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C. A. (Charles Ambrose) Van Velzer.

A course in algebra. Being course one in mathematics in the University of Wisconsin online

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IN



ALGEBRA



Being Course One in Mathematics

IN THE

UniveRvSity of 'Wisconsin.



BY-

C. A. VAN VHI.ZER and CHAS. S. SLICHTER.



MADISON, WIS.:
Capital City Pub. Co., Printers,

1888.



Copyrighted 1888

BY

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






PREFACE.



The present volume originated in a desire on the part of the
authors to furnish a text of Course I. as was previously mapped
out by the department of mathematics at the University of
Wisconsin.

The orginal intent was to produce a syllabu."? for the use of
students in this institution, but it was subsequently thought that
a work which would be useful here might also be found useful
elsewhere, and hence it w^as decided to give the w^ork more the
character of a treatise than a syllabus. To insure the best results
it was thought desirable to print the present preliminary edition
and put it to the test of class room work, and at the same time to
invite criticism and suggestions from teachers and others inter-
ested in mathematics, and then from the results of the authors'
tests, and from the experience of others, to rewrite the work,
changing it freely. For these reasons the treatment of many
subjects in the following pages should be understood as merely
tentative. The final form will depend entirely upon the results
of experience.

An examination of the text will reveal many deviations from
the beaten path, but the idea was not to deviate simpl}^ for the
sake of being different from others ; on the contrary the authors
have freely drawn from other works. The sources from which
material has been most largeh^ drawn are the following: For
problems, Christie's Test Questions and Wolstenholm's Collec-
tion ; for various matters in the text, Kempt's Lehrbuch in die
Moderne Algebra, and the algebras of Chrystal; Aldis; Hall and
Knight; Oliver, Wait and Jones; and Todhunter ; for historical
notes, Marie's Histoire des Sciences Mathematiques et Physiques,
and Matthiesen'sGrundzuge der Antiken und Modernen Algebra.



M.106055



IV

Part II. of the present work, containing chapters on Iniagin-
aries, the Rational Integral Function of x, Solution of Numerical
Equations of Higher Degree, Graphic Representation of Equa-
tions, and Determinants, has already appeared and for this part,
as well as for the present volume, suggestions are invited.

Several modifications have already suggested themselves to the
authors, but it is hoped that any into whose hands either volume
may fall will communicate with the authors with reference to any
changes before the work is put in permanent form.

University of Wisconsin,
Madison, Wis, 1888.



TABLE OF CONTENTS.



Pagb.

CHAPTER I.
Introduction, - - - - i

CHAPTER II.
Theory of Indices, - - - - i$

CHAPTER III.
Radical Quantities and Irrational Expressions, 35

CHAPTER IV.

Quadratic Equations Containing One Unknown
Quantity, - - - - - 51

CHAPTER V.

Theory of Quadratic Equations and Quadratic
Functions, - - - - 64

CHAPTER VI.
Single Equations, - - - - 74

CHAPTER VII.
Systems of Equations, - - - 89

CHAPTER VIII.
Progressions, - - - - - ic6

CHAPTER IX.
Arrangements and Groups, - - - 114

CHAPTER X.
Binomial Theorem, - - - - 130



VI

CHAPTER XI.
Theory of Limits, - - - - 139

CHAPTER Xn.
Undetermined Coefficients, - - - 153

CHAPTER XIII.
Derivatives, - - - - 164

CHAPTER XIV.

Series, - - - - - - 183

CHAPTER XV.
Logarithms, . . . . 1^5



ALGEBRA



CHAPTER I.

INTRODUCTION.

1, Definitions. When we wish to use a general term which
shall inclnde in its meaning any intelligible combination of alge-
braic symbols and quantities, the word Expression will be adopted.
Thus

^2 j\ r 2 I i I \ c^ -\- d^ -\- abed

ix^—d) {ax^-\-bx-\-e)\ ■ — — ^Ai,_l / —

may be called expressioris. It includes the words polynomial,
fraetio7i, and radieal and more besides.

When we wish to call attention to the fact that certain specified
quantities appear in an expression it may be called a Funetion of
those quantities. Thus if we desire to point out that x appears in
the first expression above, it would be called a fnnetion of x. If
we wish to state that a, b, e, and d occur in the second expression,
we would call it o. fu7ietion of a, b, e, and d. If we wivSh to say
that y occurs in the last expression, it may be called Sifunetioti of
y, or if we wish to say that a, b, and y occur in it, we would
speak of it as 3.fu?iefio?i of a, b, a?idy. A formal definition of the
word function would be :

A Function of a quantity is a name applied to any mathemati-
cal expression in which the quantity appears.

