CHAPTER IX. FRACTIONS. Definitions. 155. An algebraic fraction is the indicated quotient of two expressions written in the form α b 156. The dividend a is called the numerator; the divisor b is called the denominator; the numerator and denominator are called the terms of the fraction. ac α b = ас bc Now is obtained by multiplying both terms of by c; b bc 157. If the numerator and denominator of a fraction are both multiplied by the same number, or both divided by the same number, the value of the fraction is not altered. Reduction of Fractions to their Lowest Terms. 158. A.fraction is in its lowest terms when the numerator and denominator have no common factor. Hence, To Reduce a Fraction to Lowest Terms, Resolve the numerator and denominator into their prime factors, and cancel all the common factors; or, divide the numerator and denominator by their highest common factor. Reduce the following fractions to their lowest terms: We find by the Factor Theorem the H.C.F. of the numerator and denominator to be x 1. x3- 4x2 + 4x 2x2+4x 159. There are three signs to consider in a fraction, the sign before the fraction, the sign of the numerator, and the sign of the denominator. any two of the three signs may be changed without changing the value of the fraction. The sign of a compound expression is changed by changing the sign of every term of the expression. Hence, 1. We may change the sign of every term of the numerator and denominator of a fraction without changing the value of the fraction. 2. We may change the sign before a fraction and the sign of every term of either the numerator or denominator without changing the value of the fraction. 160. From the Law of Signs, therefore, 1. We may change the signs of an even number of factors of the numerator, or of the denominator, or of both, without changing the sign of the fraction. 2. We may change the signs of an odd number of factors of the numerator, or of the denominator, or of both counted together, if we change the sign before the fraction. In the last fraction change the sign of the numerator, the sign of the fraction, and the order of the terms of the denominator, and we have NOTE. Factors and terms must not be confounded. 161. Mixed Expressions. A mixed expression is an integral expression and a fraction. Reduction of Fractions to Integral or Mixed By the distributive law of division, § 43, Therefore, a fraction whose numerator is of a degree equal to, or higher than, the degree of the denominator is reduced to an integral or mixed expression by division. Hence, 162. To Reduce a Fraction to an Integral or Mixed Expression, Divide the numerator by the denominator. |