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IN MEMORIAM

FLOR1AN CAJORI

COLLEGE ALGEBRA

BY

J. M. TAYLOR, A.M., LL.D.,

PROFESSOR OF MATHEMATICS IN COLGATE UNIVERSITY

SIXTH EDITION

ALLYN AND BACON

Boston antJ Chicago

Copyright, 1889,

By Allyn and Bacon.

Berwick & Smith, Norwood, Mass., U.S.A.

PREFACE

THIS work originated in the author's desire for a

course in Algebra suited to the needs of his

own pupils. The increasing claims of new sciences

to a place in the college curriculum render necessary

a careful selection of matter and the most direct

methods in the old. The author's aim has been to

present each subject as concisely as a clear and

rigorous treatment would allow.

The First Part embraces an outline of those fun-

damental principles of the science that are usually

required for admission to a college or scientific

school. The subjects of Equivalent Equations and

Equivalent Systems of Equations are presented more

fully than others. Until these subjects are more

scientifically understood by the average student, it

will be found profitable to review at least this por-

tion of the First Part.

In the Second Part a full discussion of the Theory

of Limits followed by one of its most important ap-

plications, Differentiation, leads to clear and concise

IV PREFACE.

proofs of the Binomial Theorem, Logarithmic Series,

and Exponential Series, as particular cases of Mac-

laurin's Formula. It also affords the student an easy

introduction to the concepts and methods of the

higher mathematics.

Each chapter is as nearly as possible complete in

itself, so that the order of their succession can be

varied at the discretion of the teacher; and it is

recommended that Summation of Series, Continued

Fractions, and the sections marked by an asterisk

be reserved for a second reading.

In writing these pages the author has consulted

especially the works of Laurent, Bertrand, Serret,

Chrystal, Hall and Knight, Todhunter, and Burnside

and Panton. From these sources many of the prob-

lems and examples have been obtained.

J. M. TAYLOR.

Hamilton, N. Y., 1889.

PREFACE TO THIRD EDITION.

In this edition a number of changes have been

made in both definitions and demonstrations. In

the Second Part, derivatives, but not differentials,

are employed. Two chapters have been added ; one

on Determinants, the other on the Graphic Solution

of Equations and of Systems of Equations.

J. M. TAYLOR.

Hamilton, N, Y-, 1895.

CONTENTS.

FIRST PART.

CHAPTER I.

Page

Definitions and Notation 1-9

CHAPTER II.

Fundamental Operations 10-21

CHAFTER III.

Fractions 22-24

CHAPTER IV.

Theory of Exponents 25-32

CHAPTER V.

Factoring ' . . 33

Highest Common Divisor 38

Lowest Common Multiple 4 1

VI CONTENTS.

CHAPTER VI.

Page

Involution, Evolution 42

Surds and Imaginaries 46

CHAPTER VII.

Equations 56-77

Equivalent Equations 57

Linear Equations 63

Quadratic and Higher Equations 65

CHAPTER VIII.

Systems of Equations 78-91

Equivalent Systems 79

Methods of Elimination 80

Systems of Quadratic Equations 85

CHAPTER IX.

Indeterminate Equations and Systems .... 92

Discussion of Problems 98

Inequalities 101

CHAPTER X.

Ratio, Proportion, and Variation .... .104-114

CHAPTER XI.

The Progressions 115-121

CONTENTS. Vll

SECOND PART.

CHAPTER XII.

Page

Functions and Theory of Limits 122-133

Function's and Functional Notation 123

Theory of Limits . 125

Vanishing Fractions 132

Incommensurable Exponents 132

CHAPTER XIII.

Derivatives i34- T 47

Derivatives 134

Illustration of D x (ax*) 136

Rules for finding Derivatives ........ 137

Successive Derivatives 145

Continuity 146

CHAPTER XIV.

Development of Functions in Series .... 148-170

Development by Division 149

Principles of Undetermined Coefficients 150

Resolution of Fractions into Partial Fractions . . . 154

Reversion of Series 159

Maclaurin's Formula 161

The Binomial Theorem 163

viii CONTENTS.

CHAPTER XV.

Page

CONVERGENCY AND SUMMATION OF SERIES . . . I7I-I92

Convergency of Series 171

Recurring Series 177

Method of Differences 182

Interpolation 188

CHAPTER XVI.

Logarithms 193-214

General Principles 193

Common Logarithms 197

Exponential Equations 203

The Logarithmic Series 205

The Exponential Series 211

CHAPTER XVII.

Compound Interest and Annuities 215-223

CHAPTER XVIII.

Permutations and Combinations 224-236

CHAPTER XIX.

Probability 237-253

Single Events 237

Compound Events 241

CONTENTS. IX

CHAPTER XX.

Page

Continued Fractions 254-265

Conversion of a Fraction into a Continued Fraction . 255

Convergents 256

Periodic Continued Fractions 263

CHAFFER XXI.

Theory of Equations 266-317

Reduction to the Form F(x) = 266

Divisibility of F (x) 267

Horner's Method of Synthetic Division 268

Number of Roots 272

Relations between Coefficients and Roots 274

Imaginary Roots 276

Integral Roots 280

Limits of Roots 281

Equal Roots 284

Change of Sign of F{x) 286

Sturm's Theorem 288

Transformation of Equations 295

Horner's Method of Solving Numerical Equations . 300

Reciprocal Equations 307

Binomial Equations 310

Cubic Equations 312

Biquadratic Equations 315

CONTENTS.

CHAPTER XXII.

Page

Determinants 318-346

Determinants of the Second Order 318

Determinants of the Third Order 322

Determinants of the Â«th Order 328

Properties of Determinants 330

Minors and Co-factors 336

Expression of A in Co-factors 337

Eliminants 340

Multiplication of Determinants 343

CHAPTER XXIII.

Graphic Solution of Equations and of Systems 347-363

Co-ordinates of a Point 348

Graphic Solution of Indeterminate Equations . . . 349

Graphic Solution of Systems of Equations .... 353

Properties of F(x) and of F{x) = illustrated by the

Graph of F (x) 357

Geometric Representation of Imaginary and Complex

Numbers - . 360

ALGEBRA

FIRST PART.

