Font size

UC-NRLF

III

$B 531 fiM^

IN MEMORIAM

FLORIAN CAJORI

v/^ ^^->-^_^ ^4vx

i.^.^.^^ /^jLu^ly^

C7^yCn^^^>'^^-^ ^ ^^

er>^

s

SCHOOL ALGEBRA.

BY

C. A. VAN VELZER,

AND

CHAS. S. SLIGHTER,

PROFESSORS IN THE UNIVERSITY OF WISCONSIN.

MADISON, WIS.:

TRACY, GIBBS & CO.

COPYRIGHT,

C. A. VAN VELZER, CHAS. S. SLIGHTER,

1890.

Tracy, Gibbs & Co., Printers and Stereotypers.

PREFACE.

The present volume is icsued in the hope that it will assist the

teacher in the effort to get pupils to think and that it will induce pupils

to place the subject of Algebra on a rational instead of an arbitrary

basis, to work from principles rather than from rules.

In the first part of the work we have pursued what is termed the

inductive method, but we do not wish this understood as that method

which infers general principles from an accumulation of particular

cases. This is the inductive reasoning of the natural sciences, but

we believe it is never legitimate in mathematics. Induction, as we

use the term, means that method which proceeds from the particular

to the general. By particular cases, which gradually increase in

generality, the mind of the learner is prepared to appreciate the gen-

eral case, but this general case must so present itself to the learner's

mind that he sees that the truth stated must be so and cannot possibly

be otherwise.

It will be noticed that we have not thought it necessary to complete

one subject before taking up another, but subjects have sometimes

been treated in an elementary way at first and more completely at

some subsequent part of the book. We believe that by this plan

students can follow the work more easily and with more profit, and at

the same time we are enabled to treat some subjects, especially

Factors, Multiples, and Fractions, more fully than is ordinarily done.

We have placed the principles governing the use of parentheses

before the four fundamental operations of addition, subtraction, mul-

tiplication, and division, and have made the latter depend upon the

former. This we think enables us to treat the four fundamental

operations in a way which is more rational to beginners than is given

when the usual order is pursued.

'm^r^^-^r^*^ r\.

IV PREFACE.

The subject of equations is early introduced, and is distributed

through the book instead of being given together in one place. This

keeps up the interest in the subject and prevents the student from

getting the idea that he is learning a mass of theory which has no

practical application.

Indices and Surds appear after Quadratics for the reason that these

subjects are more difficult than Quadratics.

Additive and subtractive terms are distinguished from positive and

negative quantities, and the latter are postponed until after the four

fundamental operations.

This work contains about 3000 examples besides several hundred

problems and inductive exercises. Many of these are original and

many are taken from the German and French collections and from

the English examination papers. The answers are not printed in the

book for a reason that every teacher of Algebra can readily assign, but

the answers are issued in pamphlet form for the use of teachers only.

C. A. Van Velzer.

University of Wisconsin, Chas. S. Slighter.

December, 1890.

TABLE OF CONTENTS.

Bold face figures refer to the exercises and

light face figures to the pages.

CHAPTER I.

First Principles, . - - -.

1, Illustrating how a letter may be used to represent a

number, i. 2, Leading to the idea of an algebraic ex-

pression, 3. 3, Leading to the idea of an algebraic equa-

tion and to the distinction between known and unknown

numbers, 4. 4, Symbolic expressions, 6. 5, Problems, 7.

6, Expressions in which several letters arc used to stand

for numbers, 10.

CHAPTER II.

Union of Terms and Removal of Parentheses,

15

7, Union of similar terms, 15. 8, Examples, 19. 9, Paren-

theses and their removal, 21. lO, Removal of parentheses;

general form a-\-[b-\-c'), 22 ; general form a-\-{bâ€”c)^ 22 ; gen-

eral form aâ€”{b-\-c\ 23 ; general form a â€” [bâ€”c), 24 ; paren-

theses preceded by plus sign, 24 , parentheses preceded by

minus sign, 26. 11, Miscellaneous examples on the removal

of parentheses, 29.

CHAPTER III.

Addition, - - 30

12, Addition of expressions, 30; examples, 31. 13, Arrange-

ment of work in addition, 31. 14, Examples, 33.

CHAPTER IV.

Subtraction, - -jg

15, Subtraction of expressions, 36; examples, 37. 16, Ar-

rangement of work in subtraction, 38. 17, Examples, 38.

1'8, Addition and subtraction of equals, 40. 19, Examples,

42. 20, Transposition in equations, 43. 21, Examples, 44.

22, Problems, 46.

vi CONTENTS.

CHAPTER V.

Multiplication, _._ - - - 5o

23, General definition of multiplication, 50 ; examples, 51.

24, Multiplication of monomials, 52. 25, Examples, 53.

26, Law of exponents in multiplication, 54. 27, Examples,

55. 28, Multiplication of polynomials by monomials, 56.

29, Examples, 59. 30, Arrangement of work in the mul-

tiplication of a polynomial by a monomial, 59; examples, 60.

31, Multiplication of polynomials by polynomials, 62. 32,

Examples, 63. 33, Arrangement of work in the multipli-

cation of two polynomials, 64. 34-, Examples, 69. 35,

Equations involving multiplication, 71.

CHAPTER VI.

Division, - - â€¢ - Ti

36, Division of monomials by monomials, 73 ; the law of

exponents, 74. 37, Examples, 74. 38, Division of poly-

nomials by monomials, 76. 39, Examples, 77. 40, Di-

vision of polynomials by polynomials, 77. 41, Examples,

79. 42, Arrangement of work in division of polynomials

by polynomials, 81. 43, Examples, 85.

CHAPTER Vn.

Negative Quantities, - -88

44, Number and quantity, 88. 45, Opposite directions,

89. 46, How directions are distinguished, 90. 47, Posi-

tive and negative numbers, 92. 48, Illustrative examples,

94. 49, Addition, 97. 50, Subtraction, 99. 51, Multi-

plication, 100. 52, Division, loi.

CHAPTER VIII.

Parentheses, - - 103

53, Removal of parentheses, 103. 54, Insertion of paren-

theses, 104.

CHAPTER IX.

Elementary Factors, Multiples, and Fractions, - - 106

55, Factors, 106. 56, Highest common factor, 107. 57,

Lowest common multiple, 108. 58, Fractions, iii. 59,

Addition of fractions, 113; subtraction of fractions, 115.

