CHAPTER VII.

FRACTIONS.

1. The quotient of a division can be expressed as an integer or an integral expression only when the dividend is a multiple of the divisor; as a2b÷ab = a; (ax2 + 2 bx) ÷ x = ax + 2b.

If the dividend be not a multiple of the divisor, the quotient is called a Fraction; as a÷b; (ax2 + 2 bx) ÷ 203.

2. The notation for a fraction in Algebra is the same as in ordinary Arithmetic.

Thus, (ax2+2 bx)÷23 is written ax2 + 2 bx.

The Solidus,/, is frequently used instead of the horizontal ax2 + bx line to denote a fraction; as (ax2 + bx)/x3 for X3

3. As in Arithmetic, the dividend is called the Numerator of the fraction, the divisor the Denominator, and the two are called the Terms of the fraction.

4. An integer or an integral expression can be written in a fractional form with a denominator 1.

It is important to notice that an algebraic fraction may be arithmetically integral for certain values of its terms.

E.g., when a 4 and b = 2, the fraction a/b becomes 4/2

= = 2.

5. By the definition of a fraction, a/b is a number which, multiplied by b, becomes a; that is,

(a/b) × b = a, or

xb = a

(1)

6. The Sign of a Fraction. The sign of a fraction is written before the line separating its numerator from its denominator;

Since a fraction is a quotient, the sign of a fraction is determined by the rule of signs in division.

7. From the rule of signs we derive:

(i.) If the signs of the numerator and the denominator of a fraction be reversed, the sign of the fraction is unchanged.

This step is equivalent to multiplying or dividing both terms of the fraction by - 1.

(ii.) If the sign of either the numerator or the denominator of a fraction be reversed, the sign of the fraction is reversed; and conversely.

(iii.) If the signs of an even number of factors in the numerator and denominator, either or both, of a fraction be reversed, the sign of the fraction is unchanged; but, if the signs of an odd number of factors be reversed, the sign of the fraction is reversed.

Reduction of Fractions to Lowest Terms.

8. A fraction is said to be in its lowest terms when its numerator and denominator have no common integral factor.

9. The value of a fraction is not changed if both numerator and denominator be divided by the same number, not 0.

Dividing by n, vb ÷ n = a ÷n, or v(b ÷ n) = a ÷ n.

The factor 2 a2b2 is the H. C. F. of the numerator and denominator.

We therefore have

A fraction is reduced to its lowest terms by dividing its numerator and denominator by the H. C. F. of its terms.

This step is called cancelling common factors, and can usually be done mentally, if the terms of the fraction are first resolved into their prime factors.

Changing the sign of the first factor in the numerator and the sign of the fraction, we have

We find x 2 to be the H. C. F. of numerator and denominator by Ch. VI., Art. 33.

Reduce each of the following fractions to its lowest terms:

Reduction of Two or More Fractions to a Lowest Common

Denominator.

11. Two or more fractions are said to have a common denominator when their denominators are the same.

The Lowest Common Denominator (L. C. D.) of two or more fractions is the L. C. M. of their denominators.

12. The value of a fraction is not changed if both numerator and denominator be multiplied by the same number, not 0.