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;

and generally, the reciprocal of the denominator is the frac tional 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 fracticnal unit and then writing the product over the denominator; thus, the quantity c may be reduced to a fractional

1

with the fractional unit by multiplying c by 5 and

b

bc

div

the product by b, which gives

b

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

sometimes reduce the fract

65. If the numerator a of the

any quantity, 9, the resulting fraction

as many fractional units as are expressed by b

Multiplying the numerator of a fraction by any quality 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 q times, and the

α

resulting fraction will express a quantity q times less than

qb

the given fraction; hence:

Multiplying the denominator of a fraction by any quanity, 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 numerator 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 num ator and denominator do not contain a common factor. since both terms of a fraction may be divided by the quantity without altering its value, we have for reduction of a fraction to its simplest form the following

RULE.

Resolve both numerator and nominato into their simple fao tors (Art. 59); then, suppress all factors common to both terms, and the fraction w be simplest form.

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

resort

EXAMPLES.

3ab+6ac

1. Reduce the fraction

trts simplest form

3ad + 12a

hence,

3ab + 6ac

merator, hence,

3ab+6ac3a (b + 2c)

he domina

We also see, that 3 and a are factors

3ad + 12a 3a (d + 4).

Hence,

2. Reduce

to its simplest form.

9ab+3ad

II. From what was shown in Art. 63, it follows that we may reduce the entire part of a mixed quantity to a al form with the same fractional unit as the fractional parultiply. 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 part.

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,