60, Multiplication of fractions, 116. 61, Division of frac-

tions, 118.

CONTENTS. vii

CHAPTER X.

Simple Equations, - 121

62, Definitions and general principles, 121. 63, Exam-

ples, 126. 64, Literal equations, 127. 64-a, Symbolic

expressions, 129. 64&, Problems, 132.

CHAPTER XI.

Simultaneous Equations, - 139

65, Definitions and general principles, 139. 66, Elimina-

tion by substitution, 141. 67, Examples, 143. 68, Elim-

ination by comparison, 144. 69, Examples, 145. 70,

Elimination by addition and subtraction, 146. 71, Special

expedients, 148. 72, Examples, 149. 73, Simultaneous

equations containing three unknown numbers, 152. 74,

Examples, 155. 75, Literal simultaneous equations, 157.

76, Problems producing simultaneous equations, 158.

CHAPTER Xn.

Powers and Roots, - 163

77, Powers of monomials, 163. 78, Square of a binomial,

168. 79, Cube of a binomial, 169. 80, Square of a poly-

nomial, 171. 81, Roots of monomials, 173. 82, Examples,

177. 83, Square root of polynomials, 179. 84, Cube

root of polynomials, 184.

CHAPTER Xin.

Harder Factors, Multiples, and Fractions, - - - 187

85, Factors common to all the terms of an expression, 187.

86, Formation of certain products, 189. 87, Expressions

of the form X'â€”a^, 191. 88, Expressions of the form

x^-\-ax-\-l>, 193. 89, Expressins of the form a^â€”b^^ 195.

90, Expressions of the form a^-\-b^, 197. 91, Expressions

of the form x^-\-a-x^-\-a^, 198. 92, Expressions of the

form a^â€”b^y 200. 93, Expressions of the form a"-\-b'\ 208.

94, Miscellaneous factors, 212. 95, H.C.F. of expressions

which can be factored, 216. 96, H.C.F. of expressions not

easily factored, 217, examples, 225. 97, L.C.M. of expres-

sions that can be factored, 227. 98, L.C.M. of expressions

not easily factored, 229. 99, Fractions reduced to lowest

viii CONTENTS.

terms, 231. 100, Addition of fractions, 233. lOl, Sub-

traction of fractions, 234. 102, Multiplication of fractions,

236. 103, Division of fractions, 238. 104, Miscellaneous

fractions, 240.

CHAPTER XIV.

Quadratic Equations, 249

105, Preliminary topics, 249. 106, Pure quadratic equa-

tions, 252; examples, 253. 107, Affected quadratics, 255;

examples, 257. 108, Problems leading to quadratic equa-

tions, 261. 109, Equations solved like quadratics, 269.

110, Theory of quadratic equations, 270.

CHAPTER XV.

Simultaneous Equations Above the First Degree, - - 277

111, One equation of the first degree and one of the second

degree, 277; examples, 279. 112, Tv^^o quadratic equations,

280; examples, 282. 113, Miscellaneous equations, 283;

examples, 286. 1 14, Problems, 287.

CHAPTER XVI.

Theory of Indices, 291

115, Meaning of fractional exponents, 291, 116, Examples,

295. 117, Properties of fractional exponents, 296. 118,

Examples, 299. 119, Meaning of zero and negative ex-

ponents, 303. 120, Examples, 305. 121, Properties of

negative exponents, 306. 122, Examples, 310.

CHAPTER XVII.

Surds, - -314

123*, Definitions and general principles, 314. 124, To re-

move a factor from beneath the radical sign, 316. 125, To

introduce the coefficient of a surd under the radical sign,

317. 126, To integralize the expression under the radical

sign, 318. 127, To lower or raise the index of a surd, 320.

128, To reduce a surd to its simplest form, 320. 129,

Addition and subtraction of surds, 321. 130, Multiplica-

tion and division of surds, 323. 131, Powers and roots of

sures, 326. 132, Rationalization of expressions containing

quadratic surds, 327. 133, Rationalization of equations, 330.

CONTENTS. IX

CHAPTER XVIII.

Ratio, Proportion and Variation, 332

134, Ratio, 332; problems, 337. 135, Proportion, 339;

problems, 344. 136, Variation, 345; examples and prob-

lems, 348.

CHAPTER XIX.

Progressions, 350

137, Arithmetical progressions, 350. 138, Examples and

problems, 354. 139, Geometrical progressions, 357. 140,

Examples and problems, 360.

CHAPTER XX.

Binomial Theorem, 363

141, Laws of exponents and coefficients, 363. 142, Ex-

amples, 370.

SCHOOL ALGEBRA.

CHAPTER L

FIRST PRINCIPLES.

EXERCISE 1.

Illustrating how a Letter may be used to Represent a Number.

1. In Algebra, as in Arithmetic, the symbol for Plus

is -f- and the symbol for Minus is â€” .

1. How many dozen are 6 dozen + 4 dozen 4- 2

dozen ?

2. How many score are 6 score + 4 score + 2 score ?

3. How many hundred are 6 hundred + 4 hundred

+ 2 hundred ?

4. How many times 100 are 6 times 100 + 4 times

100 + 2 times 100 ?

5. How many times 10 are 6 times 10 + 4 times 10

+ 2 times 10 ?

6. How many times 7 are 6 times 7 + 4 times 7 +

2 times 7 ?

7. Six times a7ty number plus four times the same

n^imbcr plus two times the same number are how many

times that number f

2, In Algebra letters are often used to represent or stand

for numbers.

.2 FIRST PRINCIPLES.

3. In Algebra, as in Arithmetic, the symbol for

times is x .

In any statement like 6X100+3X100â€”5X100, /. c, in any state-

ment where addition, or subtraction, and multiplication occur to-

gether, the multiplications must always be performed before any

addition or subtraction takes place. Thus, 6X100-(-3 does not mean

6X103, but means 600+3.

8. If / stands for 10, how many times 10 are 4x/+

7x/â€” cSx/?

g. If /stands for 2, how many times 2 are 4x/+7x/

â€” 8x/?

ID. If t stands for 3, how many times 3 are 4x/+

Tx/â€” 8x/?

11. If 5 stands for 6, how many times 6 are 7x^+

5X5â€” 3x^?

12. If 5 stands for 7, how many times 7 are 7X5+

5X5â€”3X5?

