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OSMANIA UNIVERSITY
"
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Accession
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Title
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This book should be returned on or before the date
las^
marked
bclo*v.
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ELEMENTS OF ALGEBRA
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THE MACM1LLAN COMPANY
NKVV YORK
-
PAI-I.AS
BOSTON CHICAGO
SAN FRANCISCO
MACMILLAN & CO,
LONDON
LIMITKU
HOMBAY CALCUTTA
MELUCK'KNK
THE MACMILLAN
CO. OF
TORONTO
CANADA,
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LTD.
ELEMENTS OF ALGEBRA
BY
ARTHUR
SCJBULIi/TZE,
PH.D.
FORMERLY ASSISTANT PROFESSOR OF MATHEMATICS, NKW YORK ITNIVEKSITT
HEAD OF THK MATHEMATICAL DKI'A KTM EN T, HIH
SCHOOL OF COMMERCE, NEW 1 ORK CUT
THE MACMILLAN COMPANY
1917
All rights reserved
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COPYRIGHT,
BY
1910,
THE MACMILLAN COMPANY.
Set up and electrotyped.
Published
May,
1910.
Reprinted
February,
January, 1911; July, IQJS
January, 1915; May, September, 1916; August, 1917.
September, 1910
.
;
;
Berwick & Smith Co.
Norwood, Mass., U.S.A.
J. 8. Cushlng Co.
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1913,'
PREFACE
IN
this
book the attempt
in algebra,
with
all
while
still
made
to shorten the usual course
giving to the student complete familiarity
the essentials of the subject.
similar to the author's
to its peculiar aim,
"
While
in
Elementary Algebra,"
many
respects
this book,
has certain distinctive features, chief
which are the following
1.
is
owing
among
:
All unnecessary methods
and "cases" are
omitted.
These
omissions serve not only practical but distinctly pedagogic
" cases "
ends. Until recently the tendency was to multiply
as far as possible, in order to make every example a
social
case of a memorized method.
Such a large number of methods,
however, not only taxes a student's memory unduly but in variably leads to mechanical modes of study. The entire study
of algebra becomes a mechanical application of memorized
rules,
while the cultivation of the student's reasoning power
is neglected.
Typical in this respect is the
and ingenuity
treatment of factoring in
methods which are of
many
text-books
In this book
all
and which are applied in
advanced work are given, but "cases" that are taught only
on account of tradition, short-cuts that solve only examples
real value,
manufactured for this purpose, etc., are omitted.
All parts of the theory whicJi are beyond the comprehension
specially
2.
of
the student or wliicli are logically
practical
teachers
know how few
unsound are
omitted.
All
students understand and
appreciate the more difficult parts of the theory, and conse-
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PREFACE
vi
quently hardly ever emphasize the theoretical aspect of alge
bra.
Moreover, a great deal of the theory offered in the averis logically unsound ; e.g. all proofs for the sign
text-book
age
two negative numbers, all elementary proofs
theorem for fractional exponents, etc.
of the product of
of the binomial
3.
TJie exercises are slightly simpler than in the larger look.
The best way to introduce a beginner to a new topic is to offer
Lim a large number of simple exercises. For the more ambitious student, however, there has been placed at the end of
the book a collection of exercises which contains an abundance
of
more
difficult
cises in this
work.
book
With very few
differ
bra"; hence either book
4.
from those
may
exceptions
in the
all
the exer
"Elementary Alge-
be used to supplement the other.
Topics of practical importance, as quadratic equations and
graphs, are placed early in the course.
enable students
This arrangement will
of time to
who can devote only a minimum
algebra to study those subjects which are of such importance
for further work.
In regard
may
to
some other features of the book, the following
be quoted from the author's "Elementary Algebra":
"Particular care has been bestowed upon those chapters
in the customary courses offer the greatest difficulties to
which
the beginner, especially problems and factoring. 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, and it is hoped that this treatment will materially diminish the difficulty of this topic for young students.
" 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.
This made it necessary to introduce the theory of proportions
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PREFACE
vii
and graphical methods into the first year's work, an innovation
which seems to mark a distinct gain from the pedagogical point
of view.
"
By studying proportions during the first year's work, the
student will be able to utilize this knowledge where it is most
needed,
viz. in
geometry
;
while in the usual course proportions
are studied a long time after their principal application.
"
Graphical methods have not only a great practical value,
but they unquestionably furnish a very good antidote against
'the tendency of school algebra to degenerate into a mechanical application of
memorized
rules.'
This topic has been pre-
sented in a simple, elementary way, and
of the
modes of representation given
it is
hoped that some
will be considered im-
provements upon the prevailing methods. The entire work in
graphical methods has been so arranged that teachers who wish
a shorter course
may omit
these chapters."