CHAPTER I.

DEFINITIONS AND NOTATION.

1. Quantity is anything that can be increased, di-

minished, or measured; as any portion of time or

space, any distance, force, or weight.

2. To measure a quantity is to find how many

times it contains some other quantity of the same

kind taken as a unit, or standard of comparison.

Thus, to measure a distance, we find how many times it con-

tains some other distance taken as a unit. To measure a por-

tion of time, we find how many times it contains some other

portion of time taken as a unit.

3. By counting the units in a quantity, we gain the

idea of ' how many ' ; that is, of Arithmetical Number.

If, in the measure of any quantity, we omit the unit

of measure, we obtain an arithmetical number. It

may be a whole number or a fraction. Thus by

omitting the units ft., lb., hr., in 6 ft., 3 lbs., and

2/3 hr., we obtain the whole numbers 6 and 3, and

the fraction fz.

2 ALGEBRA.

4. Positive and Negative Quantities. Two quanti-

ties of the same kind are opposite in quality, if when

united, any amount of the one annuls or destroys an

equal amount of the other. Of two opposites one is

said to be Positive in quality, and the other Negative.

Thus, credits and debits are opposites, since equal amounts

of the two destroy each other. If we call credits positive,

debits will be negative. Two forces acting along the same line

in opposite directions are opposites ; if we call one positive, the

other is negative.

5. Algebraic Number. The sign +, read positive,

and â€” , read veg t've, are used with numbers, or their

symbols, to denote their quality, or the quality of the

quantities which they represent.

Thus, if we call credit positive, + $5 denotes $5 of

credit, and â€” $4 denotes $4 of debt. If + 8 in. de-

notes 8 in. to the right, â€”9 in. denotes 9 in. to the

left. By omitting the particular units $ and in., in

+ $5, â€” $4, + 8 in., â€”9 in., we obtain the algebraic

numbers + 5, â€” 4, + 8, â€” 9. + 5 is read ' positive 5/

â€” 4 is read ' negative 4.' Each of these numbers has

not only an arithmetical value y but also the quality of

one of two opposites ; hence

An Algebraic Number is one that has both an

arithmetical value and the quality of one of two

opposites.

Two algebraic numbers that are equal in arith-

metical value but opposite in quality destroy each

other when added.

DEFINITIONS AND NOTATION. 3

The element of quality in algebraic number doubles

the range of number.

Thus, the integers of arithmetic make up the simple

series,

o, i, 2, 3, 4, 5> 6 > 7> â€¢â€¢â€¢> *>'> (0

while the integers of algebra make up the double

series,

-3o,.. .,â€”4, -3,-2, -I, Â±0, +1, +2, +3, +4, â€¢ â€¢ â€¢,+ *>â€¢ (2)

An algebraic number is said to be increased by-

adding a positive number, and decreased by adding a

negative number.

If in series (2) we add + 1 to any number, we ob-

tain the next right-hand number. Thus, if to + 3 we

add +1, we obtain +4; if to â€”4 we add -f 1, we

obtain â€” 3 ; and so on. That is, positive numbers

increase from zero, while negative numbers decrease

from zero.

Hence positive numbers are algebraically the

greater, the greater their arithmetical values; while

negative numbers are algebraically the less, the

greater their arithmetical values.

All numbers are quantities, and the term quantity

is often used to denote number.

6. Symbols of Number. Arithmetical numbers are

usually denoted by figures. Algebraic numbers

are denoted by letters, or by figures with the signs

+ and â€” prefixed to denote their quality. A letter

4 ALGEBRA.

usually represents both the arithmetical value and

the quality of an algebraic number. Thus a may

denote +5, â€” 5, â€” 8, + 17, or any other algebraic

number. When no sign is written before a symbol

of number, the sign + is understood.

Known Numbers, or those whose values are known,

or supposed to be known, are denoted by figures, or

the first letters of the alphabet, as a, b, c, a', b', c\

&u b i% c x .

Unknown Numbers, or those whose values are to

be found, are usually denoted by the last letters of

the alphabet, as, x, y, z, x', y' t z', x lt y u %

Quantities represented by letters are called literal ;

those represented by figures are called numerical.

7. Signs of Operation. The signs, + (read phis),

â€” (read minus), X (read multiplied by), -r- (read

divided by), are used in algebra to denote algebraic

addition, subtraction, multiplication, and division,

respectively. The use of the signs + and â€” to indi-

cate operations must be carefully distinguished from

their use to denote quality. In the literal notation,

multiplication is usually denoted by writing the mul-

tiplier after the multiplicand. Thus, a b = a X b.

Sometimes a period is used; thus, 4-5 = 4 X 5.

Algebraic division is often denoted by a vinculum ;

thus T = a -5- b*

b

8. Signs of Relation and Abbreviation. The sign of

equality is =. The sign of identity is =. The sign

DEFINITIONS AND NOTATION. 5

of inequality is > or < , the opening being toward

the greater quantity.

The signs of aggregation are the parentheses ( ),

the brackets [ ], the brace { }, the vinculum ,

and the bar | . They are used to indicate that two

or more parts of an expression are to be taken as a

whole. Thus, to indicate the product of c â€” d multi-

plied by x, we may write (c â€” d) x, [c â€” d] x, {c â€” d\ x,

c \x

c â€” dx, or â€” d\

The sign .*. is read hence y or therefore; the sign

V is read since, or because.

The sign of continuation is three or more dots . . . ,

or dashes â€” , either of which is read and so on.

9. The result obtained by multiplying together

two or more numbers is called a Product. Each

of the numbers which multiplied together form a

product, is called a Factor of the product.

10. A Power of a number is the product obtained

by taking that number a certain number of times

as a factor. If ;/ is a positive integer, a" denotes

aa a a... to n factors, or the ?/th power of a. In

a", n denotes the number of equal factors in the

power, or the Degree of the power, and is called an

Exponent.

11. A Root of a quantity is one of the equal factors

into which it may be resolved.

