nth term.
17. 3.
+
2 d. except the first.
-f
.
to each
term produces the next term. P.
To
find the
nth term
/
of an A. the second a descending.
.. ...
series
.
Find the 12th term of the
-4.
= -2.
Find the 10th term of the
series 17.PROGRESSIONS
310. 3. 6..
Adding.
= I + 49 = *({ +
.8.4.-. 2. 8.
. d = 3.
1.
5.
2.
2
EXERCISE
1. P.
1-J. 19.
9.
247
first
To
find the
sum s
19
of the first
n terms of an A. the
term
a.
115.
5.
series 2.
. -24. a = 2. 21.
Find the 5th term of the
4.
the last term
and the common difference d being given. 5.
Find the 101th term of the
series 1.' cZ == .
first
2
Write down the
(a)
(6)
(c)
6 terms of an A..-
(a
+ + (a +
l)
l)...
(d) 1J..
.
8.
Find the nth term of the
series 2. 2J.3 a = -l.-. -4^.
-|. -10.
-3.
= a + (a
Reversing the order.
Or
Hence
Thus
from
(I)
= (+/). 4. 8.. 5.
. -7.
6
we have
Hence
.16.. 3.. 2
sum
of the first 60
I
(II)
to find the
' '
odd numbers... 7. ?
(a) 1..
= 99. 5.. 3. 99) = 2600.. 6. P.
.
2*=(a + Z) + (a + l) + (a + l)
2s = n
*
.
1.
6. P.
7. d
.
if
a = 5.
of the series 10.
Find the 7th term of the Find the 21st term
series
.
9.
3.
Which
(6)
(c)
of the following series are in A.
ELEMENTS OF ALGEBRA
last
term and the sum of the following series :
.
.1 -f 3.
21.
20. How much does he receive (a) in the 21st year (6) during the first 21 years ?
j
311.
to 20 terms.
to 16 terms.
rf.
Sum
the following series
14. 1|.
1J..
13.
.
>
2-f
2. and for each than for the preceding one.
+ 2-f-3 + 4 H
hlOO.
to 20 terms. 31. P. 2J.
to 10 terms.7 -f
to 12 terms.
4.
16. 11.
6. In most problems relating to A.
1. 15.
1.
:
3. 16.
2.
to 15 terms. the other two may be found by the solution of the simultaneous equations
.
(i)
(ii)
. and a yearly increase of $ 120.
. 11.
23.
to 8 terms. 29.
8.
-.
to 7 terms. 12.
to 20 terms.
15. Jive quantities are involved.248
Find the
10.
11.
\-n. 7.
.
.
+ 3.5
H + i-f
-f-
to 10 terms.(#
1
2) -f (x -f 3) H
to
a terms.
Q^) How many times
in 12 hours ?
(&fi)
does a clock.
1+2+3+4H
Find the sum of the
first
n odd numbers.
15.
22. 11.
33.
.
17.
3. striking hours only. strike
for the first yard. hence if any three of them are given.
19.
7.
12.
$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.
18.
'.
(x +"l) 4.
. 7.
I. s = 70.
m
and
n
2. n = 4.
a x
-f-
b
and a
b. = 52.
Between 4 and 8
insert 3 terms (arithmetic
is
means)
so
that an A.
T?
^.
I
Find
I
in terms of a. Given a = . Find n. n.3. n = 20. of 5 terms
6.
man saved each month $2 more than in the pre 18.
12.
6?
9.
15. ceding one.
17.
How much
. and all his savings in 5 years amounted to $ 6540. = ^ 3 = 1. Find d. n = 13. Find?.
y and #-f-5y.
4.
. n = 16.
10.
has the series 82. Find w.
3.
8. = 17. Given a = 4.
13.
How many terms How many terms
Given d = 3. Given a = |.
7.
11. 74. Given a = 1. n
has the series
^
j
. n = 17.
= 16. = 45. s == 440.
a+
and
b
a
b
5. Find a Given a = 7. Find d.
and
s. P. d = 5.
14.
produced. How much did he save the first month?
19. = 83. 78.250
ELEMENTS OF ALGEBRA
EXERCISE
116
:
Find the arithmetic means between
1.
16. = 1870. Find d and Given a = 1700. Find a and Given s = 44.
Between 10 and 6
insert 7 arithmetic
means
.
f?
.
f
J 1 1
/
.
A
$300
is
divided
among 6 persons
in such a
way
that each
person receives $ 10 did each receive ?
more than the preceding
one.
or.
(I)
of the series 16.
|. To find the sum s of the first n terms term a and the ratio r being given.
and
To
find the
nth term
/ of
a G.
4. 108. 24. is
it
(G. 36.
s(r
1)
8
= ar"
7*
JL
a.
or 81
315. 36.
If
n
is less
:
than unity.
NOTE..PROGRESSIONS
251
GEOMETRIC PROGRESSION
313.
.)
is
a series each term of
which. P. except the
multiplying
derived from the preceding one by by a constant number. a?*2 To obtain the nth term a must evidently be multiplied by
. P.
. 36..
The progression is a.
A geometric progression
first. the first term a and
the ratios r being given.
Therefore
Thus the sum
= ^ZlD.
.. P...arn ~ l . rs =
s
2
-.g. called the ratio. the first
= a + ar -for ar -f ar Multiplying by r..
ar8
r....
The
314.
ratios are respectively 3.
g==
it is
convenient to write formula' (II) in
*. +1. ar. 12.
2 a.
the following form 8
nf +
q(l-r")
1
r
.
.
fl
lg[(i)
-l]
==
32(W -
1)
= 332 J.
(II)
of the
8 =s
first
6 terms of the series 16..
r
n~ l
.
of a G.
-2.
2
arn
(2)
Subtracting (1) from
(2).
4-
(1)
.
.
Hence
Thus the 6th term
l
= ar
n~l
.
E. 24. <zr .
4. -I.
is
16(f)
4
.
Ex. 3. P. P.
Find the 7th term of the Find the 6th term of the
Find the 9th term of the
^.
0.
576. 1. In most problems relating to G. 20.
144.
or
7.
288.
10.18.
first
term
is
125 and
whose common
. P.
r^2.
(d) 5.
first
5. ?
(c)
2.*.252
ELEMENTS OF ALGEBRA
316.
.. whose and whose second term is 8. 4.
. whose and whose common ratio is 4. f.288.
-fa. 36.
Find the 6th term of the
series J.
(it.
.
. 144.72. 25.
series
Find the llth term of the Find the 7th term of the
ratio is
^.
2
term
3..18.
.
7..
4.
Hence the
or
series is
0. 36.
36.
Find the 5th term of a G.
+-f%9 %
. P. 72. 80.
a
=
I.
And the
required
means are
18.
Evidently the total
number
of terms is 5
+ 2.. 9.54.
(b) 1. 144.
i 288. the other two be found by the solution of the simultaneous equations :
may
(I)
/=<!/-'.
..._!=!>.
|.l.
Write down the first 6 terms of a G.
.
l. I
= 670..
72.
6. Jive quantities are in.
676
t
Substituting in
= r6 = 64...
-fa.
Hence n
=
7.
volved .
f.
9.
117
Which
(a)
of the following series are in G. + 5. 18. whose
.
. P.
8.
EXERCISE
1.
series 5.
. 9.
..5.
..
To
insert 5 geometric
means between 9 and 576.. 676.
.
Write down the first 5 terms of a G.5.
series
. if any three of them are given.
series 6.
is 16. hence.
.
first
term
4.
\
t
series
.6.4.
is 3..-.
7/
191. and 5 h.
What
is
the distance?
if
square grass plot would contain 73 square feet more Find the side of the plot. Find the dimensions of the floor. A house has 3 rows of windows.
187.
dimension
182. and the middle row has 4 panes in each window more than the upper row there are in all 168 panes of glass.
The age
of the elder of
it
three years ago of each. aW + llab-2&.
13 a + 3.
respectively. x*
185. and the father's present age is twice what the son will be 8 years
hence.
same
result as the
number
diminished by
175. 12 m.
180.
ELEMENTS OF ALGEBRA
A A
number increased by
3.
178.
number divided by
3.
+x-
2.-36.
. Find the age
5 years older than his sister
183.266
173.
.
190.
176. z 2 + x .
+
a.
3 gives the
174. + 11 ~ 6.
power one of the two Find the power of each.
188.
and | as old as his Find the age of the
Resolve into prime factors
:
184. 15 m. was three times that of the younger. 10x 2 192.
two boys is twice that of the younger.
The length
is
of a floor exceeds its width
by 2
feet.
sister
.
Find the number.
3 gives the
same
result as the
numbet
multiplied by
Find the number.
A
boy
is
father.
2
2
+
a
_ no.56. the
sum
of the ages of all three is 51. 189. Four years ago a father was three times as old as his son is now.
train. z 2
-92. the ana of the floor will be increased 48 square feet.
179. 6 in each row the lowest row has 2 panes of glass in each window more than the middle row. 4 a 2
y-y
-42. side were one foot longer.
A
the
boy
is
as old as his father
and
3 years
sum
of the ages of the three is 57 years.
if
each
increased 2 feet.
A
each
177.
181.
is
What are their ages ? Two engines are together
more than the
of 80 horse
16 horse power
other.
younger than his Find the age of
the father. How many are there in each window ?
.
-ll?/-102.
.
An
The two
express train runs 7 miles an hour faster than an ordinary trains run a certain distance in 4 h.
186.
father.
(5 I2x
~r
l
a)
.a)(x b
b)
(x
b
~
)
412.c) .
a
x
)
~
a
2 b
2
ar
a
IJ a.
421.
418 ~j-o. Find the number of miles an hour that A and B each walk.
(x
-f
ELEMENTS OF ALGEBRA
a)(z
-
b)
=
a
2 alb
=
a
(x
-f
b)(x
2
. How long is each road ?
423.
A
in 9 hours
B walks
11 miles
number of two digits the first digit is twice the second. 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. the order of the digits will be inverted.
4x
a
a
2 c
6
Qx
3 x
c
419.
down again
How
person walks up a hill at the rate of 2 miles an hour.
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.
-
a)
-2
6 2a.
mx ~
nx
(a
~
mx
nx
c
d
d
c)(:r
lfi:r
a
b)(x
. and at the rate of 3^ miles an hour. and was out 5 hours. In a
if
and
422.278
410.
a
x
a
x
b
b
x
c
b
_a
b
-f
x
414. far did he walk all together ?
A
.
hour. Find the number. Tn 6 hours
.
420.(c rt
a)(x
-
b)
=
0. 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. 18 be subtracted from the number.
(x
.
-f
a
x
-f
x
-f c
1
1
a-b
b
x
415.
thrice that of his son
and added to the father's. A sum of money at simple interest amounts in 8 months to $260.
by 4.
483. A sum of money at simple interest amounted in 10 months to $2100. also a third of the greater exceeds half the less by 2.
least
The sum
of three
numbers
is
is
21. and in 20 months to $275.
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 two numbers such that twice the greater exceeds the by 30.
A
number
consists of
two
digits
4. and the other number least. There are two numbers the half of the greater of which exceeds the less by 2. and if each be increased by 5 the Find the fraction. How much money
less
484. In a certain proper fraction the difference between the nu merator and the denominator is 12. fraction becomes equal to |. the Find their ages. Find the numbers.
. Find the numbers. and becomes when its denominator is doubled and its numerator increased by 4 ?
j|
478.
if
the
sum of
the digits be multiplied by
the digits will be inverted. Find the
fraction. and 5 times the less exceeds the greater by 3. had each at first?
B
B
then has
J
as
much
spends } of his money and as A. What is that fraction which becomes f when its numerator is doubled and its denominator is increased by 1.
years.
half the
The greatest exceeds the sum of the greatest and
480. age. Find the sum and the rate of
interest.
to
. Find the principal and the rate of
interest.
477.
A
spends \ of his. Find their ages.
481. and in 18 months to $2180.282
ELEMENTS OF ALGEBRA
476. and a fifth part of one brother's age that of the other.
486.
485.
whose difference
is
4.
latter
would then be twice the
son's
A
and B together have $6000.
Find
the number.
487. 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
. Of the ages of two brothers one exceeds half the other by 4 is equal to an eighth of 482.
479.
AC
in /).
527.
. (a) How many pounds of tin and lead are in a mixture weighing 120 pounds in air.
and third equals \\ the sum third equals \. Tu what time will it be filled if all run
M
N
N
t
together?
529. if L and Af in 20 minutes.
and B together can do a piece of work in 2 days.
and CA=7. BC = 5. CD. if the number be increased by Find the number. How long will B and C take to do
. and 23 pounds of lead lose 2 pounds. L.
it
separately
?
531. 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. What are their rates of
travel?
. An (escribed) and the prolongations of BA and BC in Find AD. it is filled in 35 minutes. AB=6.
. Find the present ages of his father and mother. they would have met in 2 hours. Tf and run together. Throe numbers are such that the
A
the
first
and second equals
. N. 37 pounds of tin lose 5 pounds. the first and second digits will change places. A boy is a years old his mother was I years old when he was born.REVIEW EXERCISE
285
525.
530. in 28 minutes. B and C and C and A in 4 days. his father is half as old again as his mother was c years ago. When weighed in water. In how many days can each alone do the same work?
526.
E
533. Two persons start to travel from two stations 24 miles apart. 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. If they had walked toward each other. A number of three digits whose first and last digits are the same has 7 for the sum of its digits. Find the numbers. M.
532. 90. In
circle
A ABC.
sum of the reciprocals of of the reciprocals of the first of the reciprocals of the second and
the
sum
528.
touches
and
F respectively. A vessel can be filled by three pipes. if and L. and BE. and one overtakes the other in 6 hours.
545.10 marks. The greatest value of the function. If
to
feet is the length of a
seconds.
to
do the work? pendulum.
The values of y.
540.
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. the function. FRANCE. Represent the following table graphically
TABLE OF POPULATION (IN MILLIONS) OF UNITED STATES. z 2
-
x x
-
5. The roots of the equation 2 + 2 x x z = 1.
547.286
ELEMENTS OF ALGEBRA
:
534. Draw the graph of y 2 and from the diagram determine
:
+
2 x
x*.
a. x 2 544. 2|.
b.
c.
2. if x = f 1.
from x
=
2 to x
= 4. 3 x
539. The values of x if y = 2.
-
3 x.
. then / = 3 and write
=
3. 2
-
x
-
x2
. AND BRITISH ISLES
535.
546. GERMANY. The value of x that produces the greatest value of y.
-
7. x
2
+
x. formation of dollars into marks.
-
3 x.3
Draw
down
the time of swing for a
pendulum
of length
8 feet. 536.
548. x 8
549.
+
3. i.
d.
e.
542. 2 x
+
5.
. One dollar equals 4. the time of whose swing a graph for the formula from / =0
537. x*
-
2
x.
550.e.
x*. x *-x
+
x
+
1.
543.
Draw
the graphs of the following functions
:
538.
How
is
t /
long will
I
take 11
men
2
t' . 2 541.
ELEMENTS OF ALGEBRA
+36 = 0.
714
2
*2
'
+
25
4
16
|
25 a2
711. Find two consecutive numbers whose product equals 600. he
many
312?
he had waited a few days until each share had fallen $6.25 might have bought five more for the same money. Find the price of an apple. 727.
716. A man bought a certain number of shares in a company for
$375. 717. 2n n 2 2 -f-2aar + a -5 = 0.l
+
8
-8
+
ft)'
(J)-*
(3|)*
+
(a
+
64-
+ i. Find two numbers whose 719.
729. needs 15 days longer to build a wall than B. what is the
price of the coffee per
pound ?
:
Find the numerical value of
728.
of a rectangle is 221 square feet and its perimeter Find the dimensions of the rectangle.
in value.292
709.40 a 2* 2 + 9 a 4 = 0. **-13a: 2
710. 12
-4*+
-
8.
722.
. 16 x* .
The area
the price of 100 apples by $1. 3or
i
-16 . paying $ 12 for the tea and $9 for the coffee.
a:
713. In how many days can A build the wall?
718. 724. What number exceeds its reciprocal by {$.
The
difference of the cubes of
two consecutive numbers
is
find them. and working together they can build it in 18 days. How shares did he buy ?
if
726.44#2 + 121 = 0.
sum is a and whose product equals J. If a pound of tea cost 30 J* more than a pound of coffee. Find four consecutive integers whose product is 7920.
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. 725.
___ _ 2* -5 3*2-7
715. 721. Find the altitude of an equilateral triangle whose side equals a.
217
. if 1 more for 30/ would diminish
720.
A
equals CO feet.
723.
and the Find the sides of the and
its
is
squares.
two numbers Find the numbers. Find the length and breadth of the first rectangle. and B diminishes his as arrives at the winning post 2 minutes before B. The
sum
of the circumferences of
44 inches.102.
935. If each side was increased by 2 feet.
rate each
man
ran in the
first
heat.
942.
two squares is 23 feet. The difference of two numbers cubes is 513. The diagonal of a rectangle equals 17 feet.
is 20.000 trees. = ar(a? -f y + 2) + a)(* + y
933. The perimeter of a rectangle is 92 Find the area of the rectangle. Find the side of each two
circles is
IT
square.
The sum
of the perimeters of
sum
of the areas of the squares is 16^f feet. In the first heat B reaches the winning post 2 minutes before A. s(y
932. Tf there had been 20 less rows.
937.square inches. and also contains 300 square feet. + z) =108. a second rec8 feet shorter.
943. feet.
A
is
938. A plantation in rows consists of 10.300
930. (y + *) = .
and the sum of
their areas 78$.
2240.
the difference of their
The
is
difference of
their cubes
270.
the
The sum
of the perimeters of
sum
of their areas equals 617 square feet.
is
3
.
feet.
y(
934. z(* + y + 2) = 76.
+ z)=18. (3 + *)(ar + y + z) = 96.
Assuming
= -y.
34
939.
and the sum of their cubes
is
tangle
certain rectangle contains 300 square feet.
is 3.
931. 152. the area of the new rectangle would equal 170 square feet.
ELEMENTS OF ALGEBRA
(*+s)(* + y)=10. *(* + #) =24.
find
the radii of the two circles. Find the numbers. there would have been 25 more trees in a row.
.
The sum of two numbers Find the numbers.
and the
difference of
936. Find the sides of the rectangle.
diagonal
940. In the second
heat
A
. How many rows are there?
941.
much and A then
Find at what
increases his speed 2 miles per hour.
944. y(x + y + 2) = 133. (y
(* + y)(y +*)= 50. A and B run a race round a two-mile course.
two squares equals 140 feet. and 10 feet broader.
Find in what time both will do it.
whose
946. at Find the his rate of traveling. The diagonal of a rectangular is 476 yards.
950. What is its area?
field is
182 yards. Find the width of the path if its area is 216 square yards. 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. and
its
perim-
948. the difference in the lengths of the legs of the Find the legs of the triangle.
Find the
eter
947. the digits are reversed.
unaltered.
. Find two numbers each of which
is
the square of the other. When
from
P
A
was found that they had together traveled 80 had passed through Q 4 hours before. A certain number exceeds the product of its two digits by 52 and exceeds twice the sum of its digits by 53. that
B
A
955. The area of a certain rectangle is 2400 square feet.
. its area will be increased 100 square feet. Two men can perform a piece of work in a certain time one takes 4 days longer. and that B. The sum of the contents of two cubic blocks
the
of the heights of the blocks is 11 feet.
952. triangle is 6.
overtook
miles. A rectangular lawn whose length is 30 yards and breadth 20 yards is surrounded by a path of uniform width. If the breadth of the rectangle be decreased by 1 inch and its
is
length increased by 2 inches. and the other 9 days longer to perform the work than if both worked together. Find the number. if its length is decreased 10 feet and its breadth increased 10 feet. Find its length and breadth. and if 594 be added to the number.
P and
Q.
949.
. The square described on the hypotenuse of a right triangle is 180 square inches. A number consists of three digits whose sum is 14. at
the same time
A
it
starts
and
B
from
Q
with the design to pass through Q. set out from two places. the square of the middle digit is equal to the product of the extreme digits. each block.
953.
Two
starts
travelers.
is
407 cubic feet.
sum
Find an edge of
954.REVIEW EXERCISE
301
945. was 9 hours' journey distant from P. distance between P and Q. the area lengths of the sides of the rectangle.
951. and travels in the same direction as A.
Find the number.
A and
B.
REVIEW EXERCISE
978. Insert 22 arithmetic means between 8 and 54.. and so on.
986. The Arabian Araphad reports that chess was invented by amusement of an Indian rajah.
980..
How many
sum
terms of 18
+
17
+
10
+
amount
.
v/2
1
+ +
+
1
4
+
+
3>/2
to oo
+
+
. such that the product of the and fourth may be 55. 4 grains on the 3d.. The sum
982.
1.
to
n terms.
to 105?
981. doubling the number for each successive square on the board.2
.
992. is 225.
of n terms of 7
+
9
+ 11+
is
is
40.3 '
Find the 8th
983.
Find four perfect
numbers.---
:
+
9
-
-
V2
+
.
Find the number of grains which Sessa should have received.+ lY L V. and the sum of the first nine terms is equal to the square of the sum of the first two.-.
.1
+
2.
first
984. Find the
first
term. named Sheran. and the
common
difference.04
+
. The 21st term of an A. Find four numbers in A.
989.
all
A
perfect number
is
a number which equals the sum
divisible.
990.
987. Insert 8 arithmetic means between
1
and
-.001
+
.01
3. 2 grains on the 2d. P. The
term.
to infinity
may
be 8?
. P..
to oo. then this
sum multiplied by
(Euclid.
303
979.
985.
"(.
Find
n.
What
2 a
value must a have so that the
sum
of
+
av/2
+
a
+
V2
+
.
0. Find the sum of the series
988. who rewarded the inventor by promising to place 1 grain of wheat on
Sessa for the
the 1st square of a chess-board.
of n terms of an A.
If
of
2
of
integers + 2 1 + 2'2
by which
is
it
is
the
sum
of
the series
2 n is prime. and of the second and third 03... Find the value of the infinite product 4
v'i
v7-!
v^5
.)
the last term
the series
a perfect number.-. P. 5
11.001
4.
ABC
A A
n
same
sides.
1000. prove that they cannot be in A.
999.
512
996.
is 4. are 28 and find the numbers.
997. P.304
ELEMENTS OF ALGEBRA
993. in this circle a square. and G. (I) the sum of the perimeters of
all
squares. areas of all triangles. The sides of a second equilateral triangle equal the altitudes of the first. Find (a) the sum of all circumferences. ft. Under the conditions of the preceding example. at the same time. c. many days will the latter overtake the former?
. One of them travels uniformly 10 miles a day. P. P. find the series. 995. the sides
of a third triangle equal the altitudes of the second. Each stroke of the piston of an air
air contained in the receiver. Insert 4 geometric means between 243 and 32.
The sum and sum
. after how strokes would the density of the air be xJn ^ ^ ne original
density ?
a circle is inscribed. 1003. (6) after n
What
strokes?
many
1002.
pump removes
J
of the
of air
is
fractions of the original amount contained in the receiver. The other travels 8 miles the first day and After how increases this pace by \ mile a day each succeeding day.
third circle touches the second circle and the
to infinity.
and
if
so forth
What
is
the
sum
of the areas of all circles. (6) the sum of the infinity.
1001.
.
of squares of four
numbers
in
G. In an equilateral triangle second circle touches the first circle and the sides AB and AC. and so forth to Find (a) the sum of all perimeters. Insert 3 geometric means between 2 and 162.
998. Two travelers start on the same road. In a circle whose radius is 1 a square is inscribed. are unequal. The side of an equilateral triangle equals 2. If a. in this square a circle. P.
AB =
1004. The sum and product of three numbers in G. and so forth to infinity. are
45 and 765
find the
numbers.
and the
fifth
term
is
8 times
the second . P.
994. The
fifth
term of a G. inches. (a) after 5 strokes.
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.
which have been omitted from the body of the work Indeterminate Equahave been relegated to the Appendix.
xi 4-
373 pages.
A
examples are taken from geometry. 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.
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Half
leather.ELEMENTARY ALGEBRA
By ARTHUR SCHULTZE. great many
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methods given
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time for the teaching of physics or geometry. comparatively few methods are heretofore.
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duction into Problem
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time for the teaching of physics or geometry.10
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of elementary algebra here
is
simple and practical. To meet the requirements of the College Entrance Examination Board. book is a thoroughly practical and comprehensive text-book. Logarithms.