13. Seven times a certam number plus five times the

safne number minus three times the same number are how

many times that mimber f

14. If n stands for a certain number, how ma^y times

that number are 7 X ?^ + 5 X ;zâ€” 3 X ;^ ?

15. In question 14 can n stand for 17? for 100? for

25? fori? for J? for li?

In qjiestion z^, 71 ca7i stand for Ki^Y number WHATEVER,

provided it stands for the same niunber throughoitt question

and answer.

16. If a stands for a certain number, 8xÂ« + 4x<2â€”

5 X <2 are how many times that number ?

17. If b stands for a certain number, 8x<^+4x/^â€”

hy.b are how many times that number ?

18. Could some other letter than a, b, n, 5, or / be

used to represent a number ?

ALGEBRAIC EXPRESSIONS. 3

4. The preceding questions suggest the following

principle :

A7iy letter may be icsed to represent or stand for any

number, provided that the same letter represents the sa?ne

number throughout the saine question and answer.

6. In Algebra it is usual to omit the sign X in a

product like 7 X w and write merely 7Â«. It is then read

*' seven n,'' instead of "seven times ^^," but of course it

always mea^is seven times ?^ ; and this meaning the learner

must keep in mind. Thus, if b stands for 31^, then 7^

stands for 7 times Z\\.

Evidently when the sign X occurs between figtires it

cannot be omitted. Thus we can write ^ib for 7x<^,

but we cannot write 731| for 7x31|-, for 731| has a dif-

ferent meaning already given to it.

6. The number written before a letter to show how

many times the number represented by the letter is taken,

is called the Coefficient * of the letter. Thus, in "b, 7

is called the coefficient of b.

ig. What does hb mean ? How much is this if b equals

10 ? What is the 5 called ?

20. What does 10?/ mean ? How much is this if n

equals 8 ? What is the 10 called ?

EXERCISE 2.

Leading to the Idea of an Algebraic Expression.

1. What does 7aâ€” 3^ + 5<2 equal, \i a stands for 7?

2. What does 10i^4-4aâ€” 9a equal, if a stands for 6?

* A more general definition of coefficient is given in Art. 14.

4 FIRST PRINCIPLES.

3. What does S5aâ€”7aâ€”Sa equal, if a stands for y\?

4. What does 9<2 + 6aâ€” 7aâ€” 4a + 2a equal, if a stands

for A?

5. What does 25;^ + | equal, if 7z stands for |-?

6. What does 2o?iâ€”6nâ€”^ equal, if n stands for ^?

7. Anything, whether short and simple or long and

complicated, which is or may be considered to be equal

to some number, is called an Expression.

The number to which the expression is equal is called

the Value of the Expression.

The number which a letter stands for is called the

Value of the Letter.

7. What is the value of the expression 10c-\-2câ€”5câ€”

6c-hSc, if the value of ^ is 4 ?

8. What is the value of the expression 12mâ€”bm~Q>7}t

-\-Sm, if the value of m is 12 ?

9. What is the value of the expression 7aâ€” 4aâ€” 3a + a,

if the value of a is 2 ?

10. What is the value of the expression 16^+24^â€”

10^+25, if the value of d is yV?

EXERCISE 3.

Leading to the Notion of an Algebraic Equation and the Dis-

tinction BETWEEN Known and Unknown Numbers.

8. In Algebra, as in Arithmetic, the symbol for

Equals is =.

1. What is the value of 12a, if a= 2? if a=5? ifa=7?

ifa=|? if a=6i?

2. What must a equal if the value of 12a is 36 ? if the

value of 12a is 48 ? if the value of 12a is 72 ? if the value

of 12a is 8 ? if the value of 12a is 20 ?

ALGEBRAIC EXPRESSIONS. 5

3. If ^=8, what does Ix equal? If 7jtr=56, what

does X equal ? If lx=4S), what does x equal ?

4. If ;Â«r=9, what does \\x equal? If ll;r=99, what

does X equal ? If lljtr=121, what does x equal ?

5. If a=12, what does 3^-h5Â« equal? If 3^-f-5Â«=96,

what does a equal? If 3Â«-f5a=4, what does a equal ?

6. If ;z=6, what does 6?i-^5n-\-7i equal? If G7i-\-d?i

+ Â« = 72, what does n equal? If G?^ + 5;^^- 7^=108, what

does ?i equal ?

7. If 7<^â€” 6<^ + 3^=20, what is the value of d?

9. The statement of equality which exists between

two expressions is called an Equation, and the parts on

either side of the sign = are called the Members of the

equation. The expression on the left-hand side of the sign

= is called the Left or First Member, and the expres-

sion on the right-hand side of the sign = is called the

Right or Second Member.

8. If 3>&4-2/l' + /t=120, what is the value of k?

9. If 4;r-f-5;fâ€” 3;f=3, what is the value of -r?

10. If 7jt:â€” 2jf-f-;f=6, what is the value of;*;?

11. If2j'=2^, what is the value of jk?

12. If 3^+2^=12, what is the valve of -^?

13. If r=f, what is the value of 12c-{-^â€”0c}

14. lfd=l^, what is the value of 8^^+5^â€”10 ?

15. If/=1.2, what is the value of 20/â€” 7/â€” 3/'?

16. If lOOze^â€” 78ze/+15ze'=74, what is the value of rr?

17. If 17Â«â€”5?<â€” 4/^=66, what is the value of 7^?

18. If ^=5, what is the value ot 25^â€”5^+5 ?

19. If /i=100, what is the value qf 20/i-6/i-3/i + 100?

20. If 18z'-|-llz;4-2z/4-9z/=80, what is the value of -j?

6 FIRST PRINCIPLES.

10. A careful inspection of the questions of this ex-

ercise shows that when there is only one letter in an

expression, two cases may arise : first, the value of the

letter may be given and the value of the expression re-

quired ; second, the value of the expression may be given

and the value of the letter required.

In the first case the value of the letter is known or

give?i, and in the second case the value of the letter is

unktiown or required.

Thus w^e see that in Algebra there are two kinds of

numbers, called respectively Known and Unknown,

either of which may be represented by a letter; and as it

is possible that both kinds of numbers will appear in the

same question, it is customary to distinguish between

them by representing the known numbers by the first and

intermediate letters of the alphabet, and the unknown

numbers by the last letters of the alphabet.

EXERCISE 4.

Symbolic Expressions.

1. A coat and hat cost |24 ; the hat cost $4. What

does $24- $4 stand for?