The wth root of a is denoted by y/a. In %/a,

m denotes the number of equal factors into which

6 ALGEBRA.

a is to be resolved, and is called the Index of the

root. The sign ^/~ (a modification of r, the first

letter of the word radix) denotes a root. If no in-

dex is written, 2 is understood.

12. Any combination of algebraic symbols which

represents a number is called an Algebraic Expression.

13. When an algebraic expression consists of two

or more parts connected by the signs + or â€” , each

part is called a Term. Thus, the expression

a 1 + (c â€” x) y + b z 1 + c â€” d

consists of four terms.

A Monomial is an algebraic expression of one term;

a Polynomial is one of two or more terms. A poly-

nomial of two terms is called a Binomial; one of

three terms a Trinomial.

14. The Degree of a term is the number of its lite-

ral factors. But we often speak of the degree of a

term with regard to any one of its letters. Thus,

8 a 2 fix*, which is of the ninth degree, is of the

second degree in a, the third in b, and the fourth

in x.

The degree of a polynomial is that of the term of

the highest degree. An expression is homogeneous

when all its terms are of the same degree.

A Linear expression is one of the first degree; a

Quadratic expression is one of the second degree.

DEFINITIONS AND NOTATION. 7

15. Any algebraic expression that depends upon

any number, as x, for its value is said to be a Function

of x. Thus, 5 x s is a function of x ; 5 x 2 + a s â€” 7 x

is a function of both x and a; but if we wish to con-

sider it especially with reference to x, we may call it

a function of x simply.

A Rational Integral Function of X is one that can

be put in the form

Ax" + Bx"- 1 + Cx n ~ 2 + ... + F,

in which 11 is a whole number, and A,B, ..., .F denote

any expressions not containing x.

Thus, a X s â€” 4 x' 2 â€” b x + c and x 8 â€” \ x are rational integral

functions of x of the third degree.

16. The Reciprocal of a number is one divided by

that number.

17. If a term be resolved into two factors, ei-

ther is the Coefficient of the other. The coeffi-

cient may be either numerical or literal. Thus, in

4a be 2 , 4 is the coefficient of a be 2 , 4 a of be 2 , and

4 a b of e 2 . When no numerical coefficient is

written, 1 is understood; thus, a = (+ 1) a, and

-Â« = (-!)*

18. Like or Similar Terms are such as differ only in

their coefficients. Thus, 4a be 2 and 10 a be 2 are like

terms ; 6 a 2 fty 2 and 4 a IPy 2 are like, if we regard* 6 a 2

and 4a as their coefficients, but unlike if 6 and 4 be

taken as their coefficients.

8 ALGEBRA.

19. A Theorem is a proposition to be proved.

20. A Problem is something to be done.

21. To solve a problem is to do what is required.

22. An Axiom is a self-evident truth.

The axioms most frequently used in Algebra are

the following:

1. Numbers which are equal to the same number

or to equal numbers are equal to each other.

2. If the same number or equal numbers be added

to, or subtracted from, equal numbers, the

results will be equal.

3. If equal numbers be multiplied by the same

number or equal numbers, the products will

be equal.

4. If equal numbers be divided by the same num-

ber, except zero, or by equal numbers, the

quotients will be equal.

5. Like powers or like roots of equal numbers are

equal.

23. Identical Expressions. Equal expressions that

contain only figures, or expressions that are equal

for all values of their letters, are called Identical

Expressions.

Thus, 4 + 6 and 5X2 are identical expressions; so also are

(a + b) (a- b) and a* - b 2 .

24. An Equality is a statement that two expressions

represent the same number. The two expressions are

called the Members of the equality.

DEFINITIONS AND NOTATION. 9

25. Identities and Equations. Equalities are of two

kinds, identities and equations.

The statement that two identical expressions are

equal is called an Identity. In writing identities, the

sign =, read ' is identical with,' is often used instead

of the sign =.

Thus the equality 5 + 7 = 4X3 is an identity; so also is

a' 2 â€” x 2 = (a + r) (a â€” x), since it holds true for all values of a

and x. To indicate that these equalities are identities, they may

be written 5 + 7=4X3 and a 2 â€” x 2 = (a + -r) {a â€” -*')â€¢

If two expressions are not identical, and one or

both of them contains a letter or letters, the state-

ment that they are equal is called an Equation.

Thus the equalities $x â€” 6 = o,ja = 2a + 5, and 2 y â€” 4 x = 6,

are equations. The first holds true for xâ€” 2, and the second

for a â€” 1. The equation 2y â€” 4x=6 holds true iov y = 2^ + 3;

hence if x â€” 1, y â€” 5 ; \ix = 2,y = 7) and so on.

26. Algebra is that branch of mathematics which

treats of the equation, its nature, the methods of solv-

ing it, and its use as an instrument for mathematical

investigation.

The history of algebra is the history of the equation. The

notation of algebra, including symbols of operation, rel ition,

abbreviation, and quantity, was invented to secure conciseness,

clearness, and facility in the statement, transformation, and solu-

tion of equations. The number of algebra, which has quality as

well as arithmetical value, was conceived in the effort to interpret

results obtained as solutions of equations. Hence the study of

the nature and laws of algebraic number and of the methods

of combining, factoring, and transforming algebraic expressions

should be pursued as auxiliary to the study of the equation.

This will lend interest and profit to what might otherwise be

regarded as dull and useless.

10 ALGEBRA,

CHAPTER II.

FUNDAMENTAL OPERATIONS.

27. Addition is the operation of finding the result

when two or more numbers are united into one. The

result, which must always be expressed in the sim-

plest form, is called the Sum.

28. Subtraction is the operation of taking from one

number, called the Minuend, another number, called

the Subtrahend. The result, which must be expressed

in the simplest form, is called the Remainder.

The subtrahend and the remainder are evidently

the two parts of the minuend; hence, since the whole

is equal to the sum of all its parts, we have

minuend = subtrahend + remainder.

29. To multiply one number by another is to treat

the first, called the Multiplicand, in the same way that

we would treat I to obtain the second, called the

Multiplier.

Thus, 3=1 + 1 + i ; .-.4X3 = 4 + 4 + 4-

Again, ^=1-^3 X 2; .-. 9X M = 9 + 3 X 2.