Applications taken from geometry, physics, and commercial
are numerous, but the true study of algebra has not been
sacrificed in order to make an impressive display of sham
life
applications.
to solve a
It is
undoubtedly more interesting for a student
problem that results in the height of Mt.
McKinley
than one that gives him the number of Henry's marbles. But
on the other hand very few of such applied examples are
genuine applications of algebra,
nobody would find the length
Etna by such a method,
of the Mississippi or the height of Mt.
and they usually involve difficult numerical calculations.
Moreover, such examples, based upon statistical abstracts, are
frequently arranged in sets that are algebraically uniform, and
hence the student is more easily led to do the work by rote
than when the arrangement
is
based principally upon the alge-
braic aspect of the problem.
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PREFACE
viii
It is true that
problems relating to physics often
offer
a field
The average
pupil's knowlso small that an extensive use of
for genuine applications of algebra.
edge of physics, however, is
such problems involves as a rule the teaching of physics by the
teacher of algebra.
Hence the
field of
genuine applications of elementary algebra
work seems to have certain limi-
suitable for secondary school
tations,
give as
but within these limits the author has attempted to
many
The author
simple applied examples as possible.
desires to acknowledge his indebtedness to Mr.
William P. Manguse for the careful reading of the proofs and
for
many
NEW
valuable suggestions.
ARTHUR SCHULTZE.
YORK,
April, 1910.
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CONTENTS
CHAPTER
I
PAGB
INTRODUCTION
1
Algebraic Solution of Problems
Negative Numbers
1
3
Numbers represented by Letters
Factors, Powers, and Hoots
.......
...
Algebraic Expressions and Numerical Substitutions
CHAPTER
15
........
....
Subtraction
III
...
MULTIPLICATION
Numbers
Monomials
Multiplication of a Polynomial by a
10
22
29
CHAPTER
Multiplication of
15
27
Signs of Aggregation
Exercises in Algebraic Expression
Multiplication of Algebraic
7
10
II
ADDITION, SUBTRACTION, AND PARENTHESES
Addition of Monomials
Addition of Polynomials
6
Monomial
31
31
....
34
35
Multiplication of Polynomials
36
Special Cases in Multiplication
39
CHAPTER IV
46
46
DIVISION
Division of Monomials
Division of a Polynomial by a Monomial
Division of a Polynomial by a Polynomial
Special Cases in Division
ix
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47
48
61
X
CONTENTS
CHAPTER V
PAGE
,63
LINEAR EQUATIONS AND PROBLEMS
.....,.
Solution of Linear Equations
Symbolical Expressions
Problems leading
to
55
67
63
Simple Equations
CHAPTER VI
FACTORING
Type
76
I.
Type II.
Type III.
Type IV.
Type V.
Type VI.
Summary
Polynomials, All of whose Terms contain a
mon Factor
Quadratic Trinomials of the
Quadratic Trinomials of the
Com77
Form x'2 -f px -f q
Form px 2 -f qx + r
The Square of a Binomial x 2
Ixy
The Difference of Two Squares
Grouping Terms
.
.
....
-f
/^
.
.
.
78
80
83
84
86
87
of Factoring
CHAPTER
VII
HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE
.
.
Common Factor
Lowest Common Multiple
CHAPTER
89
89
Highest
91
VIII
93
FRACTIONS
Reduction of Fractions
Addition and Subtraction of Fractions
93
97
102
Multiplication of Fractions
Division of Fractions
104
Complex Fractions
*
,
*
.
105
CHAPTER IX
FRACTIONAL AND LITERAL EQUATIONS
......
108
108
Fractional Equations
112
Literal Equations
Problems leading to Fractional and Literal Equations
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.
.114
CONTENTS
XI
CHAPTER X
RATIO AND PROPORTION
.........
PAGE
120
Ratio
120
Proportion
121
CHAPTER XI
SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE
Elimination by Addition or Subtraction
Elimination by Substitution
Literal Simultaneous Equations
Simultaneous Equations involving More than
....
129
130
133
138
Two Unknown
....
140
....
148
Graphic Solution of Equations involving One Unknown Quantity
Graphic Solution of Equations involving Two Unknown Quan-
168
Quantities
Problems leading to Simultaneous Equations
CHAPTER
143
XII
GRAPHIC REPRESENTATION OF FUNCTIONS AND EQUATIONS
Representation of Functions of One Variable
.
.
164
160
tities
CHAPTER
XIII
INVOLUTION
165
Involution of Monomials
165
Involution of Binomials
166
EVOLUTION
...
CHAPTER XIV
169
Evolution of Monomials
170
.
Evolution of Polynomials and Arithmetical Numbers
.