2. A coat and hat cost $24 ; the hat cost %x. What

does 124 -Ix stand for ?

3. A coat and hat cost $24 ; the hat cost %x and the

coat 5 times as much as the hat. What does %hx stand

for?

What does $24-|5x stand for ?

What does %x-\-%bx, or %^x, stand for?

4. If n stands for a certain number, what exprcssicn

will stand for a number which is 10 larger ?

PROBLEMS. 7

5. 1{ a stands for a certain number, what v/ill stand

for a number which is 25 smaller ?

6. There are two numbers ; the second number is five

more than twice the first number. If 71 represents the first

number, what expression will represent the second num-

ber?

7. Of two numbers the second is 12 less than 5 times

the first. If n stands for the first number, what will

stand for the second number ?

8. How many feet in n yards ?

9. How manj^ feet in ?i yards plus 5 A-ards ?

10. How many feet in 71 yards plus 5 feet ?

11. A room is a yards wide and twice as long as it is

wide. How many yards long is the room ? How many

feet long is the room ?

12. In jr years a man will be 36 years old. What is

his present age ?

13. A man is now 40 years old. How old will he be

a years from now ? How old will he be '2a years from now ?

How old was he Sd years ago ?

EXERCISE 5.

Problems.

I. A coat and hat cost $24. The coat cost 5 times as

much as the hat. What was the cost of each ?

SOLUTION BY ARITHMETIC.

Five times ^/le cost of the hat = the cost of the coat.

Once the cost of the hat = the cost of the hat.

Therefore, six times the cost of the hat â€” the cost of coat and hat.

But the cost of the coat and hat is $24.

Hence, six times the cost of the hat = $24.

Therefore, the cost of the hat = ^ of $24, or $1

Consequently the cost of the coat was $20.

8 FIRST PRINCIPLES.

SOLUTION BY ALGEBRA.

The cost of the hat is a certain number of dollars, and that number,

whatever it is, we may represent by a letter. For example, we may

say,

Let X = number of dollars the hat cost.

Then 5x = number of dollars the coat cost.

Hence, 5x-^x, or Qx, = number of dollars that both hat and coat cost.

But 24 = number of dollars that both hat and coat cost.

Therefore 6^ = 24.

Then x = 4,

and 5x =z 20.

Therefore the hat cost $4, and the coat cost $20.

The student should compare very carefully the two solutions of this

problem above given. It will be noticed that the arithmetic and

algebraic solutions are not so different from each other as would

appear at first sight. In fact, to change the first solution to the sec-

ond nothing need be done except to replace the words " i/ie cost of the

hat " by $x. In the arithmetic solution the phrase " the cost of the hat''

stands for a certain unknown number of dollars, which number of

dollars is represented by a single letter, x, in the algebraic solution.

2. The sum of two numbers is 72 and one number is

twice as large as the other. What are the numbers ?

Let jr=:the smaller number.

Then 2x=the larger number.

Therefore 'lx-^rxâ€”Tl,

i. e. 3x=72.

Hence .^â€¢â€” 24, the smaller number,

and 2.*"=48, the larger number.

3. Divide $65 between A and B so that B shall receive

4 times as much as A.

4. A rectangle is 3 times as long as it is broad, and

the distance around it is 64 feet. Find the length and

breadth of the rectangle.

5. A piece of timber 18 feet long must be cut so as to

give one piece 2 feet long and two other pieces, one of

which must be 3 times the length of the other. Find

how long each one of these pieces will be.

PROBLEMS. 9

6. A father said to his son, " Neiit year the sum of our

ages will be 70 years, and I will be 4 times as old as you

will be." What is the present age of each?

7. A man has $48, consisting of an equal number of

bank notes of the denominations of $1, $2, and 85. What

number has he of each ?

Let j:=the number he has of each. Then in $1 bills he has $x,

in $2 bills he has $2x, and in $5 bills he has $5^-.

8. John, Henry, and Mary paid 13.60 for their books.

Henry's cost twice as much as John's, and Mary's cost

three times as much as John's. Find the cost of each

scholar's books.

9. If you add together 3 times and 5 times and 7 times

a certain number you will obtain 315. What is the

number ?

10. Three persons subscribed 150000 to build a hospital.

The first two subscribed equal amounts, but the third party

subscribed 2 times as much as either of the others. How

much did each person subscribe ?

11. A man bought 10 turkeys, 10 chickens, and 10

ducks, paying Â§10 for all. A chicken cost the same as a

duck, and a turkey cost 3 times as much as a dnck. What

was the cost of each ?

12. A, B, and C form a partnership to do business. A

furnishes 4 times as much capital as C, and B furnishes 3

times as much as C. Altogether the three men put in

$24000. Wliat amount is furnished by each ?

13. A man and three boys did a piece of work for $16.

How should this money be divided among them, if we

suppose that the man did twice as much work as each

boy?

lO FIRST PRINCIPLES.

14. A man has a farm of 240 acres, of which there is

twice as much marsh land as wood land, but the rest of

the farm is 3 times as large as the marsh land and wood

land taken together. Find the number of acres each of

marsh land and wcod land in the farm.

15. Divide $1100 among A, B, and C so that A may

have twice as much as B, and B three times as much

as C.

16. A man raised 1730 bushels of grain, of which there

was 3 times as much oats as wheat, and 2 times as much

corn as oats. Find the amount of each kind that he

raised.

17. On a certain day a storekeeper took in $3G0, of

which there was 4 times as much in bank notes as in

coin, and 5 times as much in silver as in gold. Find the

amount of each kind of money that he received.

18. Of three brothers the middle one is 4 times and

the oldest one 5 times as old as the youngest. The sum

of the ages of the two oldest is 36 year?. Find the age

of each of the brothers.

ig. If each year I should double the money that I had

at the beginning of that year, in five years from now I

would have $63000. How much money have I now ?

20. There are three numbers, the sum of the last two

of which is 63. The second is five times the first, and

the third equals the difference between the Jfirst and

second. What are the numbers ?

EXERCISE C.

Expressions in which Several Letters ars used to stand

FOR Numbers.

11. We have already learned that any letter may be

used to represent any number, but it often happens that

EXPRESSIONS WITH SEVERAL LETTERS. II

different numbers occur in the same question. In this

case different letters may be used to represent these num-

bers, but here, as before, any 07ie letter must represent

the sa7ne number throughout question .an^ answer.