30. Having given a product and one factor, Division

is the operation of finding the other factor. The

given product is called the Dividend ; the given fac-

tor, the Divisor ; and the required factor, the Quotient.

From their definitions, the divisor and quotient are

evidently the two factors of the dividend ; hence,

FUNDAMENTAL OPERATIONS. II

since any number is equal to the product of its

factors, we have

quotient X divisor = dividend. (i)

Let D denote the dividend, and d the divisor; then

the expression D -5- d will denote the quotient, and

by (i) we shall have

(D + d)xd=D. (2)

31. Law of Order of Terms. Numbers to be added

may be arranged in any order ; that is,

a + b = b + a. (A)

For let there be any two quantities of the same kind,

one containing a units and the other b units. Now if

we put the second quantity with the first, the measure

of the resulting quantity will be a + b units ; and if we

put the first quantity with the second, the measure of

the resulting quantity will be b + a units. It is self-

evident that these two resulting quantities will be

equal; hence their measures will be equal;

.*. a + b = b + a.

A similar proof would apply to an expression of any

number of terms.

32. Law of Grouping of Terms. Numbers to be added

may be grouped in any manner; that is,

a + l, + c- = a+(b + <;). (B)

For by the lazv of order we have

a + b + câ€”b + c+ a

= (b + c)+a = a+{b + c).

A similar proof would apply in any other case.

12 ALGEBRA.

33. Law of Quality in Products. Two like signs give

+ ; tzvo unlike signs give â€” .

By the definition of multiplication we have

+ 3 = 0+ (+i) + (+ i) + (+i).

.*â€¢ (+4)X(+3)=+(+4)+(+4) + (+4)=+i2, (i)

and

(-4) X (+3) = + (-4) + (-4) + (-4) = - 12. (2)

Again,

-3=0-(+i)-(+i)-(+i).

.-. ( +4 )x(-3)=-(+4)-(+4)-(+4)=-i2, (3)

and

(-4)X(-3)=- (-4)-(-4)-(-4)=+i2. ( 4 )

From (i ) and (4) it follows that two factors like in

quality give a positive product ; and from (2) and (3)

it follows that two factors opposite in quality give a

negative product.

Thus the product ab is positive or negative according as a

and b are like or unlike in quality.

34. COR. I. Any product containing an odd number

of negative factors will be negative ; all other products

will be positive.

Hence, changing the quality of an even number of

factors will not affect the product; but changing the

quality of an odd number of factors will change the

quality of the product.

35. COR. 2. The quality of any term is changed by

changing the quality of any one of its factors, or by

multiplying it by â€” 1.

The quality of any expression is changed by changing

the sign before each of its terms.

FUNDAMENTAL OPERATIONS. 1 3

36. Law of Order of Factors. Factors may be ar-

ranged in any order ; that is,

ab = ba. (A')

For from arithmetic we know that any change in the

order of factors will not change the aritJimctica! value

of their product; and from the law of quality, any

change in the order of factors will not change the

quality of their product.

Thus, (+ 4) (- 3) (- 5) = (- 5) (+ 4) (- 3) = (- 3) (~ 5) (+ 4).

37. Law of Grouping of Factors. Factors may be

grouped in any manner ; that is,

abc = a(bc). (B')

For by the law of order we have

abc â€” bca.

â€” {be) a â€” a (b c).

Since a (bcd)= bc(da), a product is multiplied by

any number a by multiplying one of its factors by a.

Note. The laws of order and grouping are often called the

commutative and associative laws of addition and multiplication.

38. Equimultiples of two or more expressions are

the products obtained by multiplying each of them

by the same expression.

Thus, A m and B m are equimultiples of A and B.

39. If Am = Bm (i)

and m is not zero, thc7i

A = B. (2)

For dividing each member of identity (i) by m, we

obtain identity (2).

14 ALGEBRA.

40. Distributive Law. The product of two expres-

sions is equal to the sum of the products obtained by

multiplying each term of either expression by the other,

and conversely. That is,

(a + b + c+ ...) x = ax + bx + cx + .... (C)

Let m and n denote any positive integers, and a and

b any numbers whatever; then we have

(a + b) m = (a + b) + {a + b) + . . . to m terms

= am + bm. (i)

Again,

(a + b) (m -i- n) = a (m -7- ti) + b(m-r-n). (2)

For multiplying each member of (2) by n, we obtain

the identity (1) ; hence, by Â§ 39, (2) is an identity.

Hence,

(a + b)z = az + bz, (3)

in which z is any positive number.

Again,

(a + b)(-z)=a(-z) + b (- *). (4)

For changing the quality of the members of (4), we

obtain identity (3) ; hence, by Â§ 39, (4) is an identity.

The same reasoning would apply to any polynomial

as well as to a + b ; hence (C) is proved for all values

of x.

By this law similar terms are united into one; thus ^ax

41. Law of Exponents. Let m and H be any positive

integers, then by definition we have

a m a n = (aaa ... m factors) (a a a ... ton factors)

= aaa ... m + Â« factors = a m+ "-

FUNDAMENTAL OPERATIONS. 1 5

42. From the laws (A), (B), (C) of Â§Â§ 31, 32, 40,

we have the following Rule for Addition:

Write the expressions under each other, so that like

terms shall be in the same column ; then add the col-

umns separately.

43. Rule for Subtraction. To subtract one algebraic

expression from another, add to the minuend the sub-

trahend with its quality changed.

For let 5 denote the subtrahend, and R the re-

mainder; then (R + S) will denote the minuend, and

â€” 5 the subtrahend with its quality changed. But

(X+S) + (-S) = R.

Hence, parentheses preceded by the sign â€” may be

removed if the sign before each of the included terms

be changed from + to â€” or from â€” to +.

Thus, a c â€” (;;/ â€” 2 c n + 3 a x) = a c â€” m + 2 c n â€” 3 a x.

In arithmetic addition implies increase, and subtraction de-

crease ; but in algebra addition may cause decrease, and sub-

traction increase. To solve any problem of subtraction in

algebra, we first reduce it to one of addition.