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The Advanced Algebra is an amplification of the Elementary. The Exercises are very numerous and well graded. save Inequalities. The more important subjects
which have been omitted from the body of the work Indeterminate Equahave been relegated to the Appendix. which has been retained to serve as a basis for higher work. and commercial life. 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. comparatively few methods are
given. so that the tions.ELEMENTARY ALGEBRA
By ARTHUR Sen ULTZE.
. 64-66 FIFTH AVENUE. aoo pages. Proofs that are special cases of general principles obtained from the Exercises are not given in detail. SCHULTZE.
NEW YORK
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.
.
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.
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Secondary Schools
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BOSTON
CHICAGO
ATLANTA
. which have been selected with great care and can be found in the average high school library.
is
an excellent example of the newer type of
school histories.
$1. and a full index are provided.
THE MACMILLAN COMPANY
64-66 Fifth Avenue.
diagrams. All
smaller movements and single events are clearly grouped under these general movements.
i2mo. 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.40
is distinguished from a large number of American text-books in that its main theme is the development of history the nation.
Cloth. The book deserves the attention
of history teachers/'
Journal of Pedagogy. Studies and Questions at the end of each chapter take the place of the individual teacher's lesson plans. Maps. | 677.169 | 1 |
Hard 7th grade math problems
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Algebra 2 Worksheets (wonderful Algebra Sections #1)
Algebra Sections was posted on August 28, 2017 at 7:06 am. It is published at the Sectional category. Algebra Sections is tagged with Algebra Sections, Algebra, Sections..
Algebra
al•ge•bra
(al′jə brə),USA pronunciation n.
the branch of mathematics that deals with general statements of relations, utilizing letters and other symbols to represent specific sets of numbers, values, vectors, etc., in the description of such relations.
any of several algebraic systems, esp. a ring in which elements can be multiplied by real or complex numbers (linear algebra) as well as by other elements of the ring.
any special system of notation adapted to the study of a special system of relationship: algebra of classesALGEBRA COURSE SCHEMATIC: Algebra Schematic Shaded
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Wikibooksβ
Cambridge O Level Mathematics (Syllabus D)
Welcome to the Cambridge O Level Mathematics (Syllabus D) (4024) study guide. There are a total of 39 topics to be covered in this syllabus. This syllabus is assessed by two components: Paper 1 (non-calculator paper) (80 marks weighed at 50% of total) and Paper 2 (calculator paper) (100 marks weighed at 50% of total). Paper 1 consists of approximately 25 short-answer questions. Paper 2 consists of two sections: Section A (52 marks) consisting of six compulsory structured questions, and Section B (48 marks) where four of the five structured questions must be answered. This syllabus is available for examination in both the May/June and the October/November examination series. | 677.169 | 1 |
1,001 Math Problems (1001 Series)
The most sensible method to grasp math is to perform, perform, practice—and 1,001 Math Problems deals "mathophobes" and others who simply desire a little math tutoring the perform they should prevail. even if scholars need assistance calculating a tip or dealing with a standardized math attempt that may ensure their destiny, the 1,001 math questions during this helpful handbook presents them with the ability units they should grasp math, algebra, and geometry challenges.
Thought of to be the toughest mathematical difficulties to unravel, note difficulties proceed to terrify scholars throughout all math disciplines. This new identify on the earth difficulties sequence demystifies those tough difficulties as soon as and for all by means of exhibiting even the main math-phobic readers uncomplicated, step by step advice and methods.
This approachable textual content experiences discrete items and the relationsips that bind them. It is helping scholars comprehend and follow the ability of discrete math to electronic computers and different glossy functions. It offers very good coaching for classes in linear algebra, quantity idea, and modern/abstract algebra and for desktop technology classes in facts buildings, algorithms, programming languages, compilers, databases, and computation.
Focus inequalities for features of self sustaining random variables is a space of chance concept that has witnessed an outstanding revolution within the previous few many years, and has functions in a large choice of components corresponding to desktop studying, facts, discrete arithmetic, and high-dimensional geometry.
Five 5/7 ft c. three 2/7 toes d. 2 5/7 ft 722. Karl is 4 instances as previous as Pam, who's one-third as previous as Jackie. If Jackie is eighteen, what's the sum in their a long time? a. sixty four b. fifty four c. forty eight d. 24 723. Lee used to be 1/4 as younger as Keenan 5 years in the past. If the sum in their a while is a hundred and ten, how previous is Lee? a. 20 b. 25 c. sixty five d. eighty five 724. 3 coolers of water according to video game are wanted for a baseball workforce of 25 gamers. If the roster is extended to forty avid gamers, what number coolers are wanted? a. four b. five c. 6 d. 7 725. A pancake recipe demands 1 half cups of flour with the intention to make 14 pancakes.
27,530. 15 b. $28,601. 50 c. $28,701. 50 d. $29,610. 50 558. The temperature in solar Village reached a hundred levels or extra approximately 15% percentage of the prior 12 months. approximately what number days did the temperature in sunlight Village climb to a hundred or climb to a hundred or extra? (1 yr = one year) around your solution. a. forty five b. fifty four c. fifty five d. sixty seven 559. a definite radio station performs classical track in the course of 20% of its airtime. If the station is at the air 24 hours an afternoon, what percentage hours every day is the station now not enjoying classical tune? a. eight b. 15. 6 c. | 677.169 | 1 |
other areas of mathematics, geometry is a continually growing and evolving field. Computers, technology, and the sciences drive many new discoveries in mathematics. Suitable for middle and high school students on the subject of geometry, this title highlights various developments in the subject.Read more...
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Editorial reviews
Publisher Synopsis
"...[an] excellent reference book...Highly recommended." - Choice "The definitions are clear and easy to understand...useful to teachers...Recommended for academic, public, and some high-school libraries." - Booklist "...a thorough resource for math students and instructors and would also provide excellent reference service...Recommended." - Library Media Connection"Read more... | 677.169 | 1 |
The Official SAT Question of the Day
Thursday, October 24, 2013
Quadratics - The Basics
We have started investigating quadratics and will soon be adding a new number system to the mix to help us capture concepts that quadratics give rise to that linear and exponential models do not. Stay tuned on that. As for the basics of quadratics here are the concept development lessons around their graphs and behavior as well as the PowerPoint that marches us toward unknown territory.
Practice on the basics can be found on Khan Academy and TenMarks. Here is the link to Khan Academy and we will be onboarding TenMarks in the coming weeks. | 677.169 | 1 |
Algebra: Form and Function by William G. McCallum, Eric Connally, Deborah Hughes-Hallett
This publication deals a clean method of algebra that specializes in educating readers tips on how to really comprehend the rules, instead of viewing them simply as instruments for different kinds of arithmetic. It is dependent upon a storyline to shape the spine of the chapters and make the fabric extra enticing. Conceptual workout units are integrated to teach how the data is utilized within the genuine global. utilizing symbolic notation as a framework, enterprise execs will come away with a tremendously more suitable ability set.
"Presenting the lawsuits of a convention held lately at Northwestern college, Evanston, Illinois, at the social gathering of the retirement of famous mathematician Daniel Zelinsky, this novel reference presents up to date insurance of subject matters in commutative and noncommutative ring extensions, specifically these regarding problems with separability, Galois conception, and cohomology.
On the center of this brief creation to classification thought is the assumption of a common estate, vital all through arithmetic. After an introductory bankruptcy giving the fundamental definitions, separate chapters clarify 3 ways of expressing common houses: through adjoint functors, representable functors, and bounds.
5 71. 53f REVIEW EXERCISES AND PROBLEMS FOR CHAPTER 1 In Problems 73–75 assume v tickets are sold for $p each and w tickets are sold for $q each. 73. What does the expression vp + wq represent in terms of ticket sales? 94. Group expressions (a)–(f) together so that expressions in each group are equivalent. Note that some groups may contain only one expression. 2 3 5 10 (a) + (b) (c) k k 2k 2k 74. Write an expression for the average amount spent per ticket. 75. Suppose v = 2w and q = 4p. Rewrite the expression in Problem 73 in terms of w and q.
There could be other solutions as well. With some equations, it is possible to see from their structure that there is no solution. Example 6 Solution For each of the following equations, why is there no solution? 3x + 1 (a) x2 = −4 (b) t = t + 1 (c) =1 3x + 2 (d) √ w + 4 = −3 (a) Since the square of any number is positive, this equation has no solutions. (b) No number can equal one more than itself, so there are no solutions. (c) A fraction can equal one only when its numerator and denominator are equal.
26. Quabbin Reservoir in Massachusetts provides much of Boston's water. At the start of 2009 the reservoir contained 412 billion gallons of water. 20. If x + y + z = 25, find the value of (y − 10) + (z + 8) + (x − 5). 21. If xyz = 100, find the value of (3x)(2y)(5z). x 22. If xyz = 20, find the value of (2z)( )(6y). 4 23. Rewrite the expression a + 2(b − a) − 3(c + b) without using parentheses. Simplify your answer. 24. A car travels 200 miles in t hours at a speed of r mph. If the car travels half as fast but three times as long, how far does it travel? | 677.169 | 1 |
AccessmathAccessMaths is a geometrical drawing program designed to provide a range of input options for students with upper extremity or fine motor disabilities. The program can be operated using the keyboard, mouse, track ball, or Concept Keyboard. The program approaches drawing tasks using geometrical principles. The user defines reference points on the screen, such as the ends of lines or the center points of arcs, and the program automatically draws the shape. The program's drawing tools allow the swift and accurate construction of lines, squares, rectangles, circles, semi-circles, ellipses, triangles (equilateral, isosceles and scalene), polygons (regular and irregular), parallelograms, arcs, and pie charts, all of which may be annotated by arrows and text. There are three preset Toolbox configuration levels to suit different ability levels of use, and users can customize these by adding or deleting tools, or creating completely new configurations. The drawing tools are complemented by a full set of editing facilities, including scale and rotation tools. The user can create grids to work on, import a background image, and create and edit text for annotating drawings. Other features incude three on-screen measuring tools (ruler, protractor, and set square) that can be manipulated freely around the drawing area; a zoom tool; a transform tool that enables users to modify an existing shape; a fill tool for coloring drawings; and a joined-line tool for the creation of complex polygons. Users can scroll around pages, move shapes simply by clicking on them, and export all or part of a worksheet or picture as a bitmap file. Three simple educational activities (an abacus, clock and number line) are included to give younger students a friendly introduction to the world of mathematics. Images from all three can be pasted directly into the main program as part of the drawing or for printing. Users decide which tools and features appear on screen, so that the appearance of the screen matches the ability of the user. Switch users can access the program through specialized Switch Access for Windows (SAW) Selection Sets. On-screen help is available via a Help dialogue box and an extensive set of help files. This program provides users with access to the tools necessary for the geometry components of key stages 2, 3, and 4 of the United Kingdom's National Curriculum. COMPATIBILITY: For use with IBM and compatible computers. SYSTEM REQUIREMENTS: 32 megabytes (Mb) random access memory (RAM); 5 Mb of hard disk space; Windows 95, 98, ME, NT4, or 2000. | 677.169 | 1 |
Elementary Statistics: A Step by Step Approach
McGraw-Hill
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Publish Date: 2008-10-28
Binding: Hardcover
Author: Allan G. Bluman
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ELEMENTARY STATISTICS: A STEP BY STEP APPROACH is for general beginning statistics courses with a basic algebra prerequisite. The book is non-theoretical, explaining concepts intuitively and teaching problem solving through worked examples and step-by-step instructions. This edition places more emphasis on conceptual understanding and understanding results. This edition also features increased emphasis on Excel, MINITAB, and the TI-83 Plus and TI-84 Plus graphing calculators; computing technologies commonly used in such courses. | 677.169 | 1 |
Beginning with a brief introduction to algorithms and diophantine equations, this volume aims to provide a coherent account of the methods used to find all the solutions to certain diophantine equations, particularly those procedures which have been developed for use on a computer. The study is divided into three parts, the emphasis throughout being on examining approaches with a wide range of applications. The first section considers basic techniques including local methods, sieving, descent arguments and the LLL algorithm. The second section explores problems which can be solved using Baker's theory of linear forms in logarithms. The final section looks at problems associated with curves, mainly focusing on rational and integral points on elliptic curves. Each chapter concludes with a useful set of exercises. A detailed bibliography is included. This book will appeal to graduate students and research workers, with a basic knowledge of number theory, who are interested in solving diophantine equations using computational methods.
Recensioner i media
'... should certainly establish itself as a key reference for established researchers and a natural starting point for new PhD students in the area.' E. V. Flynn, Bulletin of the London Mathematical Society | 677.169 | 1 |
Algebra I at CMS is a rigorous, fast-paced course which covers the content for both Algebra I and 8th grade math. This course provides the foundation for students as they prepare to enter high school and take Geometry, Algebra II, and later, more advanced mathematics courses. The core content taught in this course will provide the foundation for these advanced mathematics courses. It is essential that students study and complete homework daily. Students must show mastery (average of 87% or higher) throughout the year and pass the end-of-course assessment at the end of the year to earn their Algebra I credit.
Announcements
Unit 6 Test-Systems of Equations and Inequalities
We are currently working on Module 9 in the algebra 1 book. This module is over systems of equations and inequalities. The test will be sometime between the 15th and 22nd of January. Students should study the unit nightly using module 9 and handouts. | 677.169 | 1 |
MAS 5311 Introduction to Algebra 1
The text for the course is Dummit and Foote, but the topics should be essentially the same whichever text, if any, is used.
This exam is on Group theory. The exam will test for:
Ability to write clear, rigorous proofs.
A solid understanding of the basic definitions.
Thorough understanding of the statements and proofs of theorems.
Ability to apply theorems to specific situations.
Familiarity with a broad range of examples and ability to do computations with them.
All topics are examinable except for those explicitly excluded. In the case of very long proofs, exam questions may involve only a part of the argument. In exams it is generally allowed to apply main theorems by quoting them correctly, as long as doing so does not result in circular logic, as would be the case if the question is really asking for part of the proof of the theorem quoted.
The modules below (in order) correspond to the first six chapters of Dummit and Foote.
The section on generators and relations is not examinable. Topics: Definition of groups, basic examples, dihedral groups, symmetric groups, matrix groups, the quaternion group, homomorphisms and isomorphisms, group actions.
Subgroup lattices of particular groups should be studied but not memorized Topics: Subgroups, centralizers and normalizers, cyclic groups, subgroup generated by a subset, the subgroup lattice.
The lists of small groups in 5.3 should be studied, but not memorized. Topics: Direct products, the fundamental theorem for finite abelian groups (statement and applications are examinable, but not the proof), semidirect products, examples of groups of small order.
Sections 6.2 and 6.3 are optional and not examinable. Topics: p-groups, nilpotent groups, solvable groups; Not for Examination: Free groups, generators and relations | 677.169 | 1 |
Inverse Trigonometric Functions (Review)
Students investigate the concepts related to inverse trigonometric functions and then review the graphs for the inverse. They apply the functions using the context of the example problems provided in the lesson plan. The lesson plan includes directions for the teacher to follow for direct instruction. | 677.169 | 1 |
Mathematical Reasoning: Writing and Proof is a text for the first college mathematics course that introduces students to the processes of constructing and writing proofs and focuses on the formal development of mathematics.
The primary goals of the text are to help students: • Develop logical thinking skills and to develop the ability to think more abstractly in a proof oriented setting. • Develop the ability to construct and write mathematical proofs using standard methods of mathematical proof including direct proofs, proof by contradiction, mathematical induction, case analysis, and counterexamples. • Develop the ability to read and understand written mathematical proofs. • Develop talents for creative thinking and problem solving. • Improve their quality of communication in mathematics. This includes improving writing techniques, reading comprehension, and oral communication in mathematics. • Better understand the nature of mathematics and its language. Another important goal of this text is to provide students with material that will be needed for their further study of mathematics. Important features of the book include: • Emphasis on writing in mathematics • Instruction in the process of constructing proofs • Emphasis on active learning. • Includes material needed for further study in mathematics.
About the Authors
Richard Hammack is an associate professor of mathematics at Virginia Commonwealth University in Richmond, Virginia. A native of rural southern Virginia, he studied painting at Rhode Island School of Design before an interest in computer graphics and visualization led him to mathematics. He works mostly in the areas of combinatorics and graph theory. | 677.169 | 1 |
Algebra I: Stem and Leaf Plots I
CTC Math
Join with more than 217,000 students now confident in math because finally they can do it!
Learn at your pace, not somebody else's
Stop and rewind the teacher until you get it
Builds confidence
24 x 7 unlimited access when you want it
Catch up or you can even get ahead
Start getting much better grades
View the Tutorial
Each of the 1,400+ CTC Math tutorials last around 4-9 minutes and presents the concepts of the math lesson step-by-step. Using synchronised audio and animation which harnesses both audio and visual learning styles simultaneously. CTC math lessons can be studied at home or at school and even on some modern mobile phones. Obviously no one can force a young person to study math, but what we have done here is to provide materials that are interesting and stimulating in themselves, and which will encourage the student, once started, to continue studying.
Completing the Interactive Questions or Worksheet
Following each math tutorial there are interactive questions or an optional printable worksheet which tests the understanding of key concepts. Answers are entered into CTC Math automated marking system which then stores the results in each individual student's ongoing progress report. This brings in the third learning style - Kinaesthetic, which is the process of actually doing the math yourself.
One Page Summaries
A printable one page summary of each tutorial provides the student with concise and complete notes from the math tutorial. Ideal as a reminder for homework, revision and review.
View Worked Solutions
After answers have been submitted, printable fully worked solutions become available showing every step which should be taken to reach the correct answer, just as it should be done in a math exam.
Marking and Reports
Ongoing progress reports are built for each individual. This helps parents and students in identifying weaker areas requiring further study. Every math lesson a student attempts is logged and the results stored along with statistical information detailing when they passed, what they scored and how much time and effort they put in to pass that lesson. Parents can instantly see the areas where their children may need additional help. | 677.169 | 1 |
I want to advance my knowledge of mathematics to the point where I can look at math PhD research papers and appreciate them, because they currently look like foreign gibberish to me.
I have a PhD in math and most of the research literature still looks like foreign gibberish to me, unless you are talking about my sub-sub-sub-field that I studied in grad school, and even then, it's not that easy to read. I know the some of the basics of all the main branches of math, but generally not more than the basics.
I made a few mind map things like that for certain branches of topology because I didn't necessarily agree with the order I was taught things in. But overall, it would get to be sort of convoluted and a bit ambiguous at some points, if you wanted all those subjects. I don't think I would be brave enough to even attempt it. I'm not even sure about what order I would learn the undergraduate material in, in hindsight because I'm always changing the way I think about things and what looks like the right way to me now will probably be different a year from now.
It's hard to compare a subject like topology with calculus because although, technically calculus could be seen as a bigger field, it's usually covered in a set way in the curriculum with 3 or 4 standard classes. Topology breaks down into point-set, algebraic topology (including homotopy theory, homology, cohomology, each of which could be its own class), differential topology, and it's less standardized. Each of those topology classes also would have to be its own part of the flow chart because algebraic topology will require abstract algebra, for example, but other topology classes might not. There are other problems, such as the issue with point-set topology that it doesn't have any logical prerequisites, but it would be a bad idea to take it without having some familiarity with proofs and preferably real analysis.
Also, you'd run into issues like what books you are going to learn from and how that might affect the order in which you'd learn things. Something like Visual Complex Analysis would be a good book to read before taking real analysis because it's good preparation for the kind of thinking you need to do in order to really understand proofs, but normally complex analysis would probably be best taken after real analysis because it's a little more complicated. So, the flow chart would get very subjective and very convoluted.
Yes, this will no doubt be a lifelong pursuit, but part of the attraction to me is to be able to see connections between various branches of math and to be able to see symmetries is seemingly unrelatedThat being said, if anyone was to solve the Riemann hypothesis it certainly wouldn't be me because I am no Gauss or Euler, but I want to be able to at least attempt to tackle these mathematical mysteries with a vast and broad mathematical understanding in many differentI also attribute a lot of my interest in math to Euler's equation, but the reason it attracted my attention was that I found it puzzling and even irritating when I first saw it, since I didn't know the motivation for it. That lead me to discover complex analysis to dispel the mystery of Euler's equation. The thing is, you can't just think of more advanced math as just more of the same. It's very different from the math that you've probably learned so far. The scale is much greater. It doesn't take that long to learn stuff like complex numbers and trig and exponents. When you think of research level math today, you have to think in terms of thousands and thousands of pages of books and journal articles. Then, suddenly, it doesn't seem so easy to make connections between different subjects. Using Euler's equation as a model gives you a very misleading picture I thinkWell, it is good to know something about different fields, but most mathematicians are not capable of learning more than one thing in depth. They learn a little bit about a lot of subjects, which is helpful in their research, but they only have real expertise in one thing. Part of the problem is "publish or perish". That, together with the extreme complexity of today's world leads to a bit of a specialization contest because no one really has time to master a lot of different subjects in depth. Over the course of a whole career, you might get a little bit more flexibility, but especially at the beginning, there are too many constraints to be able to branch out much. Freeman Dyson had some article he wrote about birds and frogs, where the birds cover a wide area, but not in much depth, and the frogs just knew their little area very well. He said we needed both birds and frogs. But even a bird has limitations in how far they can see.
Most people end up studying a good chunk of your list, but usually not all the more advanced topics. | 677.169 | 1 |
Name: ___________________________ Date: ______________
Math 2 Course Information Worksheet
Directions:
Explore the Math 2 Course Information area.
Open and print this worksheet.
Recall the Course Information to answer the questions below. Refer back to the Course
Information area as needed.
Write your answers in the spaces provided.
Place this worksheet in your Math 1 notebook.
1. When entering the course, how do you get to the area where you would view the lessons for the
course?
___________________________________________________________________________
___________________________________________________________________________
2. What is your teacher's name? _________________________________________________
3. My teacher's contact information is:
Telephone number: _____________________________
E-mail address: _____________________________
4. I printed out a copy of my pace guide and added dates to it.
My start date is: ________________ My end date is: ________________
5. I know how to submit an assignment. (circle one) Yes No
6. In order to find my current grade, I would go to _____________________________.
7. I can find my teacher's comments and grades on assignments I submitted in the
__________________________.
8. I would go to _____________________ to resubmit an assessment to my instructor.
9. I know how to use my course e-mail. (circle one) Yes No
10. List any other questions you have about the Course Information area. Then practice using your
course e-mail by sending these questions to your instructor.
___________________________________________________________________________
_________________________________________________________________________
__
___________________________________________________________________________ | 677.169 | 1 |
MATHEMATICS course SYLLABUS...
More students to topics in Mathematics that
are often used in scientific approaches especially in economic and business
management .
The mathematic course contains many problems from the
basic to the advanced, to supply student with a wide range of abilities and
interests.
Beside some of the examples are simply designed to build skills,
every effort has been made to generalize problems, so that students can see
common uses and practical applications of the mathematics they are
studying, and appreciate the power of mathematics.
We hope that students will be able to seize the essence of all the matters and
also make use of mathematical concepts to the realities .
Objectives
There are three objectives :
1.
To introduce
Less | 677.169 | 1 |
Pre-Algebra
—————————————————————— Tuesday, September 27:
In class today, you completed your 6 TRAITS OF WRITING ESSAY FOR MATH. Homework: Do all odd-numbered problems under Practice & Apply on pages 101 and 102 in Chapter 3, lesson one.
——————————————————- Friday, September 23:
In class today, you completed a test on Chapter 2, and were handed-out a worksheet for you to try that is related to simplifying expressions:
——————————————————- Wednesday, September 21:
To help you prepare for the upcoming Chapter 2 Test,
Go to: THIS LINK (CLICK!!) Then click on Chapter 2, and work through each of the self-quizzes. Record your score. And, if you're unhappy with your score, try the quiz again.
Homework:
——————————————————- Monday, September 19: Learning Goals: Study Guide and Review of Chapter 2
Homework: Practice Test on page 93
————————————- Thursday, September 8:
In class today, you completed the Chapter 1 test. For homework, please do page 55 in your textbook titled "Getting Started". This is a pre-requisite skills check to be used as an introduction to our next chapter, Chapter two.
———————————————————————————- Tuesday, September 6: Learning Goals: Chapter 1 Review and Test Preparation
At home assignment – Complete this CHAPTER 1 PRACTICE TEST…
————————————————————————————— Friday, September 2: Learning Goals: Scatter Plots-Constructing and Interpreting them.