P^ind the value of the following expressions, if Â«=5,

III

$B 531 fiM^

IN MEMORIAM

FLORIAN CAJORI

v/^ ^^->-^_^ ^4vx

i.^.^.^^ /^jLu^ly^

C7^yCn^^^>'^^-^ ^ ^^

er>^

s

SCHOOL ALGEBRA.

BY

C. A. VAN VELZER,

AND

CHAS. S. SLIGHTER,

PROFESSORS IN THE UNIVERSITY OF WISCONSIN.

MADISON, WIS.:

TRACY, GIBBS & CO.

COPYRIGHT,

C. A. VAN VELZER, CHAS. S. SLIGHTER,

1890.

Tracy, Gibbs & Co., Printers and Stereotypers.

PREFACE.

The present volume is icsued in the hope that it will assist the

teacher in the effort to get pupils to think and that it will induce pupils

to place the subject of Algebra on a rational instead of an arbitrary

basis, to work from principles rather than from rules.

In the first part of the work we have pursued what is termed the

inductive method, but we do not wish this understood as that method

which infers general principles from an accumulation of particular

cases. This is the inductive reasoning of the natural sciences, but

we believe it is never legitimate in mathematics. Induction, as we

use the term, means that method which proceeds from the particular

to the general. By particular cases, which gradually increase in

generality, the mind of the learner is prepared to appreciate the gen-

eral case, but this general case must so present itself to the learner's

mind that he sees that the truth stated must be so and cannot possibly

be otherwise.

It will be noticed that we have not thought it necessary to complete

one subject before taking up another, but subjects have sometimes

been treated in an elementary way at first and more completely at

some subsequent part of the book. We believe that by this plan

students can follow the work more easily and with more profit, and at

the same time we are enabled to treat some subjects, especially

Factors, Multiples, and Fractions, more fully than is ordinarily done.

We have placed the principles governing the use of parentheses

before the four fundamental operations of addition, subtraction, mul-

tiplication, and division, and have made the latter depend upon the

former. This we think enables us to treat the four fundamental

operations in a way which is more rational to beginners than is given

when the usual order is pursued.

'm^r^^-^r^*^ r\.

IV PREFACE.

The subject of equations is early introduced, and is distributed

through the book instead of being given together in one place. This

keeps up the interest in the subject and prevents the student from

getting the idea that he is learning a mass of theory which has no

practical application.

Indices and Surds appear after Quadratics for the reason that these

subjects are more difficult than Quadratics.

Additive and subtractive terms are distinguished from positive and

negative quantities, and the latter are postponed until after the four

fundamental operations.

This work contains about 3000 examples besides several hundred

problems and inductive exercises. Many of these are original and

many are taken from the German and French collections and from

the English examination papers. The answers are not printed in the

book for a reason that every teacher of Algebra can readily assign, but

the answers are issued in pamphlet form for the use of teachers only.

C. A. Van Velzer.

University of Wisconsin, Chas. S. Slighter.

December, 1890.

TABLE OF CONTENTS.

Bold face figures refer to the exercises and

light face figures to the pages.

CHAPTER I.

First Principles, . - - -.

1, Illustrating how a letter may be used to represent a

number, i. 2, Leading to the idea of an algebraic ex-

pression, 3. 3, Leading to the idea of an algebraic equa-

tion and to the distinction between known and unknown

numbers, 4. 4, Symbolic expressions, 6. 5, Problems, 7.

6, Expressions in which several letters arc used to stand

for numbers, 10.

CHAPTER II.

Union of Terms and Removal of Parentheses,

15

7, Union of similar terms, 15. 8, Examples, 19. 9, Paren-

theses and their removal, 21. lO, Removal of parentheses;

general form a-\-[b-\-c'), 22 ; general form a-\-{bâ€”c)^ 22 ; gen-

eral form aâ€”{b-\-c\ 23 ; general form a â€” [bâ€”c), 24 ; paren-

theses preceded by plus sign, 24 , parentheses preceded by

minus sign, 26. 11, Miscellaneous examples on the removal

of parentheses, 29.

CHAPTER III.

Addition, - - 30

12, Addition of expressions, 30; examples, 31. 13, Arrange-

ment of work in addition, 31. 14, Examples, 33.

CHAPTER IV.

Subtraction, - -jg

15, Subtraction of expressions, 36; examples, 37. 16, Ar-

rangement of work in subtraction, 38. 17, Examples, 38.

1'8, Addition and subtraction of equals, 40. 19, Examples,

42. 20, Transposition in equations, 43. 21, Examples, 44.

22, Problems, 46.

vi CONTENTS.

CHAPTER V.

Multiplication, _._ - - - 5o

23, General definition of multiplication, 50 ; examples, 51.

24, Multiplication of monomials, 52. 25, Examples, 53.

26, Law of exponents in multiplication, 54. 27, Examples,

55. 28, Multiplication of polynomials by monomials, 56.

29, Examples, 59. 30, Arrangement of work in the mul-

tiplication of a polynomial by a monomial, 59; examples, 60.

31, Multiplication of polynomials by polynomials, 62. 32,

Examples, 63. 33, Arrangement of work in the multipli-

cation of two polynomials, 64. 34-, Examples, 69. 35,

Equations involving multiplication, 71.

CHAPTER VI.

Division, - - â€¢ - Ti

36, Division of monomials by monomials, 73 ; the law of

exponents, 74. 37, Examples, 74. 38, Division of poly-

nomials by monomials, 76. 39, Examples, 77. 40, Di-

vision of polynomials by polynomials, 77. 41, Examples,

79. 42, Arrangement of work in division of polynomials

by polynomials, 81. 43, Examples, 85.

CHAPTER Vn.

Negative Quantities, - -88

44, Number and quantity, 88. 45, Opposite directions,

89. 46, How directions are distinguished, 90. 47, Posi-

tive and negative numbers, 92. 48, Illustrative examples,

94. 49, Addition, 97. 50, Subtraction, 99. 51, Multi-

plication, 100. 52, Division, loi.

CHAPTER VIII.

Parentheses, - - 103

53, Removal of parentheses, 103. 54, Insertion of paren-

theses, 104.

CHAPTER IX.

Elementary Factors, Multiples, and Fractions, - - 106

55, Factors, 106. 56, Highest common factor, 107. 57,

Lowest common multiple, 108. 58, Fractions, iii. 59,

Addition of fractions, 113; subtraction of fractions, 115.

