CHA P. I.

PART II.

OF FRACTIONS.

بير

1.

DEFINITIONS.

WHE THEN

HEN a quotient is expreffed by a fraction, the divifion above

the line is called the numerator; and the divifor below it is called the denominator.

II. If the numerator is lefs than the deno

minator, it is called a proper fraction. III. If the numerator is not lefs than the

denominator, it is called an improper fraction.

IV.

J

IV. If one part of a quantity is an integer, and the other a fraction, it is called a mixt quantity,

V. The reciprocal of a fraction, is a frac

tion whofe numerator is the denominator of the other; and whofe denominator is the numerator of the other. The reciprocal of an integer is the quotient of 1

The diftinctions in Def. II. III. IV. properly belong to common arithmetic, from which they are borrowed, and are scarcely ufed in Algebra.

The operations concerning fractions are founded on the following propofition:

If the divifor and dividend be either both multiplied, or both divided, by the fame quantity, the quotient is the Jame; or, if both the numerator and denominator of the fraction be either multiplied or divided by the

Jame

a

the nature of divifion, if the quotient b

(c) be multiplied by the divifor b, the product must be the dividend a.

Hence

(xb =) tea, and likewise ma=mbc,

and dividing both by mb,

ma

mb

c. Converse

C.

Cor. 1. Hence a fraction may be reduced to another of the fame value, but of a more fimple form, by dividing both numerator and denominator by any common measure.

Cor. 2. A fraction is multiplied by any integer, by multiplying the numerator, or dividing the denominator by that integer: and conversely, a fraction is divided by any

integer,

integer, by dividing the numerator, or mul tiplying the denominator by that integer.

Prob. I. To find the greatest common measure of two quantities.

Rule. Divide the greater by the lefs: and, if there is no remainder, the less is the greatest common measure required. If there is a remainder, divide the laft divifor by it; and thus proceed continually dividing the laft divifor by its remainder, till no remainder is left, and the last divifor is the greatest common measure required...

The greatest common measure of 45 and 63 is 9; the greatest common measure of

From the nature of this operation, it is plain, that it may always be continued, till there be no remainder. The rule depends on the two following principles.

1. A quantity which measures both divifor and remainder, muft measure the dividend.

2. A quantity which measures both divisor and dividend, must also measure the remainder.

For a quantity which measures two other quantities, muft alfo measure both their fum and difference; and, from the nature of divifion, the dividend confifts of the divifor repeated a certain number of times, together with the remainder. By the first it appears, that the number found by this rule is a common measure; and, by the fecond, it is plain there can be no greater common measure; for, if there were, it muft neceffarily measure the quantity alrea dy found less than itself, which is abfurd.

When the greatest common measure of algebraical quantities is required, if either of them be simple, any common fimple di