Homework: Complete worksheets that were begun in class; and do all even numbered problems on pages 43 and 44.
Important dates ahead…Practice Test on Chapter 1 next Tuesday, Sept 6, and the real Chapter 1 Test is scheduled for Thursday!
NEED HELP? Click HERE!
Homework for Monday, August 29:
Learning Goal: Variables and Equations; Identify and solve open sentences, and translate verbal sentences into equations
Complete: Textbook Pages 31 and 32: Odd numbers for problems 1-73. This assignment was started in class.
Heads-up!… There will be a Quiz based upon the content learned in lessons four and five. | 677.169 | 1 |
Most important and scoring topics in JEE advanced mathematics
"People will rate you, shake you and break you. But how strong you stand is what makes you. "
Mathematics remains to be a mystery for a lot of JEE Advanced aspirants. Over the years, it has been considered to be the most challenging subject. Here we discuss the different branches of IIT JEE Mathematics and how much importance they carry.
1) Algebra: The topics in this branch of Mathematics are:
a) Complex Numbers
b) Theory of Equations
c) Sequence and Series
d) Permutations and Combinations
e) Binomial Theorem
f) Probability
g) Matrices and Determinants
h) Functions and Relations
i) Trigonometry
A straightforward and high scoring branch, Algebra related questions are generally easy. The only thing one needs to hang around is the amount of practice. This is a high scoring topic and generally gives a boost to the final mathematics result.
2) Calculus: The topics in this branch of Mathematics are:
Limits
Continuity and Differentiability
Differential Calculus
Application of Derivatives
Indefinite Integration
Definite Integration
Area under a Curve
Differential Equations
A lengthy topic, Calculus challenges problem solving techniques of a JEE aspirant. This topic carries a lot of weightage. Generally, moderate-level questions are asked from this topic.
3) Co-Ordinate Geometry: The topics in this branch of Mathematics are:
Point
Straight Lines
Properties of triangle
Circles
Parabola
Ellipse
Hyperbola
Vectors
3D Geometry
One of the most important topics of Mathematics, Co-ordinate geometry challenges a student's ability to interpret, solve and apply concepts. A lengthy, conceptual and a bit difficult topic, one can master this branch only by a lot of day-to-day topic. Never skip this or you can face a lot of difficulties in attaining a respectable rank | 677.169 | 1 |
Background geometry and notation
Abstract
Projective geometry [7, 38, 106] provides this book with the basic mathematical background, on top of which an effective and robust metrology framework is developed. This chapter presents the notation conventions and specific details of projective geometry which will be employed in the later chapters. | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
Coverage of sequences and series are central to the author's approach, and as such, analyticity is defined using convergence of power series throughout. The result is that readers view analytic functions fundamentally different than differentiable functions from calculus, and in addition, the differences between complex and real analysis become apparent. Sequences and series of functions are foundational to understanding complex analysis, and uniform coverage of compact sets is included to interchange certain limit processes. While this is often glossed over in competing books, the author presents this discussion with rigor. Since power series is discussed initially, the definitions of exponential and trigonometric functions are delayed, allowing readers' definitions to be in terms of series. This approach emphasizes that definitions are natural extensions of their real counterparts. Each section ends with a "Summary and Notes" essay that gives readers an overview of the topics discussed as well as the historical background. Topical coverage includes: complex numbers; complex functions and mappings; analytic functions; Cauchy's integral theory; the Residue Theorem; harmonic functions and Fourier series; sets and functions; and results from advanced calculus. | 677.169 | 1 |
BUCKHALTER, STACI
Welcome to Osceola and to Algebra 1 Honors! Here are a few procedures that may answer your questions about my class and help the school year run smoothly.
BINDER
Each student is responsible for bringing his/her binder to school each day and keeping it up to date and organized. During the first week, students will learn how to set up and use the math section of his/her binder. Parents, I encourage you to review your child's binder often.
PLANNER/AGENDA
Students are required to purchase a planner/agenda from the school and bring it to school each day. They will write in the planner/agenda each day. Students and parents can refer to it for information on homework and upcoming tests.
SUPPLIES
A school wide supply list was sent home before school started. Students need to bring their binders to school each day with all the requested supplies so that they can complete school work. I strongly encourage students to have a calculator at home that he/she can access when completing homework. The recommended calculator is a TI-30Xa, as this is the one they will be allowed to use on the EOC and in my classroom. If you have any questions about the supplies, please let me know. Each day for math, students will need notebook paper, graph paper, pencils (and possibly lead), an eraser, and a composition notebook.
Textbooks
There will be a hardcopy textbook issued to each student that may be kept at home during the year. Homework assignments will often come from this book. Each student will also be issued a consumable textbook that will be kept at school for the duration of the school year. Students will receive one Module (chapter) at a time. When the Module is complete, I will recommend to the students to take it out of their binder and keep it in a safe place at home until the end of the year so it is available to use in preparing for Quarterly Assessments and EOC in the spring.
GRADES
Grades will be calculated according to this scale:
Homework/Participation 10%
Class Work/Quizzes/Learning Checks 40%
Tests/Projects 50%
HOMEWORK (HW) ASSIGNMENTS
There will be homework assignments throughout each week. I will be checking for completion (which means all work shown) on all homework assigned. Students earn a chapter homework grade. This homework grade will be determined by dividing the number of homework assignments completed on time by the number of homework assignments assigned for that chapter. Students will not receive credit for late homework.
ABSENCES/MAKE-UP WORK
Students are required to complete all missed work. It is the student's responsibility to complete his/her planner for any/all days absent. As well, it is the student's responsibility to find out if there was graded class work turned in and/or if there was homework checked; and if so he/she needs to get those papers turned in as soon as possible. All grades from absences will be a 0 until the work has been completed and turned in. If a quiz or test was missed due to an absence, it would be best for students to make arrangements to stay afterschool to make it up. Please check my website for the day's assignment when absent.
TESTS/QUIZZES/Quarterly Assessments
Quizzes will be given after every 3-4 lessons and chapter tests will be given every couple of weeks. Quarterly Assessments will be given according to the Marion County Algebra 1 Honors Focus Calendar. Assessment dates will be posted ahead of time. Some tests may be assigned to be completed online at home.
PROJECTS
We will have in-class and out-of-class projects. Projects will be real world application of math skills taught. Projects will be in the same grading category as tests.
CLASSROOM WISH LIST
If you would like to contribute to our classroom needs, here are some helpful items.
w0.7 lead wbox of tissues w notebook paper
wgraph paper w#2 pencils wBand-Aids
windividually wrapped candy (nut free) wgraph paper wexpo markers
COMMUNICATION
Communication is key when creating a successful year for students. Parents, review your child's agenda/planner and binder often. Use my website to check out what we are doing in class and to keep up with assignments in case your child missed class. Email me any concerns you have: Staci.Buckhalter@marion.k12.fl.us | 677.169 | 1 |
Introduction
Chapter 1. Introduction
KGeo is a program for interactive Geometry just like programs such as Euklid, Zirkel und Lineal or Kseg. It was
my wish to write an open source version of this software that is free for schools and combines the best of both (all)
worlds.
This manual describes KGeo Version 1.0.1. Not only do I want to give an introduction on how to
use use KGeo generally, but also to point to possible applications in the Math classroom. Whoever wants to add some
application of KGeo is very welcome: <marc.bartsch@web.de>.
Another aim is to publish KGeo in different languages. So far, it is only German and Enlgish, but I hope that French and
Spanish translation will part of KGeo soon. If you feel like translating KGeo into your language, do not hesitate to
write an e-mail to: <marc.bartsch@web.de>. | 677.169 | 1 |
Description
This interactive maths resource will identify trouble areas and suggest further studies, and is full of video tutorials and real-life examples to help you revise and practise, progress and get top exam results.
With Collins Revision Algebra you can now hone your mathematics skills wherever you are.
• Be inspired by interactive animations, exciting video clips of students teaching a problem, real-life examples of maths at work, worked exam questions, tutorials and more • Choose precisely which topics to revise and practise with material corresponding to the Collins New GCSE Maths scheme • Test yourself with interactive assessment questions that identify trouble areas and suggest relevant further revision
Videos are downloaded separately by the app, directly to your device, upon installation.
Collect all four Apps covering all four GCSE maths strands, for a total of 900 practice questions, 300 assessment questions and 130 video clips!
Reviews
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Chloë Milbourne February 18, 2013 | 677.169 | 1 |
Course Description
Description
This proof-based geometry course, based on a popular classic textbook, covers concepts typically offered in a full-year honors geometry course. To supplement the lessons in the text book, videos, online interactives, assessments and projects provide students an opportunity to develop mathematical reasoning, critical thinking skills, and problem solving techniques to investigate and explore geometry. Students are also introduced to a dynamic software tool, GeoGebra, through projects that they create. Additionally, students are invited to participate and ask questions in an open Help Room guided by instructors.
Transformations
Geometry Help Room
Each week, all active students are invited to an open Geometry Help Room, which is led by a rotating staff of instructors. Students are encouraged to come with questions or just to meet other online students. Topics reviewed vary each week.
The Geometry Help Room meets each Wednesday from 7 – 8 p.m. ET.
Sample Video Lecture
Sample Video
Sample GeoGebra Interactive classroom for individual or group discussions with the instructor. The classroom works on standard computers with the Adobe Flash plugin, and also tablets or handhelds that support the Adobe Connect Mobile app. | 677.169 | 1 |
So, it is advised to explore these books to make your knowledge foundation better. Central Board of Secondary Education as a supplementary practice for Science and maths. Books and their solutions of other subjects as Science, Social Science and Hindi are also available to download in PDF format. If you are having any suggestion for the improvement, you are welcome. The improvement of the website and its contents are based on your suggestion and feedback. In class 9, we have studied rational numbers and irrational numbers collectively forming real numbers. Getting LCM and HCF using prime factorisation and Euclid's division lemma.
Proving square roots of 2, 3, 5 are irrational numbers. We have learnt Degree and coefficient of terms of polynomial and their factorization in class 9. Now we will find the relationship between the zeroes and their coefficient. Use of factor theorem to find the zeroes of the polynomials with degree 2, 3 or 4.
You have studied a linear equation in one variable and two variables in previous class. Now there is solution of pair of linear equations in class 10, using various methods like graphically, elimination, substitution and cross multiplication. Factorization, Completing the square method and using quadratic formula. Checking the nature of roots using discriminant.
Questions based on daily life. To check whether the given sequence is in Arithmetic Progressions or not. To find the missing term of the AP and sum of the given sequence. To determine AP when sum to n terms is given.
Some application base questions from daily life. Proof of Pythagoras theorem and its converse. Area theorem and its applications. Representation of points in Cartesian plane and finding the distance between two points using distance formula. Section formula to find the ratio and mid points. Formula for area of triangle and to prove the collinearity of three points. Trigonometry is the oldest branch of mathematics. | 677.169 | 1 |
-calculus concepts. Can identify student short | 677.169 | 1 |
With Safari, you learn the way you learn best. Get unlimited access to videos, live online training,
learning paths, books, tutorials, and more.
The finite element method is capable of producing accurate approximate solutions for a wide variety of differential equations. The domain of the problem is broken into a finite number of geometrically simple subdivisions. These subdivisions, known as finite elements, are shown in Fig. 1.1. The exact solution is approximated on an individual element by low-order polynomial interpolation functions that attempt to represent the actual displacements that exist on the domain of the finite element.
Figure 1.1. A finite element mesh.
Errors are produced by the finite element model when the low-order ...
With Safari, you learn the way you learn best. Get unlimited access to videos, live online training,
learning paths, books, interactive tutorials, and more. | 677.169 | 1 |
Better than most books on the subject
This text is excellent. Easy to read, Geometry, by Ray C. Jurgensen far surpasses the content found in many of today's Geometry textbooks. Each chapter begins with a visual, tied into the subject of the text itself. Exercises are simple, yet effective in teaching the material in print form to readers .Each chapter contains an algebra review, which is a skill that is lacking in most high school geometry students. Each chapter also contains a challenge questions, along with application questions.
I firmly believe that schools and teachers should buy up these texts and use them as supplementary material for Geometry classes in high school. For the money, you cannot find a better book. I would wager, as well, that the books authored by the co-writers, Richard G. Brown, and John W. Jurgensen, would be worth investigating. Good writers flock together, and a well written math textbook is worth its weight in gold | 677.169 | 1 |
module 1 - MAC 1140 Module 1 Introduction to Function and...
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Unformatted text preview: MAC 1140
Module 1 Introduction to Function and Graphs Learning Objectives
Upon completing this module, you should be able to 1 2 Learning Objectives 3 Introduction to Functions and Graphs
There are four sections in this module: 1.1 1.2 1.3 1.4 Numbers, Data, and Problem Solving Visualization of Data Functions and Their Representations Types of Functions and Their Rates of Change
Click link to download other modules. Rev.S08 4 Let's get started by recognizing some common set of numbers. Rev.S08 Click link to download other modules. 5 What is the difference between Natural Numbers and Integers?
Natural Numbers (or counting numbers) are numbers in the set N = {1, 2, 3, ...}. Integers are numbers in the set I = {... -3, -2, -1, 0, 1, 2, 3, ...}. Rev.S08 Click link to download other modules. 6 What are Rational Numbers? Rational Numbers are real numbers which can be expressed as the ratio of two integers p/q where q 0 Examples: 0.5 = 3 = 3/1 -5 = -10/2 0 = 0/2 0.52 = 52/100 0.333... = 1/3 Note that: Every integer is a rational number. Rational numbers can be expressed as decimals which either terminate (end) or repeat a sequence of digits.
Click link to download other modules. Rev.S08 7 What are Irrational Numbers? Irrational Numbers are real numbers which are not rational numbers. Irrational numbers Cannot be expressed as the ratio of two integers. Have a decimal representation which does not terminate and does not repeat a sequence of digits. Examples: 2, 3 5, ! , 0.01001000100001.... Rev.S08 Click link to download other modules. 8 Classifying Real Numbers Classify each number as one or more of the following: natural number, integer, rational number, irrational number.
22 , ! 11 7 25 = 5 so it is a natural number, integer, rational number 25 ,
3 8, 3.14, .01010101..., 3 8 = 2 so it is a natural number, integer, rational number 22 3.14, .01010101...., and are rational numbers. 7
Click link to download other modules. ! 11 is an irrational number.
Rev.S08 9 Let's Look at Scientific Notation A real number r is in scientific notation when r is written as c x 10n, where and n is an integer. Examples: The distance to the sun is 93,000,000 mi. In scientific notation this is 9.3 x 107 mi. Rev.S08 The size of a typical virus is .000005cm. In scientific notation this is 5 x 10-6 cm.
Click link to download other modules. 10 Example
Example 1 Evaluate (5 x 106) (3 x 10-4), writing the result in scientific notation and in standard form. (5 x 106) (3 x 10-4) = (5 x 3) x (106 x 10-4 ) (- = 15 x 106 + (-4) = 15 x 102 = 1.5 x 103 (scientific notation) = 1500 (standard form) Rev.S08 Click link to download other modules. 11 Another Example
Example 2 Evaluate writing the answer in scientific notation and in standard form. 5 " 106 , !4 2 " 10 5 ! 106 5 106 = ! "4 = 2.5 ! 106"( "4) = 2.5 ! 1010 (scientific notation) 2 ! 10"4 2 10 = 25,000,000,000 (standard form)
Rev.S08
Click link to download other modules. 12 Problem-Solving Strategies
Problem: A rectangular sheet of aluminum foil is 20 centimeters by 30 centimeters and weighs 4.86 grams. If 1 cubic centimeter of foil weighs 2.7 grams, find the thickness of the foil. Possible Solution Strategies Make a sketch. Apply formulas. Rev.S08 Click link to download other modules. 13 Example
Problem: A rectangular sheet of aluminum foil is 20 centimeters by 30 centimeters and weighs 4.86 grams. If 1 cubic centimeter of aluminum foil weighs 2.70 grams, find the thickness. Solution: Start by making a sketch of a rectangular sheet of aluminum, as shown above. Since Volume = Area x Thickness we need to find Volume and Area. Then we will calculate the Thickness by Thickness = Volume/Area Because the foil weighs 4.86 grams and each 2.70 grams equals 1 cubic centimeter, the volume of the foil is 4.86/2.70 = 1.8 cm3 The foil is rectangular with an area of 20 centimeters x 30 centimeters = 600 cm2. The thickness is 1.8 cm3/600 cm2 =.003 cm
Rev.S08
Click link to download other modules. 14 Analyzing One Variable Data Given the numbers -5, 50, 8, 2.5, -7.8, 3.5 find the maximum number, minimum number, range, median, and mean. Arranging the numbers in numerical order yields -7.8, -5, 2.5, 3.5, 8, 50. Minimum value is -7.8; maximum value is 50. Range is 50 (-7.8) = 57.8 Median is the middle number. Since there is an even number of numbers, the median is the average of 2.5 and 3.5 or 3. The mean is ! 7.8 + (!5) + 2.5 + 3.5 + 8 + 50 = 8.53 5
Rev.S08
Click link to download other modules. 15 What is a Relation? What are Domain and Range? A relation is a set of ordered pairs. If we denote the ordered pairs by (x, y) The set of all x - values is the DOMAIN. The set of all y - values is the RANGE. Example The relation {(1, 2), (-2, 3), (-4, -4), (1, -2), (-3,0), (0, -3)} has domain D = {-4, -3, -2, 0, 1} and range R = {-4, -3, -2, 0, 2, 3} Rev.S08 Click link to download other modules. 16 How to Represent a Relation in a Graph?
The relation {(1, 2), (-2, 3), (-4, -4), (1, -2), (-3, 0), (0, -3)} has the following graph: Rev.S08 Click link to download other modules. 17 When do we use the Distance Formula?
We use the distance formula when we want to measure the distance between two points. The distance d between two points (x1, y1) and (x2, y2) in the xy-plane is d = ( x2 ! x1 ) 2 + ( y2 ! y1 ) 2 Rev.S08 Click link to download other modules. 18 Example of Using the Distance Formula.
Use the distance formula to find the distance between the two points (-2, 4) and (1, -3). d = (1 " ("2)) 2 + ("3 " 4) 2 = 32 + ("7) 2 = 9 + 49 = 58 ! 7.62
Rev.S08
Click link to download other modules. 19 Midpoint Formula
The midpoint of the segment with endpoints (x1, y1) and (x2, y2) in the xy-plane is & x1 + x2 y1 + y2 # , $ ! 2 " % 2 Rev.S08 Click link to download other modules. 20 Example of Using the Midpoint Formula
Use the midpoint formula to find the midpoint of the segment with endpoints (-2, 4) and (1, -3). Midpoint is: & ' 2 + 1 4 + ('3) # & ' 1 1 # , $ !=$ , ! 2 " % 2 2" % 2 Rev.S08 Click link to download other modules. 21 Is Function a Relation? Recall that a relation is a set of ordered pairs (x,y) . If we think of values of x as being inputs and values of y as being outputs, a function is a relation such that for each input there is exactly one output. This is symbolized by output = f(input) or y = f(x) Rev.S08 Click link to download other modules. 22 Function Notation y = f ( x) Is pronounced "y is a function of x." Means that given a value of x (input), there is exactly one corresponding value of y (output). x is called the independent variable as it represents inputs, and y is called the dependent variable as it represents outputs. Note that: f(x) is NOT f multiplied by x. f is NOT a variable, but the name of a function (the name of a relationship between variables). Rev.S08 Click link to download other modules. 23 What are Domain and Range? The set of all meaningful inputs is called the DOMAIN of the function. The set of corresponding outputs is called the RANGE of the function. Rev.S08 Click link to download other modules. 24 What is a Function? A function is a relation in which each element of the domain corresponds to exactly one element in the range. Rev.S08 Click link to download other modules. 25 Here is an Example Suppose a car travels at 70 miles per hour. Let y be the distance the car travels in x hours. Then y = 70 x. Since for each value of x (that is the time in hours the car travels) there is just one corresponding value of y (that is the distance traveled), y is a function of x and we write y = f(x) = 70x Evaluate f(3) and interpret. f(3) = 70(3) = 210. This means that the car travels 210 miles in 3 hours. Rev.S08 Click link to download other modules. 26 Here is Another Example
Given the following data, is y a function of x? Input x 3 4 8 Output y 6 6 -5 Note: The data in the table can be written as the set of ordered pairs {(3,6), (4,6), (8, -5)}. Yes, y is a function of x, because for each value of x, there is just one corresponding value of y. Using function notation we write f(3) = 6; f(4) = 6; f(8) = -5. Rev.S08 Click link to download other modules. 27 One More Example Undergraduate Classification at Study-Hard University (SHU) is a function of Hours Earned. We can write this in function notation as C = f(H). Why is C a function of H? For each value of H there is exactly one corresponding value of C. In other words, for each input there is exactly one corresponding output. Rev.S08 Click link to download other modules. 28 One More Example (Cont.)
Here is the classification of students at SHU (from catalogue): No student may be classified as a sophomore until after earning at least 30 semester hours. No student may be classified as a junior until after earning at least 60 hours. No student may be classified as a senior until after earning at least 90 hours. Rev.S08 Click link to download other modules. 29 One More Example (Cont.)
Remember C = f(H) Evaluate f(20), f(30), f(0), f(20) and f(61): f(20) = Freshman f(30) = Sophomore f(0) = Freshman f(61) = Junior What is the domain of f? What is the range of f? Rev.S08 Click link to download other modules. 30 One More Example (Cont.) Domain of f is the set of non-negative integers { ,1,2,3,4...} 0 Alternatively, some individuals say the domain is the set of positive rational numbers, since technically one could earn a fractional number of hours if they transferred in some quarter hours. For example, 4 quarter hours = 2 2/3 semester hours. Some might say the domain is the set of non-negative real numbers [0, !) , but this set includes irrational numbers. It is impossible to earn an irrational number of credit hours. For example, one could not earn 2 hours. Range of f is {Fr, Soph, Jr, Sr}
Click link to download other modules. Rev.S08 31 Identifying Functions Referring to the previous example concerning SHU, is hours earned a function of classification? That is, is H = f(C)? Explain why or why not. Is classification a function of years spent at SHU? Why or why not? Given x = y2, is y a function of x? Why or why not? Given x = y2, is x a function of y? Why or why not? Given y = x2 +7, is y a function of x? Why, why not? Rev.S08 Click link to download other modules. 32 Identifying Functions (Cont.) Is hours earned a function of classification? That is, is H = f(C)? That is, for each value of C is there just one corresponding value of H? No. One example is if C = Freshman, then H could be 3 or 10 (or lots of other values for that matter) Rev.S08 Click link to download other modules. 33 Identifying Functions (Cont.) Is classification a function of years spent at SHU? That is, is C = f(Y)? That is, for each value of Y is there just one corresponding value of C? No. One example is if Y = 4, then C could be Sr. or Jr. It could be Jr if a student was a part time student and full loads were not taken. Rev.S08 Click link to download other modules. 34 Identifying Functions (Cont.) Given x = y2, is y a function of x? That is, given a value of x, is there just one corresponding value of y? No, if x = 4, then y = 2 or y = -2. Rev.S08 Click link to download other modules. 35 Identifying Functions (Cont.) Given x = y2, is x a function of y? That is, given a value of y, is there just one corresponding value of x? Yes, given a value of y, there is just one corresponding value of x, namely y2. Rev.S08 Click link to download other modules. 36 Identifying Functions (Cont.) Given y = x2 +7, is y a function of x? That is, given a value of x, is there just one corresponding value of y? Yes, given a value of x, there is just one corresponding value of y, namely x2 +7. Rev.S08 Click link to download other modules. 37 Five Ways to Represent a Function Verbally Numerically Diagrammaticly Symbolically Graphically Rev.S08 Click link to download other modules. 38 Verbal Representation Referring to the previous example: If you have less than 30 hours, you are a freshman. If you have 30 or more hours, but less than 60 hours, you are a sophomore. If you have 60 or more hours, but less than 90 hours, you are a junior. If you have 90 or more hours, you are a senior. Rev.S08 Click link to download other modules. 39 Numeric Representation H 0 1 ? ? ? ? 29 30 31 ? ? ? 59 60 61 ? ? ? 89 90 91 ? ? ? C Freshman Freshman Freshman Sophomore Sophomore Sophomore Junior Junior Junior Senior Senior Rev.S08 Click link to download other modules. 40 Symbolic Representation
"Freshman # Sopho if # C = f (H ) = $ # Junior if # Senior & if 0 ! H < 30 30 ! H < 60 60 ! H < 90 if H % 90 Rev.S08 Click link to download other modules. 41 H
0 1 2 29 30 31 59 60 61 89 90 91 Click link to download other modules. C
Freshman Di Re agra pre mm sen at tat ic ion Sophomore Junior Senior Rev.S08 42 Graphical Representation
In this graph the domain is considered to be [0, !) instead of {0,1,2,3...}, and note that inputs are typically graphed on the horizontal axis and outputs are typically graphed on the vertical axis. Rev.S08 Click link to download other modules. 43 Vertical Line Test
Another way to determine if a graph represents a function, simply visualize vertical lines in the xy-plane. If each vertical line intersects a graph at no more than one point, then it is the graph of a function. Rev.S08 Click link to download other modules. 44 What is a Constant Function? A function f represented by f(x) = b, where b is a constant (fixed number), is a constant function. Examples: f ( x) = 2 !1 f ( x) = 2 f ( x) = 2
Click link to download other modules. f(x) = 2 Note: Graph of a constant function is a horizontal line.
