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56. Divide ++xTMy+xy"+y^+1 by y+x

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It has been observed (Art. 31) that algebraical division is usually expressed by means of fractions; that is, by making the quantity to be divided the numerator, and its divisor the denominator of a fraction. If the dividend is not exactly divisible by the divisor, or there are few or no quantities common to both; this mode of expressing the result is generally more convenient than that of actually dividing each term of the dividend by the divisor, unless there is some particular object in view which can be more readily effected by so doing. Or the division may be performed upon one or more of the quantities of the dividend, and the remaining part of the quotient expressed, as in Art. 36.

FRACTIONS.

Algebraical Fractions are subject to Rules and Operations similar to those used in Vulgar Fractions in common arithmetic.

37. A fraction may be converted into another of equal value, by either multiplying or dividing both the numerator and denominator by the same

D

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mx

m. Conversely, the fraction , may be reduced

x

to

my'

, by dividing the numerator and denominator each by m. (Art. 32.)

Hence a fraction may be reduced to its lowest terms, by dividing both its numerator and deno-` minator by their greatest common measure or divisor.

A fraction whose numerator and denominator

consist of one term only, as

4ar

7ax y z

,, may easily

In this case the

be reduced to its lowest terms. greatest common divisor is at once seen to be

ar; therefore the reduced fraction is

4.x

In

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x3-y3 x3-y3 cx+x2

3x+3y' 5x-5y' a2c+a3x' (x − y)2 '

lowest terms.

to their

38. A fraction may have the signs of each quantity in its numerator and denominator changed without altering its value.

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39. Any quantity may be represented as a fraction, by placing 1 or unity under it for a denominator; thus x, and x+y, may be repre

X

sented as and

x+y
1

respectively, since they

1

are not changed in value by being divided by unity.

40. To reduce fractions to others of the same value, which have the same, or a common denominator. Multiply each numerator of the given fractions, into all the denominators except its own for new numerators, and all the denominators together, for a common denominator; as in common arithmetic.

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xxbx c bcx

yxa xc=acy The new numerators.
zx axb=abz

axbxc=abc, the common denominator.

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In many simple cases, there is no necessity to apply the foregoing rule, since the required fractions can be more easily found, and in lower terms, by multiplying the numerator and denominator of each fraction by the smallest quan. tities that will bring them to the same denominator.

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