60, Multiplication of fractions, 116. 61, Division of frac-

tions, 118.

CONTENTS. vii

CHAPTER X.

Simple Equations, - 121

62, Definitions and general principles, 121. 63, Exam-

ples, 126. 64, Literal equations, 127. 64-a, Symbolic

expressions, 129. 64&, Problems, 132.

CHAPTER XI.

Simultaneous Equations, - 139

65, Definitions and general principles, 139. 66, Elimina-

tion by substitution, 141. 67, Examples, 143. 68, Elim-

ination by comparison, 144. 69, Examples, 145. 70,

Elimination by addition and subtraction, 146. 71, Special

expedients, 148. 72, Examples, 149. 73, Simultaneous

equations containing three unknown numbers, 152. 74,

Examples, 155. 75, Literal simultaneous equations, 157.

76, Problems producing simultaneous equations, 158.

CHAPTER Xn.

Powers and Roots, - 163

77, Powers of monomials, 163. 78, Square of a binomial,

168. 79, Cube of a binomial, 169. 80, Square of a poly-

nomial, 171. 81, Roots of monomials, 173. 82, Examples,

177. 83, Square root of polynomials, 179. 84, Cube

root of polynomials, 184.

CHAPTER Xin.

Harder Factors, Multiples, and Fractions, - - - 187

85, Factors common to all the terms of an expression, 187.

86, Formation of certain products, 189. 87, Expressions

of the form X'â€”a^, 191. 88, Expressions of the form

x^-\-ax-\-l>, 193. 89, Expressins of the form a^â€”b^^ 195.

90, Expressions of the form a^-\-b^, 197. 91, Expressions

of the form x^-\-a-x^-\-a^, 198. 92, Expressions of the

form a^â€”b^y 200. 93, Expressions of the form a"-\-b'\ 208.

94, Miscellaneous factors, 212. 95, H.C.F. of expressions

which can be factored, 216. 96, H.C.F. of expressions not

easily factored, 217, examples, 225. 97, L.C.M. of expres-

sions that can be factored, 227. 98, L.C.M. of expressions

not easily factored, 229. 99, Fractions reduced to lowest

viii CONTENTS.

terms, 231. 100, Addition of fractions, 233. lOl, Sub-

traction of fractions, 234. 102, Multiplication of fractions,

236. 103, Division of fractions, 238. 104, Miscellaneous

fractions, 240.

CHAPTER XIV.

Quadratic Equations, 249

105, Preliminary topics, 249. 106, Pure quadratic equa-

tions, 252; examples, 253. 107, Affected quadratics, 255;

examples, 257. 108, Problems leading to quadratic equa-

tions, 261. 109, Equations solved like quadratics, 269.

110, Theory of quadratic equations, 270.

CHAPTER XV.

Simultaneous Equations Above the First Degree, - - 277

111, One equation of the first degree and one of the second

degree, 277; examples, 279. 112, Tv^^o quadratic equations,

280; examples, 282. 113, Miscellaneous equations, 283;

examples, 286. 1 14, Problems, 287.

CHAPTER XVI.

Theory of Indices, 291

115, Meaning of fractional exponents, 291, 116, Examples,

295. 117, Properties of fractional exponents, 296. 118,

Examples, 299. 119, Meaning of zero and negative ex-

ponents, 303. 120, Examples, 305. 121, Properties of

negative exponents, 306. 122, Examples, 310.

CHAPTER XVII.

Surds, - -314

123*, Definitions and general principles, 314. 124, To re-

move a factor from beneath the radical sign, 316. 125, To

introduce the coefficient of a surd under the radical sign,

317. 126, To integralize the expression under the radical

sign, 318. 127, To lower or raise the index of a surd, 320.

128, To reduce a surd to its simplest form, 320. 129,

Addition and subtraction of surds, 321. 130, Multiplica-

tion and division of surds, 323. 131, Powers and roots of

sures, 326. 132, Rationalization of expressions containing

quadratic surds, 327. 133, Rationalization of equations, 330.

CONTENTS. IX

CHAPTER XVIII.

Ratio, Proportion and Variation, 332

134, Ratio, 332; problems, 337. 135, Proportion, 339;

problems, 344. 136, Variation, 345; examples and prob-

lems, 348.

CHAPTER XIX.

Progressions, 350

137, Arithmetical progressions, 350. 138, Examples and

problems, 354. 139, Geometrical progressions, 357. 140,

Examples and problems, 360.

CHAPTER XX.

Binomial Theorem, 363

141, Laws of exponents and coefficients, 363. 142, Ex-

amples, 370.

SCHOOL ALGEBRA.

CHAPTER L

FIRST PRINCIPLES.

EXERCISE 1.

Illustrating how a Letter may be used to Represent a Number.

1. In Algebra, as in Arithmetic, the symbol for Plus

is -f- and the symbol for Minus is â€” .

1. How many dozen are 6 dozen + 4 dozen 4- 2

dozen ?

2. How many score are 6 score + 4 score + 2 score ?

3. How many hundred are 6 hundred + 4 hundred

+ 2 hundred ?

4. How many times 100 are 6 times 100 + 4 times

100 + 2 times 100 ?

5. How many times 10 are 6 times 10 + 4 times 10

+ 2 times 10 ?

6. How many times 7 are 6 times 7 + 4 times 7 +

2 times 7 ?

7. Six times a7ty number plus four times the same

n^imbcr plus two times the same number are how many

times that number f

2, In Algebra letters are often used to represent or stand

for numbers.

.2 FIRST PRINCIPLES.

3. In Algebra, as in Arithmetic, the symbol for

times is x .

In any statement like 6X100+3X100â€”5X100, /. c, in any state-

ment where addition, or subtraction, and multiplication occur to-

gether, the multiplications must always be performed before any

addition or subtraction takes place. Thus, 6X100-(-3 does not mean

6X103, but means 600+3.

8. If / stands for 10, how many times 10 are 4x/+

7x/â€” cSx/?

g. If /stands for 2, how many times 2 are 4x/+7x/

â€” 8x/?

ID. If t stands for 3, how many times 3 are 4x/+

Tx/â€” 8x/?

11. If 5 stands for 6, how many times 6 are 7x^+

5X5â€” 3x^?

12. If 5 stands for 7, how many times 7 are 7X5+

5X5â€”3X5?