Rev.S08 45 What is a Linear Function? A function f represented by f(x) = ax + b, where a and b are constants, is a linear function. Examples: f ( x) = 2 x + 3 f ( x) = #5 x # f ( x) = 2 1 2 (Note:a = 2 and b = 3) 1" ! Note:a = #5 and b = # % $ & 2' (Note:a = 0 and b = 2) f(x) = 2x + 3 2x Note that a f(x) = 2 is both a linear function and a constant function. A constant function is a special case of a linear function.
Rev.S08
Click link to download other modules. 46 Rate of Change of a Linear Function
x y
Table of values for f(x) = 2x + 3. Note throughout the table, as x increases by 1 unit, y increases by 2 units. In other words, the RATE OF CHANGE of y with respect to x is constantly 2 throughout the table. Since the rate of change of y with respect to x is constant, the function is LINEAR. Another name for rate of change of a linear function is SLOPE. -2 -1 -1 1 0 1 2 3
Rev.S08 3 5 7 9 Click link to download other modules. 47 The Slope of a Line
The slope m of the line passing through the points (x1, y1) and (x2, y2) is Rev.S08 Click link to download other modules. 48 Example of Calculation of Slope Find the slope of the line passing through the points (-2, -1) and (3, 9).
(3, 9) m= "y 9 ! (!1) 10 = = =2 "x 3 ! (!2) 5
(-2, -1) The slope being 2 means that for each unit x increases, the corresponding increase in y is 2. The rate of change of y with respect to x is 2/1 or 2.
Rev.S08
Click link to download other modules. 49 Example of a Linear Function The table and corresponding graph show the price y of x tons of landscape rock.
1 2 3 4 25 5 75 100 X (tons) y (price in dollars) y is a linear function of x and the slope is The rate of change of price y with respect to tonage x is 25 to 1. This means that for an increase of 1 ton of rock the price increases by $25.
Click link to download other modules. "y 50 ! 25 = = 25 "x 2 !1 Rev.S08 50 Example of a Nonlinear Function
x y 0 0 1 1 2 4 Table of values for f(x) = x2 Note that as x increases from 0 to 1, y increases by 1 unit; while as x increases from 1 to 2, y increases by 3 units. 1 does not equal 3. This function does NOT have a CONSTANT RATE OF CHANGE of y with respect to x, so the function is NOT LINEAR. Note that the graph is not a line.
Rev.S08
Click link to download other modules. 51 Average Rate of Change Let (x1, y1) and (x2, y2) be distinct points on the graph of a function f. The average rate of change of f from x1 to x2 is y2 ! y1 x2 ! x1
Note that the average rate of change of f from x1 to x2 is the slope of the line passing through (x1, y1) and (x2, y2) Rev.S08 Click link to download other modules. 52 What is the Difference Quotient? The difference quotient of a function f is an expression of the form f ( x + h) ! f ( x ) where h is not 0. h Note that a difference quotient is actually an average rate of change.
Click link to download other modules. Rev.S08 53 What have we learned?
We have learned to:
1 54 What have we learned? (Cont.) 55 Credit
Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: Rockswold, Gary, Precalculus with Modeling and Visualization, 3th Edition Rev.S08 Click link to download other modules. 56 ...
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The book "Quantitative Aptitude for Competitive Exams" contains specific topics in Quantitative Aptitudewhich form a part of most of the Competitive Exams. The book contains to the point theory in all the chapters with illustrations followed by an exercise with detailed solutions | 677.169 | 1 |
Sets (Introductory Module in Algebra)
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A set is a group of individuals or entities which usually share a common characteristic. Sets are used in mathematics to group and collectively evaluate objects. When talking of sets, two important concepts emerge: cardinality and subsets. The cardinality of a set refers to its number of elements. An empty set contains not a single element. On the other hand, a subset is a set which is composed of some of the elements of a larger set. | 677.169 | 1 |
Algebra geometry
Created with Infinite Pre-Algebra. Algebra 1 Worksheets. Created with Infinite Algebra 1. Geometry Worksheets. Created with Infinite Geometry. Algebra 2 Worksheets. Learn high school geometry for free—transformations, congruence, similarity, trigonometry, analytic geometry, and more. Full curriculum of exercises and videos. Learn algebra 1 for free—linear equations, functions, polynomials, factoring, and more. Full curriculum of exercises and videos. Come here to see how well you know geometry problems where you have to know algebra.
Mathematics (from Greek μάθημα máthēma, knowledge, study, learning; often shortened to maths or math) is the study of topics such as quantity , structure. Each topic listed below can have lessons, solvers that show work, an opportunity to ask a free tutor, and the list of questions already answered by the free tutors. Introduction to Algebraic Geometry Donu Arapura Blow up of y 2 =x 3 In a sentence, algebraic geometry is the study of solutions to algebraic equations. IXL is the world's most popular subscription-based learning site for K-12. Used by over 6 million students, IXL provides unlimited practice in more than 7,000 topics.
Algebra geometry
Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents. Algebraic geometry is the study of geometries that come from algebra, in particular, from rings. In classical algebraic geometry, the algebra is the ring of. Mathematics (from Greek μάθημα máthēma, knowledge, study, learning; often shortened to maths or math) is the study of topics such as quantity , structure. IXL is the world's most popular subscription-based learning site for K-12. Used by over 6 million students, IXL provides unlimited practice in more than 7,000 topics. Pre-Algebra, Algebra I, Algebra II, Geometry: homework help by free math tutors, solvers, lessons. Each section has solvers (calculators), lessons, and a place where.
Software for math teachers that creates exactly the worksheets you need in a matter of minutes. Try for free. Available for Pre-Algebra, Algebra 1, Geometry, Algebra. Free math lessons and math homework help from basic math to algebra, geometry and beyond. Students, teachers, parents, and everyone can find solutions to their math. Welcome to IXL's Geometry page. Practice math online with unlimited questions in more than 200 Geometry math skills. Free math lessons and math homework help from basic math to algebra, geometry and beyond. Students, teachers, parents, and everyone can find solutions to their math.
Cool Math has free online cool math lessons, cool math games and fun math activities. Really clear math lessons (pre-algebra, algebra, precalculus), cool math games. Learn algebra 1 for free—linear equations, functions, polynomials, factoring, and more. Full curriculum of exercises and videos. Pre-Algebra, Algebra I, Algebra II, Geometry: homework help by free math tutors, solvers, lessons. Each section has solvers (calculators), lessons, and a place where. Geometry. Geometry is all about shapes and their properties. If you like playing with objects, or like drawing, then geometry is for you! Geometry can be divided into. Geometry. Geometry is all about shapes and their properties. If you like playing with objects, or like drawing, then geometry is for you! Geometry can be divided into.
Created with Infinite Pre-Algebra. Algebra 1 Worksheets. Created with Infinite Algebra 1. Geometry Worksheets. Created with Infinite Geometry. Algebra 2 Worksheets. Introduction to Algebraic Geometry Donu Arapura Blow up of y 2 =x 3 In a sentence, algebraic geometry is the study of solutions to algebraic equations. Welcome to IXL's Geometry page. Practice math online with unlimited questions in more than 200 Geometry math skills. Algebraic geometry is the study of geometries that come from algebra, in particular, from rings. In classical algebraic geometry, the algebra is the ring of. | 677.169 | 1 |
Sets
Here, we're gonna talk about sets. What will these sets contain? Other sets! Like a bunch of cardboard boxes that you open only to find more cardboard boxes, and so on all the way down.
You might ask "how is this relevant to a book on quantum computing?"
Well, hopefully we'll see a few answers later. For now, suffice it to say that math is the foundation of all human thought, and set theory – countable, uncountable, etc. – that's the foundation of math. So regardless of what a book is about, it seems like a fine place to start.
I probably should tell you explicitly that I'm compressing a whole math course into this chapter. On the one hand, that means I don't really expect you to understand everything. On the other hand, to the extent you ...
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MATH 1351 - Mathematics for Teachers II
Prerequisites: MATH 1350 with a grade of "C" or better, or equivalent
This course is intended to build or reinforce a foundation in fundamental mathematics concepts and skills. It includes the concepts of geometry, measurement, probability, and statistics with an emphasis on problem solving and critical thinking. | 677.169 | 1 |
Description
Mathematics is present in all aspects of engineering. An understanding of key mathematical concepts together with an ability to apply them successfully to solve engineering problems are vital skills that every engineering student must acquire. Mathematics for Engineers provides the easiest, most effective solution to teaching and studying mathematics for today's engineering student. MODERN and INTERACTIVE are the key themes of this accessible, step-by-step approach to the subject. Presented in an informal and accessible manner, the material is structured so that the students' knowledge and understanding are built up gradually. The interactive examples have been carefully designed to encourage students to engage fully in the problem solving process. Highlighted key points and use of icons throughout the book aid further understanding of the mathematical concepts being presented. show more
About Tony Croft
Dr Anthony Croft has many years experience of teaching mathematics to students of engineering gained both at De Montfort University Leicester and, more recently, at Loughborough University. He is currently the manager of the Mathematics Learning Support Centre at Loughborough which is the focus for a number of initiatives such as large-scale diagnostic testing of engineering students, and the formation of an 'open-learning' engineering mathematics environment. He is committed to ensuring that students have available the widest possible range of resources with which to enhance their mathematical skills. He has taken a particular interest in developing resources for those many students now embarking upon engineering degree programmes in the UK who have not had the opportunity to develop a solid foundation in mathematics through the traditional 'A' Level route. These include web-based support and CAL material as well as paper-based materials. He is currently working with the Engineering Council and the Institute of Mathematics and its Applications, together with colleagues from other universities, to produce new curricula for Chartered and Incorporated Engineers. Dr Croft is the author of two other very successful textbooks on engineering mathematics and a general textbook on Foundation Maths. He is a series editor of the Essential Maths for Students Series. Dr Robert Davison is a Principal lecturer in the Department of Mathematical Sciences at De Montfort University. He has worked in both further and higher education, with 20 years of teaching experience. He has specialised in the teaching of mathematics to engineering students, having taught on mechanical, electrical, and electronic engineering courses, and information technology courses. Dr Davison has taken an active interest in student learning and for many years represented his department at the Open learning Foundation. He presented a paper at the Open learning Foundation conference on 'Supported Self Study' and delivered a related paper at the IMA Conference on 'Teaching Mathematics to Engineers' at Loughborough in 1994. He has co-authored two other engineering mathematics books, a general foundation book, two books on discrete mathematics and he is editor of the Longman 'Essential Maths for Students' series | 677.169 | 1 |
Metallbautechnik Lernsituationen Technologie Technische Mathematik Lernfelder 5 Und 6 Lernsituation | 677.169 | 1 |
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Description
The two books in this series provide complete coverage of Units I and II of the new CAPE Pure Mathematics syllabus. They offer a sound platform for students pursuing courses at tertiary institutions throughout the Caribbean. Each topic is covered in depth with additional material in areas that students find most challenging.
Key features:
• Objectives at the beginning of each chapter aid planning, focus learning and confirm syllabus coverage
• Key terms are highlighted to develop students' vocabulary throughout the course
• A wide variety of exercises develops students' knowledge, application and ability in all areas of the syllabus
• Worked solutions throughout the text provide students with easy-to-follow examples of new concepts
• Graded exercises at the end of each section can be used to check students' understanding and monitor progress
• Checklists and at-a-glance summaries at the end of each chapter encourage students to review their understanding and go back over areas of weakness
• Examination-style questions at the end of each module give students plenty of practice in the types of questions they'll meet in the examinations
About the author
Dipchand Bahall has over 20 years' experience teaching Advanced Level Mathematics at Presentation College, Chaguanas, and St Joseph's Convent, St Joseph, in Trinidad and Tobago, and at Cayman Prep and High School. He holds a Masters Degree in Statistics, a Diploma in Education (Teaching of Mathematics) and a BSc in Mathematics. Dipchand is presently a Senior Instructor at The University of Trinidad and Tobago, Point Lisas Campus.
Please note that this is an eBook version of this title and can NOT be printed. For more information about eBooks, including how to download the software you'll need, see our FAQs page. | 677.169 | 1 |
ACC7
Monday: 2/26/18
Objective:8.F.B Use functions to model relationships between quantities.8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y)values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Essential Question: How do we make use of and understand data? I Can collect data about a function and represent it as a graph. I can describe the graph of a function in words.
Tuesday: 2/27/18
Objective:8.G.C Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. Essential Question: How do we determine volume? I Can recognize the 3D shapes cylinder, cone, rectangular prism, and sphere. I know that volume is the amount of space contained inside a three-dimensional figure.
Wednesday: 2/28/18
Objective:8.G.C.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. Essential Question: How do we find the volume of a cylinder? I Can find the volume of a cylinder in mathematical and real-world situations. I know the formula for volume of a cylinder.
Thursday: 3/1/18
Objective:8.G.C.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. Essential Question: What strategies do you use for finding the missing dimensions of a cylinder when given volume? I Canfind missing information about a cylinder if I know its volume and some other information. | 677.169 | 1 |
By connecting applications, modeling, and visualization, Gary Rockswold motivates students to learn mathematics in the context of their experiences. In order to both learn and retain the material, students must see a connection between the concepts and their real lives. In this new edition, connections are taken to a new level with "See the Concept" features, where students make important connections through detailed visualizations that deepen understanding.
Rockswold is also known for presenting the concept of a function as a unifying theme, with an emphasis on the rule of four (verbal, graphical, numerical, and symbolic representations). A flexible approach allows instructors to strike their own balance of skills, rule of four, applications, modeling, and technology.
Author Biography
Dr. Gary Rockswold has taught mathematics for 25 years at all levels from seventh grade to graduate school, including junior high and high school students, talented youth, vocational, undergraduate and graduate students, and adult education classes. He graduated with majors in mathematics and physics from St. Olaf College in Northfield, Minnesota, where he was elected to Phi Beta Kappa. He received his Ph.D. in applied mathematics from Iowa State University. He has an interdisciplinary background and has also taught physical science, astronomy, and computer science. Outside of mathematics, he enjoys spending time with his wife and two children. | 677.169 | 1 |
Tensor Calculus while presenting the concepts and Techniques begins with a brief introduction and history of tensors, followed by the study of systems of different orders, Einstein summation convention, kronecker symbol leading to the concepts of tensor algebra and tensor calculus. The authors conclude with a stimulating study in Riemannian geometry.
Key Features
∑ Self-contained
∑ Basic Concepts of Vectors and Matrices Explained
∑ Unique Presentation to Understand the Theory
∑ Illuminating Examples
NEW TO THE SECOND EDITION:
∑ Several new problems alongwith a stimulating study in Riemannian geometry. | 677.169 | 1 |
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1 Creating a Transition CCSSM Unit Example of a unit on expressions and equationsSVMI
2 Think in Terms of UnitsPhil Daro has suggested that it is not the lesson or activity, but rather the unit that is the "optimal grain-size for the learning of mathematics". Hence that was the starting point for our Scope and Sequence.Developers of High School: Patrick Callahan, Dick Stanley, David Foster, Brad Findell,Phil Daro, and Marge Cappo
3 New K-12 Math Curriculum Inspired by The Common Core State Standards The Gates Foundation and the Pearson Foundation are funding a large scale project to create a system of courses to support the ELA and Mathematics CCSS. These will be a modular, electronic curriculum spanning all grade levels. A Santa Cruz based company, Learning In Motion, is working to write the lessons.
7 The Big Idea of the UnitThe seventh grade unit "Expressions and Equations" addresses the mathematical concepts of variables, expressions and equations. Students learn that variables can represent unknown quantities. An algebraic expression is a generalization of number(s), operation(s) and/or variable(s) that represents one or more quantities. An algebraic equation is a statement that indicates two expressions are equal. Students represent real situations or mathematical relationships with expressions and equations. Students solve equations using mathematical methods such as, guess and check, using tables, graphs, deductive reasoning, and algebraic methods. Student use properties of operations to change expressions and equations into different forms. They model real life situations with expressions and equations. Algebraic methods of solving equations are learned and applied to everyday and mathematical situations.
13 Math Talks Math Talks are used to: • A daily ritual with the entire class for the purpose of developing conceptual understanding of and efficiency with numbers, operations and other mathematics such as geometry and algebra. (no more than 10 minutes per day)Math Talks are used to:• Support active student engagement through signaling• Review and practice procedures and concepts• Introduce a concept before diving into the lesson of the day• Support students in deepening their understanding of the Properties of Arithmetic and our Place Value System• Explore mathematical connections and relationships• Encourage students to construct viable arguments and critique the reasoning of others• Support students in using precise mathematical language in sharing their different strategies and approaches
14 Math TalkIf the lime and cherries are single digits, what are their values?
15 Math TalkIf the grape is a single digit, what values could the grape be?
16 Math TalkIf the plum and orange are single digits What are their values?
18 Problems of the MonthA program to foster school-wide participation in math and problem solving.
19 CCSS Mathematical Practices REASONING AND EXPLAINING2. Reason abstractly and quantitatively3. Construct viable arguments and critique the reasoning of othersOVERARCHING HABITS OF MIND1. Make sense of problems and persevere in solving them6. Attend to precisionMODELING AND USING TOOLS4. Model with mathematics5. Use appropriate tools strategicallySEEING STRUCTURE AND GENERALIZING7. Look for and make use of structure8. Look for and express regularity in repeated reasoning
20 How are the POM be used?The POM are used school wide to promote problem solving.Each problem is divided into five levels, A-E, to meet the learning development needs of all students.A great tool for Differentiated Instruction.Students, teachers and parents learn to ask questions and persevere in solving non-routine problems.The whole school celebrates doing mathematics at school.
21 It's better to solve one problem five different ways than to solve five different problems. George Polya
27 "A problem is not a problem if you can solve it in 24 hours." George Polya
28 Creating a PosterYour concluding thoughts on an explanation poster for a level you feel you have completedANDYour current thoughts on a status poster for a level you are still exploring.
29 Explanation Poster: The focus of your poster should be on how your findings can be justified mathematically and how your findings make sense. Include words and visuals (such as drawings) as a part of your justification.Status Poster: The focus of your poster should be on your processes so far and where you think you want to go next and/or questions/wonderings you have about this level. Include words and visuals as a part of your justification.*Remember to justify or explain your processes you have used so far and why they make mathematical sense as clearly as you can.
30 Gallery Walk Each group will display their poster. Each group selects a group member to be the docent to answer questions or provide clarifications/explanations.The other group members examines, explores, reviews the other groups' posters.There will be time for your group to re-assemble and discuss the information shared in the groups' posters.Please mind gallery walk norms and be respectful of the work and information shared.Share the organizing schemes in a gallery walk (whole group gallery walk – 10 minutes)Whole group reflecting on findings from gallery walk (whole group – 8 minutes)
38 Writing Algebraic Expressions Which two expressions are equivalent?
39 Which Equations Describe The Story? A pencil costs $2 less than a notebook.A pen costs 3 times as much as a pencil.The pen costs $9Which of the four equations opposite describe this story?Let x represent the cost of notebook.
40 Writing Equations To Describe A Story? Write an equation for Story 1 (S-1).Let x represent the number you are trying to find.(Solo, Partner, Class)
41 Writing Equations To Describe A Story? Write equations for Stories 2-6 (S-2S-6).Let x represent the number you are trying to find.Individual Think TimePartner Talk
42 Matching Stories and Equations Work together to match each story with an equation.Be sure to check to see whether any of the equations you and your partner have written down "match" the equations on the cards.
43 Sharing Matched Stories and Equations Docent/Data Collector Protocol Before we begin sharing our work, have one partner write down the agreed upon matches on the dry/erase board.Docent (Remains)Data Collectors (Visits)Discuss, revise, edit, and extend your mathematical thinking to create a Presentation Poster.
45 Showing Steps to Solving Equations Material ManagementDivide your construction paper into four quadrants.Glue cards E1 to E4 on the front of your posterGlue cards E5 and E6 on the back of your poster
46 Showing Steps to Solving Equations Matching "E1-E4" with "Steps to Solving"Select the steps needed to solve each of the equation cards E1-E447 Showing Steps to Solving Equations Once you've finished matching E1-E4, begin showing the steps and justifications for E5 and E648 Follow-Up Lesson Original pre-assessment, "Express Yourself" Carefully look over your original work.Please write about what you have learned since you did this task."Express Yourself (Re-visited)"Please use what you have learned to answer these questions. | 677.169 | 1 |
334OutcomesF11 - Math 334 Fall 2011 Outcome Statements...
Math 334 Fall 2011Outcome StatementsHomework AssignmentsSupplemental ProblemsIntegrals Expected KnownGraphs for Selected Homework ProblemsHow to read the Outcome Statements. The sections from the book we will use are listing in chronologyorder.In parenthesis following the title of each section is listed the expected number of class days to bespent on that section. For each section there are listed four categories of information.•Outcomes: these are specific learning objectives, the things you are expected to learn.•Reading: this indicates the section from the book that you should read before class.•Homework: this gives a list of assigned homework problems from the book.•Outcome Mapping: this associates the learning outcome with the assigned homework problems. Forexample, on the next page in the outcome mapping for the section "Some Basic Mathematical Models;Direction Fields" you will find "A. 22" which means that problem 22 is associated with outcomestatement A, which is "Model physical processes using differential equations."You are encouraged to use these outcome statements in your preparation for exams. All of the questions onthe exams will be designed to test your achieving of a randomly selected subset of the learning objectives asdescribed in the outcomes statements. You can test yourself on each outcome statement by checking if youcan correctly do the assigned homework associated with that outcome statement.
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Mathematics A Level at Ysgol Gyfun Penweddig
Course description
- Studying mathematics to Advanced Level sets a challenge to pupils who enjoy dealing with algebra, trigonometry, forces and statistics.
- The Advanced Level results of the Mathematics Department have been consistently excellent, with about 60% of candidates gaining grade A.
- Surveys have shown that those who study mathematics to 'A' level can earn a wage of up to 10% more than their contemporaries.
- Mathematics is taught through the medium of english and welsh.
Future opportunities
- Aberystwyth and Swansea universities are particularly enthusiastic in attracting Welsh students. Aberystwyth University can offer some opportunities to study through the medium of Welsh, and this department co-operates closely with the Physics Department.
Recent research there concentrated on errors in weather forecasting! Statistics is a strength here as well.
- Swansea University also offer tutorials through the medium of Welsh , and they offer courses which combine mathematics and the financial world , by using calculus to model market tendencies. 'WIMCS', the organisation established to encourage mathematics in
Wales, has its headquarters within this department.
- There are several attractive career options for students who wish to study mathematics. For those choosing to remain in education, extra money is available to train as mathematics teachers.
- There is an opportunity to study engineering, and work in the world of planning and the dynamic construction industry.
- Mathematicians are much in demand for work in the banking and trade sector, with world-wide openings.
How to apply
If you want to apply for this course, you will need to contact Ysgol Gyfun Penweddig directly. | 677.169 | 1 |
College Algebra—Modeling the City
This teaching manual was developed by Cathy Evins, an Institutional Partner of Engaging Mathematics and Mathematics Lecturer at Roosevelt University, with contributions by Barbara Gonzalez and Mary Williams. The manual provides a curricular guide to college algebra taught through Chicago-based problems and data modeling. Guidance is provided on "flipping" your classroom and incorporating a more active learning classroom environment. | 677.169 | 1 |
Maths data coursework
GCSE single science – GOV. Subject content for GCSE in single science which was introduced in 2016. Subject content for GCSE in single science which was introduced in 2016. Updated document with amendments to appendix 1 and 3 and a new appendix 4 listing maths data coursework and techniques.
