CHAPTER III.

OF ALGEBRAIC FRACTIONS.

62. AN ALGEBRAIC FRACTION is an expression of one or more equal parts of 1.

One of these equal parts is called the fractional unit. Thus, α is an algebraic fraction, and expresses that 1 has been divided

b

into b equal parts and that a such parts are taken.

The quantity a, written above the line, is called the numer utor; the quantity b, written below the line, the denominator; and both are called terms of the fraction.

1

One of the equal parts, as is called the fractional unit;

b

and generally, the reciprocal of the denominator is the fractional unit.

The numerator always expresses the number of times that the fractional unit is taken; for example, in the given fraction, the

fractional unit is taken a times.

1 b

63. An entire quantity is one which does not contain any fractional terms; thus,

a2bcx is an entire quantity.

A mixed quantity is one which contains both entire and frac tional terms; thus,

Every entire quantity can be reduced to a fractional form having a given fractional unit, by multiplying it by the denominator of the fractional unit and then writing the product over the denominator; thus, the quantity c may be reduced to a fractional

1

form with the fractional unit by multiplying c by and

bc

dividing the product by b, which gives ठ

64. If the numerator is exactly divisible by the denominator, a fractional expression may be reduced to an entire one, by simply performing the division indicated; if the numerator is not exactly divisible, the application of the rule for division will sometimes reduce the fractional to a mixed quantity.

a

65. If the numerator a of the fraction be multiplied by

b

any quantity, q, the resulting fraction will

aq b

express q times as many fractional units as are expressed by ; hence:

α

b

Multiplying the numerator of a fraction by any quantity is equivalent to multiplying the fraction by the same quantity.

66. If the denominator be multiplied by any quantity, q, the value of the fractional unit, will be diminished 9 times, and the

α

resulting fraction will express a quantity q times less than

gb

the given fraction; hence:

Multiplying the denominator of a fraction by any quan ́ity, is equivalent to dividing the fraction by the same quantity.

67. Since we may multiply and divide an expression by the same quantity without altering its value, it follows from Arts 65 and 66, that:

Both numerator and denominator of a fraction may be multiplied by the same quantity, without changing the value of the fraction. In like manner it is evident that:

Both numerater and denominator of a fraction may be divided by the same quantity without changing the value of the fraction.

68. We shall now apply these principles in deducing rules for the transformation or reduction of fractions.

I. A fractional is said to be in its simplest form when the numer. ator and denominator do not contain a common factor. Now, since both terms of a fraction may be divided by the same quantity without altering its value, we have for the reduction of a fraction to its simplest form the following

RULE.

Resolve both numerator and denominator into their simple factors (Art. 59); then, suppress all the factors common to both terms, and the fraction will be in its simplest form.

REMARK.-When the terms of the fraction cannot be resolved into their simple factors by the aid of the rules already given, resort must be had to the method of the greatest common divi sor, yet to be explained.

EXAMPLES.

3ab + 6ac

1. Reduce the fraction

tots simplest form.

3ad + 12a

We see, by inspection, that 3 and a factor of the numerator, hence,

3ab+6ac3a (b + 2c)

We also see, that 3 and a are factors the denominat

II. From what was shown in Art. 63, it follows that we may reduce the entire part of a mixed quantity to a fractional form with the same fractional unit as the fractional part, by multiply. ing and dividing it by the denominator of the fractional part. The two parts having then the same fractional unit, may be reduced by adding their numerators and writing the sum obtained over the common denominator.

Hence, to reduce a mixed quantity to a fractional form, we have the

RULE.

Multiply the entire part by the denominator of the fraction: then add the product to the numerator and write the sum over the denominator of the fractional purt.

REMARK. We shall hereafter treat mixed quantities as though they were fractional, supposing them to have been reduced to a fractional form by the preceding rule.

III. From Art. 64, we deduce the following rule for reducing a fractional to an entire or mixed quantity.

RULE.

Divide the numerator by the denominator, and continue the oper ation so long as the first term of the remainder is divisible by the first term of the divisor: then the entire part of the quotient found, added to the quotient of the remainder by the divisor, will be the mixed quantity required.

If the remainder is 0, the division is exact, and the quotient is an entire quantity, equivalent to the given fractional expression.