Edward Olney.

# The complete algebra : embracing simple and quadratic equations, proportion, and the progressions, with an elemenary and practial view of logarithms, a brief treatment of numerical higher equations, and a chapter on the business rules of arithmetic treated algebraically online

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Sheldon & Compa?iy's Text-Hooks,

The Scieiiee of Government in Connection tvith
American Institutions, By Joseph Alden, D.D., LL.D.,
Pres. of State Normal School, Albany. 1 vol 12mo.
Adapted to the wants of High Schools and Colleges.

Allien' s Citizen's 3Ianual: a Text-Book on Government, in
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Common Schools. It is in the form of questions and answers.
By Joseph Alden, D.D., LL.D. 1 vol. ICmo.

Hereafter no American can he said to he educated who does not thoroughly

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University of California • Berkeley

The Theodore P. Hill Collection

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Early American Mathematics Books

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OLNEY'S MATHEMATICAL SERIES.

THE

COMPLETE ALGEBRA

EMBRACING

SIMPLE AND QUADRATIC EQUATIONS, PROPORTION AND

THE PROGRESSIONS, WITH AN ELEMENTARY AND

PRACTICAL VIEW OF LOGARITHMS; A BRIEF

TREATMENT OF NUMERICAL HIGHER

EQUATIONS.

AND A CHAPTER ON THE

TREATED ALGEBRAICALLY

Designed to be sufficiently elementary for beginners, and sufficiently

THOROUGH and COMPREHENSIVE TO MEET THE WANTS OF OUR BETTER
HIGH SCHOOLS, ACADEMIES, AND ORDINARY COLLEGES.

BY

EDWARD OLNEY,

Professor of Mathematics in the University of Michigan,

NEW YORK :

SHELDON & COMPANY,

8 MURRAY STREET.

OLNEY'S MATHEMATICAL SERIES

EMBRACES THE EOLLOWING BOOKS.

"Primary Arithmetic ^

Jb'leme7its of jirithmetic,

Practical jirithmetic.

Teacher's Mand-^ook of

A.rithmetical £Jxercises,

Science of Arithmetic,

Send for ftill Circular of Olney's Arithmetics.

fntrodicctio7i to Algebra,
Co7nptete Algebra,

JGCey to Complete Algeb?'a,
^nipersitjy Algebra,

£!ey to University Algebra,

2'est £^xamples in Algebra,
^leme?its of Geometry,
(Separate.)
Elements of Trig 07iome try,

(Separate.)

introduction to Geometry, ^a7*t I,

(Bound separate.)
Elements of Geometry a7?d Trigonometry,

(Bound in one Vol.)
Geometry a7id Trigo7iomet7y,
(University Edition.)
General Geometry a7id Calculus,

Copyright ^y 1870, 1875, 1878, by Sheldon <&» Co,

Blectrotyped by Smith & McDotraAL, 82 Beekman St., N. Y.

PREFACE.

Three main purposes have controlled the author in the preparation
of this work : — First, to provide, in a single volume, of convenient
size, an elementaiy treatise upon Algebra, sufl&ciently simple for the
use of beginners in the science — sufl&ciently full on the subjects em-
braced to render their rediscussion in a subsequent volume unnecessary,
and to aflford a range of topics comprehensive enough to meet the
wants of our Common and Pubhc High Schools, and for a good pre-
paration to enter any of our Colleges or Universities. Second, to train
the pupil to methods of reasoning, rather than in mere methods of
operating. Third, to present the whole in such a form as to make the
book a convenient one to use in the class room.

As the mathematical studies are at present pursued in our schools,
pupils are expected to have a fair knowledge of Mental, and some
knowledge of what is variously styled Practical or "Written Arithmetic,
before entering upon the study of Algebra. "Without expressing any
opinion upon the propriety of this course, the author has accepted it
as a fact likely to exist for some time to come, and has attempted to
adapt his Algebra to it. This book is, therefore, not a mere child's
book, but is designed as a First Book in Algebra for pupils who have
the knowledge of Arithmetic usually considered requisite to a com-
mencement of this study.