13. Seven times a certam number plus five times the

safne number minus three times the same number are how

many times that mimber f

14. If n stands for a certain number, how ma^y times

that number are 7 X ?^ + 5 X ;zâ€” 3 X ;^ ?

15. In question 14 can n stand for 17? for 100? for

25? fori? for J? for li?

In qjiestion z^, 71 ca7i stand for Ki^Y number WHATEVER,

provided it stands for the same niunber throughoitt question

and answer.

16. If a stands for a certain number, 8xÂ« + 4x<2â€”

5 X <2 are how many times that number ?

17. If b stands for a certain number, 8x<^+4x/^â€”

hy.b are how many times that number ?

18. Could some other letter than a, b, n, 5, or / be

used to represent a number ?

ALGEBRAIC EXPRESSIONS. 3

4. The preceding questions suggest the following

principle :

A7iy letter may be icsed to represent or stand for any

number, provided that the same letter represents the sa?ne

number throughout the saine question and answer.

6. In Algebra it is usual to omit the sign X in a

product like 7 X w and write merely 7Â«. It is then read

*' seven n,'' instead of "seven times ^^," but of course it

always mea^is seven times ?^ ; and this meaning the learner

must keep in mind. Thus, if b stands for 31^, then 7^

stands for 7 times Z\\.

Evidently when the sign X occurs between figtires it

cannot be omitted. Thus we can write ^ib for 7x<^,

but we cannot write 731| for 7x31|-, for 731| has a dif-

ferent meaning already given to it.

6. The number written before a letter to show how

many times the number represented by the letter is taken,

is called the Coefficient * of the letter. Thus, in "b, 7

is called the coefficient of b.

ig. What does hb mean ? How much is this if b equals

10 ? What is the 5 called ?

20. What does 10?/ mean ? How much is this if n

equals 8 ? What is the 10 called ?

EXERCISE 2.

Leading to the Idea of an Algebraic Expression.

1. What does 7aâ€” 3^ + 5<2 equal, \i a stands for 7?

2. What does 10i^4-4aâ€” 9a equal, if a stands for 6?

* A more general definition of coefficient is given in Art. 14.

4 FIRST PRINCIPLES.

3. What does S5aâ€”7aâ€”Sa equal, if a stands for y\?

4. What does 9<2 + 6aâ€” 7aâ€” 4a + 2a equal, if a stands

for A?

5. What does 25;^ + | equal, if 7z stands for |-?

6. What does 2o?iâ€”6nâ€”^ equal, if n stands for ^?

7. Anything, whether short and simple or long and

complicated, which is or may be considered to be equal

to some number, is called an Expression.

The number to which the expression is equal is called

the Value of the Expression.

The number which a letter stands for is called the

Value of the Letter.

7. What is the value of the expression 10c-\-2câ€”5câ€”

6c-hSc, if the value of ^ is 4 ?

8. What is the value of the expression 12mâ€”bm~Q>7}t

-\-Sm, if the value of m is 12 ?

9. What is the value of the expression 7aâ€” 4aâ€” 3a + a,

if the value of a is 2 ?

10. What is the value of the expression 16^+24^â€”

10^+25, if the value of d is yV?

EXERCISE 3.

Leading to the Notion of an Algebraic Equation and the Dis-

tinction BETWEEN Known and Unknown Numbers.

8. In Algebra, as in Arithmetic, the symbol for

Equals is =.

1. What is the value of 12a, if a= 2? if a=5? ifa=7?

ifa=|? if a=6i?

2. What must a equal if the value of 12a is 36 ? if the

value of 12a is 48 ? if the value of 12a is 72 ? if the value

of 12a is 8 ? if the value of 12a is 20 ?

ALGEBRAIC EXPRESSIONS. 5

3. If ^=8, what does Ix equal? If 7jtr=56, what

does X equal ? If lx=4S), what does x equal ?

4. If ;Â«r=9, what does \\x equal? If ll;r=99, what

does X equal ? If lljtr=121, what does x equal ?

5. If a=12, what does 3^-h5Â« equal? If 3^-f-5Â«=96,

what does a equal? If 3Â«-f5a=4, what does a equal ?

6. If ;z=6, what does 6?i-^5n-\-7i equal? If G7i-\-d?i

+ Â« = 72, what does n equal? If G?^ + 5;^^- 7^=108, what

does ?i equal ?

7. If 7<^â€” 6<^ + 3^=20, what is the value of d?

9. The statement of equality which exists between

two expressions is called an Equation, and the parts on

either side of the sign = are called the Members of the

equation. The expression on the left-hand side of the sign

= is called the Left or First Member, and the expres-

sion on the right-hand side of the sign = is called the

Right or Second Member.

8. If 3>&4-2/l' + /t=120, what is the value of k?

9. If 4;r-f-5;fâ€” 3;f=3, what is the value of -r?

10. If 7jt:â€” 2jf-f-;f=6, what is the value of;*;?

11. If2j'=2^, what is the value of jk?

12. If 3^+2^=12, what is the valve of -^?

13. If r=f, what is the value of 12c-{-^â€”0c}

14. lfd=l^, what is the value of 8^^+5^â€”10 ?

15. If/=1.2, what is the value of 20/â€” 7/â€” 3/'?

16. If lOOze^â€” 78ze/+15ze'=74, what is the value of rr?

17. If 17Â«â€”5?<â€” 4/^=66, what is the value of 7^?

18. If ^=5, what is the value ot 25^â€”5^+5 ?

19. If /i=100, what is the value qf 20/i-6/i-3/i + 100?

20. If 18z'-|-llz;4-2z/4-9z/=80, what is the value of -j?

6 FIRST PRINCIPLES.

10. A careful inspection of the questions of this ex-

ercise shows that when there is only one letter in an

expression, two cases may arise : first, the value of the

letter may be given and the value of the expression re-

quired ; second, the value of the expression may be given

and the value of the letter required.

In the first case the value of the letter is known or

give?i, and in the second case the value of the letter is

unktiown or required.

Thus w^e see that in Algebra there are two kinds of

numbers, called respectively Known and Unknown,

either of which may be represented by a letter; and as it

is possible that both kinds of numbers will appear in the

same question, it is customary to distinguish between

them by representing the known numbers by the first and

intermediate letters of the alphabet, and the unknown

numbers by the last letters of the alphabet.

EXERCISE 4.

Symbolic Expressions.

1. A coat and hat cost |24 ; the hat cost $4. What

does $24- $4 stand for?