IN MEMORIAM

FLOR1AN CAJORI

COLLEGE ALGEBRA

BY

J. M. TAYLOR, A.M., LL.D.,

PROFESSOR OF MATHEMATICS IN COLGATE UNIVERSITY

SIXTH EDITION

ALLYN AND BACON

Boston antJ Chicago

Copyright, 1889,

By Allyn and Bacon.

Berwick & Smith, Norwood, Mass., U.S.A.

PREFACE

THIS work originated in the author's desire for a

course in Algebra suited to the needs of his

own pupils. The increasing claims of new sciences

to a place in the college curriculum render necessary

a careful selection of matter and the most direct

methods in the old. The author's aim has been to

present each subject as concisely as a clear and

rigorous treatment would allow.

The First Part embraces an outline of those fun-

damental principles of the science that are usually

required for admission to a college or scientific

school. The subjects of Equivalent Equations and

Equivalent Systems of Equations are presented more

fully than others. Until these subjects are more

scientifically understood by the average student, it

will be found profitable to review at least this por-

tion of the First Part.

In the Second Part a full discussion of the Theory

of Limits followed by one of its most important ap-

plications, Differentiation, leads to clear and concise

IV PREFACE.

proofs of the Binomial Theorem, Logarithmic Series,

and Exponential Series, as particular cases of Mac-

laurin's Formula. It also affords the student an easy

introduction to the concepts and methods of the

higher mathematics.

Each chapter is as nearly as possible complete in

itself, so that the order of their succession can be

varied at the discretion of the teacher; and it is

recommended that Summation of Series, Continued

Fractions, and the sections marked by an asterisk

be reserved for a second reading.

In writing these pages the author has consulted

especially the works of Laurent, Bertrand, Serret,

Chrystal, Hall and Knight, Todhunter, and Burnside

and Panton. From these sources many of the prob-

lems and examples have been obtained.

J. M. TAYLOR.

Hamilton, N. Y., 1889.

PREFACE TO THIRD EDITION.

In this edition a number of changes have been

made in both definitions and demonstrations. In

the Second Part, derivatives, but not differentials,

are employed. Two chapters have been added ; one

on Determinants, the other on the Graphic Solution

of Equations and of Systems of Equations.

J. M. TAYLOR.

Hamilton, N, Y-, 1895.

CONTENTS.

FIRST PART.

CHAPTER I.

Page

Definitions and Notation 1-9

CHAPTER II.

Fundamental Operations 10-21

CHAFTER III.

Fractions 22-24

CHAPTER IV.

Theory of Exponents 25-32

CHAPTER V.

Factoring ' . . 33

Highest Common Divisor 38

Lowest Common Multiple 4 1

VI CONTENTS.

CHAPTER VI.

Page

Involution, Evolution 42

Surds and Imaginaries 46

CHAPTER VII.

Equations 56-77

Equivalent Equations 57

Linear Equations 63

Quadratic and Higher Equations 65

CHAPTER VIII.

Systems of Equations 78-91

Equivalent Systems 79

Methods of Elimination 80

Systems of Quadratic Equations 85

CHAPTER IX.

Indeterminate Equations and Systems .... 92

Discussion of Problems 98

Inequalities 101

CHAPTER X.

Ratio, Proportion, and Variation .... .104-114

CHAPTER XI.

The Progressions 115-121

CONTENTS. Vll

SECOND PART.

CHAPTER XII.

Page

Functions and Theory of Limits 122-133

Function's and Functional Notation 123

Theory of Limits . 125

Vanishing Fractions 132

Incommensurable Exponents 132

CHAPTER XIII.

Derivatives i34- T 47

Derivatives 134

Illustration of D x (ax*) 136

Rules for finding Derivatives ........ 137

Successive Derivatives 145

Continuity 146

CHAPTER XIV.

Development of Functions in Series .... 148-170

Development by Division 149

Principles of Undetermined Coefficients 150

Resolution of Fractions into Partial Fractions . . . 154

Reversion of Series 159

Maclaurin's Formula 161

The Binomial Theorem 163

viii CONTENTS.

CHAPTER XV.

Page

CONVERGENCY AND SUMMATION OF SERIES . . . I7I-I92

Convergency of Series 171

Recurring Series 177

Method of Differences 182

Interpolation 188

CHAPTER XVI.

Logarithms 193-214

General Principles 193

Common Logarithms 197

Exponential Equations 203

The Logarithmic Series 205

The Exponential Series 211

CHAPTER XVII.

Compound Interest and Annuities 215-223

CHAPTER XVIII.

Permutations and Combinations 224-236

CHAPTER XIX.

Probability 237-253

Single Events 237

Compound Events 241

CONTENTS. IX

CHAPTER XX.

Page

Continued Fractions 254-265

Conversion of a Fraction into a Continued Fraction . 255

Convergents 256

Periodic Continued Fractions 263

CHAFFER XXI.

Theory of Equations 266-317

Reduction to the Form F(x) = 266

Divisibility of F (x) 267

Horner's Method of Synthetic Division 268

Number of Roots 272

Relations between Coefficients and Roots 274

Imaginary Roots 276

Integral Roots 280

Limits of Roots 281

Equal Roots 284

Change of Sign of F{x) 286

Sturm's Theorem 288

Transformation of Equations 295

Horner's Method of Solving Numerical Equations . 300

Reciprocal Equations 307

Binomial Equations 310

Cubic Equations 312

Biquadratic Equations 315

CONTENTS.

CHAPTER XXII.

Page

Determinants 318-346

Determinants of the Second Order 318

Determinants of the Third Order 322

Determinants of the Â«th Order 328

Properties of Determinants 330

Minors and Co-factors 336

Expression of A in Co-factors 337

Eliminants 340

Multiplication of Determinants 343

CHAPTER XXIII.

Graphic Solution of Equations and of Systems 347-363

Co-ordinates of a Point 348

Graphic Solution of Indeterminate Equations . . . 349

Graphic Solution of Systems of Equations .... 353

Properties of F(x) and of F{x) = illustrated by the

Graph of F (x) 357

Geometric Representation of Imaginary and Complex

Numbers - . 360

ALGEBRA

FIRST PART.