In some geometries triangles add up to more than 180 degrees, every even number greater than 2 can be expressed as maths data coursework sum of two primes. Work out the lengths of the adjacent sides as you would to calculate gradient. Poisson distribution can predict volcanic eruptions, how many handshakes are required so that everyone shakes hands maths data coursework everyone else? Please ma20013 coursework this error screen to sharedip, while the world holds its breath in anticipation of the Age of the Petabyte, then the number represented by the expression is 'irrational'. At the forefront of a new educational journey with Harbour. This is simply the average of the x, individual shop hours will vary.
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In the test for rational and irrational numbers; with a bit of ingenuity you can enrich even quite simple topics to bring in a range of mathematical skills.
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Maths data coursework
maths data courseworkThe first of its kind in the world and includes Data Science, this is a puzzle that was posed over 1500 years ago by a Ma20013 coursework mathematician. Maths data coursework ipsum dolor sit amet, tracker software to model the motion of a bullet. I have supplemented these with some more possible areas for investigation. The University of Maths data coursework in Exeter, why does the twin paradox work? A professor at the Moscow Institute of Physics and Technology and an early experimenter in data science, my name is Eden and this is so good resource. Examples of parallel lines, solve all the clues in a level to make it onto the leaderboard.
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Number Power Review is designed to help students brush up on basic computation and problem-solving skills--from whole numbers to beginning algebra. This book presents the core of mathematical skills considered most important in today's changing world. These are also the skills most likely to appear on state high school competency tests, on adult high school program tests, on the GED Test, and on pre-employment tests. Because of the importance of problem solving in all areas of math, Number Power Review discusses fifteen problem-solving strategies. These strategies are designed to help build the critical thinking skills students need today. | 677.169 | 1 |
Systems of linear equations are ubiquitous to mathematicians. A recent author claims that "[w]ell over 75 percent of all mathematical problems encountered in scientific or industrial applications involve solving a linear system at some stage." At Adams State College, they appear in the course descriptions of at least seven mathematics courses, including two general education classes. We will look at some of the historical problems that spurred the development of solution algorithms, investigate some of the theory that has been derived from the study of these systems, and describe some of their modern uses. | 677.169 | 1 |
Course Information
Meeting Time and Location
Course Description
This is a course in algebra, similar to high school courses in algebra except that the pace will be faster. We will begin with some review of real-number concepts, and proceed into linear equations in one variable, mathematical modeling, polynomials, rational expressions, functions, lines, exponents, and radicals, equations, inequalities, and polynomial and rational functions.
Course Objectives
By the end of this course, you will have acquired many mathematical tools which include the ability to:
identify, distinguish, perform algebraic operations and find solutions to equations using the integer, rational, real and complex number systems.
use common algebraic methods to solve linear, quadratic, polynomial, radical, and absolute value equations and inequalities.
translate the written problem and create algebraic models to solve real-life problems.
use and create algebraic functions.
demonstrate your understanding of introductory language of mathematics through the use of proper mathematics notation.
Required Materials
Textbook
Intermediate Algebra for College Students, 7 Edition, by Robert Blitzer. Be sure to check the edition when purchasing your textbook; other editions have similar material, but the assigned problems may be different. No other materials are required for this class. You do NOT need to purchase a subscription to MyMathLab or pay to access any other online resources. You will not be expected to bring your textbook to class. If you prefer to purchase an electronic version of the text, you're welcome to do so.
Calculators
You are not required to purchase a calculator for this course, and you will not be permitted to use a calculator or other electronic device on any quizzes or exams. You are strongly encouraged to avoid using a calculator while working on homework.
Quizzes and Exams
Every Friday, we will have either a quiz or an exam. Both will be based on the suggested homework problems. All exams are cumulative; each exam will include some material from the previous exams. Use of notes, textbooks, calculators, electronic devices, or other materials will not be permitted during an exam. The exact topics on each quiz and exam will be announced in class.
If you need to miss class during a scheduled exam for a documented, excused reason (illness, family emergency, athletics), you must schedule a time to retake any exam within one week of the day the exam was given in class.
Grades
Basis of Final Grade
75%: Average of exam grades
25%: Average of quiz grades
Grading Scale
F
D
D+
C
C+
B
B+
A
0-59
60-64
65-69
70-77
78-83
84-89
90-93
94-100
Learning Differences
In keeping with college policy, any student with a disability who needs academic accommodations must call Learning Differences Program secretary at 824-3017, to arrange a confidential appointment with the director of the Learning Differences Program during the first week of classes.
Mercy Mission
This course supports the mission of Mercyhurst University by creating students who are intellectually creative. Students will foster this creativity by: applying critical thinking and qualitative reasoning techniques to new disciplines; developing, analyzing, and synthesizing scientific ideas; and engaging in innovative problem solving strategies.
Schedule
The exact topic covered on a particular date is subject to change. Exams and quizzes will be given on the day they are scheduled.
Homework
When we finish a section in the book, you should immediately begin working on the homework problems, listed in this syllabus. Stay up to date with homework, and get help if you cannot understand a problem after trying it on your own. Do not ignore a problem that you are struggling with. A weak spot in this foundation will lead to a bigger problem in the future.
Your work will not be collected. However, actually working through these problems is the key to your success in this class. Mathematics can only be learned through practice.
If you are having trouble with a topic, please come talk to me during office hours, ask questions in class, seek help from a classmate, or request a tutor. You are expected to try to work on all problems on your own first; when coming to my office, be prepared to show me what you've already tried. | 677.169 | 1 |
Copy of Math Editor tutorial
One of the new features of Symbaloo Lessonplans is the Math editor. This editor is designed to create mathematical functions in an easy way. In this tutorial we will show you how to use it.
This lessonplan is intended for creators of lessonplans, not for students | 677.169 | 1 |
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Mathematics - Advanced Level at Corby Business Academy
Course description
The course builds on all of the work you have done at GCSE and is examined through end of unit exams. There is no coursework at all!
Some of the work in C1 and C2 overlaps with the work done at GCSE at A/A* Grade.
Is this for me?
If you are a hard working well motivated student who enjoys algebra and problem solving. If you are going to get a B or higher GCSE grade and you want to earn 10% more money than other people. Then Advanced Level Mathematics is the course for you and you will then be able to choose from a vast range of related degree courses to study at university.
What courses does it go with?
Mathematics is a great course to support almost any other A Level subject. It works particularly well with Business and Science where the courses are complementary but also provides a balanced curriculum alongside English or other essay writing subjects. You can go onto Higher Education with just an A Level in Maths and University Admissions Tutors are always impressed by good maths grades.
Course content
What subjects will I study?
During the two year course you will extend your knowledge of Number, Algebra, Geometry and Statistics and learn explore new areas such as Calculus, (Differentiation and Integration). You will study six units. Three units in Y12 make up the Advanced Supplementary Level and a further three in Y13 make up the Advanced Level.
1. Pure Mathematics 1
2. Pure Mathematics 2
3. Statistics 1 or Mechanics 1
4. Pure Mathematics 3
5. Pure Mathematics 4
6. Statistics 1 or Mechanics 1 or Decision Mathematics.
There's more.....
In addition to your core studies you will also develop essential employment related skills in:
Application of number
Communication
ICT
Problem solving
Working as a team
Entry requirements
You will need a minimum of a B in Mathematics and 5 C's at GCSE preferably including English.
Assessment
NO coursework!
Each unit is assessed by a short exam and graded as E - A with points awarded. At the end of the course the points add up to give a final grade.
There are literally thousands of courses which are maths related and all future employers want staff with mathematical skills.
Careers
Mathematics is relevant to almost every career, including:
Actuary
Administrator
Archeology
Architect
Armed Forces
Bank Manager
Business Manager
Building Surveyor
Chemist
Customer Service
Dentist
Doctor
Designer
Electrical Engineer
Finance
Gas Engineer
Geologist
Human Resources
Marketing
Mathematician
Mechanical Engineer
Nurse
Optician
Physicist
Pilot
Police
Psychologist
Tax Manager
Teacher
Scientist
Self-Employed -Own Business Manager Social Worker
Statistician
Veterinarian
Further information
What kind of extra support will I get from Corby Business Academy?
Corby Business Academy is noted for its excellent student support with a focus on individual needs. All students have a personal tutor who guides and monitors them, helping them achieve their full potential. In addition, you also have access to specialist support from the study support team, student services and the learning centre. Mathematics staff are available for additional support from 3.30pm to 5.00pm on Tuesdays and Thursdays.
For more information:
If you wish to find out more information please contact Mrs Anderson or Miss Rowe | 677.169 | 1 |
Department of Mathematics
Math Notes/GT/ECS 505/Unit-1 NOTE: This handout is not a replacement of the prescribed textbooks. You are recommended to refer the following prescribed textbooks for further reading. You may find good number of examples for practice in these textbooks together with applications of graph theory in engineering problems and computer science. 1. 2. 3. Graph Theory With Applications to Engineering & Computer Science, Narsingh Deo, Prentice Hall of India Graph Theory, Frank Harary, Narosa Publishing House Graph Theory & Applications, Bondy & Murthy, Addison Wesley
The matter contained in will be discussed/explained in classroom lectures. I will appreciate your active participation. Wish you all success. Your course begins here! 1. GRAPH THEORY BASICS: DEFINITIONS & TYPES OF GRAPHS
The word graph refers to a specific mathematical structure usually represented as diagram consisting of points joined by lines. In applications the points may, for instance, correspond to chemical atoms, towns, electrical terminals or anything that can be connected in pairs. The lines may be chemical bonds, roads, wires or other connections. Applications of graph theory are found in communications, structures and mechanisms, electrical networks, transport systems, social
networks and computer science etc. A graph is a mathematical structure comprising a set of vertices, V, and a set of edges, E, which connect the vertices. It is usually represented as a diagram consisting of points, representing the vertices (or nodes), joined by lines, representing the edges (figure 1.1). It is formally denoted by G = (V, E). If in the graph the vertices are connected by directed lines, the graph is called directed graph or a digraph (figure 1.4). The graph is called labeled graph, if the vertices have labels or names (figure 1.2). If each edge has a weight associated with it, it is then called a weighted graph, (figure 1.3). The two vertices joined by an edge are said to be adjacent. The vertices are said to be incident with the edge that joins them and an edge is said to be incident with the vertices it joins. The two or more edges joining the same pair of vertices are called multiple/ parallel edges (figure 1.5). An edge joining a vertex to itself is called a loop (figure 1.5). A graph with no multiple edges and no loops is a simple graph(figure 1.1).
MATH NOTES/GT/ECS 505/UNIT-1
Prepared by Dr S Ahmad Ali, Math Department, BBDNITM 2011
1
4). Try to write the degree sequence of graphs in figures 1.6 .
MATH NOTES/GT/ECS 505/UNIT-1
Prepared by Dr S Ahmad Ali. 2. A vertex of degree 1 is pendent. is the length of a shortest path joining them. A loops count as 2. The two subgraphs are said to be vertex-disjoint subgraphs if there is no common vertex in the two subgraphs. b). Note that edge-disjoint subgraphs may have common vertices but vertex-disjoint subgraphs cannot have common edges. indegree of a vertex is the number of edges incident on it and the outdegree of a vertex is the number of edges incident from it. The degree of a vertex v is defined as the number of edges incident with v. then G –v denotes a subgraph obtained by deleting the vertex v from G. the degrees of the vertices of G. If v is a vertex in a graph G.7. A vertex having no edge being incident on it is isolated vertex. G – e denoted a subgraph obtained by deleting the edge e from G.5 and 1. By deleting a vertex. If G1 = (V!. In a digraph. in figure 1. G2 is an induced subgraph of G. The degree sequence of a graph G is the sequence obtained by listing. denoted by dist (a. The degree of isolated vertex is 0. Any two subgraphs G1 and G2 of the graph G are edge-disjoint if they do not have any edge in common.7 the degree sequence of G is (1. Math Department. Similarly. E) then V! ⊆V and E1 ⊆ E. In figure 1. G3 is a spanning subgraph of G. b). In figure 1. In figure 1. denoted diam (G). E1) is a subgraph of G = (V. BBDNITM 2011
2
. all edges incident on that vertex are also deleted.7 below. G3 and G4 are subgraphs of G. G2. in ascending order with repeats. For example.The distance between two vertices a and b. An induced (generated) subgraph is a subset of the vertices of the graph together with all the edges of the graph between the vertices of this subset. A subgraph of G is a graph all of whose vertices and edges are vertices and edges of G. is max dist (a. A subgraph H is a spanning subgraph if V (H) = V (G). The diameter of a connected graph.7. 2. The deletion of an edge does not delete the vertices of graph. G1. 3.
s . An infinite graph with finite number of edges has an infinite number of isolated vertices. If the degree is r the graph is r-regular.The Handshaking Theorem states that the sum of the degrees of the vertices of a graph is equal to twice the number of edges. BBDNITM 2011
3
. (Can u prove it?) A walk of length k in a graph is a succession of k edges joining two vertices. Nn is 0-regular graph.11 shows some bipartite graphs. Figure 1. Note that an edge cannot occur more than once in a walk. Math Department. a walk is an alternating sequence of vertices and edges. Figure 1. Notice that the allocation of the nodes to the sets A and B can sometimes be done in several ways. Kn is (n – 1)-regular and so has
n(n – 1) /2 edges. A graph whose vertex set is and edge set is a finite set is called a finite graph otherwise an infinite is an infinite graph (figure 1. A cycle graph consists of a single cycle of vertices and edges. all vertices have the same degree.
MATH NOTES/GT/EC S505/UNIT-1
Prepared by Dr S Ahmad Ali. See the systematic proof in next Section. The cycle graph with n vertices is denoted by Cn.3). The complete bipartite graph with r vertices in A and s vertices in B is denoted Kr. A bipartite graph is a graph whose vertices can be split into two subsets A and B in such a way that every edge of G joins a vertex in A with one in B. In a regular graph.12 shows some complete bipartite graphs.
A complete bipartite graph is a bipartite graph in which every vertex in A is joined to every vertex in B by exactly one edge. In other words. then it has ½ n r edges (from the Handshaking Theorem). If G is r-regular with n vertices. such that each edge is incident with the vertices preceding and following it. The null graph with n vertices is denoted Nn. Verify it! A complete graph is a graph in which every vertex is joined to every other vertex by exactly one edge. This follows readily from the fact that each edge joins two vertices (not necessarily distinct) and so contributes 1 to the degree of each of those vertices. The complete graph with n vertices is denoted by Kn. Verify it!
A null graph is a graph with no edges.
a b c d a e is a walk of length 5 between a and e. NOTE: For more definitions and graph theory terminology. … .5: The PROOF. The walk d a b c d is a circuit.
THEOREM 2.
THEOREM 2. A path is a walk in which all the edges and all the vertices are distinct. is called an open walk. In figure 1.
Sum of even degree vertices = Sum of odd degree vertices = No. vn . E) be a graph with vertices v1.
MATH NOTES/GT/ECS 505/UNIT-1
Prepared by Dr S Ahmad Ali. of edges in G.4: The
sum of all degrees of a graph is even.2:
(Handshaking Theorem): The sum of degrees of all vertices is two times the number of
edges. or e = n ( n − 1 ) /2. A walks.
Prove it yourself. which are not closed. If a walk begins and ends at same vertex.and e edges. Hence the theorem. Thus.1: The maximum
number of degree of any vertex in a simple graph of n vertices is n − 1 . if there are e edges in G then sum of degrees of all vertices of G will be equal to 2e. a walk which does not intersects it self is a path. By handshaking theorem Sum of degrees of all n vertices = 2 e
We know that maximum number of degree of any vertex in a simple graph of n vertices is n − 1 . therefore
n −1+ n −1+ n −1+
….14. The walk a b c d e is a path of length 4 between a and e. A circuit is a closed walk in which no vertex appears more that once.
THEOREM 2.
THEOREM 2. refer to classroom discussions and prescribed textbooks. a vertex may appear more than once in a walk. SOME IMPORTANT RESULS
THEOREM 2. Math Department.
Let G = (V. except the initial and final vertex. v2.
Prove it yourself. Prove it yourself. BBDNITM 2011
4
. deg (v1) + deg (v2) + … + deg (vn) = 2e.
2. That is.
maximum number of edges in a simple graph with n vertices is n ( n − 1 ) 2 . n term = 2 e
n ( n − 1 ) = 2 e . it is called a closed walk.
Let G be a graph with n vertices and e edges.3: In
a digraph. v3.
PROOF. Since each edge is
associated with two vertices and hence it contributes two degree in the sum of degree of all vertices. In other words.However.
PROBLEM 1.
Sum: Let G1 = (V1 . then u has n –1 adjacent vertices. …. 2.
a graph G. n − 1 . Intersection: The intersection of two graphs G1 ∩ G 2 is a graph whose vertex set and edge sets are
V1 ∩ V2 and E1 ∩ E 2 respectively.THEOREM 2. 4
(ii) 1. 2. Sum of odd degree vertices is an even number. 4.Sum of even degree vertices
This implies that. Let u be a vertex of degree n –1.
a simple graph with at least 2 vertices. Hence. 2. of edges in G) Sum of even degree vertices + Sum of odd degree vertices = 2 (no. Hence the degree of each vertex is less than or equal to n − 1 .2 2.2: What is the largest number of vertices in a graph with 35 edges if all vertices are of degree 3. the number of vertices of odd degree is even. 5
PROBLEM 1. this contradiction proves that a simple graph contains two vertices of same degree. 2.4: Show that compliment of a bipartite graph need not to be a bipartite.5: Find the size of the k – regular graph. 3
(iii) 2. Hence v is an isolated vertex. E1 ) and G 2 = (V2 .6: In PROOF. Hence v cannot be an isolated vertex. Math Department. OPERATIONS ON GRAPHS
Union: Let G1 = (V1 . The sum G1 + G 2 is defined as a graph whose vertex set is V1 ∪ V2 and edge set consists of those edges that are in G1 and in
G2 and the edges obtained by joining each vertex of G1 to each vertex of G2 . 3. Let v be a vertex of degree 0.3: Show that there exists no graph corresponding to the following degree sequence:
(i) 0. n ≥ 2.1: Find the number of vertices in a graph with 21 edges and 3 vertices of degree 4 and other vertices
each of degree 3. In PROOF. Hence possible degrees for n vertices of G are 0. PROBLEM 1. 3.
THEOREM 2. there are at least 2 vertices of the same degree. 1.
Ring Sum: The ring sum G1 ⊕ G2 is defined as a graph whose vertex set is V1 ∪ V2 and edge set is
( E1 ∪ E 2 ) − ( E1 ∩ E 2 ) . BBDNITM 2011
MATH NOTES/GT/ECS 505/UNIT-1
5
. of edges in G) . it has no loops or parallel
edges. The graph G is simple. Hence the theorem. PROBLEM 1. Suppose all vertices of G are of different degree.6: Show that there exists there exists no cycle of odd length in a bipartite graph. E 2 ) be two graphs such that V1 ∩ V2 = φ .
PROBLEM 1. of edges in G) Sum of odd degree vertices = 2 (no.
We know that in a graph G Sum of degrees of all vertices = 2 (no. E1 ) and G 2 = (V2 .
Let G be the simple graph with n vertices. number of odd degree vertices is even. PROBLEM 1. 5. E 2 ) be two graphs.
Prepared by Dr S Ahmad Ali.
3. 1. 3. The union of two graphs G1 ∪ G 2 is a graph whose vertex set and edge sets are V1 ∪ V2 and E1 ∪ E 2 respectively..7.
Further. In figure 4. then every
edge of G occurs in G1 or in G2 . For example. In other words. denoted as G ′ . It is easy to observe that a disconnected graph is union of two or more connected graphs. the complement of a graph G = (V.8: Find the compliment of the first graph in Problem 1.Compliment: The compliment of a graph G .1 the graph is a connected graph and that in figure 4.
4. a graph with m edges can be decomposed in 2m—1 — 1 different ways into pair of
subgraphs.2. the disconnected graph has two connected components. each connected subgraph of a graph G is a connected component
MATH NOTES/GT/ECS 505/UNIT-1
Prepared by Dr S Ahmad Ali.1. It may be noted that if G is decomposed into two graphs G1 and G2 . Math Department. intersection and ring sum of the following graphs:
PROBLEM 1. CONNECTED & DISCONNECTED GRAPHS AND COMPONENTS
A graph is said to be connected if its every vertex is reachable from every vertex. some of the vertices may occur in both
G1 and G2 . E) is a graph with vertex set V and edge set E' such that e ∈ E' if and only if e ∉ E.2 is a disconnected graph. BBDNITM 2011
6
.
Decomposition: A graph G is said to be decomposed in to two graphs G1 and G 2 if G1 ∪ G 2 = G and
G1 ∩ G2 = a null graph. Moreover. However. which are called connected components. in figure 4.
PROBLEM 1. A graph is self-complimentary if it is isomorphic to its compliment (see the definition of isomorphism in next section). A graph that is not connected is a disconnected graph. but not in both.7: Find the union. is a simple graph with same vertex set as that of G and in which any two vertices are adjacent if they are not adjacent in G .
THEOREM 4. say V2. Therefore.. Consider now the set V1 of all vertices of G that are joined by a path to a. there exists a path between v1 and v2. n k be the number vertices
in k components.3:
A simple graph with n vertices and k components can have at most edges.
THEOREM 4... An edge in a connected graph is called a bridge if its removal leaves a disconnected graph. .
PROOF.of the vertex set V of the graph G. therefore maximum number of edges in G is
1 2
k
∑n
i =1
i
( n i − 1) =
1 2
k
∑
i =1
n i2 −
1 2
k
∑
i =1
ni =
1 2
k
∑
i =1
n i2 −
n 2
MATH NOTES/GT/ECS 505/UNIT-1
Prepared by Dr S Ahmad Ali. n k = n
We know that the maximum number of edges in i th component can be n ( n − 1 ) 2 . . Thus there exists a partition. If G is a connected graph. Let there exists partition V1 and V2. Let us now suppose that G is a disconnected graph. . the set V1 does not contain all vertices of G as G is disconnected. there must
be a path joining two vertices.
( n − k )( n − k + 1 ) 2
PROOF. Obviously. i. therefore v1 and v2 must belong to the same component. Conversely. This completes the proof.of
G .
PROOF.. Math Department. BBDNITM 2011
7
. Let a ∈ V1 and b ∈ V2
such that no edge exists that connects a and b.e. In other words.. As each component is a connected graph... The theorem
is proved.
THEOREM 4.1: A graph G is disconnected if and only if its vertex set V can be partitioned into two non-
empty disjoint sets V1 and V2 such that there exists no edge in whose one vertex in V1 and other in V2. Thus the remaining vertices will be contained in another set... This completes the proof. Since the number of odd degree vertices in a graph is always even. Let n 1 . If such a partition exists then G is disconnected. n 2 . let us now suppose that G is disconnected. A vertex in a connected graph is called a cut-vertex if its removal makes the graph disconnected. Let a be any vertex of G. . n 2 . each graph can be expressed as the union of one or more
pair-wise disjoint connected graphs.2: If a graph (connected or disconnected) has exactly two vertices of odd degree. in G there exists a path between every two vertices.
n 1 . Let v1 and v2 be the vertices of odd degree and all other vertices are of even degree.. Let G be the graph with n vertices and k components.
More precisely. Thus.12: What is the total number of subgraphs and spanning subgraphs of K6. For example. Math Department. BBDNITM 2011
8
. and the pattern of vertex degrees. then the union of two paths has
atleast one circuit. and then the two graphs can not be isomorphic. here are two different ways of drawing C5 (figure 5. The correspondence itself is called an isomorphism. two graphs can be proved to be non-isomorphic by identifying some property that one possesses and the other does not.9: Show that if G is not connected then G' is connected. It is important to note that the converse is not true. The graphs in figure 5. if one graph has two vertices of degree 5 and another has three vertices of degree 5.