2. Definition. An expression is Integral with respect to any
quantity or quantities, that is, is an integral function of those
quantities, when the quantities named do not appear in any man-
ner as divisors. Thus 5,r^-f|jf— >/2 is integral with respect to
x ; that is, is an integral function of x.

a—b-\-a-b ab

x^-^xy X

is integral with respect to a and b, but fractional with respect to



2 Algebra.

jc and J/ ; that is, is an integral function of a and d, but 2i fractional
function of jr and r, the word fractional meaning just the opposite
to integral.

3. Definition. An Expression is Rational with respect to
any quantity or quantities, or is a rational function of those quan-
tities, when the quantities referred to are not involved in any
manner by the extraction of a root. Thus

X

is rational with respect to x, but irrational with respect to c and
d\ that is, it is a rational function of x, but an irrational function
of c and d, the term irrational being used in just the opposite
sense from rational.

4. An expression may be both rational and integral with re-
spect to certain quantities, in which case it may be spoken of as a
Rational Integral Expression with respect to those quantities, or
as a rational integral function of the quantities. In the sam6
way we may speak of an expression which is rational and frac-
tional with respect to certain quantities as a Rational Fractional
Expression with reference to those quantities, or as a rational
fractional function of the quantities. In like manner we may use
the terms Irrational Integral Expression and Irrational Fractional
Expression, or Irrational Integral Function or Irrational Frac-
tional Function.

In the following examples the student is expected to answer
the question, What kind of an expression? with reference to the
quantities specified opposite each.

J. ax^-\-a^x''-\-a^x. With respect to Jt* ? to a ? to Jt: and a?

2. — {\-\-~y \—c\ With respect to ^? to r?



a



J. bx'' 7-^xy. With respect to a'? tOJ^'? to x and j?

v/^ x'-^^ b-^Wc With respect to Jt:? to^y?

ay^-\-by-\-c ' to x andjj'? to a, b, and r?

5. Definition. If by any operation we render an expres-
sion integral with reference to certain quantities, in respect to



Introduction. 3

which it was previously fractional, we are said to Integralize the
expression with respect to those quantities. Thus the expression

a'x^ 2ab b'x'
b x^ a
is integralized with respect to a and b if it is multiplied by ab.

Similarly, if, by any operation, we render an expression
rational with reference to certain quantities, in respect to which
it was previously irrational, we are said to Rationalize the expres-
sion with respect to the quantities named. Thus if we square
the irrational expression

'^ x'-^V~ab xy^-y-"
it is rationalized with respect to x and y.

6. Definitions. The Degree of a term with respect to any
quantity or quantities is the sum of the exponents of the quan-
tities named. Thus ab'x'^y is of the third degree with reference to
X, of the first degree with reference to y, of the fourth degree with
reference to x and y, of the third degree with reference to a and
b, etc. But the degree with reference to any quantities is not
spoken of unless the term is rational and integral with respect to
those quantities. Thus we do not speak of the degree of such a



\^'



term ^s — , with respect to either a or x.

The Degree of a polynomial with respect to any specified quan-
tities is the degree of that one of its tenus whose degree (with re-
spect to the same quantities) is highest. Thus, x^—alx'^y^-\-cxy
is of the third degree with respect to x, of the second degree with
respect to y, and of the fourth degree with respect to x and y.
But the degree of a polynomial is not spoken of unless the poly-
nomial is rational and integral with respect to the quantities
specified.

It can easily be seen that the degree of the product of several
polynomials is the sum of their separate degrees. Thus

(^'+^^+y) {xy^bx-y)
is of the fifth degree with respect to x and y ; of what degree is it
with respect to jf ? with respect to_y?

The Degree of an Equation is the degree of the term of highest
degree with respect to the unk7iotvn quantities. But both mem-



4 Algebra.

bers of the equation must be rational and integral with respect to
the unknown quantities and the indicated operations must be per-
formed ; otherwise the degree is not spoken of.
What is the degree of

(jt" -f y ) (jry + 1 ) = 208-ry ?

Instead of speaking of expressions as being of the first or of the
second or of the third degree, they are commonly designated by
adjectives borrowed from geometry as linear or quadratic or cubic
expressions respectively. An expression of the fourth degree is
sometimes called bi-quadratic, meaning twice squared.

In place of the expression, "of the second degree in respect to
.ar," it is common to say, "of the second degree in x/'

7, DkfinitioNvS. a polynomial is Homogeneous with respect
to certain quantities when all its terms are of the same degree
with respect to those quantities. Thus a^-\-d'b-\-ab~-\-b^ is homo-
geneous with respect to a and b.