CHAPTER I.

DEFINITIONS AND NOTATION.

1. Quantity is anything that can be increased, di-

minished, or measured; as any portion of time or

space, any distance, force, or weight.

2. To measure a quantity is to find how many

times it contains some other quantity of the same

kind taken as a unit, or standard of comparison.

Thus, to measure a distance, we find how many times it con-

tains some other distance taken as a unit. To measure a por-

tion of time, we find how many times it contains some other

portion of time taken as a unit.

3. By counting the units in a quantity, we gain the

idea of ' how many ' ; that is, of Arithmetical Number.

If, in the measure of any quantity, we omit the unit

of measure, we obtain an arithmetical number. It

may be a whole number or a fraction. Thus by

omitting the units ft., lb., hr., in 6 ft., 3 lbs., and

2/3 hr., we obtain the whole numbers 6 and 3, and

the fraction fz.

2 ALGEBRA.

4. Positive and Negative Quantities. Two quanti-

ties of the same kind are opposite in quality, if when

united, any amount of the one annuls or destroys an

equal amount of the other. Of two opposites one is

said to be Positive in quality, and the other Negative.

Thus, credits and debits are opposites, since equal amounts

of the two destroy each other. If we call credits positive,

debits will be negative. Two forces acting along the same line

in opposite directions are opposites ; if we call one positive, the

other is negative.

5. Algebraic Number. The sign +, read positive,

and â€” , read veg t've, are used with numbers, or their

symbols, to denote their quality, or the quality of the

quantities which they represent.

Thus, if we call credit positive, + $5 denotes $5 of

credit, and â€” $4 denotes $4 of debt. If + 8 in. de-

notes 8 in. to the right, â€”9 in. denotes 9 in. to the

left. By omitting the particular units $ and in., in

+ $5, â€” $4, + 8 in., â€”9 in., we obtain the algebraic

numbers + 5, â€” 4, + 8, â€” 9. + 5 is read ' positive 5/

â€” 4 is read ' negative 4.' Each of these numbers has

not only an arithmetical value y but also the quality of

one of two opposites ; hence

An Algebraic Number is one that has both an

arithmetical value and the quality of one of two

opposites.

Two algebraic numbers that are equal in arith-

metical value but opposite in quality destroy each

other when added.

DEFINITIONS AND NOTATION. 3

The element of quality in algebraic number doubles

the range of number.

Thus, the integers of arithmetic make up the simple

series,

o, i, 2, 3, 4, 5> 6 > 7> â€¢â€¢â€¢> *>'> (0

while the integers of algebra make up the double

series,

-3o,.. .,â€”4, -3,-2, -I, Â±0, +1, +2, +3, +4, â€¢ â€¢ â€¢,+ *>â€¢ (2)

An algebraic number is said to be increased by-

adding a positive number, and decreased by adding a

negative number.

If in series (2) we add + 1 to any number, we ob-

tain the next right-hand number. Thus, if to + 3 we

add +1, we obtain +4; if to â€”4 we add -f 1, we

obtain â€” 3 ; and so on. That is, positive numbers

increase from zero, while negative numbers decrease

from zero.

Hence positive numbers are algebraically the

greater, the greater their arithmetical values; while

negative numbers are algebraically the less, the

greater their arithmetical values.

All numbers are quantities, and the term quantity

is often used to denote number.

6. Symbols of Number. Arithmetical numbers are

usually denoted by figures. Algebraic numbers

are denoted by letters, or by figures with the signs

+ and â€” prefixed to denote their quality. A letter

4 ALGEBRA.

usually represents both the arithmetical value and

the quality of an algebraic number. Thus a may

denote +5, â€” 5, â€” 8, + 17, or any other algebraic

number. When no sign is written before a symbol

of number, the sign + is understood.

Known Numbers, or those whose values are known,

or supposed to be known, are denoted by figures, or

the first letters of the alphabet, as a, b, c, a', b', c\

&u b i% c x .

Unknown Numbers, or those whose values are to

be found, are usually denoted by the last letters of

the alphabet, as, x, y, z, x', y' t z', x lt y u %

Quantities represented by letters are called literal ;

those represented by figures are called numerical.

7. Signs of Operation. The signs, + (read phis),

â€” (read minus), X (read multiplied by), -r- (read

divided by), are used in algebra to denote algebraic

addition, subtraction, multiplication, and division,

respectively. The use of the signs + and â€” to indi-

cate operations must be carefully distinguished from

their use to denote quality. In the literal notation,

multiplication is usually denoted by writing the mul-

tiplier after the multiplicand. Thus, a b = a X b.

Sometimes a period is used; thus, 4-5 = 4 X 5.

Algebraic division is often denoted by a vinculum ;

thus T = a -5- b*

b

8. Signs of Relation and Abbreviation. The sign of

equality is =. The sign of identity is =. The sign

DEFINITIONS AND NOTATION. 5

of inequality is > or < , the opening being toward

the greater quantity.

The signs of aggregation are the parentheses ( ),

the brackets [ ], the brace { }, the vinculum ,

and the bar | . They are used to indicate that two

or more parts of an expression are to be taken as a

whole. Thus, to indicate the product of c â€” d multi-

plied by x, we may write (c â€” d) x, [c â€” d] x, {c â€” d\ x,

c \x

c â€” dx, or â€” d\

The sign .*. is read hence y or therefore; the sign

V is read since, or because.

The sign of continuation is three or more dots . . . ,

or dashes â€” , either of which is read and so on.

9. The result obtained by multiplying together

two or more numbers is called a Product. Each

of the numbers which multiplied together form a

product, is called a Factor of the product.

10. A Power of a number is the product obtained

by taking that number a certain number of times

as a factor. If ;/ is a positive integer, a" denotes

aa a a... to n factors, or the ?/th power of a. In

a", n denotes the number of equal factors in the

power, or the Degree of the power, and is called an

Exponent.

11. A Root of a quantity is one of the equal factors

into which it may be resolved.