The volume contains a more than ordinarily thorough and full treat-
ment of the processes of Literal Arithmetic, viz., the Fundamental
Rules, Fractions, Involution, Evolution, and the Calculus of Radicals.
It is in these processes that pupils usually find the most difficulty;
and in them comparatively few become proficient, especially in the
Calculus of Radicals. Yet no one can become an algebraist, and, if
not an algebraist, not a mathematician, without being perfectly mas-
ter of these processes. Hence they are treated so much at length and
with so much care and thoroughness. It is idle to deceive the pupil
with the notion that he is mastering the subject of algebra by obtain-
ing a superficial acquaintance with the nature and uses of some of the
more simple forms of the equation. The pupil who cannot attain a
good degree of familiarity with the literal arithmetic, cannot appre-
ciate mathematical reasoning, or profit much by this study.

VI PREFACE.

The treatment of Simple and Quadratic Equations will be found as
full as is requisite for the foundation of a good mathematical educa-
tion. It is not the author's purpose to rediscuss the themes here con-
sidered in his second volume. The matter presented on these subjects
does not differ much from what is usually contained in our so-called
higher algebras ; and, in fact, is more thoroughly treated than in most
of them. Katio, Proportion, and the Progressions have received a
proper share of attention. The transformation of a Proportion will
be found to be discussed in a somewhat different manner from the
usual one, and, it is hoped, in such a way that the pupil who enters
into the spirit of the treatment, and becomes familiar with the meth-
ods, will feel that this elegant and important mathematical instrument
is his own.

It is thought that the chapter on the Business Bules of Arithmetic
will be particularly acceptable to a large class of students. Some
pri^blems in Interest, Common Discount, and Alligation, which are
ordinarily considered intricate, will be found quite within the reach of
ordinary minds. Other applications of the equation to the solution of
practical questions, are not less extended than in our common treatises.
The chapter on Logarithms will give the student a clear conception of
the nature of this kind of number, and enable him to understand and
use the common tables.

No attempt is made at the Discussion of Equations, or the Devel-
opment of Series. To discuss an equation well, is a mathematical
accomplishment. It is not a thing which the tyro can master. He
may follow another through such a discussion ; but he will do little
more. The Development of Functions is reserved for the second
volume, where the Binomial Formula, and the Logarithmic Series will
be seen to be but deductions from a more general, and more easily
produced formula. The Binomial Formula is given (though not pro-
duced) in this treatise, and much care is bestowed in showing how to
use it in developing the common forms to which it is appUcable. It is
thought that these topics, with the discussion of them given in the
text, will meet the author's first purpose.

The attempt to accomplish the second end, viz. , to train the pupil
to methods of reasoning, rather than in mere methods of operating,
has given character to the presentation of every topic. Propositions
are clearly stated at the outset, and demonstrations are given in form,
and with the rigor of a geometrical argument. That there is some
defect in our methods of instruction, in this regard, must be jminfully
evident to every one who has been called to examine large numbers of
our youth in this study. The author has examined for admission to

PREFACE. Vll

college, from 25 to 150 different students from all parts of our country,
each year, for the last 16 years, and he has almost invariably found
little or no knowledge of the processes as arguments, even when a
good degree of skill in the use of the processes had been attained.
Perhaps a majority of those examined could multiply the square root
of 2 by the cube root of 3, but scarce one in 50 could develop the
process in a logical form, or, in most cases, give any rational account
of it. Now, it need not be said that, in a course of education, this is
a fundamental defect ; it is failure just where success is vital. The
processes of a mathematical science are of comjoaratively little worth
to a great majority of those who study them ; the development of tho
reasoning powers to which such studies are addressed, is of the high-
est importance to all. By teachers who cannot appreciate these truths,
this book will very probably be misunderstood ; but to such as do feel
the force of them, the author appeals with the fullest conj&dence, not
indeed that his book will meet the exigency, but that it will be wel-
comed as an effort in the right direction, and as a help in remedying

To such as use the book the author commends the following

SUGGESTIONS TO TEACHERS.

