TABLE OF CONTENTS

PREFACE

Воок І

THE FUNDAMENTAL OPERATIONS OF COMMON ALGEBRA

THEIR LAWS AND APPLICATIONS

INTRODUCTION

The Nature of Numbers and the Fundamental Postulate of Arithmetic

The Equality of Two Groups .

Symbolic Representation of Numbers

The Equation and Inequalities

Counting

11

12

13

14

15

CHAPTER I

ADDITION AND MULTIPLICATION

The Sum of Two Groups-Parenthesis

Addition of Positive Integers and its Fundamental Laws

Multiplication of Positive Integers and its Fundamental Laws .

The Exponent and Index Law of Multiplication

Addition and Multiplication of Monomials and Polynomials

17

18

19

21

24

CHAPTER II

POSITIVE AND NEGATIVE NUMBERS - ADDITION AND SUBTRACTION

Definition

Series of Natural Numbers

Positive and Negative Numbers

The Absolute Value of Numbers — Unlike Signs

Addition of Algebraic Numbers, Rule

Addition of Monomials

Subtraction of Algebraic Numbers, Rule

Subtraction of Monomials

27

28

29

30

31

32

35

36

CHAPTER III

SUBTRACTION AND THE NEGATIVE INTEGER GENERALIZED DISCUSSION

.

Numerical Subtraction

39

Determinateness of Numerical Subtraction-Formal Rules of Subtraction 40

Zero and the Negative

42

Limitation of Subtraction - Symbolic Equations

45

Principle of Permanence

46

Subtraction of Polynomials

48

Use of Parentheses

50

CHAPTER IV

PositivE AND NEGATIVE NUMBERS MULTIPLICATION

53.

Multiplication of Monomials

Multiplication of Polynomials

Polynomials in Ascending and Descending Powers of x

55

CHAPTER V

PositivE AND NEGATIVE NUMBERS - DIVISION

Numerical Division

57

Determinateness of Division

58

Formal Rule of Division—The Index Law

59

Exponent Zero — Division of Monomials

60

Division of Polynomials — First and Second Rules

61

Third and Fourth Rules of Division

67

Indeterminateness of Division by 0-Determinateness of Symbolic Division 69

The Vanishing of a Product

70

CHAPTER VI

APPLICATIONS OF THE FUNDAMENTAL OPERATIONS — SIMPLE EQUATIONS

An Identity-An Equation of Condition-The Unknown Quantity

Axioms

Transposition

71

72

73

CHAPTER VII

APPLICATION OF ADDITION AND MULTIPLICATION-POWERS OF MonomiALS-

BINOMIALS

Definitions: Rational Integral Polynomials, etc.

79

Power Formulae - The Double Sign

80

Binomial Theorem

82

CHAPTER VIII

FACTORING AND SOLUTION OF EQUATIONS BY FACTORING

To Factor a Polynomial

To Factor a Trinomial .

To Factor the Difference of Two Squares

Case IV: To Factor the Sum and Difference of Two Cubes

Case V: To Factor the Trinomial 22 + px +9

Case VI: To Factor Polynomials.

Remainder Theorem

Factor Theorem

Case VII: To Factor Polynomials which have Binomial Factor

Case VIII: To Factor a Polynomial of Four Terms

Solution of Equations - Imaginary Numbers

89

90

92

93

94

96

98

99

CHAPTER IX

101

GREATEST COMMON DIVISOR AND LEAST COMMON Multiple

G. C. D. of Quantities Readily Factored

G. C. D. of Two or more Algebraic Expressions

Least Common Multiple of Quantities readily Factored and those not

readily Factored .

102

109

CHAPTER X

FRACTIONS

Definitions: Rational Fractions, etc.

Rule of Signs

Reduction of Fractions to Lowest Terms

Reduction of Fractions to a Lowest Common Denominator

Addition and Subtraction of Fractions

Multiplication of Fractions

Powers of Fractions .

Division of Fractions

Complex Fractions

Continued Fractions

Special Theorems in Fractions

113

114

115

121

124

129

131

133

134

136

139

Воок II

SOLUTION OF EQUATIONS OF THE FIRST DEGREE

CHAPTER I

EQUATIONS OF THE FIRST DEGREE

Identity — The Equation

Root of an Equation - Degree of an Equation

Equivalent Equations

Theorem on Transformation of Equations

Removal of Denominator

The Solution of an Equation of the First Degree

145

146

147

147

151

154

Formulae for the Solution of an Equation of the First Degree in Ono

Unknown Quantity

158

INTERPRETATION OF THE SOLUTION OF PROBLEMS

Positive and Negative Solutions

Zero Solutions

Indeterminate Solutions

Infinite Solutions

187

190

192

193

CHAPTER VI

SIMULTANEOUS LINEAR EQUATIONS IN Two UNKNOWN QUANTITIES

Indeterminate Equations

Independent Equations

Incompatible Equations

Equivalent Systems of Equations .

The Solution of two Equations of the First Degree – Elimination by

Addition and Subtraction

Elimination by Substitution .

Elimination by Comparison

Elimination by Undetermined Multiplier

198

199

200

201

204

207

212

218

CHAPTER VII.

GENERAL SOLUTION OF A SYSTEM OF Two EQUATIONS

General Solution

Composition of Formulae

Symmetry of Equations

Discussion of Cases abi – ba' +0, etc.

Homogeneous Equations.

Two Equations of the First Degree which have a Common Root

CHAPTER VIII

GENERAL SOLUTION OF A SYSTEM OF THREE EQUATIONS

Examples of Three Equations of the First Degree

Number of Solutions of a System of n Linear Equations

Problems involving Three or More Unknown Numbers

238

242

251

DIOPHANTIAN EQUATIONS OF THE FIRST DEGREE

Simple Indeterminate Equations

Rule for Solution of ax + by=0

Given One Solution of ax — by=c in Positive Integers, to Find the

General Solution

General Solution of Two Equations

268

270

272

274

Воок III

INVOLUTION-EVOLUTION-FRACTIONAL AND NEGATIVE EXPONENTS

CHAPTER I

INVOLUTION

Index Law of Involution .

Powers of Monomials, Fractions, Binomials, and Polynomials

277

278

CHAPTER II

EVOLUTION

Definition of a Root .

Law of Signs of Roots of Quantities

Principal Root.

Theorems in Evolution

Square Root of Compound Quantities

Square Root of Arithmetical Numbers

Cube Root of a Polynomial

Cube Root of Arithmetical Numbers