PROBLEM 1. such as the number of vertices. two graphs are isomorphic then (i) they have same number of edges. GRAPH ISOMOSRPHISM
The graphs G1 and G2 are said to be isomorphic if there exists a one-to-one correspondence between vertices in G1 and vertices in G2 such that there is an edge between two vertices in G1 if and only if there is an edge between the two corresponding vertices in G2.Using the following inequality
k
∑n
i =1
2 i
≤ n 2 − ( k − 1 )( 2 n − k )
. two isomorphic graphs may be drawn to look quite different.10: Show that if a graph (connected or disconnected) has exactly two vertices of odd degree. the two graphs are isomorphic. c → C. number of edges.1. and (iii) their degree sequence are same. (ii) they have
same number of vertices.3 further illustrate this idea. but if two graphs satisfy these conditions they are not necessarily be isomorphic. Consider the following correspondence between vertices in the two graphs shown in figure 5.
5. Further. The isomorphic graphs share a great many properties. For example. C corresponds to 3. The two graphs are isomorphic under the correspondence a → A. then there
must be a path joining two vertices.
PROBLEM 1.e.
MATH NOTES/GT/ECS 505/UNIT-1
Prepared by Dr S Ahmad Ali. PROBLEM 1. D corresponds to 4. b → B. i. Since there is an edge between two vertices in the first graph if and only if there is an edge between the two corresponding vertices in the second graph on the right. B corresponds to 2. 2
PROBLEM 1.11: Show that if the intersection of two paths is a disconnected graph.. if two graphs are isomorphic then all above conditions are satisfied. Therefore.
We get
1 2
k
∑n
i =1
i
( n i − 1) ≤
1 ( n − k )( n − k + 1 ) .2). A corresponds to 1.
. For example. A graph that contains an Euler line is called an Euler graph.
THEOREM 6. It is important to note that an Euler graph is always connected.2 are not Euler graphs because no Euler line exists.18: Show that there are 11 non-isomorphic graphs with 4 vertices. the graphs in figure 6. vn.16: Show that the two simple connected graphs with n vertices each of degree 2 are isomorphic.17: If a graph with n vertices is isomorphic to its compliment G'. v2. f → F.
PROBLEM 1. number of edges and their degree sequences). e → E. e1.15: Construct three non-isomorphic spanning subgraphs of the following graph
PROBLEM 1. EULER GRAPH
A closed walk in a graph that contains all edges of the graph exactly once is called an Euler line. v1.
PROOF. Let the Euler line starting from the
vertex v1 and traversing all edges of graph be v1.
The following are the important results for Euler graphs. PROBLEM 1. en.1 are both Euler graphs. show that n or (n -1) must be
divisible by 4.14: Show that the following graphs are isomorphic:
PROBLEM 1. …….13: Which of the following graphs are isomorphic:
(i)
(ii)
PROBLEM 1.
6. e2. The close walk containing all edges (Euler line) in two graphs are a b c d e a and A a B d C e D f E c B b E g D h A respectively. Therefore. The vertices in this circuit
MATH NOTES/GT/ECS 505/UNIT-1
Prepared by Dr S Ahmad Ali.
PROBLEM 1. Math Department.
Let G be an Euler graph.1: A connected graph is an Euler graph if and only if its all vertices are of even degree. v3.d → D. an Euler line exist in G. (also check their number of vertices. The graphs in figure 6. BBDNITM 2011
9
.
Consider v be any vertex. say u. so we can always exit from every vertex we enter. thus forming a circuit.19: Which of the following graphs are Euler graph.
(i) the complete graph K5. Let
the connected graph G is decomposed in to circuits. the remove the edges included in this circuit from G. it is an Euler line. start making a circuit from u. Since the degree of each vertex of the circuit is 2. Conversely. then it cannot have an Euler circuit. 3
PROBLEM 1. H must touch G – H at one vertex. Since all vertices in H are even. G is connected. Math Department.. it must have another edge incident of v.may not be all distinct but all edges are distinct. Hence all vertices are of even degree. KONIGSBERG BRIDGE PROBLEM
The Seven Bridges of Königsberg is a notable historical problem in mathematics. i. If this circuit does not contain all edges of G. there are atleast two edges incident on u as G is Euler graph and all vertices are of even degree. Since the vertex v is also of even degree. say H.
THEOREM 6. If we continue this process we arrive at the vertex already traversed. Since every vertex is even. G is the union of edge-disjoint
circuits. In this circuit every pair of edge ei and ei+1 contributes 2 to the degree of vertex vi+1. Thus. BBDNITM 2011
10
. let connected graph G has all vertices of even degree.2: The connected graph G is an Euler graph if and only if G can be decomposed into
circuits. The vertex v is also even so we can arrive at v so as to complete the circuit. Hence G is an Euler graph.
7. we get an Euler line. ……. suppose that G is an Euler graph. (ii) the complete bipartite graph K2. If this circuit contains all edges. the circuit must terminate at u. Thus v2. we get a subgraph. This completes the proof of the theorem. Therefore.21: Show that if a graph had a vertex of odd degree. All vertices in the remaining graph are of even degree. We repeat this process until all edges are covered. hence the degree of each vertex of G is even. G is an Euler graph..20: Which of the following graphs are Euler graphs. Let us start constructing a circuit from v and going through edges of G so that no edge is traversed twice. The city of Königsberg in
MATH NOTES/GT/ECS 505/UNIT-1
Prepared by Dr S Ahmad Ali. repeat the above process and remove another circuit formed from G until no edge is left. Now remove this circuit from G.e.
PROOF. Moreover. v3. Conversely. Thus the graph is decomposed into circuits. with w as the vertex on its other end. Consider a vertex u. Let other end of the edge incident on u is v. Now. . Again all vertices in H are even.
PROBLEM 1.. Its negative resolution by Leonhard Euler in 1735 laid the foundations of graph theory. and v1 gets 2 degree from e1 and en. Obliviously. vn are all of even degree. also find the Euler line if it is Eulerian:
(i)
(ii)
PROBLEM 1. of G.
a connected graph with a Hamiltonian path may or may not have a Hamiltonian circuit. we get a path that is called a Hamiltonian path.3. b. and included two large islands (A and D) which were connected to each other and the mainland (B and C) by seven bridges. BBDNITM 2011
11
.
The problem came to the attention of a Swiss mathematician named Leonhard Euler. But. HAMILTONIAN GRAPH
A closed walk in a connected graph that traverses every vertex exactly once (except initial and terminal vertices) is called a Hamiltonian circuit.
8. which only serves to record which pair of vertices (land masses) is connected by that bridge (see figure 7. trying to devise a route which crossed every bridge once only and returned to its starting point. f. g) as shown in figure 7. on the other hand. He finally proved (during the 1730s) that the task was impossible. e. Euler replaced each land mass with an abstract "vertex" or node. there are now five bridges in Königsberg (modern name Kaliningrad).1. and each bridge with an abstract connection. Math Department. What the citizens of Königsberg were seeking was an Euler line through this graph. Two others were later demolished and replaced by a modern highway. if any one edge from a Hamiltonian circuit is removed. it has Hamiltonian circuit a b c d e f g h a (note that all
MATH NOTES/GT/ECS 505/UNIT-1
Prepared by Dr S Ahmad Ali. It is obvious that the number of edges in a Hamiltonian path in a connected graph with n vertices is n –1. Russia) was set on both sides of the Pregel river. It is also important to note that in a connected graph. c. The degree sequence of the graph in figure 7. a. although only two of them are from Euler's time (one was rebuilt in 1935). The connected graph is figure 8. A connected graph that has a Hamiltonian circuit is called Hamiltonian graph.5) so it is not Eulerian and no Euler line exists. d.3. a Hamiltonian path is a subgraph of a Hamiltonian circuit. Thus. We can see this readily using the result of theorem 6. Thus. The three other bridges remain.2 is (3. Theorem 6. an "edge". It is because of this fact that a connected graph with a Hamiltonian circuit has a Hamiltonian path also.2). Something of your interest: Two of the seven original bridges were destroyed by bombs during World War II.Prussia (now Kaliningrad. but no one could find a way to do it.1 is a Hamiltonian graph.1.1 tells us that the graph is Eulerian if and only if every vertex has even degree. during their Sunday afternoon strolls. (namely. Further. The citizens of Königsberg used to amuse themselves. a Hamiltonian circuit in a connected graph with n vertices has exactly n edges.
2) /2 = (n2 – 5 n + 6) /2. .
(i) Hamiltonian but not Euler. Let H is a subgraph obtained on removing u and
v from G.5 is both Euler and Hamiltonian (Can you give the reasons. (iii) both Hamiltonian and Euler graph. By the hypothesis of the theorem. n ≥ PROBLEM 1. a connected graph that is an Euler graph may or may not be a Hamiltonian graph and vice versa. The graph is figure 8.22.
PROBLEM 1. Math Department.3 is a Hamiltonian graph as it has a Hamiltonian circuit a b c d e f a. Also.
THEOREM 8. it is not a Hamiltonian graph. the graph in figure 8. the graph in figure 8. The maximum number of edges in H may be
n -2
C2 = (n-1) (n .
PROOF. Show that the complete graph Kn.25: If G be a graph with n vertices. then show that G has a Hamiltonian circuit with each vertex
of degree ≥ n/2.(n2 – 5 n + 6) /2 = n. but this graph is not Euler graph because all vertices are not of even degree.1 (Dirac's Theorem): A simple connected graph with n (n ≥ 3) vertices is Hamiltonian if
deg v ≥ n/2 for every vertex v in G.24: Which of the following graphs are Hamiltonian graphs.(n2 – 5 n + 6) /2.
THEOREM 8.1) (n . m – deg u – deg v ≤ (n2 – 5 n + 6) /2 deg u + deg v ≥ m . This completes the proof.vertices have been used but not all edges). deg u + deg v ≥ (n2 – 3 n + 6) /2 . Lastly.2 has a Hamiltonian path a b c d but does not have a Hamiltonian circuit.
Therefore. (ii) Euler graph but not Hamiltonian graph.
It may be noted that an Euler line visits each edge of the graph exactly once whereas a Hamiltonian circuit visits each vertex of the graph exactly once. BBDNITM 2011
12
. Next. n ≥ 3. also find the Hamiltonian circuit
(i)
(ii)
PROBLEM 1.2: A simple connected graph with n vertices and m edges is Hamiltonian if
m ≥ 1/2 (n . Try it !). A complete graph is Hamiltonian graph.4 is Euler graph but not a Hamiltonian graph.
PROBLEM 1. The subgraph H has n – 2 vertices and number of edges in it is m – deg u – deg v or less than this. Let u and v be two non-adjacent vertices in G. Total number of Hamiltonian circuits in a complete graph with n vertices is (n-1) !/2.2) + 2.23: Draw a graph which is:
3. is Hamiltonian.
MATH NOTES/GT/ECS 505/UNIT-1
Prepared by Dr S Ahmad Ali. The connected graph is figure 8. or.
one possible approach to get the shortest rout is to calculate the weights of all the (n -1)! /2 Hamiltonian circuits and find the shortest one. n . BBDNITM 2011
13
.5. graph is not possible because maximum degree cannot exceed one less than the number of vertices. 23 1.1. the problem is to find a shortest Hamiltonian circuit. We can try to approach this problem of finding a shortest rout by constructing a graph. G is a disconnected graph.PROBLEM 1.. Given the distance between cities. e=nk/2
1. we do not know any efficient algorithm for solving this problem. in what order should he travel so as to visit every city precisely once and return home covering shortest distance. then u and v are not adjacent in G. we need to show that there exists a path between u and v.
9. Given that G is not connected. As we can notice that the approach discussed above is practically impossible because of the cumbersome calculation involved. Note. therefore.. We know that Sum of degree of all n vertices = 2e i. TRAVELLING SALESMAN PROBLEM
A sales man is required to visit a number of cities during a trip. the number of Hamiltonian circuits are 181440.9. Let G be a k – regular graph with n vertices and e edges. Let u and v be any two vertices. such a graph is not possible.e. Unfortunately. The traveling salesman has to visit each city (vertex in graph) exactly once and has to return home (the vertex from where he starts). that is. k + k + k + …. Therefore. For example. We know that in a complete graph of n vertices there exist (n -1)! /2 Hamiltonian circuits. so this graph is a weighted graph. 1. Let us represent cities by vertices in a graph and roads by edges connecting cities.e. k = 2 e.27: Solve the traveling salesman problem in the following graph:
ANSWERS & HINTS TO PROBLEMS
1. To show that G' is connected.
PROBLEM 1. n times = 2 e.2.3. 13 1. Hence they are adjacent
MATH NOTES/GT/ECS 505/UNIT-1
Prepared by Dr S Ahmad Ali.26: In a complete graph with n vertices there are (n . the graph is a complete graph. This implies that G has more than one component. Since we are given lengths of roads.. Math Department. (iii) Thee r four vertices given and maximum degree is 6. In (i) and (ii) there are odd number of odd degree vertices. if n = 10. If u and v are vertices in different components. i. Let us consider that there exists a road between every two city. which are connected.1)/2 edge-disjoint Hamiltonian circuits. if n is an
odd number ≥ 3.
1.12. deg v ≥ n/2. It now follows from above that n (n – 1) /4 is a positive integer.13. and total number of spanning subgraphs is 215. so they have same number of edges. that is .10. let us suppose that u and v are in the same component of G. BBDNITM 2011
14
.
MATH NOTES/GT/ECS 505/UNIT-1
Prepared by Dr S Ahmad Ali. 3 is not even.17. of vertices is 6 and no. Obiviously. Hence there is a path between u and v. and therefore.19.in G. If n > 2. (i) Degree of each vertex in K5 is even. Let in a graph G. Hence Kn is Hamiltonian. (i) Hamiltonian (ii) Not Hamiltonian 1. Therefore. Hence G' is connected. Let u and v be two vertices of G. Also total nuber of edges in G and G' together equals to number of edges in Kn. Since G and G' are isomorphic. u w v make a path u v in G'.1) 215 . we at once get deg u ≥ n/2. (ii) Each vertex in K2.
3.27. (i) Isomorphic (ii) Not isomorphic
1. The graph has two Hamiltonian circuits: A B C D E F H G A and A B C D E F G H A.20. Math Department. Let w be a vertex of G but in other component of G. therefore total number of subgraphs is (26 .25. 1. Thus. n or (n – 1) must be divisible by 4. We know that deg u + deg v ≥ n or.24. salesman should travel according to the first circuit A B C D E F H G A.15. Since we know that in any graph the number of odd vertices are always even in number. 1. the degree of each vertex is n –1.
1. In K6. it is Eulerian. 1. of edges is 15. . 1. therefore the two odd vertices of G must belong to the same component of G.
1. the vertices u and v are of odd degree and rest of the vertices are of even degree. no. Next. in Kn . n ≥ or n –2 > 2 n or n –1 > n /2.22. In Kn. 1. The total weight of the circuits are 22 and 25 respectively. we have n –2 > 0 Thus. each G and G' have n (n – 1) /4 edges. deg u + deg v ≥ n/2 + n/2
By comparison (comparison test in binomial expressions) for n =1. (i) Not Eulerian (ii) Eulerian 1. the degree of every vertex is greater than n /2. n (n – 1) /2. it is not Eulerian. | 677.169 | 1 |
Algebra 2 or Advanced Algebra is a class that is designed to help with college and career readiness.
Announcements
Day 2 Notes
Graphing exponential Functions:
I posted the notes and I would like the students to complete as much as possible. I really want them to do the first page. Use google and youtube to help with understanding. The students can also email me if they are not sure what to do. I will also post a key to the notes to help the students.
Unit 4 Part B Test
Unit 4 Part B test is still this Thursday. There is a study guide available online.
Unit 3 Test
Tomorrow there is a Unit test.
Unit 3 Quiz
There will be a quiz on Tuesday, October 17th. We will review Monday for the quiz.
The quiz is over:
Fundamental theorem of algebra
Difference/sums of squares and cubes
Alternate factoring methods
There is a Quiz Review under the files. The review will help witht the quiz. There is also an answer key available. | 677.169 | 1 |
Weber 7th and 8th Grade Math
Monday – 03/0506/18 – We will review and students will be given a study guide for common assessment 22 (Test) on Wednesday.
Wednesday 03/07/18 – We will begin class with a brief review of vocabulary, processes, and calculations involved with statistics sampling. The second half of class will be spent on CA #22.
Thursday 03/07/18 – Students will begin learning about dot plots. The La Math Course 2 book, page 576 will be used for practice problems.
Friday 03/08/18 – Students will begin working on box and whisker plots. The LA Course 2 book, page 588Begin 4th nine weeks – | 677.169 | 1 |
Description: The basic theme of this book is to study precalculus within the context of problem solving. The pace of this course will be faster than a high school class in precalculus. Above that, we aim for greater command of the material, especially the ability to extend what we have learned to new situations.3 views) | 677.169 | 1 |
Developmental Math Course Outcomes and Objectives
Transcription
1 Developmental Math Course Outcomes and Objectives I. Math 0910 Basic Arithmetic/Pre-Algebra Upon satisfactory completion of this course, the student should be able to perform the following outcomes and supporting objectives: A. Add, subtract, multiply and divide whole numbers and solve application problems using whole numbers. B. Add, subtract, multiply and divide fractions and solve applications problems using fractions. C. Add, subtract, multiply and divide decimals and solve application problems using decimals. D. Demonstrate an understanding of ratio and proportion and their applications. E. Demonstrate an understanding of percents and solve percent application problems. F. Use the English system and the Metric system to measure length, mass, and capacity. G. Demonstrate an understanding of and use the order of operations. H. Evaluate a variable expression given the values for the variables. I. Simplify a variable expression by collecting like terms. J. Solve simple linear equations. K. Estimate a reasonable answer to an application problem. L. Display several standard problem-solving techniques. M. Demonstrate an understanding of the study skills necessary for success in mathematics courses. II. Math 0950 Beginning Algebra I Upon satisfactory completion of Math 0950 Beginning Algebra I, the student should be able to perform the following outcomes and supporting objectives: A. Apply knowledge of real numbers, their operations and basic properties. B. Solve linear equations and linear inequalities with one variable. C. Identify components of the rectangular coordinate system, determine the equations of lines and graph lines. D. Solve systems of linear equations in two variables. III. Math 0960 Beginning Algebra II Upon satisfactory completion of MATH Beginning Algebra II, the student should be able to perform the following outcomes and supporting objectives: A. Simplify, evaluate, and perform basic operations on polynomials and exponential expressions. 1. Define polynomial, standard form, degree, coefficient, monomial, binomial, trinomial, and degree of polynomial in one variable.
2 2. Perform addition and subtraction with polynomials in one or more variables. 3. Multiply monomial expressions. 4. Use the product rule for exponents. 5. Use the power rule for exponents. 6. Use the products-to-powers rule for exponents. 7. Multiply a monomial and a polynomial. 8. Multiply two polynomials when neither is a monomial. 9. Use special-product formulas to multiply binomials with one or more variables. 10. Determine the product of two general binomials. 11. Determine the product of conjugate binomials. 12. Determine the square of a binomial sum. 13. Determine the square of a binomial difference. 14. Evaluate polynomials in several variables given values for each variable. 15. Divide monomial expressions. 16. Use the quotient rule for exponents. 17. Use the zero-exponent rule. 18. Use the quotients-to-powers rule. 19. Use the negative exponent rule. 20. Divide polynomials by a monomial. 21. Simplify expressions with exponents involving multiple properties of exponents. 22. Perform conversion from scientific notation to decimal notation. 23. Perform conversion from decimal notation to scientific notation. 24. Use properties of exponents to perform computations with numbers in scientific notation. 25. Solve applications using scientific notation. B. Factor polynomials and solve equations by factoring. 1. Factor the greatest common factor (GCF) from polynomial. 2. Factor four termed polynomials using the method of GCF. 3. Factor perfect square trinomials. 4. Factor the difference of two perfect squares. 5. Factor trinomials with leading coefficient of one. 6. Factor trinomials with leading coefficients not equal to one. 7. Factor the sum and difference of two perfect cubes. 8. Factor four termed polynomials using the difference of squares method. 9. Factor four termed polynomials involving the difference of squares with a sum or difference of cubes 10. Identify prime polynomials. 11. Write polynomial equations in standard form. 12. Solve polynomial equations by factoring. 13. Solve applications involving factoring. C. Simplify rational expressions and solve rational equations. 1. Define and recognize rational expressions. 2. Determine the values for variables for which the rational expressions are undefined. 3. Evaluate rational expressions at given numerical values. 4. Simplify rational expressions into lowest terms.
3 5. Determine the least common denominator for rational expressions. 6. Rewrite rational expressions to have a common denominator. 7. Perform algebraic operations on rational expressions. 8. Define and simplify complex rational expressions. 9. Solve equations involving rational expressions. 10. Solve applications involving rational equations Course Outcomes & Objectives Upon satisfactory completion of this course, the student should be able to perform the following outcomes and supporting objectives: A. Demonstrate the capacity to engage in logical thinking B. Critically read technical information. C. Exercise sound judgment in making personal and social decisions. D. Demonstrate an appreciation for the power of mathematics. E. Create mathematical models for a variety of problems. F. Use analytic and quantitative means to solve problems using mathematical models. G. Find the best methods for solving real-life problems. H. Create linear programming models for business/management problems. I. Use geometric means to find the optimal solution for business/management problems. J. Demonstrate the ability to produce and interpret data and draw conclusions about the world around us. K. Demonstrate the ability to determine simple probabilities and solve related problems. L. Demonstrate an understanding of the role of mathematics in measuring and predicting growth in: 1. the study of finance 2. the biological sciences 3. population M. Demonstrate an appreciation for the role of mathematics in individual and societal choices. N. Use the calculator/computer as a tool in computation and problem solving Course Outcomes and Objectives Upon satisfactory completion of MATH Intermediate Algebra, the student should be able to perform the following outcomes and supporting objectives: Outcome: Solve systems of linear equations in three variables. A. Verify solutions of systems of linear equations in three variables. B. Identify consistent, inconsistent, and dependent systems in three variables and learn to write the solutions for each type. C. Discuss visually what the solutions to linear equations in three variables look like in terms of planes in three-dimensional space. D. Solve systems of linear equations in three variables using the Elimination Method. E. Use systems of linear equations in three variables to model application problems and then solve those systems using the Elimination Method.
4 Outcome: Define, evaluate and perform operations on functions. A. Define relation, domain, range, function, and inverse function. B. Identify the domain and range of a relation and determine whether a relation is a function. C. Identify the domain and range of a relation and determine whether a relation is a function. D. Evaluate functions algebraically and graphically. E. Identify the domain and range of a function from its graph and the domain of a function algebraically. F. Use the vertical line test to identify functions. G. Perform operations (sum, difference, product, and quotient) on functions and determine their new domain. H. Form composite functions and find their domain. I. Find inverse functions algebraically. J. Use the horizontal line test to determine if a function has an inverse function. K. Graph inverse functions given the graph of a one-to-one function. Outcome: Solve, simplify, and graph linear Inequalities in two variables. A. Find the intersection and union of two sets. B. Solve compound inequalities involving and & or and express solutions in interval notation, in set-builder notation and by graphing on a number line. C. Graph linear inequalities in two variables. D. Graph systems of linear inequalities. E. Use compound inequalities and linear inequalities to solve application problems. Outcome: Rewrite and solve equations dealing with radical expressions. A. Define square root, cube root, nth root, principal root, conjugate, and non real roots. B. Evaluate radical expressions (square root, cube root, and higher roots) and radical functions. C. Use a calculator to approximate irrational radical expressions. D. Find the domain of radical functions. E. Simplify radical expressions involving the nth root of a and where a is any real number and n can be an even or odd natural number. F. Simplify exponential expressions and radical expressions using rational exponents.
5 G. Define and apply the properties need to multiply, divide, add, subtract, and simplify radical expressions. H. Multiply radical expressions with more than one term and using polynomial special products. I. Rationalize denominators containing one radical term including square roots, cube roots, and higher roots. J. Rationalize denominators containing two terms including one or more square roots. K. Solve radical equations with one and two radical terms and with rational exponents resulting in real solutions. L. Identify radical equations that have no solutions or extraneous solutions. M. Solve application problems involving radical equations and radical functions. Outcome: Solve quadratic equations and graph quadratic functions. A. Define quadratic functions and equations. B. Use the four ways to solve a quadratic equations including Zero-Product Property, Square Root Property, Completing the Square, and the Quadratic Formula and discuss when to use the methods. C. Use the discriminant to find the number and types of solutions to quadratic equations and to determine when expressions of the form ax2 + bx + cm> factor using rational numbers. D. Write quadratic equations from given solutions. E. Graph quadratic functions and find intercepts, vertex, axis of symmetry, and range using the f(x)=ax2 + bx + c and f(x)=a(x-h)2 + k forms. F. Maximize or minimize quadratic functions using the vertex. G. Solve equations quadratic in form. H. Find real and complex solutions to quadratic equations and equations quadratic in form. I. Find solutions to polynomial and rational inequalities. J. Model applications related to polynomial and rational inequalities. Outcome: Simplify expressions and solve equations involving exponential and logarithmic expressions. A. Define and graph exponential and logarithmic functions. B. Determine the domain and ranges of exponential and logarithmic functions. C. Identify properties of logarithmic functions including the Product, Quotient, Power, and Change of Base Rules and use the properties to rewrite logarithmic functions. D. Define and evaluate common and natural logarithms.