The wth root of a is denoted by y/a. In %/a,

m denotes the number of equal factors into which

6 ALGEBRA.

a is to be resolved, and is called the Index of the

root. The sign ^/~ (a modification of r, the first

letter of the word radix) denotes a root. If no in-

dex is written, 2 is understood.

12. Any combination of algebraic symbols which

represents a number is called an Algebraic Expression.

13. When an algebraic expression consists of two

or more parts connected by the signs + or â€” , each

part is called a Term. Thus, the expression

a 1 + (c â€” x) y + b z 1 + c â€” d

consists of four terms.

A Monomial is an algebraic expression of one term;

a Polynomial is one of two or more terms. A poly-

nomial of two terms is called a Binomial; one of

three terms a Trinomial.

14. The Degree of a term is the number of its lite-

ral factors. But we often speak of the degree of a

term with regard to any one of its letters. Thus,

8 a 2 fix*, which is of the ninth degree, is of the

second degree in a, the third in b, and the fourth

in x.

The degree of a polynomial is that of the term of

the highest degree. An expression is homogeneous

when all its terms are of the same degree.

A Linear expression is one of the first degree; a

Quadratic expression is one of the second degree.

DEFINITIONS AND NOTATION. 7

15. Any algebraic expression that depends upon

any number, as x, for its value is said to be a Function

of x. Thus, 5 x s is a function of x ; 5 x 2 + a s â€” 7 x

is a function of both x and a; but if we wish to con-

sider it especially with reference to x, we may call it

a function of x simply.

A Rational Integral Function of X is one that can

be put in the form

Ax" + Bx"- 1 + Cx n ~ 2 + ... + F,

in which 11 is a whole number, and A,B, ..., .F denote

any expressions not containing x.

Thus, a X s â€” 4 x' 2 â€” b x + c and x 8 â€” \ x are rational integral

functions of x of the third degree.

16. The Reciprocal of a number is one divided by

that number.

17. If a term be resolved into two factors, ei-

ther is the Coefficient of the other. The coeffi-

cient may be either numerical or literal. Thus, in

4a be 2 , 4 is the coefficient of a be 2 , 4 a of be 2 , and

4 a b of e 2 . When no numerical coefficient is

written, 1 is understood; thus, a = (+ 1) a, and

-Â« = (-!)*

18. Like or Similar Terms are such as differ only in

their coefficients. Thus, 4a be 2 and 10 a be 2 are like

terms ; 6 a 2 fty 2 and 4 a IPy 2 are like, if we regard* 6 a 2

and 4a as their coefficients, but unlike if 6 and 4 be

taken as their coefficients.

8 ALGEBRA.

19. A Theorem is a proposition to be proved.

20. A Problem is something to be done.

21. To solve a problem is to do what is required.

22. An Axiom is a self-evident truth.

The axioms most frequently used in Algebra are

the following:

1. Numbers which are equal to the same number

or to equal numbers are equal to each other.

2. If the same number or equal numbers be added

to, or subtracted from, equal numbers, the

results will be equal.

3. If equal numbers be multiplied by the same

number or equal numbers, the products will

be equal.

4. If equal numbers be divided by the same num-

ber, except zero, or by equal numbers, the

quotients will be equal.

5. Like powers or like roots of equal numbers are

equal.

23. Identical Expressions. Equal expressions that

contain only figures, or expressions that are equal

for all values of their letters, are called Identical

Expressions.

Thus, 4 + 6 and 5X2 are identical expressions; so also are

(a + b) (a- b) and a* - b 2 .

24. An Equality is a statement that two expressions

represent the same number. The two expressions are

called the Members of the equality.

DEFINITIONS AND NOTATION. 9

25. Identities and Equations. Equalities are of two

kinds, identities and equations.

The statement that two identical expressions are

equal is called an Identity. In writing identities, the

sign =, read ' is identical with,' is often used instead

of the sign =.

Thus the equality 5 + 7 = 4X3 is an identity; so also is

a' 2 â€” x 2 = (a + r) (a â€” x), since it holds true for all values of a

and x. To indicate that these equalities are identities, they may

be written 5 + 7=4X3 and a 2 â€” x 2 = (a + -r) {a â€” -*')â€¢

If two expressions are not identical, and one or

both of them contains a letter or letters, the state-

ment that they are equal is called an Equation.

Thus the equalities $x â€” 6 = o,ja = 2a + 5, and 2 y â€” 4 x = 6,

are equations. The first holds true for xâ€” 2, and the second

for a â€” 1. The equation 2y â€” 4x=6 holds true iov y = 2^ + 3;

hence if x â€” 1, y â€” 5 ; \ix = 2,y = 7) and so on.

26. Algebra is that branch of mathematics which

treats of the equation, its nature, the methods of solv-

ing it, and its use as an instrument for mathematical

investigation.

The history of algebra is the history of the equation. The

notation of algebra, including symbols of operation, rel ition,

abbreviation, and quantity, was invented to secure conciseness,

clearness, and facility in the statement, transformation, and solu-

tion of equations. The number of algebra, which has quality as

well as arithmetical value, was conceived in the effort to interpret

results obtained as solutions of equations. Hence the study of

the nature and laws of algebraic number and of the methods

of combining, factoring, and transforming algebraic expressions

should be pursued as auxiliary to the study of the equation.

This will lend interest and profit to what might otherwise be

regarded as dull and useless.

10 ALGEBRA,

CHAPTER II.

FUNDAMENTAL OPERATIONS.

27. Addition is the operation of finding the result

when two or more numbers are united into one. The

result, which must always be expressed in the sim-

plest form, is called the Sum.

28. Subtraction is the operation of taking from one

number, called the Minuend, another number, called

the Subtrahend. The result, which must be expressed

in the simplest form, is called the Remainder.

The subtrahend and the remainder are evidently

the two parts of the minuend; hence, since the whole

is equal to the sum of all its parts, we have

minuend = subtrahend + remainder.

29. To multiply one number by another is to treat

the first, called the Multiplicand, in the same way that

we would treat I to obtain the second, called the

Multiplier.

Thus, 3=1 + 1 + i ; .-.4X3 = 4 + 4 + 4-

Again, ^=1-^3 X 2; .-. 9X M = 9 + 3 X 2.