.
171
.
1*78
CHAPTER XV
QUADRATIC EQUATIONS INVOLVING ONB UNKNOWN QUANTITY
Pure Quadratic Equations
178
Complete Quadratic Equations
Problems involving Quadratics
181
Equations in the Quadratic
Character of the Roots
Form
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189
191
193
CONTENTS
xii
CHAPTER XVI
PAGK
195
THE THEORT OP EXPONENTS
Fractional and Negative Exponents
Use of Negative and Fractional Exponents
....
195
200
CHAPTER XVII
RADICALS
205
206
Transformation of Radicals
Addition and Subtraction of Radicals
210
.212
Multiplication of Radicals
Division of Radicals
.....
Involution and Evolution of Radicals
219
Square Roots of Quadratic Surds
Radical Equations
CHAPTER
214
218
221
XVIII
THE FACTOR THEOREM
227
CHAPTER XIX
SIMULTANEOUS QUADRATIC EQUATIONS
I.
II.
......
Equations solved by finding x +/ and x
/
One Equation Linear, the Other Quadratic
III.
Homogeneous Equations
IV.
Special Devices
232
.
.
.
232
.
.
.
234
236
237
Interpretation of Negative Results
and the Forms
i
-,
.
.
241
243
Problems
CHAPTER XX
PROGRESSIONS
246
.
Arithmetic Progression
Geometric Progression
Infinite
24(j
251
263
Geometric Progression
CHAPTER XXI
BINOMIAL THEOREM
.
BEVIEW EXERCISE
.
.
.
.
.
.
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.
.
255
268
ELEMENTS OF ALGEBRA
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ELEMENTS OF ALGEBRA
CHAPTER
I
INTRODUCTION
1.
Algebra
may
it
arithmetic,
be called an extension of arithmetic. Like
numbers, but these numbers are fre-
treats of
quently denoted by
problem.
letters,
as illustrated in
the following
ALGEBRAIC SOLUTION OF PROBLEMS
2.
Problem.
is five
The sum
two numbers is 42, and the greater
Find the numbers.
the smaller number.
of
times the smaller.
'
x
Let
5 x = the greater number,
6x
the sum of the two numbers.
Then
and
6x
Therefore,
= 42,
x = 7, the smaller number,
5 x = 35, the greater number.
and
3.
A problem
4.
An
is
a question proposed for solution.
equation is a statement expressing the equality of
quantities; as,
6 a?
two
= 42.
In algebra, problems are frequently solved by denoting
numbers by letters and by expressing the problem in the form
of an equation.
5.
6.
Unknown numbers
letters of the alphabet
are employed.
B
;
are usually represented
as, x, y,
z,
by the
last
but sometimes other letters
1
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ELEMENTS OF ALGEBRA
2
EXERCISE
1
Solve algebraically the following problems
1.
The sum
numbers is 40, and the greater
Find the numbers.
of two
times the smaller.
A man
:
is
four
and a carriage for $ 480, receiving
for the horse as for the carriage.
much
did he receive for the carriage ?
2.
twice as
3.
A
sold a horse
How
much
and
B own
vested twice as
a house worth $ 14,100, and
much
capital as B.
How much
A
has
in-
has each
invested ?
4.
The population
of
South America
is
9 times that of
Australia, and both continents together have 50,000,000 inFind the population of each.
habitants.
The
and fall of the tides in Seattle is twice that in
their sum is 18 feet.
Find the rise and fall
and
Philadelphia,
5.
rise
of the tides in Philadelphia.
6.
Divide $ 240 among A, B, and C so that A may receive
much as C. and B 8 times as much as C.
6 times as
A pole 56 feet high was broken so that the part broken
was 6 times the length of the part left standing. .Find the
length of the two parts.
7.
off
8.
If
The sum
two
of the sides of a triangle equals 40 inches.
sides of the triangle are equal, and each is twice the
A
remaining side, how long is each side ?
A
9.
The sum
triangle is
are equal,
of the three angles of any
180. If 2 angles of a triangle
and the remaining angle is 4
times their sum,
how many
degrees are
there in each ?
B
G
10. The number of negroes in Africa
10 times the number of Indians in America, and the sum of
both is 165,000,000. How many are there of each ?
is
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INTRODUCTION
3
Divide $280 among A, B, and C, so that
much as A, and C twice as much as B.
11.
B may
receive
twice as
Divide $90 among A, B, and C, so that B may receive
much as A, and C as much as A and B together.
12.
twice as
A
13.
which
is
line 20 inches long is divided into two parts, one of
long are the parts ?
equal to 5 times the other.
How
A
travels twice as fast as B, and the
tances traveled by the two is 57 miles.