Great pains have been taken to adapt the whole to the use o£ the
class room. Definitions and all Propositions are made to stand out
clearly, and have been written with great care. These should be mem-
orized by the student, as well as thoroughly comprehended. It is
also recommended that the Hules be committed to memory, verbatim ;
not, indeed, to be recited as a mere parrot-like performance, but as a
means of acquiring the language of the science, and attaining faciHty
in clothing its thoughts in a becoming garb. But let the Demonstka-
TioN of every Proposition and Bute he inseparably connected with the state-
ment of the truth or process. The author has endeavored to make these
demonstrations to their propositions just what the demonstrations in
Geometry are to theirs ; and the method of use should be the same.

The Model Solutions are samples of what the pupil should give at
the blackboard in the class room, repeating the rcasonhig in the ^^expla-
nation " of every example, till it is perfectly familiar. These solutions
are not mere statements of hoio the thing is done, but are designed to
be a logical presentation of the process as an argument. It is just at
'this point that the great, and well-nigh universal failure referred to
above occurs. By a close adherence to the plan here presented, it is
hoped something may be done to remedy the evil.

The Synopses should be made the basis of reviews. No student

VUl PEEFACE.

should be allowed to tMnk he has mastered a subject, till he can go to
the blackboard, put down the Synopsis, and discuss the whole
theme therefrom, stating every principle, in good language, and dem-
onstrating, in good style, every proposition.

The Test Questions are not designed to be full or exhaustive ; they
are usually only a few detached questions, such as, if well answered,
would satisfy an examiner that a student understood the subject under
consideration.

The Examples are more than ordinarily numerous, and have been
selected from the common sources. In the Key there will be found a
very large number of additional examples for the teacher's use in class
room drill . It will often be found serviceable to assign these in the
class room in preference to those in the text, for reasons which will be
apparent to every teacher.

As to originahty, though nothing may have been developed which is
absolutely new, there are many features to which attention might be
called. Among these are the exposition of the doctrine of the signs -}-
and — , the theory of signs in multipUcation, the treatment of subtrac-
tion, some things in factoring, and the whole spirit of the treatment of
the Calculus of Radicals, the exposition of the nature of a Literal Frac-
tion, the treatment of Proportion, etc. The very idea of the character
and province of the science, as given in the definition of Algebra, may
not be without interest. But the Author is in no fear of being called
a plagiarist. That he has not borrowed more freely, may be his chief
misfortune.

Valuable suggestions have been received from many practical teach-
ers ; and the Author feels under special obligation to Mr. Thos. Hunter,
President of the Female Normal and High School, New York City, who
has read the entire work in manuscript, and whose words of criticism,
not less than his generous appreciation and approbation, have done
much to foster the hope that the work may not be without merit.

"With these remarks, the Author commits his effort to the judgment
of his fellow teachers, not indeed without the keenest sense of its im-
perfections, but with the humble hope that it may be found suggestive,
and to some extent serviceable in promoting a much needed reform in
our methods of teaching this fundamental branch of mathematical

science.

EDWARD OLNEY.

IJNrvEBsrrT of Michigan,
Ann Arbor, January^ 1870.

N. B. — For scope and purpose of the Appendix, see foot-note, p. 3^1.

CONTENTS.

inthod uction,

SECTION L

A BRIEF SURVEY OF THE OBJECTS OF PURE MATHEMAT-
ICS AND OF THE SEVERAL BRANCHES.

PAGE

Puke Mathematics. — Definition [I); branches en\imerated
{2,3) 1

Quantity. — Definition (4) 1

Number. — Definition {5}', Discontinuous and continuous {6,
7,8) 2,3

Definition OF THE Seveeal Branches. — Arithmetic \P)\ Al-
gebra {10); Calculus {11)', Geometry {12) 3, 4:

Synopsis 4

SECTION II

L0GIC0-MATHE3kLiTICAL TERMS.