30. Having given a product and one factor, Division

is the operation of finding the other factor. The

given product is called the Dividend ; the given fac-

tor, the Divisor ; and the required factor, the Quotient.

From their definitions, the divisor and quotient are

evidently the two factors of the dividend ; hence,

FUNDAMENTAL OPERATIONS. II

since any number is equal to the product of its

factors, we have

quotient X divisor = dividend. (i)

Let D denote the dividend, and d the divisor; then

the expression D -5- d will denote the quotient, and

by (i) we shall have

(D + d)xd=D. (2)

31. Law of Order of Terms. Numbers to be added

may be arranged in any order ; that is,

a + b = b + a. (A)

For let there be any two quantities of the same kind,

one containing a units and the other b units. Now if

we put the second quantity with the first, the measure

of the resulting quantity will be a + b units ; and if we

put the first quantity with the second, the measure of

the resulting quantity will be b + a units. It is self-

evident that these two resulting quantities will be

equal; hence their measures will be equal;

.*. a + b = b + a.

A similar proof would apply to an expression of any

number of terms.

32. Law of Grouping of Terms. Numbers to be added

may be grouped in any manner; that is,

a + l, + c- = a+(b + <;). (B)

For by the lazv of order we have

a + b + câ€”b + c+ a

= (b + c)+a = a+{b + c).

A similar proof would apply in any other case.

12 ALGEBRA.

33. Law of Quality in Products. Two like signs give

+ ; tzvo unlike signs give â€” .

By the definition of multiplication we have

+ 3 = 0+ (+i) + (+ i) + (+i).

.*â€¢ (+4)X(+3)=+(+4)+(+4) + (+4)=+i2, (i)

and

(-4) X (+3) = + (-4) + (-4) + (-4) = - 12. (2)

Again,

-3=0-(+i)-(+i)-(+i).

.-. ( +4 )x(-3)=-(+4)-(+4)-(+4)=-i2, (3)

and

(-4)X(-3)=- (-4)-(-4)-(-4)=+i2. ( 4 )

From (i ) and (4) it follows that two factors like in

quality give a positive product ; and from (2) and (3)

it follows that two factors opposite in quality give a

negative product.

Thus the product ab is positive or negative according as a

and b are like or unlike in quality.

34. COR. I. Any product containing an odd number

of negative factors will be negative ; all other products

will be positive.

Hence, changing the quality of an even number of

factors will not affect the product; but changing the

quality of an odd number of factors will change the

quality of the product.

35. COR. 2. The quality of any term is changed by

changing the quality of any one of its factors, or by

multiplying it by â€” 1.

The quality of any expression is changed by changing

the sign before each of its terms.

FUNDAMENTAL OPERATIONS. 1 3

36. Law of Order of Factors. Factors may be ar-

ranged in any order ; that is,

ab = ba. (A')

For from arithmetic we know that any change in the

order of factors will not change the aritJimctica! value

of their product; and from the law of quality, any

change in the order of factors will not change the

quality of their product.

Thus, (+ 4) (- 3) (- 5) = (- 5) (+ 4) (- 3) = (- 3) (~ 5) (+ 4).

37. Law of Grouping of Factors. Factors may be

grouped in any manner ; that is,

abc = a(bc). (B')

For by the law of order we have

abc â€” bca.

â€” {be) a â€” a (b c).

Since a (bcd)= bc(da), a product is multiplied by

any number a by multiplying one of its factors by a.

Note. The laws of order and grouping are often called the

commutative and associative laws of addition and multiplication.

38. Equimultiples of two or more expressions are

the products obtained by multiplying each of them

by the same expression.

Thus, A m and B m are equimultiples of A and B.

39. If Am = Bm (i)

and m is not zero, thc7i

A = B. (2)

For dividing each member of identity (i) by m, we

obtain identity (2).

14 ALGEBRA.

40. Distributive Law. The product of two expres-

sions is equal to the sum of the products obtained by

multiplying each term of either expression by the other,

and conversely. That is,

(a + b + c+ ...) x = ax + bx + cx + .... (C)

Let m and n denote any positive integers, and a and

b any numbers whatever; then we have

(a + b) m = (a + b) + {a + b) + . . . to m terms

= am + bm. (i)

Again,

(a + b) (m -i- n) = a (m -7- ti) + b(m-r-n). (2)

For multiplying each member of (2) by n, we obtain

the identity (1) ; hence, by Â§ 39, (2) is an identity.

Hence,

(a + b)z = az + bz, (3)

in which z is any positive number.

Again,

(a + b)(-z)=a(-z) + b (- *). (4)

For changing the quality of the members of (4), we

obtain identity (3) ; hence, by Â§ 39, (4) is an identity.

The same reasoning would apply to any polynomial

as well as to a + b ; hence (C) is proved for all values

of x.

By this law similar terms are united into one; thus ^ax

41. Law of Exponents. Let m and H be any positive

integers, then by definition we have

a m a n = (aaa ... m factors) (a a a ... ton factors)

= aaa ... m + Â« factors = a m+ "-

FUNDAMENTAL OPERATIONS. 1 5

42. From the laws (A), (B), (C) of Â§Â§ 31, 32, 40,

we have the following Rule for Addition:

Write the expressions under each other, so that like

terms shall be in the same column ; then add the col-

umns separately.

43. Rule for Subtraction. To subtract one algebraic

expression from another, add to the minuend the sub-

trahend with its quality changed.

For let 5 denote the subtrahend, and R the re-

mainder; then (R + S) will denote the minuend, and

â€” 5 the subtrahend with its quality changed. But

(X+S) + (-S) = R.

Hence, parentheses preceded by the sign â€” may be

removed if the sign before each of the included terms

be changed from + to â€” or from â€” to +.

Thus, a c â€” (;;/ â€” 2 c n + 3 a x) = a c â€” m + 2 c n â€” 3 a x.

In arithmetic addition implies increase, and subtraction de-

crease ; but in algebra addition may cause decrease, and sub-

traction increase. To solve any problem of subtraction in

algebra, we first reduce it to one of addition.