An equation is Homogeneous when all the terms are of the same
degree with reference to the unkjiown quantities. Thus the equa-
tion xy-\-j/^-\-x^=o is homogeneous, but xy-\-y^-\-x^=2o is not
homogeneous.

It is to be noted ^ here that we use the term homogeneous equa-
tion in the strict .sense, following the established use of the term.
But in some American text books homogeneous equation includes
equations like x-y-{-y^-\-x^=20, that is, no account is taken of
terms involving nothing but known quantities.

8. Definition. An expression is Symmetrical with respect
to two quantities if the expression is unaltered when the two
quantities are interchanged. Thus x^-{-y^ is symmetrical with re-
spect to X and J' ; for putting j/ for x and x for j we oh\.3.h\y^-\-x^^
which is the same as the original. Also x''-\-ax+a'' is symmet-
rical with respect to a and x. Is x^-\-2xy—y^ symmetrical with
reference to x and y ?

An equation of two unknown quantities is symmetrical when
the interchange of the unknown quantities throughout does not
modify the equation. Such is

X -\- xy -f- xy'^ +y= 1 024.



Introduction. 5

9. Incommensurable Numbers. Algebraic numbers* may
be divided into two kinds, depending upon the relation which
they bear to the unit or unity. If a number has a common
measure with unity, it is called a commensurable number. Thus
7 is a commensurable number ; also f is a commensurable number,
since one quarter of the unit is a common measure between f and
unity. Commensurable numbers thus include both integers and
fractions. If a number has no common measure with unity, it is
called an incommensurable number. Thus v^ 2 is incommen-
surable. A little consideration will show that v^ 2 cannot be an
integer nor a fraction. It is not an integer because (0)^=0,
(1)^=1, and (2)^=4, and there are no integers intermediate be-
tween these. It cannot be a fraction, for if possible suppose that

some irreducible fraction, represented by-^, equals v^ 2 . Then

^ 2 — -r-, or squaring, 2=-^, which is absurd, for an integer

cannot equal an irreducible fraction. Therefore >/ 2 is not a
fraction. But it is an exact quantity, for we can draw a geomet-
rical representation of it. Take each of the two sides, CA and
CB, of a right angled triangle equal to i. Then AB, the hypoth-
enuse, will equal v^( i )M-(i)'= ^ ~2 Thus A^
v^ 2 is the exact distance from A to B,
which is a perfectly definite quantity.
Thus the idea that incommensurables are
indefinite or inexact must be avoided, (l)
This notion has arisen because the frac-
tions we often use in place of incommen-
surables, such as 1. 4 1 42+ for >/ 2, are
merely approximatioyis to the true value. (^ (x) ~J^

We now give a property of incommensurable numbers which
will serve to make their separation from the class of commensur-
able numbers (integers and fractions) more apparent. It is that
an incommensurable yiumber whe^i expressed in the decimal scale
never repeats, while a commensurable number so expressed always
repeats.

* As here used the term Algebraic number does not include the so-called imaginaries,
which, strictly speaking, are not numbers at all. Imaginaries ai-e treated in Chapter 1,
Part 11.



6 Algebra,. ^^ ... .^^^^. ;

Thus, 75=75.0000000000+ repeati7ig the o.

|-= .5000000000+ repeating ihe o.
i= -3333333333+ repcati7ig the 3.
yYT= .279279279279+ repeating the 279.
\/ 3 = 2.7320508+ never repeating ,
"^20= 2. 71 44 177+ 7tever repeating.
-= T). 1/^1 ^^26+ never repeating.
The student should endeavor to get a fair notion of what is
meant by an incommensurable number. It is a difficult idea to
grasp at once, but it is one which the student should continue to
consider until the conception takes a definite and rational shape.

POSITIVE AND NEGATIVE QUANTITIES.

10. In Algebra we are often called upon to distinguish between
quantities which are directly opposite each other ; as, for instance,
degrees above zero from degrees below zero on a thermometer scale,,
distance north of the Equator from distance south of the Equator,
distance east of a given point from distance west of the same given
point, etc.

The distinction is made by means of the signs + and — , e. g.^
if +10° means a temperature of 10° above zero, then —10° would
mean a temperature of 10° beloiv zero, and if + 10 miles means 10
miles «^r/// of the Equator, then — 10 miles would mean 10 miles
south of the Equator, and if +10 rods means 10 rods east of a
given point, then — 10 rods would mean 10 rods west of the same
given point, and if +10 be ten units of any kind in a7iy sense,
then —10 would be ten units of the sa7ne kind in just the opposite
sense.