6 E. Write exponential equations as logarithmic equations and logarithmic equations as exponential equations using the definition of a logarithm: ay = x if and only if logax = y. F. Use the one-to-one property of exponential functions to solve one-to-one exponential equations. G. Use the one-to-one property of logarithmic functions to solve one-to-one logarithmic equations. H. Use the inverse relationship between the exponential function and logarithmic function to solve exponential and logarithmic equations. I. Evaluate applications involving exponential and logarithmic functions and equations including exponential growth and decay and periodic and continuous compounding Course Outcomes & Objectives Upon satisfactory completion of this course, the student should be able to perform the following outcomes and supporting objectives: A. Demonstrate decision making skills to solve applications from a variety of disciplines with an emphasis on health careers. B. Apply dimensional analysis and proportions to perform unit conversions C. Demonstrate an understanding of the metric system and how this system relates to the US Customary System of Measurement D. Interpret and analyze mathematical models, and decide on appropriate techniques and methods to obtain solution(s). E. Determine whether solutions are reasonable and appropriate to the application. F. Manipulate and solve equations generated by the mathematical models related to a variety of disciplines. G. Create and interpret graphs H. Demonstrate a working understanding of functions I. Evaluate and manipulate formulas J. Use technology to assist in problem solving
SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of
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Algebra I Pacing Guide Days Units Notes 9 Chapter 1 (1.1-1.4, 1.6-1.7) Expressions, Equations and Functions Differentiate between and write expressions, equations and inequalities as well as applying order
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Algebra I COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics, with an emphasis
Course Title: Math A Elementary Algebra Unit 0: Course Introduction In this unit you will get familiar with important course documents, such as syllabus and communication policy. You will register on MyMathLabALGEBRA 1/ALGEBRA 1 HONORS CREDIT HOURS: 1.0 COURSE LENGTH: 2 Semesters COURSE DESCRIPTION The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical
Course Objectives The Duke TIP course corresponds to a high school course and is designed for gifted students in grades seven through nine who want to build their algebra skills before taking algebra in
CD 1 Real Numbers, Variables, and Algebraic Expressions The Algebra I Interactive Series is designed to incorporate all modalities of learning into one easy to use learning tool; thereby reinforcing learning
Students will take Self Tests covering the topics found in Chapter R (Reference: Basic Algebraic Concepts) and Chapter 1 (Linear Functions, Equations, and Inequalities). If any deficiencies are revealed,
First Nine Weeks SOL Topic Blocks.4 Place the following sets of numbers in a hierarchy of subsets: complex, pure imaginary, real, rational, irrational, integers, whole and natural. 7. Recognize that the
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. RationalizeREVIEW SHEETS INTERMEDIATE ALGEBRA MATH 95 A Summary of Concepts Needed to be Successful in Mathematics The following sheets list the key concepts which are taught in the specified math course. The sheets
1 Algebra Scope and Sequence of Instruction Instructional Suggestions: Instructional strategies at this level should include connections back to prior learning activities from K-7. Students must demonstrate
Topic: Expressions and Operations ALGEBRA II - STANDARD AII.1 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbersModule: Graphing Linear Equations_(10.1 10.5) Graph Linear Equations; Find the equation of a line. Plot ordered pairs on How is the Graph paper Definition of: The ability to the Rectangular Rectangular
High School Mathematics Algebra This course is designed to give students the foundation of understanding algebra at a moderate pace. Essential material will be covered to prepare the students for Geometry.
Core provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness in math. This full-year course is aligned with Florida's Postsecondary
Math 1111 Journal Entries Unit I (Sections 1.1-1.2, 1.4-1.6) Name Respond to each item, giving sufficient detail. You may handwrite your responses with neat penmanship. Your portfolio should be a collection
Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,Name: Date: Start Time : End Time : Multiplying Polynomials 5 (WS#A10436) Polynomials are expressions that consist of two or more monomials. Polynomials can be multiplied together using the distributive
STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an
Alabama Course of Study: Mathematics (Grades 9-12) NUMBER AND OPERATIONS 1. Simplify numerical expressions using properties of real numbers and order of operations, including those involving square roots,
2.1 Algebraic Expressions and Combining like Terms Evaluate the following algebraic expressions for the given values of the variables. 3 3 3 Simplify the following algebraic expressions by combining like
MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purposeCore Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
Official Math 112 Catalog Description Topics include properties of functions and graphs, linear and quadratic equations, polynomial functions, exponential and logarithmic functions with applications. A | 677.169 | 1 |
12th GradeVolume 6 of the Learn Math Fast System teaches students how to solve complicated word problems by showing them how to build an algebraic equation. Written in the same conversational tone as theJacob's Geometry textbook has guided nearly one million students through the process of developing not just knowledge about Geometry, but a lasting understanding of Geometry concepts, principles,... Read More
Volume 7 of the Learn Math Fast System teaches High School Geometry in a unique way. Comes with five "Smart Cards" and four "Postulate Cards" to help students learn and remember the most important... Read More | 677.169 | 1 |
Edexcel gcse maths coursework tasks
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Edexcel gcse maths coursework tasks
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Math for critical thinking or college algebra
These problem-solving and critical-thinking why it is important to learn algebra may 2009 math courses at a community college or. Math 143 college algebra course syllabus use critical thinking skills to solve word problems which include maximizing math 143 spring 2017 course. I must take college level math for my associates degree, and i have a choice between college algebra or math for critical thinking which one is better for. Math 1314 1 collin county community college college algebra (critical thinking and communication skills) 2.
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Math for critical thinking or college algebra
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The mathematics department offers whether you advance into differential equations or stop at college algebra, critical thinking skills learned in. Algebraic thinking is evergreen's entry-point college-level math class the course develops problem-solving and critical-thinking skills by using algebra to solve. Critical thinking: where to begin college and university faculty critical thinking and mathematical problem solving good for all levels of math and.
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Impact of critical thinking on performance in mathematics among senior secondary school students wwwiosrjournalsorg. Common course outline: course discipline/number/title: goal 2/critical thinking, goal 4/mathematics/logical reasoning this first college level algebra course. Cranium crackers book 2 - ebook critical thinking activities in math grades: 5-6 critical thinking, mathematics view sample for general mathematics, algebra. In the critical thinking worksheets ' they need to use their basic math vocabulary and thinking process to use their knowledge of algebra to solve for a. | 677.169 | 1 |
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100+ Series Algebra
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Book Description ... More polynomials, factoring, plotting coordinates, graphing, and exercises involving radicals are all part of this book. Examples of solution methods are presented at the top of each page and puzzles and riddles gauge the success of skills learned. The numbers of the related standards for each activity can be found in the table of contents. Answer key provided.
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Further Maths at Swanwick Hall School
Course description
Further Mathematics
Some A level Maths students study Further Maths as well as Maths, allowing them to study a broader range of applied and pure mathematics modules. Universities consider this to be the best preparation for some degrees courses such as maths, physics, computer science and engineering.
Course Structure:
Further Mathematics AS
Decision 1 (examined January) a very new branch of mathematics, involving mathematical modelling to create algorithms which apply in business to the computer systems that run our modern world.
Statistics 2 (examined June) extends the data handling and probability concepts further from statistics 1 Further Pure 1 (examined June) introduces you to many of the foundations to a STEM degree course including the imaginary concept of complex numbers.
Entry requirements
In order to access this course students must have achieved a minimum of A*- C in five different subjects including English and Maths at GCSE and be taking Maths at A level. (see entry requirements for Maths A level)
How to apply
If you want to apply for this course, you will need to contact Swanwick Hall School directly. | 677.169 | 1 |
Algebra II
Course Summary
What is Algebra 2 all about?
Algebra 2 develops students' conceptual understanding, fluency, and ability to apply advanced functions. Students draw connections between function types. In particular, students apply skills learned early in the year with linear, quadratic, and polynomial functions to inform their understanding later in the year when they study rational, radical, and trigonometric functions. Students choose appropriate functions and restrictions, based in solid understanding of the features of the functions, to build functions that model contextual situations. Fluency is an important part of Algebra 2, as the ability to perform procedures quickly and easily allows students to more deeply understand concepts.
How did we order the units?
In Unit 1, Linear Functions and Applications,students review the features of functions through the study of inverse functions, modeling contextual situations, and operating with functions, systems of functions, and piecewise functions. Students will increase their fluency in identifying and analyzing features of linear functions through algebraic, graphic, contextual, and tabular representations. Students will use these features to effectively model and draw conclusions about contextual situations. The skills students develop in this unit will be applied and extended to other function types throughout the year, including quadratic, polynomial, rational, exponential, logarithmic, and trigonometric functions.
In Unit 2, Quadratics, students will revisit concepts learned in Algebra 1, such as features of quadratic equations, transformation of quadratic functions, systems of quadratic functions, and moving from one equation form to another (e.g., vertex form to standard form, standard form to intercept form). Increased fluency with quadratic equations and functions provides a strong base for studying polynomials, rational functions, and trigonometric identities. In this unit, students will also be introduced to a new type of number system, the imaginary numbers, and will identify and operate with imaginary solutions. As with Unit 1, students will apply quadratic equations to contextual situations, to systems of functions, and when translating between representations. Graphing calculators are introduced heavily in this unit and will be used for the remainder of the year.
In Unit 3, Polynomials, students will apply skills from the first two units to develop an understanding of the features of polynomial functions. Analysis of polynomial functions for degree, end behavior, and number and type of solutions builds on the work done in Unit 2; these are advanced topics that will be applied to future function types. Students will write polynomial functions to reveal features of the functions, find solutions to systems, and apply transformations, building from Units 1 and 2. Students will be introduced to the idea of an "identity" in this unit as well as operate with polynomials. Division of polynomials is introduced in this unit and will be explored through the concepts of remainder theorem as well as a prerequisite to rational functions.
In Unit 4, Rational and Radical Functions, students will extend their understanding of inverse functions to functions with a degree higher than 1. Alongside this concept, students will factor and simplify rational expressions and functions to reveal domain restrictions and asymptotes. Students will become fluent in operating with rational and radical expressions and use the structure to model contextual situations. In this unit, students will also revisit the concept of an extraneous solution, first introduced in Unit 1, through the solution of radical and rational equations.
In Unit 5, Exponential Modeling and Logarithms, students will model with exponential growth and decay, including use of the continuous compounding base, e, to solve contextual problems in finance, biology, and other situations. Students will learn that logarithms are the inverse of exponentials and operate with and graph logarithms fluently. Students will discover the strength of logarithms to identify solutions, features, and patterns in functions. Students will use exponential functions and logarithmic functions as part of a system of functions in modeling contexts.
In Unit 6, The Unit Circle and Trigonometric Functions, students will review geometric trigonometry as an introduction to trigonometric functions. Students will use sketches of the trigonometric functions of sine and cosine to develop understanding of the reciprocal trig functions, inverse trig functions, and transformational identities of trig functions. Features of trigonometric functions represented graphically will be translated to algebraic representations, and the features unique to trig functions will be explored and used in mathematical and application problems. Students will be introduced to the unit circle and will be expected to derive this easily. The Pythagorean identity will be used heavily in this unit, and students will be expected to know this identity and derive other forms of the identity for use in problems. This unit concludes the formal study of transformation, inverse, systems, features of functions, and using different functions to model contexts that began in Unit 1.
In Unit 7, Descriptive Statistics, students will build on their understanding of shape center and spread developed in middle school and Algebra 1 to use the mean and standard deviation to estimate population percentages. Students will synthesize their understanding of descriptive statistics with univariate and bivariate data through identification of strength and weakness of "evidence" from the analysis of statistical models to prepare them for the next unit.
In Unit 8, Inferential Statistics, students will become critical consumers of statistical information. Students will use their understanding of descriptive statistics, developed in Unit 7, as well as their understanding of probability, developed in Geometry, to make inferences about population parameters. Students will look critically at methodology in collecting data, citing procedures and practices that lead to reliable statistical models. In this unit, students will use technology to run simulations, using sound practices in probability concepts to make decisions and predictions.
This course follows the 2017 Massachusetts Curriculum Frameworks and incorporates foundational material from Algebra 1 where it is supportive of the current standards. | 677.169 | 1 |
Put the fun into functional skills with specially written engaging real-life situations that use mathematics. Students will become confident in applying maths to a range of problems that prepare them for work and life. Ready-to-use tasks can be slotted into everyday GCSE Maths lessons to cover Level 1 and Level 2 functional skills.
* Develop problem solving skills with 40 totally new and inspiring topics, some familiar and some unfamiliar, grouped into three sections: beginner, improver, advanced * Build, apply and secure the functional and process skills that are integral to the 2010 GCSE Maths Specifications * Challenge students at all levels with open questions that can be approached in different ways * Encourage discussion and collaboration between students to encourage team work and responsibility * Make mathematics relevant and useful, with scenarios such as Coastguard Search and Rescue, and Investigating Design and Cost in Tea Bag Production * Promote self-assessment with 'How did you find these tasks?' questions at the end of each topic | 677.169 | 1 |
GCSE Mathematics for OCR Foundation Problem-solving Book
4.11 - 1251 ratings - Source
A new series of bespoke, full-coverage resources developed for the 2015 GCSE Mathematics qualifications. Endorsed for the OCR J560 GCSE Mathematics Foundation tier specification for first teaching from 2015, this Problem-solving Book contains a variety of questions for students to develop their problem-solving and reasoning skills within the context of the new GCSE curriculum. Suitable for all Foundation tier students, this resource will stretch the more able and provide support to those who need it. Questions with worked solutions will help students develop the reasoning, interpreting, estimating and communication skills required to help them effectively solve problems. Encouraging progression by promoting higher-level thinking, our Problem-solving Books will help prepare students for further study.There are four of them on the flag and you know there are eight altogether, so 1 2
of the trigrams appear on the flag. (Rather neatly ... Finding the square root of a
large number without using a calculator or computer is difficult. You could start by
anbsp;...
Title
:
GCSE Mathematics for OCR Foundation Problem-solving Book
Author
:
Tabitha Steel, Coral Thomas, Mark Dawes, Steven Watson
Publisher
:
Cambridge University Press | 677.169 | 1 |
General Education Mathematics
MAT 110 or the equivalent with minimum grade of C or appropriate score on the Mathematics Placement Test.
III. Course (Catalog) Description
Course focuses on mathematical reasoning and the solving of real-life problems. Topics include: counting techniques and probability, logic, set theory, and mathematics of finance. Calculators/computers used when appropriate.
Construct and interpret Venn diagrams to solve problems using basic set vocabulary, notations, and operations.
Determine the validity of logical arguments by translating statements into symbolic language and analyzing truth tables, applying DeMorgan's Law, and comparing the statements to standard valid forms of arguments.
Apply techniques of the mathematics of finance to common personal financial situations found in everyday life.
Analyze and solve expected value and real-life problems using a variety of counting techniques to determine probabilityProbability with the Fundamental Counting Principle, Permutations, and Combinations
Events involving Not and Or; Odds
Events Involving And; Conditional Probability
Expected Value
VII. Methods of Instruction
Methods of presentation include lecture, discussion, demonstration, group work, and regularly assigned homework. Techniques will emphasize critical thinking and applications. Calculators/computers will be used where appropriate.
Course may be taught as face-to-face, hybrid or online course.
VIII. Course Practices Required
(To be completed by instructor.)
IX. Instructional Materials
Note: Current textbook information for each course and section is available on Oakton's Schedule of Classes.
Within the Schedule of Classes, textbooks can be found by clicking on an individual course section and looking for the words "View Book Information".
(To be completed by instructor.) Evaluation methods can include assignments, quizzes, chapter or major tests, individual or group projects, computer assignments and/or a final examination.
The following applies to the online section of this course only: This online course requires that students take their exams as directed by their instructor: either on campus at Oakton's Testing Center, at an authorized testing center with a face-to-face monitor, or remotely through a pre- approved testing service. (To be customized by instructor.) | 677.169 | 1 |
Pre-algebra
Pre-algebra
Step further into the world of algebra skills. Students will continue their studies on integers, order of operations, and numeric expressions in a 32-week 90 min course. They will dive deeper into algebraic expressions, equations, inequalities and graphing on a coordinate plane. Problem solving will be a key aspect of pre-algebra.
This course will be supplemented with tutor-made video lessons twice a week and online assignments through a game-based program provided by the tutor. Class time will be spent reinforcing video lessons through hands-on activities and extending the lessons for advanced learners. Students will need to come to class having viewed the lessons and completed their daily online assignments in order to stay on track. | 677.169 | 1 |
Welcome to Algebra 2
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Transcript of Welcome to Algebra 2
Welcome to AFM Mrs. Choren Prerequisites Algebra 1 a 4 on the Algebra 1 EOC Geometry Honors at least a B average at least a 3 on the Geometry EOC Supplies Classroom Procedures 1. Put your homework in the folder. 2. Take your assigned seat. 3. Get everything out for the day: calculator, computer, etc 4. Complete the Warm - Up. Mrs. Choren About Me 9th year teaching/1st year at LCS Undergraduate degree from UNC - Chapel Hill Master of Arts in Teaching from UNC - Charlotte National Board Certified Married 8 years Have a 17 month old son Have 2 dogs: Cosmo and Toby Grading Scale EVERY day you will need Pencil Notebook paper Graphing Calculator Positive Attitude 3 - Prong Notebook Tests/Projects: 35% Quizzes: 25% Homework: 20% Classwork: 20%
Final Exam: MSL (Measure of Student Learning) - State test, will count 25% of your final grade NO CELLPHONES! Other Electronic Listening Devices MAY be allowed later in the semester If caught with one: 1. Warning 2. Taken and you can pick up in the front office along with a parent contact | 677.169 | 1 |
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What is Y, when input is 0, 2, 4, 6 and output is 3, 4, 5, 6 and domain is 0, 2, 4, 6 | 677.169 | 1 |
Material Detail
Elementary Algebra Exercise Book I
This is a free textbook from BookBoon.
'Algebra is one of the main branches in mathematics. The book series of elementary algebra exercises includes useful problems in most topics in basic algebra. The problems have a wide variation in difficulty, which is indicated by the number of stars | 677.169 | 1 |
Find a Fairmount Heights worked for over 40 years as an engineer applying math to the solution of real-world problems. Algebra is the abstraction of basic arithmetic, using letters to stand in for specific known or unknown numbers. The abstract notation of algebra often gives new students difficulty, but the concept, when properly explained, is not difficult | 677.169 | 1 |
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This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world's leading mathematicians, that introduce basic mathematical tools and vocabulary; trace the development of modern mathematics; explain essential terms and concepts; examine core ideas in major areas of mathematics; describe the achievements of scores of famous mathematicians; explore the impact of mathematics on other disciplines such as biology, finance, and music--and much, much more.
Unparalleled in its depth of coverage, The Princeton Companion to Mathematics surveys the most active and exciting branches of pure mathematics. Accessible in style, this is an indispensable resource for undergraduate and graduate students in mathematics as well as for researchers and scholars seeking to understand areas outside their specialties.
Features nearly 200 entries, organized thematically and written by an international team of distinguished contributors
Presents major ideas and branches of pure mathematics in a clear, accessible style
Defines and explains important mathematical concepts, methods, theorems, and open problems
Introduces the language of mathematics and the goals of mathematical research
Covers number theory, algebra, analysis, geometry, logic, probability, and more
Traces the history and development of modern mathematics
Profiles more than ninety-five mathematicians who influenced those working today
Editorial Reviews
Review
Winner of the 2011 Euler Book Prize, Mathematical Association of America Honorable Mention for the 2008 PROSE Award for Single Volume Reference/Science, Association of American Publishers One of Choice's Outstanding Academic Titles for 2009
"The Princeton Companion to Mathematics makes a heroic attempt to keep [abstract concepts] to a minimum . . . and conveys the breadth, depth and diversity of mathematics. It is impressive and well written and it's good value for [the] money."--Ian Stewart, The Times
"This is a panoramic view of modern mathematics. It is tough going in some places, but much of it is surprisingly accessible. A must for budding number-crunchers."--The Economist (Best Books of 2008)
"Although the editors' original goal of text that could be understood by anyone with a good background in high school mathematics provided short-lived, this wide-ranging account should reward undergraduate and graduate students and anyone curious about math as well as help research mathematicians understand the work of their colleagues in other specialties. The editors note some advantages a carefully organized printed reference may enjoy over a collection of Web pages, and this impressive volume supports their claim."--Science
"This impressive book represents an extremely ambitious and, I might add, highly successful attempt by Timothy Gowers and his coeditors, June Barrow-Green and Imre Leader, to give a current account of the subject of mathematics. It has something for nearly everyone, from beginning students of mathematics who would like to get some sense of what the subject is all about, all the way to professional mathematicians who would like to get a better idea of what their colleagues are doing. . . . If I had to choose just one book in the world to give an interested reader some idea of the scope, goals and achievements of modern mathematics, without a doubt this would be the one. So try it. I guarantee you'll like it!"--American Scientist
"Accessible, technically precise and thorough account of all math's major aspects. Students of math will find this book a helpful reference for understanding their classes; students of everything else will find helpful guides to understanding how math describes it all."--Tom Siegfried, Science News
"Once in a while a book comes along that should be on every mathematician's bookshelf. This is such a book. Described as a 'companion', this 1000-page tome is an authoritative and informative reference work that is also highly pleasurable to dip into. Much of it can be read with benefit by undergraduate mathematicians, while there is a great deal to engage professional mathematicians of all persuasions."--Robin Wilson, London Mathematical Society
"Imagine taking an overview of elementary and advanced mathematics, a history of mathematics and mathematicians, and a mathematical encyclopedia and combining them all into one comprehensive reference book. That is what Timothy Gowers, the 1998 Fields Medal laureate, has successfully accomplished in compiling and editing The Princeton Companion to Mathematics. At more than 1,000 pages and with nearly 200 entries written by some of the leading mathematicians of our time and specialists in their fields, this book is a one-of-a-kind reference for all things mathematics."--Mathematics Teacher
"Overall [The Princeton Companion to Mathematics] is an enormous achievement for which the authors deserve to be thanked. It contains a wealth of material, much of a kind one would not find elsewhere, and can be enjoyed by readers with man different backgrounds."--Simon Donaldson, Notices of the American Mathematical Society
"This is an enormously ambitious book, full of beautiful things; I would wish to keep it on my bedside table, but that could only be possible relays, since of course it is far too large. . . . To sum up, [The Princeton Companion to Mathematics] is really excellent. I know of no book that will give a young student a better idea of what mathematics is about. I am certain that this is the only single book that is likely to tell me what my colleagues are doing."--Bryan Birch, Notices of the American Mathematical Society
From the Inside Flap
"This is a wonderful book. The content is overwhelming. Every practicing mathematician, everyone who uses mathematics, and everyone who is interested in mathematics must have a copy of their own."--Simon A. Levin, Princeton University
"The Princeton Companion to Mathematics fills a vital need. It is the only book of its kind."--Victor J. Katz, professor emeritus, University of the District of Columbia
"I think that this is a wonderful book, completely different from anything that has been written before about mathematics and mathematicians."--Endre Süli, University of Oxford
"The Princeton Companion to Mathematics is a much needed--and will become a much used--reference work. In fact, it will stand alone as the reference work in mathematics."--John J. Watkins, Colorado College
Top customer reviews
There was a problem filtering reviews right now. Please try again later.
I love this book. I have the paper and kindle versions because they complement each other. It is because of books like this that make me eschew TV so I can buy more books. Don't try to understand everything the first time around, just read it and wonder at how many smart people there are and why you never seem to get to meet them. This book delivers shock and awe for people who like to think. There are no exercises to make you feel guilty about not doing themThe Princeton Companion to Mathematics is such an extraordinary book that I am still amazed that the chief editor, Timothy Gowers, managed to pull it off. The renowned mathematician Doron Zeilberger announced that if he could take only one book with him to a desert island, it would be the Princeton Companion to Mathematics.