14.
sum
of the dis-
How many
A, B, C, and
15.
does
A
much
take, if
B
and
D
as B,
miles did
4
each travel ?
D buy $ 2100 worth of goods. How much
buys twice as much as A, C three times as
six times as
much
NEGATIVE NUMBE
EXERCISE
2
Subtract 9 from 16.
1.
2.
Can 9 be subtracted from 7 ?
3.
In arithmetic
4.
The temperature
What
is
why
cannot 9 be subtracted from 7 ?
"*
\
noon is 16 ami at 4 P.M. it is 9
the temperature at 4 P.M.? State this as an
at
of subtraction.
5.
less.
6.
The temperature
8.
4 P.M.
is
7, and
at 10 P.M.
it is
10
expressing the last
below zero) ?
What then is 7 -10?
answer
7.
at
What is the temperature at 10 P.M. ?
Do you know of any other way of
(3
Can you think
of
any other
practical examples
require the subtraction of a greater
which
number from a smaller
one?
7.
Many
greater
practical examples require the subtraction of a
one, and in order to express in
number from a smaller
a convenient form the results of these, and similar examples,
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ELEMENTS OF ALGEBRA
4
it becomes necessary to enlarge our concept of number, so as to
include numbers less than zero.
8. Negative numbers are numbers smaller than zero; they
are denoted by a prefixed minus sign as
5 (read " minus 5 ").
Numbers greater than zero, for the sake of distinction, are fre;
quently called positive numbers, and are written either with a
prefixed plus sign, or without any prefixed sign as -f- 5 or 5.
;
The
fact that a
below zero
thermometer falling 10 from 7 indicates 3
may now
be expressed
7 -10
= -3.
Instead of saying a gain of $ 30, and a loss of $ 90
we may write
is
equal to a
loss of $ 60,
$30
The
9.
-$90 = -$60.
6,
It is convenient for
10.
number
absolute value of a
without regard to its sign.
5 is
The absolute value of
is
the number taken
of -f 3 is 3.
many
discussions to represent the
numbers by a succession of equal distances laid off on
from a point 0, and the negative numbers by a similar
positive
a line
series in the opposite direction.
,
I
-6
I
-5
lit
-4
-2
-3
I
I
I
+\
+2
I
-1
Thus, in the annexed diagram, the line from
the line from
to
4,
I
I
+4
4-5
y
I
+6
to 4- 6 represents 4- 5,
1
etc.
left.
equals 4, 5 subtracted from
EXERCISE
1.
3
The addition of 3 is repspaces toward the right, and the subtrac-
4 represents
resented by a motion of "three
tion of 8 by a similar motion toward the
Thus, 5 added to
I
+
If in financial transactions
we
1 equals
6, etc.
3
indicate a man's income
by
a positive sign, what does a negative sign indicate ?
2. State in what manner the positive and negative signs may
be used to indicate north and south latitude, east and west
longitude, motion upstream
and downstream.
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INTRODUCTION
3.
If north latitude
is
indicated by a positive sign, by what
is
south latitude represented ?
is
north latitude represented
4.
If south latitude
5.
What
6.
A
is
5
indicated by a positive sign, by what
?
the meaning of the year
6 yards per second ?
erly motion of
is
20 A.D. ?
merchant gains $ 200, and loses $ 350.
- 350.
(b) Find 200
Of an
(a)
east-
What
is
his total gain or loss ?
7.
If the temperature at 4 A.M. is
8 and at 9 A.M. it is 7
what is the temperature at 9 A.M. ? What, therefore,
higher,
is
8
- +7?
8. A vessel
sails
journey.
9.
sails
A
22
(6)
11.
12.
13.
14.
15.
16.
17.
26.
from a point in 25 north latitude, and
Find the latitude at the end of the
(a)
Find 25 -38.
vessel starts from a point in 15 south latitude, and
due south, (a) Find the latitude at the end of the
journey,
10.
starts
38 due south,
(b)
Subtract 22 from
From 30 subtract 40.
From 4 subtract 7.
From 7 subtract 9.
From 19 subtract 34.
From subtract 14.
From
12 subtract 20.
2 subtract 5.
From
1 subtract 1.
From
15.
24.
To
6
2
To
To
1
From 1
To - 8
To
7
From
25.
Add
18.
19.
20.
21.
22.
23.
add
1.
add
2.
subtract 2.
add
9.
add
4.
1 subtract 2.
1 and 2.
Solve examples 16-25 by using a diagram similar to
10, and considering additions and subtractions as
the one of
motions.
(a)
Which is the greater number
lor -1? (b) -2 or -4?
28.
By how much
27.
12.
add
is
:
7 greater than
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12 ?