These two kinds of quantities are called /^«VzV^ and negative.

11. The distinction between positive and negative quantities is
made by means of the same signs as are used to denote the opera-
tions of addition and subtraction, and it might seem that it is un-
fortunate and unnatural that the same signs are used in these two
ways. It may be unfortunate, but it is not unnatural, as w^e pro-
ceed to show.

12. Suppose that, by one transaction, a man gained $500, and
by another he lost $700 ; then he lost all he gained and $200 more,



Positive and Negative Quantities. 7

or his capital suffered a diminution of $200. If his original
capital was $1,000, then the first transaction increased it to
$1,500, and the second transaction diminished it to $800. Thus
an addition of $500 followed by a diminution of $700 is equivalent
to a single diminution of $200, or

$i,ooo4-$5oo— $700= $1,000— $200.

Hence $^00— $-] 00 whe7i joined to $1,000 may be replaced by
— $200 joijied to $1,000.

Now, as any other original capital would have answered as well
as $1,000, we may neglect that original capital and write
$500— $700= —$200.

Thus we see, by this illustration, that it is fiatural to prefix the
minus sign to the $200 to indicate a resultant loss of $200.

13. We might have used an illustration involving some other
kind of quantity than money, as time, distajice, etc., and have ob-
tained an equation similar to the one just written. We may then
make an abstraction of the $ sign and write simply

500— 700= — 200.

14. In Arithemetic we are concerned only with the quantities

o, I, 2, 3, 4, etc.,
and intermediate quantities, but in Algebra we consider besides

these the quantities

o, —I, —2, —3, —4, etc.,
and intermediate quantities.

15. We may represent these two classes of quantities on the
following scale,

—5.-4, —3. —2, —I, o, I, 2, 3, 4, 5,

which extends indefinitely in both directions from zero.

The sign -f perhaps ought to precede each of the quantities at
the right of o in this scale, but when no sign is written before a
quantity the -f sign is always understood.

16. Quantities to the right of o in the above scale are positive
and those to the left of o are 7iegative, or we might say Arabic
numerals preceded by a + sign or by no sign at all are positive
quantities, and Arabic numerals preceded by a — sign are 7iegative
quantities.



8 Algebra.

17. In Algebra quantities are represented by letters, but a letter
is just as apt to represent a quantity to the left of o in the above
scale as it is to represent one to the right of o ; so that, while in
the case of a numerical quantity, i. e. one represented by figures,
we can tell whether the quantity represented is positive or nega-
tive by the sign preceding it, yet, it the case of a literal quantity,
/. e. one represented by letters, we cannot tell by the sign before
it whether the quantity represented is positive or negative.

If we speak of the quantity 5 we know that it is positive, but if
we speak of the quantity a we do 7iot know by the sign before it
whether it is positive or negative.

We know that —5 is negative, but we do 7iot know that —a is
negative.

A mifius sign before a letter always represents a qna?itity of the
opposite kind from that represented by the same quantity with a plus
sign or no sig7i at all before it. Thus, if «=3, then — <2=— 3, and
if a=— 3, then —a—T).

18. Looking at the above scale it is evident that of any two
positive quantities the one at the right is greater than the other or
the one at the left is less than the other, e. g. io>6 or 6 < 10.

Now it is found convenient to extend the meaning of the words
* ' less than ' ' and ' ' greater than ' ' so that this same thing shall be
true throughout the whole scale.

Thus we would say that

— 5< — 3 and— 2<o.
It should be carefully noticed that this is a technical use of the
words ' ' greater than ' ' and ' ' less than ' ' and conforms to the pop-
ular use of these words only when the quantities are positive.

Of course it would be wrong to say that — 2 is less than o if we
use ' ' less than ' ' in the popular sense, because no quantity can be
less than nothing at all, in the popular sense of " less than."

In objecting to the use of the words " less than " in the popular
sense. Prof. De Morgan, one of the great mathematicians of Eng-
land, says : " The student should reject the definition still some-
times given of a negative quantity that it is less than nothing. It
is astonishing that the human intellect should ever have tolerated
such an absurdity as the idea of a quantity less than nothing ;



Introduction. 9

above all, that the notion should have outlived the belief in
judicial astrology, and the existence of witches, either of which
is ten thousand times more possible."

This strong language is directed against the use of the words
*'less than" in the popular sense, but let the student keep in
mind that the words are used in a technical sense and there will be
no objection to such an inequality as — 2<o.