Why such high praise? Simply put, the PCM gives a single-volume overview of all of pure mathematics, with a clarity and coherence that cannot be found anywhere else. To be sure, there do exist several good books on the history of mathematics that give a good overview of elementary mathematics and introduce the reader to some of the great mathematicians of the past. There also exist excellent "popular science" books by writers such as Martin Gardner and Ian Stewart, that explain selected topics in advanced mathematics to the lay reader in an engaging and clear manner. And there are also encyclopedias (including Wikipedia) that delineate the main branches of mathematics and give succinct definitions of all the main concepts. But only the PCM does all of these things at once, in only a thousand pages.
The PCM is all things to all people. If your mathematical background is limited, you can still learn a great deal from the more elementary sections of the book, as well as from the biographical sketches of nearly a hundred famous mathematicians of the past. At the other end of the scale, even professional mathematicians will learn something from the articles on branches of mathematics other than their own specialty. Gowers made a systematic effort to find contributors who are not only world experts in their subject, but who write extremely well. He also forced the contributors to write in as accessible and elementary a manner as possible. The result is that even highly abstruse areas of mathematics are explained here with a clarity that is difficult to find anywhere else in the mathematical literature. The PCM is thus especially valuable to mathematics majors and graduate students.
Despite the ambitious scope of the book, it retains a strong sense of unity and coherence, by consistently emphasizing the forest rather than the trees. It also gives the reader a holistic view of mathematics by devoting different sections of the book to different perspectives on the subject. For example, one section organizes mathematics by sub-discipline, while another section highlights the main results and open problems of mathematics, while yet another section picks out the most important concepts. By putting all these aspects together in one volume, the PCM gives the reader a bird's-eye view of the whole subject that is not available from Wikipedia or from a shelf full of popular books on disparate topics.
The PCM is so well-written that it can be read either cover-to-cover, or browsed at random, or consulted as a reference when needed.
One word of warning: As Gowers himself notes, the book would be more accurately titled, "The Princeton Companion to Pure Mathematics." While applications of mathematics to other fields are touched on briefly, Gowers consciously limited the book primarily to pure mathematics, in order to keep the scope of the book manageable.
Should you still have doubts about the book, you can browse parts of the book for free: Selections from the book may be found at the book's official website, and many of the contributing mathematicians have posted their own sections on their own websites (you can find these easily using Google). And for more reviews of the book, see Gowers's blog.
Got my copy a week ago. What an exceptional book! Any of the random samples I read so far provides a informative, yet pleasant read. Gowers (Rouse Ball Professor of Mathematics in Cambridge) did a fantastic job in editing the many articles into a coherent and surprisingly accessible overview of modern mathematics. From inception to publication of this book took Gowers and his associate editors some 6 years. The amount of editorial attention given to this publication clearly shows and translated into a book that is - unlike any other math book I know of - easy to read and of high quality.
This book provides lots of material that is of interest to non-mathematicians. As is mentioned in one of the other reviews here, this heavy volume does not contain a separate chapter on mathematical physics, yet as a physicist I found lots of material directly relevant to physics. There is a very interesting chapter on the general theory of relativity, and lots of material on quantum mechanics. Also fundamental concepts highly relevant in physics such as spherical harmonics, dynamical systems, deterministic chaotic behavior, phase transitions, Lie groups, etc. are covered in inviting shorter sections. Each of the subjects is introduced in such a way that the reader first gains an intuitive understanding of the concept, that subsequently gets deepened via a more rigorous approach.
This is an outstanding depiction of the mathematical landscape compiled by a team of subject matter experts. The beautifully written articles are easy to read and yet substantive. If you spend your days wading through the silos of specified texts, fragmented course handouts and assignments and wish that someone would make your life easier by describing the broader picture, this is a truly great book to buy. | 677.169 | 1 |
24 May 2016
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Julian Jeweil - Module - Drumcode - DC179
Julian Jeweil - Module - Drumcode - DC179What is a Module? (Abstract Algebra)
A module is a generalization of a vector space. You can think of it as a group of vectors with scalars from a ring instead of a field. In this lesson, we introduce the module, give a variety of examples, and talk about the ways in which modules and vector spaces are different from one another.
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UCI Math 1A/1B: Pre-Calculus
Pre-Calculus: Inverseť 236Juliet. [A novel. By Mrs. Stevenson.]?, 001911890
Author: CARTER, afterwards STEVENSON, Mary Elizabeth.
Volume: 02
Page: 7 359Sometimes the best way to understand a set of data is to sketch a simple graph. This exercise can reveal hidden trends and meanings not clear from just looking at the numbers. In this unit you will review the various approaches to sketching graphs and learn some more advanced techniques.Gleanings from the Desert of Arabia?, 003737243
Author: UPTON, Roger D.
Page: 281
Year: 1881 177 177 97 | 677.169 | 1 |
Essential Mathematics for Electronics Technicians
Synopsis
Essential Mathematics for Electronics Technicians by Fred Monaco
This introduction to applied mathematics for beginning technology students uses electronics as a vehicle for explaining mathematics in a non-threatening way. Coverage encompasses all areas of mathematics applicable to electronics technology, from basic algebra to advanced functions in trigonometry. An introduction to computers is provided through the use of prototype programs in BASIC. The approach encourages students to learn programming through use of BASIC programs to solve realistic circuit problems. | 677.169 | 1 |
Get here Class 12 Maths NCERT Textbook Answers of Chapter 11. NCERT Solutions Class 12th Maths includes answers of all the questions of Three Dimensional Geometry provided in NCERT Text Book which is prescribed for class 12 in schools
National Council of Educational Research and Training (NCERT) Book Solutions for class 12th
Subject: Maths
Chapter: Chapter 11 – Three Dimensional Geometry | 677.169 | 1 |
I'm getting really bored in my math class. It's math answer generator, but we're covering higher grade syllabus . The concepts are really complicated and that's why I usually doze off in the class. I like the subject and don't want to drop it, but I have a real problem understanding it. Can someone guide me?
Sounds like your concepts are not strong. Mastering in math answer generator requires that your concepts be concrete. I know students who actually start teaching juniors in their first year. Why don't you try Algebrator? I am pretty sure, this program will help you.
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84848)
Download the PDF for this book
Authors:
Professor Peter Dunsby
COURSE OUTLINE
General Physics
1. Mathematical Tools
2. Linear Algebra and Tensors
Relativity
1. General Relativity
2. Mathematics
Clicked 464 times. Last clicked 11/19/2014 - 14:42.
Teaching & Learning Context:
This website contains lecture notes on Special and General Relativity, with a mathematical approach. The first part is exclusively dedicated to Special Relativity and its mathematical treatment using Tensors. The last four subjects go through General Relativity, explaining its principles, the effects of gravity and how to apply Tensors to the study of curvatur 93 131 162Learning languages and finding out about other countries can be fun, as well as useful, and this free course, Why study languages?, The course is aimed at secondary school students, age approximately 1116.
First published on Thu, 24 Aug 2017 as Why study languages?. To find out more visit The Open University's Openlearn website. Creative-Commons 2017
Image from ?[Manual of Geology: treating of the principles of the science with special reference to American geological history ? Revised edition.]?, 000858023
Author: DANA, James Dwight.
Page: 691Collection: De Forest Douglas Diver Railroad Photographs, ca. 1870 - 1948, Cornell University Library
Title: Locomotive Building Order, American Locomotive Co. 1937
Place: New York (State)
Medium: Gelatin silver print
Notes: Record of Order for Class "Four-Cylinder Simple" 4664 S 582, Order No. S-1779, August 1937, Built for Union Pacific 247
RMC ID: RMC2004_1247
Persistent URI: hdl.handle.net/1813.001/5z0730
Title: YMCA Building
Photograph date: ca. 1875-ca. 1890
Location: North and Central America: United States
Materials: albumen print
Image: 8 1/8 x 6 in.; 20.6375 x 15.24 cm
Provenance: Transfer from the College of Architecture, Art and Planning
Persistent URI: hdl.handle.net/1813.001/5sn8
Title: Cambridge. Jesus College (Chapel)
Photograph date: ca. 1865-ca. 1885
Building Date: ca. 1100-ca. 1299
Location: Europe: United Kingdom; Cambridge7 163 10500338
Title: Château de Blois. Chimney (François Premier Wing)
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Building Date: 1515-1525
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Location: Europe: France; Blois
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Image: 10.5118 x 7.874 in.; 26.7 x 20 cm
Style: Renaissance
Provenance: Transfer from the College of Architecture, Art and Planning
Persistent URI: hdl.handle.net/1813.001/5sg aim of this case study was to devise a classroom exercise under 'controlled conditions' in an attempt to ensure that we were assessing social work students' own work, rather than something that they had found - or bought - on the internet. The aim was to stimulate learning beyond "acquisition of knowledge and skills" towards "changes in behaviour" (Barr et al 2000 cited in Carpenter, 2005) and the development of skills required of emerging social work practitioners.
Subjects
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["licenses_description_" not defined] ["licenses_description_" not defined] 393 89 offers an introduction to noncooperative game theory. The course is intended both for graduate students who wish to develop a solid background in game theory in order to pursue research in the applied fields of economics and related disciplines, and for students wishing to specialize in economic theory. While the course is designed for graduate students in economics, it is open to all students who have taken and passed 14.121 | 677.169 | 1 |
Khan Academy- Khan Academy is a useful, free tutorial website that has video help for most subjects and grade levels. They have many useful math videos.
Graphing Calculator- This free online graphing calculator can be used if you do not own one. It does not look the same as the TI calculators, but can still graph, plot points, make a table and do all the other features we use regularly in class.
Algebra 2 Textbook- Click the link to access the Algebra 2 textbook online. Username is ludwig304. Password is algebra 2. You will need to turn off your pop-up blocker to open the textbook.
IXL for Algebra 2- A great resource which allows to you practice up to 20 questions per day. Gives correct answers and explanations. IXL for Algebra 1- A great resource which allows you to practice up to 20 questions per day. Gives correct answers and explanations. | 677.169 | 1 |
Pre-requisite
Before studying calculus, all students should complete grade 9 and 10 academic mathematics courses with a high achievement in both. In your third year of high school, the expectation is that you complete grade 11 university (Functions) course in semester 1, followed by Grade 12 university (Advanced Functions) in Semester 2.
Mathematics Pathway for AP-Calculus
After successfully completing your mathematics courses in your third year of high school, you would be expected to complete AP-Calculus course in semester 1 in your fourth year of high school. The AP-Calculus exam is written during the first week of May, therefore, weekly AP-Calculus tutorials will be needed to prepare students for the AP Examination.
This pathway for high school mathematics is designed for university-bound students.: Thesecourses will consist of area of study that included algebra, geometry, trigonometry, analytic geometry, and elementary functions. These functions include linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined functions. In particular, before studying calculus, students must be familiar with the properties of functions, the algebra of functions, and the graphs of functions. Students must also understand the language of functions (domain and range, odd and even, periodic, symmetry, zeros, intercepts, and so on) and know the values of the trigonometric functions at the numbers 0, π/6, π/4, π/3, π/2, and their multiples. | 677.169 | 1 |
Product Description
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Harold Jacobs' Geometry: Seeing, Doing, Understanding is an authoritative math text through which nearly one million students have learned the principles of geometry. Students learn through a combination of discussions, cartoons, anecdotes, and vivid exercises that work together to develop an understanding of the concepts behind the formulas. A proof-based course, Jacobs' Geometry will help students see the main ideas and applications, put their new understanding into practice through doing the exercises, and finally understand why geometry works the way it does. Sixteen units cover: lines, angles, direct and indirect proofs, congruence, inequalities, parallel lines, quadrilaterals, tr ansformations, area, similarity, triangles, circles, concurrence theorems, non-Euclidean geometries, and more.
780 pages, indexed, hardcover. Selected answers are provided in the back of the text. | 677.169 | 1 |
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Coding the Matrix: Linear Algebra through Applicat…
An engaging introduction to vectors and matrices and the algorithms that operate on them, intended for the student who knows how to program. Mathematical concepts and computational problems are motivated by applications in computer science. The reader learns by doing, writing programs to implement the mathematical concepts and using them to carry out tasks and explore the applications. … | 677.169 | 1 |
Comp Tools for Problem Solving
An introduction to mathematical modeling and programming using software such as MATLAB and Excel, with emphasis on problem solving in mathematics and science. Some of the mathematical applications include optimization, numerical simulations, elementary probability, and differential equations. Fields of applications involve Biology, economics, and other science disciplines. Prerequisite: Grade of C or higher in MATH 151. Spring. | 677.169 | 1 |
Material Detail
Elementary Algebra and Calculus - The Whys and Hows
This is a free textbook offered by BookBoon.
'The book is based on lecture notes Larissa created while teaching large classes of STEM students at a University of widening access and embodies a systematic and efficient teaching method that marries modern evidence-based pedagogical findings with ideas that can be traced back to such educational and mathematical giants as Socrates and Euler. The courses, which incorporated Larissa's modules, had been accredited by several UK professional bodies, often after ascertaining that there was no correlation between quality of student degrees and quality of their qualifications on entry | 677.169 | 1 |
A classical introduction to modern number theory Telecharger Gratuit PDF
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K. number theory, branch of mathematics concerned with properties of whatever became of sin the positive integers (1, 2, 3, …). sometimes called "higher arithmetic," it is among the oldest and most natural of mathematical pursuits. ma2316: kristi siegel associate professor, english dept. number theory has always fascinated amateurs as well as professional mathematicians. a classical introduction to modern number theory. number theory: get this from a library! light, color, and vision. 84) 4.4/5 (34) author: trait, behavioural, contingency and transformational theory. a classical introduction to modern number theory by k. rosen is a terrific book for the ambitious student looking for a self-guided tour of the subject. buy a classical introduction to modern number theory (graduate texts in mathematics) (v. [kenneth f ireland; michael i rosen] home. ireland and rosen's a classical introduction to modern number theory hardly needs another good review from me but hey, i'm going to give it one anyway. | 677.169 | 1 |
Abstract
MathML and the Java programming language offer mathematicians and
mathematics educators powerful new tools for publishing WWW-based mathematics
materials. In particular, these technologies now make it possible to present both
geometric information and interactive geometric models over the WWW. This paper
presents an overview of these technologies and discusses their potential uses in on-campus
and distance education courses. | 677.169 | 1 |
Product Information
Basic College Mathematics
About This Book
Basic College Mathematics is a modern, affordable choice for a textbook that fits the traditional, one-semester, basic mathematics course. The normal sequence of topics is enriched with modeling applications, study skills and the built-in Tickets to Success features (as detailed below). As students regularly use these features, they build a foundation of successful studying practices that will benefit them in all their future courses. Additionally, the built-in, modern, QR code technology connects the textbook to the videos and resources on MathTV through the student's smartphones and tablets. The availability of the eBook allows them leave their print book at home, and still have access to all the books content through a computer, smartphone, or tablet. The associated Matched Problems Worksheets gives you the opportunity to flip your classroom, without the need to create new materials yourself.
Textbook Features
Progress Check — New in 2016! (eBook only)
This new Everything is recorded in a section-by-section Progress Report to help students keep track of their progress.
Getting Ready for Class
Because we want to encourage students to read the book, we place four questions at the end of every section, under the heading Getting Ready for Class, that students can answer from the reading. Even a minimal attempt to answer these questions enhances the students' in-class experience. A number of people using our books use these questions as a student's ticket to class. They collect the student answers as students come in the door.
Paying Attention to Instructions
We try to anticipate problems students have and build features into the text that address these problems. For example, we know students don't always pay attention to instructions when they are doing their homework, and it gets them into trouble on tests and quizzes. So, we build problems into the text, under the heading Paying Attention to Instructions, that require that they read instructions.
Getting Ready for the Next Section
These problems appear at the end of every problem set. They are the exact problems that students will see when they read through the next section of the text. Students who consistently work these problems will be much better prepared for class than students who don't work these problems.
Real-Data Application Problems
This book includes many new problems that use real-world applications for the mathematical subject, making the concepts easier to understand. Many times the charts and graphics in the text look like the types of charts and graphics students see in the media.
Facts from Geometry
This feature gives students a chance to see how topics from geometry are related to the algebra they are using.
Staying Current with Technology
QR codes are all integrated throughout the text. Click on each page to see a larger view. If you have a QR reader installed on you smart device, you can scan the QR codes here yourself.
Meet The Tutors
Helping you through your course
Get to know your video tutors for Basic College Mathematics! | 677.169 | 1 |
Course Summary
Math 97: Introduction to Mathematical Reasoning has been evaluated and recommended for up to 4 semester hours and may be transferred to over 2,000 colleges and universities. This course allows for a self-paced study option, with short, engaging lessons that you can use to take the first step toward earning your degree.
Course Objective
The course objective is to give you a solid foundation in mathematical concepts such as logic, math combinations, geometric proofs, math sets, number theory, and more.
Grading Policy
Your grade for this course will be calculated out of 300 points. The minimum score required to pass and become eligible for college credit for this course is 210 points, or an overall course grade of 70%. The table below shows the assignments you must complete and how they'll be incorporated into the overall grade.
Assignment
Possible Points
Quizzes
100
Proctored Final Exam
200
Total
300
Quizzes
Quizzes are meant to test your comprehension of each lesson as you progress through the course. Here's a breakdown of how you will be graded on quizzes and how they'll factor into your final score:
You will have 3 attempts to take each quiz for a score.
The highest score of your first 3 attempts will be recorded as your score for each quiz.
When you've completed the course, the highest scores from your first 3 attempts at each quiz will be averaged together and weighed against the total possible points for quizzes. For instance, if your average quiz score is 85%, you'll receive 85 out of 100 possible points for quizzes.
After your initial 3 attempts, you can take a quiz for practice as many times as you'd like.
You will need to pass each quiz with a score of at least 80% to earn course progress for the lesson. However, it is not necessary to earn 80% within the first three quiz attempts.
Proctored Final Exam
The proctored final exam is a cumulative test designed to ensure that you've mastered the material in the course.
You'll earn points equivalent to the percentage grade you receive on your proctored final. (So if you earn 90% on the final, that's 180 points toward your final grade.)
If you're unsatisfied with your score on the exam, you'll be eligible to retake the exam after a 3-day waiting period.
You can only retake the exam twice, so be sure to use your study guide and fully prepare yourself before you take the exam again.
Course Outcomes
Identify and compare different types of sets and their representation, including finite, infinite, countable, and uncountable sets
Demonstrate techniques for performing operations and solving equations with rational and irrational numbers
Differentiate between relations and functions, and determine if a function is an injection, surjection, or bijection
Write equations to calculate combinations and permutations and use those equations to solve problems
Measure the angles of a triangle and use indirect proofs to prove two lines are parallel
Construct a geometric proof to determine the validity of a statement
Calculate the area of basic geometric shapes such as triangles, quadrilaterals, polygons, and circles
Compare figures to determine if they're symmetrical
Prerequisites
There are no prerequisites for this course.
Course Format
Math 97 consists of short video lessons that are organized into topical chapters. Each video is approximately 5-10 minutes in length and comes with a quick quiz to help you measure your learning. The course is completely self-paced. Watch lessons on your schedule whenever and wherever you want.
At the end of each chapter, you can complete a chapter test to see if you're ready to move on or have some material to review. Once you've completed the entire course, take the practice test and use the study tools in the course to prepare for the proctored final exam. You may take the proctored final exam whenever you are ready.
How Credit Recommendations Work
This course has been evaluated and recommended by ACE for 2 semester hours in the lower division baccalaureate degree category and by NCCRS for 4 semester hours in the associate/certificate category. To apply for transfer credit, follow these steps:
If you already have a school in mind, check with the registrar to see if the school will grant credit for courses recommended by either ACE or NCCRS.
Complete Math 97 by watching video lessons and taking short quizzes.
Take the Math 97 final exam directly on the Study.com site.
Request a transcript to be sent to the accredited school of your choice!
Check out this page for more information on Study.com's credit-recommended courses.
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Before taking the exam, all of the following requirements must be met:
Please complete all of the pre-requirements in the Pre-Exam Checklist in order to take the exam.
Exam Process Details
1. Register For Exam
Registering for the exam is simple. First, be sure you meet the system requirements. Next, you'll need to agree to the
academic
integrity policy. Then just confirm your name and the exam name, and you're ready to go!
2. Download Software Secure
You'll receive an unique access code. Please write this down — you'll need it to take the exam. Then download
Software Secure and
follow
the installation instructions.
3. Take Exam
The exam contains 50 - 100 multiple choice questions. You will have two hours to complete the exam, so don't start until
you're
sure you
can complete the entire thing. And remember to pace yourself!
4. Get Exam Results
We will send you an email with your official exam results within 1 to 2 weeks | 677.169 | 1 |
Beginning algebra Telecharger ePub
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Summary
Algebra for College Students, fourth edition, is written for students who have had the equivalent of one year of high school algebra. The content of the book is drawn from both intermediate algebra and college algebra and provides comprehensive coverage of the topics required in a strong one-term course in intermediate algebra or a one-term algebra for college students course. The goal of the Blitzer Algebra series is to provide students with a strong foundation in Algebra. Each text is designed to develop students' critical thinking and problem-solving capabilities and prepare students for subsequent Algebra courses as well as service math courses. Topics are presented in an interesting and inviting format, incorporating real world sourced data and encouraging modeling and problem-solving. | 677.169 | 1 |
Mathematics for the Practical Man
Description: In this book mathematics, from algebra through calculus, has been treated in such a manner as to be clear to anyone. Men who wish to study a part of mathematics which they have not hitherto had, engineers who wish to refer to phases of mathematics which so easily slip from the memory, students who desire a simple reference book, will find this manual just the book for which they have been looking.
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Mathematics Illuminated by MacGregor Campbell - Annenberg Foundation Mathematics Illuminated is a text for adult learners and high school teachers. It explores major themes of mathematics, from humankind's earliest study of prime numbers, to the cutting-edge mathematics used to reveal the shape of the universe. (3715 views)7650 views) | 677.169 | 1 |
Description
The main topics in this introductory text to discrete geometry include basics on convex sets, convex polytopes and hyperplane arrangements, combinatorial complexity of geometric configurations, intersection patterns and transversals of convex sets, geometric Ramsey-type results, and embeddings of finite metric spaces into normed spaces. In each area, the text explains several key results and methods. | 677.169 | 1 |
There are four sections of algebra this year. This is a
faster-paced math where students will relate and apply algebraic
concepts to geometry, statistics, data analysis, probability, and
discrete mathematics. The curriculum follows the textbook
Discovering Algebra (2nd edition) and aligns
with the high school algebra curriculum. Students who
successfully complete 8th grade algebra may study
geometry their freshman year.
To help your child, we encourage parents to utilize the
companion website at
You'll find practice problems, enrichment resources, study guides,
and other tools to help your child succeed. There are even parent
guides for each chapter, helping you "refresh" your algebra skills
and assist your child with their homework.
Please understand that Discovering Algebra relies on
student investigations to understand algebraic
problem-solving. Rather than just learning the "steps",
students begin each lesson with an investigation that requires
independent thinking and student inquiry.
8th Grade
Pre-Algebra
There are two sections of "traditional" pre-algebra. This
course provides a standards-aligned curriculum that meets and
exceeds the Illinois 8th grade math state goals.
This is a continuation of the 7th grade math class and
will be taught in a similar way, but the content is more
challenging and complex than in 7th grade. The
curriculum loosely follows the Glencoe Pre-Algebra
textbook (2001 edition), but teachers supplement with many
activities and lessons that extend beyond the textbook.
Students who successfully complete pre-algebra may take
algebra their freshman year.
To help your child, we encourage parents to utilize the
companion website at
There are parent/student study guides and enrichment activities for
families, along with practice tests and quizzes that be taken
on-line.
There are two sections of a "pilot" pre-algebra curriculum
called Essentials for Algebra. This is a curriculum
we are trying for the first time at Manteno Middle School. It
was piloted successfully at Manteno High School last year.
The authors explain that "Essentials for Algebra offers
a unique progression for introducing and expanding problem types.
When a new skill or operation is introduced, it is presented in a
highly structured, step-by-step manner. Work on new skills and
problem types develop in small increments from lesson to lesson.
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become skilled at solving complex problems independently."
Students who successfully complete pre-algebra may study
algebra their freshman year | 677.169 | 1 |
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CBSE, Class VII, Mathematics Practical Geometry
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