Proposition. — Definition {13) 5

Varieties of Propositions. — Enumerated (j[4) ; Axiom {15)',
Theorem {16); Lemma {18); Corollary {10); Postulate
{20); Problem {21) 5-7

Definition OF — Demonstration {17); Rule {22); Solution

{23); SchoUum {24) 6, 7

Synopsis 7

♦>♦ •

PART L-LITERAL ARITIBIETIC.

CHAPTER I.

FUNDAMENTAL RULES.

SECTION I.

NOTATION.

System of Notation. — Definition {25) 8

Symbols OF Quantity.— Arabic {26); Literal {27)\ advan-
tages of latter {28) ♦. 8, 9

Examples 10, 11

Laws.— Of Decimal Notation {29); of Literal Notation (30) 12-11

X CONTENTS.

PAOK

Symbol OO. and its meaning {31) 14

Symbols of Opekation. — {32) ; Sign -f- {33)', Sign — (54);
Sign ^(55); Sign X, etc. {36)\ Sign-^, etc. (57); Sign V"

{44) 14-15

Definitions. — Power (55); Koot (59) ; Exponent (40), A
positive integer {41), A positive fraction {42)i A negative

number {43) ; Examples 15-18

Symbols of Relation : — Sign : {45^', Sign •• {4G); Sign =

{47) ; Signs > < {49), 18-19

Symbols of Aggregation. — , (),[], { }, j, {50, 51) 19

Symbols of Continuation {52) 19

Symbols of Deduction {53) 20

Positive and Negative. — How applied {54) ; Double use of
signs -f- and — {55) ; Essential sign of a quantity {50) ;
Illustrations ; Abstract quantities no sign {57) \ Less than

Zero {58) ; Increase of a negative quantity {50) 20, 21

Names of Diffeeent Forms of Expression. — Polynomial
{00) ; Monomial, Binomial, Trinomial, etc. {01) ; Coeffi-
cient {02) ; Similar Terms {03) 23, 24

Exercises in Notation 24, 25

Exercises in Eeading and Evaluating Expressions 25, 26

Synopsis and Test Questions 26, 27

SECTION II.

Definitions. — Addition {04) ; Sum or Amount {05) 28

Prop. 1. — To add similar terms {00) ; Examples ; Cor. 1, Sign
of the sum {07) ', Sch. Addition sometimes seems like Sub-
traction ; Cor. 2, Algebraic sum of a positive and negative
quantity {08} ; Cor. 3, Addition does not always imply an
increase {09) 28-31

Prop. 3.— To ndd dirsimilar tciTJis {^0); Sch. Tlie sign —
before a term; Cor, Adding a negative quantity {71))

Examples 31-^33

pROB. 1. — To add polynomials {72, ; Sch. 1, Practical method

• Examples 33-37

Prop. 3. — Terms but partially similar {73) ; Examples 37. 38

Prop. 4. —Compound terms (74) ; Examples -. oS. SO

Synopsis and Te?.t Qt'estsons 40

CONTENTS. XI

PAOB

SECTION III,

SUBTKACTION.

Definitions. — Subtraction, Minuend, Subtrahend, Difference,
or Eemainder (75, 76) 40, 41

Prob. — To perform Subtraction {77) 42

Practical Suggestions, Sch. 1, 2 ; Removing a ( ) preceded
by the sign — {7S) ; Introducing a ( ) {70) '■> Several paren-
theses inclosing each other {80) 45-48

Examples 42-49

Synopsis and Test Questions 49

SECTION IV.

MULTIPLICATION.

