formed every step of the operation will lead to a true result, whatever may be the difference of its appearance.

Neither need we conclude that we have committed errors, though the product of a compound divisor and quotient do not amount to the same identical expression as the dividend; because we have already shown that equi-multiples and like parts of any divisor and dividend will all lead to exactly the same quotient.

Before we can thoroughly understand division, and those general relations which are founded upon its principle, or rather in which its principle consists, it is necessary to have recourse to some farther explanations, which can be more conveniently made in a new section; we shall therefore close this one by subjoining the operation for the above example, as divided by the other factor.

a2ab+b2) a3+b3 (a+b

-a3+a2b-ab2

+a2b-ab2+b3

-a2b+ab2-b3

0.

SECTION VII.

NATURE AND MANAGEMENT OF FRACTIONS.

A FRACTION is a quantity viewed in its relation to some other quantity of the same kind which is considered as a whole; and as every case of division may be considered as reducible either exactly or to any degree of nicety that may be required to an expression in which the divisor is one, or, which is exactly the

same in effect, every possible case of division may be conceived as consisting of equi-multiples of 1 and the quotient by the divisor, the simple expression of division by writing the dividend over the divisor, and separating them by a line, is also the

a

general expression of a fraction. Thus is an expression for

b

any fraction, in which the quantity b is understood to mean a whole, or the number 1, and a any quantity whatever, only it must be one of the same kind with b. If a and b were expressed arithmetically it would be necessary to express them both in the same unit, in order that the numbers might express the same relation as the values; and when general expressions are used it is necessary to understand them in this manner.

Perhaps the simplest notion we can have of the nature of a fraction is the arithmetical one, which supposes that the whole is divided into as many equal parts as the under term of the fraction expresses, while the value of the fraction consists in the number of those parts which the upper term expresses. Thus,

19

in the expression the under number 20 shows that some

20

thing considered as a whole is understood to be divided into 20 equal parts; and the upper number 19 shows that the value of this particular fraction is 19 of those parts. From this it follows that the value of the fraction does not depend upon the absolute numbers in which it is expressed, but upon the relation of those numbers to each other; and that each of the two numbers has a distinct operation to perform.

One whole, by whatever number it may be expressed, may be considered as always meaning the very same quantity, unless the contrary is expressly stated; and thus, the larger number which the under term of a fraction expresses, the smaller must be the value of every individual 1 of that number; but the larger the

120

TERMS OF FRACTIONS.

upper term, the value must always be the greater. A fraction may thus be considered as having a sort of double value, or a value which may at any rate be considered as the result of two operations, a division by the under term to find the value of 1 in the upper number, and a multiplication of the value so found by the upper number.

In an arithmetical point of view, the number of the under term fixes the denomination of the fraction, in the very same way as the denominations of real quantities are fixed by the standards in which they are counted; and for this reason the under number is called the denominator of the fraction. The denominator is thus, as it were, "small change" for the integer number 1, just as shillings are small change for a pound, or yards are small change for a mile. The upper number shows how much of this small change the fraction consists of, and for this reason it is called the numerator, or "the teller of the number" of the fraction. It may be any number, equal to the denominator, or greater, or less; and it may be a number which cannot be exactly expressed in terms of the denominator, at the same time that there is between the two a relation which we can perfectly understand.

Hence the doctrine of fractions is a very general one in mathematical science, as it involves all comparisons in which the whole value of one quantity is compared with the whole value of another. There is something neat in the signs which are used to express the comparisons, ab is relation generally, and says little more than that a and b are quantities of the same

α

Ї

kind; is a more definite statement of the relation, for it points out that a is the standard with which b is compared. ab with the compound sign is more definite still, for it points out the difference of the related quantities; but the line

in it is not the sign - pointing out the difference by which the one quantity exceeds or falls short of the other, it relates to the whole of both, which is not necessary in the case of the mere difference as obtained by subtraction.

A fraction is still a quantity though the value of that quantity is expressed by a relation; and thus we must have some means of knowing when two fractions are equal and when not. We cannot tell this generally, or even in some common cases, by comparing the numerators; for these have equal values with equal expressions only when the denominators are also equal. If both consist of the same expressions their equality is of no use, as we can draw no conclusion from it.

Neither can we make the comparison generally if the terms are sums or differences indicated by the signs + or —. Thus a + c a+b b+d C- -d

we cannot tell whether

or

is the greater.

But if we have any means of showing that the two products arising from the multiplication of the numerator of each by the denominator of the other are equal, then we are in a condition for proving the equality of the two fractions; and this is important, as being the foundation of the rule of proportion, or "rule of three," which is so valuable in reasoning and calculation, both in arithmetic and in mathematics generally.

Now, from the connection that there is between a fraction and a case of division, it is evident that all equal fractions must have their terms equi-multiples of that form of the fraction

a

c

which has 1 for its denominator. Thus, if = then the b

product ad is equal to the product bc.

For, let the form of the fraction of which these terms are

equi-multiples be and no matter whether q is less than 1,

1'

equal to 1, greater than 1, or whether it can or cannot be ex

C

must But m and n stand for any multipliers whatsoever.

2

that is,

qn

n

q x n

1 x n'

Multiply qm by n, and qn by m, and we

have qmn=qnm, which is an identical proportion, the product of the same three factors; and these factors are perfectly general, and may be anything, provided that those which are expressed by the same letters are equal to each other.

Take an example: a man is entitled to

13

of a pound, would

16

16

16 shillings pay him? 16 shillings is

; and the question is,

20

13 16

is it equal to ? Multiply the numerator of each by the de

nominator of the other, and we obtain for the first 16 × 16=256, and for the second 13 × 20=260, which is more than the other, so that 16 shillings is not quite enough.

Let us see how much it wants. The numerator of each fraction has been multiplied by the denominator of the other; and if we multiply the denominator by the same we shall have equimultiples, or fractions of the same value as the original ones, and they will at the same time have equal denominations; that is,