2. A coat and hat cost $24 ; the hat cost %x. What

does 124 -Ix stand for ?

3. A coat and hat cost $24 ; the hat cost %x and the

coat 5 times as much as the hat. What does %hx stand

for?

What does $24-|5x stand for ?

What does %x-\-%bx, or %^x, stand for?

4. If n stands for a certain number, what exprcssicn

will stand for a number which is 10 larger ?

PROBLEMS. 7

5. 1{ a stands for a certain number, what v/ill stand

for a number which is 25 smaller ?

6. There are two numbers ; the second number is five

more than twice the first number. If 71 represents the first

number, what expression will represent the second num-

ber?

7. Of two numbers the second is 12 less than 5 times

the first. If n stands for the first number, what will

stand for the second number ?

8. How many feet in n yards ?

9. How manj^ feet in ?i yards plus 5 A-ards ?

10. How many feet in 71 yards plus 5 feet ?

11. A room is a yards wide and twice as long as it is

wide. How many yards long is the room ? How many

feet long is the room ?

12. In jr years a man will be 36 years old. What is

his present age ?

13. A man is now 40 years old. How old will he be

a years from now ? How old will he be '2a years from now ?

How old was he Sd years ago ?

EXERCISE 5.

Problems.

I. A coat and hat cost $24. The coat cost 5 times as

much as the hat. What was the cost of each ?

SOLUTION BY ARITHMETIC.

Five times ^/le cost of the hat = the cost of the coat.

Once the cost of the hat = the cost of the hat.

Therefore, six times the cost of the hat â€” the cost of coat and hat.

But the cost of the coat and hat is $24.

Hence, six times the cost of the hat = $24.

Therefore, the cost of the hat = ^ of $24, or $1

Consequently the cost of the coat was $20.

8 FIRST PRINCIPLES.

SOLUTION BY ALGEBRA.

The cost of the hat is a certain number of dollars, and that number,

whatever it is, we may represent by a letter. For example, we may

say,

Let X = number of dollars the hat cost.

Then 5x = number of dollars the coat cost.

Hence, 5x-^x, or Qx, = number of dollars that both hat and coat cost.

But 24 = number of dollars that both hat and coat cost.

Therefore 6^ = 24.

Then x = 4,

and 5x =z 20.

Therefore the hat cost $4, and the coat cost $20.

The student should compare very carefully the two solutions of this

problem above given. It will be noticed that the arithmetic and

algebraic solutions are not so different from each other as would

appear at first sight. In fact, to change the first solution to the sec-

ond nothing need be done except to replace the words " i/ie cost of the

hat " by $x. In the arithmetic solution the phrase " the cost of the hat''

stands for a certain unknown number of dollars, which number of

dollars is represented by a single letter, x, in the algebraic solution.

2. The sum of two numbers is 72 and one number is

twice as large as the other. What are the numbers ?

Let jr=:the smaller number.

Then 2x=the larger number.

Therefore 'lx-^rxâ€”Tl,

i. e. 3x=72.

Hence .^â€¢â€” 24, the smaller number,

and 2.*"=48, the larger number.

3. Divide $65 between A and B so that B shall receive

4 times as much as A.

4. A rectangle is 3 times as long as it is broad, and

the distance around it is 64 feet. Find the length and

breadth of the rectangle.

5. A piece of timber 18 feet long must be cut so as to

give one piece 2 feet long and two other pieces, one of

which must be 3 times the length of the other. Find

how long each one of these pieces will be.

PROBLEMS. 9

6. A father said to his son, " Neiit year the sum of our

ages will be 70 years, and I will be 4 times as old as you

will be." What is the present age of each?

7. A man has $48, consisting of an equal number of

bank notes of the denominations of $1, $2, and 85. What

number has he of each ?

Let j:=the number he has of each. Then in $1 bills he has $x,

in $2 bills he has $2x, and in $5 bills he has $5^-.

8. John, Henry, and Mary paid 13.60 for their books.

Henry's cost twice as much as John's, and Mary's cost

three times as much as John's. Find the cost of each

scholar's books.

9. If you add together 3 times and 5 times and 7 times

a certain number you will obtain 315. What is the

number ?

10. Three persons subscribed 150000 to build a hospital.

The first two subscribed equal amounts, but the third party

subscribed 2 times as much as either of the others. How

much did each person subscribe ?

11. A man bought 10 turkeys, 10 chickens, and 10

ducks, paying Â§10 for all. A chicken cost the same as a

duck, and a turkey cost 3 times as much as a dnck. What

was the cost of each ?

12. A, B, and C form a partnership to do business. A

furnishes 4 times as much capital as C, and B furnishes 3

times as much as C. Altogether the three men put in

$24000. Wliat amount is furnished by each ?

13. A man and three boys did a piece of work for $16.

How should this money be divided among them, if we

suppose that the man did twice as much work as each

boy?

lO FIRST PRINCIPLES.

14. A man has a farm of 240 acres, of which there is

twice as much marsh land as wood land, but the rest of

the farm is 3 times as large as the marsh land and wood

land taken together. Find the number of acres each of

marsh land and wcod land in the farm.

15. Divide $1100 among A, B, and C so that A may

have twice as much as B, and B three times as much

as C.

16. A man raised 1730 bushels of grain, of which there

was 3 times as much oats as wheat, and 2 times as much

corn as oats. Find the amount of each kind that he

raised.

17. On a certain day a storekeeper took in $3G0, of

which there was 4 times as much in bank notes as in

coin, and 5 times as much in silver as in gold. Find the

amount of each kind of money that he received.

18. Of three brothers the middle one is 4 times and

the oldest one 5 times as old as the youngest. The sum

of the ages of the two oldest is 36 year?. Find the age

of each of the brothers.

ig. If each year I should double the money that I had

at the beginning of that year, in five years from now I

would have $63000. How much money have I now ?

20. There are three numbers, the sum of the last two

of which is 63. The second is five times the first, and

the third equals the difference between the Jfirst and

second. What are the numbers ?

EXERCISE C.

Expressions in which Several Letters ars used to stand

FOR Numbers.

11. We have already learned that any letter may be

used to represent any number, but it often happens that

EXPRESSIONS WITH SEVERAL LETTERS. II

different numbers occur in the same question. In this

case different letters may be used to represent these num-

bers, but here, as before, any 07ie letter must represent

the sa7ne number throughout question .an^ answer.

P^ind the value of the following expressions, if Â«=5,