Definitions. — {81), Chrs. 1, 2, and Sch. Character of the

factors and product {82, 83, 84) 50

Peop. 1. — Order of factors immaterial {85) 51

Pkop. 2. — Sign of product {80) ; Cors. 1, 2, 3, Sign of pro-
duct of several factors (87, 88, 89) 51, 52

Peop. 3. — Product of quantities affected with exponents (90) ;

Examples 52-54

Prob. 1. — To multiply'- monomials {91) ; Examples 54-56

Prob. 2. — To multiply polynomials {92'*', Examples 56-59

Three Important Theorems. —Square of the sum {94);
Square of the difference {95); Product of sum and differ-
ence (96 1 ; Examples 59-61

Synopsis and Test Questions 62

SECTION V.

DIVISION.

Definitions. — {97) ; Relation to multiplication (98, 99) ;
Cors. 1-5, Deductions from definition {100-104); Can-
cellation {105) 63, 64

Lemma 1. — Sign of quotient {106) 64

Lemma 2. a^ -^ a" {107) ; Examples ; Cor. 1. Exponent
{108) ; Cor. 2, How negative exponents arise {109) ;
Cor. 3, To transfer a factor from dividend to divisor
{110) 64-66

•-•

Xn CONTENTS.

PAQB

Peob. 1. — Division of Monomials {111) ; Examples 67, 68

Peob. 2. — To divide a polynomial by a monomial {112); Ex-
amples ; Arrangement of terms {113) 68-70

Peob. 3. — ^To divide a polynomial by a polynomial {114);

Examples 70-75

Synopsis and Test Questions 76

^-^^

CHAPTEE n.

FA CTOniNG.

SECTION I.

FUNDAMENTAL PROPOSITIONS.

Definitions. — Factor [115) ; Common Divisor {116) ;

Common Multiple (117) ; Composite {118) ; Prime

{119, 120) 77

Peop. 1. — To resolve a monomial {121) ; Examples 78

Peop. 2. — To remove a monomial factor from a polynomial

{122) ; Examples 79

Peop. 3. — To factor a trinomial square {123); Examples 79-81

Peop. 4. — To factor the difference of two squares {124) ;

Examples 81, 82

Peop. 5. — Given one factor to find the other {125); Examples 82

Peop. 6. — By what the sum, and the difference of like powers

are divisible {126) ; Examples 83-85

Prop. 6 applied to the case of fractional exponents {128) ;

Examples 85, 86

Peop. 7. — To resolve a trinomial {120) ; Examples 86, 8'J

Peop. 8. — To resolve a polynomial by separating it into parts

{130); Examples 87, 88

SECTION II

GREATEST OR HIGHEST COMMON DIVISOR.

Definition (,131) ^^

Lemma I.'— The H. C. D. the product of all common factors

U32) ; Examples 89, 90

• *,•

CONTENTS. XIU

PAOE

Lemma 2. — A divisor of the polynomial of the form Ace"-}-

Bx—^-{-Cx"-^ Ex-\-F {134:) 90

Lemma 3. — A divisor of a number divides any multiple

{135) 91

Lemma 4. — A C. D. divides the sum, and the difference

{136) 9]

Genebal EuiiE for H. C. D. demonstrated and applied {137 y

138) 92-98

SECTION III.

LOWEST OK LEAST COMMON MULTIPLE.

Pkob. — To find theL. C. M. by resolving numbers into factors

{140) : Examples 98, 99

Finding the L. C. M. faciUtated by the method of H. C. D.

Sch 99, 100

Synopsis and Test Questions 101

* • «

OHAPTEE III.

F B, A C T I O N S,

SECTION L

Defenittoi^s and Fundamental Principles. — Of the
terms Fraction, Numerator, Denominator, etc, {141).. . . 103

The true conception of a literal fraction {142) 103

Value of a Fraction {143) 103

Cor's 1, 3. — Change^ in terms of a fraction {144, 145,). . 103

Uses and Definitions of various terms (140-156) 103-105

Signs of a Fraction : —

Three things to consider {157) 105

Essential character ; Examples {158). 105, 106

SECTION IL

REDUCTIONS.

Varieties of {159) 107

Prob. 1.— To lowest terms ; Examples 107, 109

Sch. 1.— The use of the H. C. D. • 108

XIV CONTENTS.