Illustrations. — If we speak of temperature as indicated by a
thermometer scale, then ''greater t/ia7i'' means higher and ''less
tha7i ' ' means lower. If we speak of distance east and west and
agree that distances measured east are positive, then "greater
thafi ' ' means ' ' east of ' , and ' ' less tha7i ' ' means * ' ivest o/'\ If we
agree that distances measured north are positive and those meas-
ured south are negative, then "greater than'' means "north of'\
and ' ' less tha?i ' ' means ' ' south of'\ etc.

THE RULE OF SIGNS IN MULTIPLICATION AND DIVISION IN

ALGEBRA.

19. If we take a and b any two positive quantities, it is easy to
see that the notion of multiplication we get from arithmetic will
enable us to deal with any case of multiplication where the vinl-
tiplier is a positive quantity, for, evidently, a can be repeated b
times, and so can —a be repeated b times, but a caymot be repeated
— b times, e. g. 3, and also —3, can be repeated 5 times, but 3
cannot be repeated —5 times.

Thus, when the mnltiplier is negative, multiplicatioii has no mean-
ing according to the arithmetical notio7i of multiplication, and so we
are obliged to broaden our ideas of multiplication in some way or
else exclude the operation when the multiplier is negative.

20. The primary definition of multiplication is repeated addi-
tion, yet, even in arithmetic, the word outgrows its original
meaning, for, by no stretch of language, can the operation of mul-
tiplying I by |- be brought under the original definition.

According to the original definition, multiplication, in arithme-
tic, is intelligible so long as the multiplier is a whole number.

3 can be repeated 4 times, and so can -J- be repeated 4 times
but 4 ca7i7iot be repeated \ a time.

A— 2-



lo Algebra.

^ repeated 4 times is ^ ^nultiplied by 4, yet, in arithmetic, 4
multiplied by ^ is a familiar operation.

Let us inquire how this comes to have a meaning, and how it
happens that 4 multiplied by \ turns out to be ^ ^4.

21. As long as a and b are positive whole numbers it is easy to
see \h2Xab=ba.

Suppose, to fix the ideas, that ^=3 and ^=5, then we may
write down 5 rows of dots with three dots in each row, thus —



and we have in all 5 times 3 dots. But we may look at vertical
rows instead of horizontal ones and we see three rows with 5 dots
in each row, and of course the number of dots is the same ; so we
may say

5x3=3x5.
Any other positive whole numbers would do as well as 3 and 5,
and so if a and b are ariy positive whole numbers,

ab=-ba,
i. e. , in the product of two numbers, it is iiidiffereyit which is the
multiplier ajid which the multiplica?id, so long as both ?iu?nbers are
integers. ^

22. Now, in arithmetic^ the operation of multiplication is so
extended that eve7i when one of the quantities is a fractio7i it shall
still be indifferent which of the two quantities is the multiplier and
which the multiplicand.

This gives a 77ieaning to multiplication when the multiplier is a
fraction, and thus it happens that 4 multiplied by ^ is taken to
mean the same as \ multiplied by 4.

23. In exactly the same way in algebra, the operation of mul-
tiplication is extended so that whatever numbers, positive or neg-
ative, integral or fractional, are represented by a and b we vShall
always have

ab=ba.



Introduction. i i

and since we know what is meant by —3 multiplied by 5, the
equation ab=ba gives a meajiing to 5 multiplied by —3.
.•.5 multiplied by —3= — 15.
From this we are led to say that when the multiplier is neg-
ative, the product is just the opposite from what it would be if
the multiplier were positive.

Therefore, if a and b are any two positive quantities, we may
wTite the following four equations :

a.b=ab (i)

{—a).b=—ab (2)

a.{—b)^—ab (3)

{^-a).{-b) = ab. (4)

From the ist and 4th we conclude that the product of two posi-
tive quantities or two iiegative quayitities is positive, and from the 2d
and 3d, the product of 07ie positive and o?ie negative quantity is neg-
ative.

24. The four equations just written are true whether a and
b are positive nor not.

Consider, for example, the second equation under the supposi-
tion that a is negative and b positive ; then {—a).b becomes the
product of two positive quantities and is therefore positive, but
—ab is also positive in this case, as it should be, rendering the
equation still true. And so of the other equations, whether a and
b are positive or not. Therefore, directing our attention to the
signs, we may say that the product of two quantities preceded b}'
like signs is a quantity preceded by the -j- sign, and the product
of two quantities preceded by tmlike signs is a quantity preceded
by a — sign.

This statement is usually shortened into the following —


1 3 4 5 6 7 8 9 10 11 12 13 14

Online LibraryC. A. (Charles Ambrose) Van VelzerA course in algebra. Being course one in mathematics in the University of Wisconsin → online text (page 1 of 14)