PAGE

ScH. 2. — The converse process ; Examples 109

Pbob. 2. — To improper or mixed form (160); Examples 109, 110

Cob. — Use of negative exponents {161) 110

Pbob. 3. — From integral or mixed to fractional form

(,162); Examples Ill, 112

Pbob. 4. — To common denominator {163)', Examples. . . 112, 113

Cob.— To L. C. D. {164); Examples 114, 115

Pbob. 5. — Complex to simple {165); Examples 115-117

SECTION III.

Pbob.— To add fractions {166) ; Examples 117-119

Cob. — To add mixed numbers {167) ; Examples 119-121

SECTION IK

SUBTKACTION.

Pbob. — To subtract fractions {168); Examples 121-123

Cob. — To subtract mixed numbers {160); Examples 123, 124

SECTION V.

MULTIPLICATION.

Pbob. 1. — A fraction by an integer {170); Examples 124, 126

Pbob. 2. —To multiply by a fraction {171); Examples 126-129

ScH. — When no common factors 127

Cob. — To multiply mixed numbers {172) 129

SECTION VL

DIVISION.

Prob. 1. — To divide by an integer {173); Examples 130, 131

Pbob. 2. — To divide by a fraction {174); Examples 131, 136

Eeason for inverting the divisor 132

ScH. 1. — To reverse the operation of multiplication 132

ScH. 2. — Using the form of a complex fraction 135

KEcrPBOCAii, what {175) 136

Synopsis of Fbactions and Test Questions 137

CONTENTS. XV

PAGE

OHAPTEE IV. ,

rOWEJRS AND ROOTS.

SECTION L

INVOLUTION.

Note. — The Fundamental Principle 138

General Definitions. — Power (J^ 76); E.oot {ITf^f); Power
and Root correlatives {178)', Names of different Powers \

and Roots {179) ; Exponent or Index ; How to read an expo-
nent {180) ; Transferring a factor from numerator to denomi-
(J85); Imaginary Quantity (J84j; Real (J^5); Similar
Radicals {180); Rationalize {187); to affect with an expo-
nent {188); Involution {189); Evolution (J90) ; Calculus

Involution :

Peob. 1. — To raise to any Power {192) ; Examples 141-143

CoE.— Signs of Powers {193} 143

Peob. 2. — To affect with any Exponent {194) 143

Examples 144-148

Peob. 3.— The Binomial Formula {195) 148

Applications 148-154

Form for expansion of (1 -j- a^)" 1^0

CoE. 1. — "When the series terminates (J 96) ... 151

CoE. 2.— Number of terms {197) 151

CoE. 3.— Equal coefficients {198) 152

CJoB. 4. — Sum of exponents in any term {199) 152

Cob. 5.— Statement of the Rule {200) 152

Cob. 6. — Signs of the terms in the expansion of
(a — h)^ {201) 153

SECTION IL

EVOLUTION.

pROB. 1. — To extract root of perfect power (;^0^) ; Examples 154

ScH.— Signs of root ^203) 155

CoE. 1. — Roots of Monomials {204:); Examples 155

CoE. 2.— Root of Product {205) 156

Cob. 3.— Root of a Quotient {206) ; Examples 156

XVI CONTENTS.

PAGE

Prob. 2. — To extract the square root of a Polynomial, Eule,

Dem. and Examples (207) 157-160

Peob. 3. — To extract the square root of a Decimal Number,

Kule, Dem. {209) ; Examples 161-164

Cob. — Eoots of Fractions {210); Examples 165

Peob. 4. — To extract the cube root of a Polynomial {211) ;

Eule, Dem., Examples 165-169

Peob. 5. — To extract the cube root of a Decimal Number

{212); Eule, Dem., Examples 169-173

Peob. 6. — To extract roots whose indices are composed of the

factors 2 and 3 {213); Examples 174

Peob. 7. — To extract the mth root of a Decimal Number {214:) 174

Peob. 8. — To extract the wth root of a Polynomial {215). . . 175

Examples 175-177

SECTION III.