a, a2b numerator by ɑ, we obtain ab, or b; and if we multiply the denominator of the same fraction by a, we obtain ab, or b; that is, the original value of the fraction ab has been divided by a. 3d. The value of a fraction is not changed if we multiply or divide both numerator and denominator by the same number. 110. The proper Sign of a Fraction.-Each term in the numer ator and denominator of a fraction has its own particular sign, and a sign is also written before the dividing line of a fraction. The relation of these signs to each other is determined by the principles already established for division. The sign prefixed to the numerator of a fraction affects merely the dividend; the sign prefixed to the denominator affects merely the divisor; but the sign prefixed to the dividing line of a fraction affects the quotient. The latter sign may be called the apparent sign of the fraction, while the real sign of the fraction is the sign of its numerical value when reduced. The real sign of a fraction depends not merely upon its apparent sign, but also upon the signs of the numerator and denominator. From Art. 73, it follows that Also, since a minus sign before the dividing line of a fraction shows that the quotient is to be subtracted, which is done by changing its sign, it follows that Hence we see that of the three signs belonging to the numer ator, denominator, and dividing line of a fraction, any two may be changed from + to, or from to +, without affecting the real sign of the fraction. 111. When the numerator or denominator of a fraction is a polynomial, it must be observed that by the sign of the numerator is to be understood the sign of the entire numerator, as distinguished from the sign of any one of its terms taken singly. a+b+c a—b—c Thus, is equivalent to + x When no sign is prefixed either to the terms of a fraction or to its dividing line, plus is always to be understood. Reduction of Fractions. 112. To reduce a Fraction to its Lowest Terms.-A fraction is in its lowest terms when the numerator and denominator contain no common factor; and since the value of a fraction is not changed if we divide both numerator and denominator by the same number (Art. 109), we have the following RULE. Divide both numerator and denominator by their greatest common divisor. Or, Cancel all those factors which are common to both numerator and denominator. 113. To reduce a Fraction to an Entire or Mixed Quantity.When any term of the numerator is divisible by some term in the denominator, the division indicated by a fraction may be at least partially performed. Hence we have the following RULE. Divide the numerator by the denominator, continuing the operation as far as possible; then write the remainder, if any, over the denominator, and annex the fraction thus formed to the entire part. EXAMPLES. 114. To reduce a Mixed Quantity to the Form of a Fraction.This problem is the converse of the last, and we may proceed by the following RULE. Multiply the entire part by the denominator of the fraction; to the product add the numerator with its proper sign, and write the result over the denominator. 7. Reduce 2a-7 4a2 -50 to the form of a fraction. 1 Ans. 2a+7° 8. Reduce (a−1)2_(a—1) to the form of a fraction. α 115. To reduce Fractions having Different Denominators to Equivalent Fractions having a Common Denominator: а с Suppose it is required to reduce the fractions and to b d a common denominator. Since, by Art. 109, both terms of a fraction may be multiplied by the same quantity without changing its value, we may multiply both terms of each fraction by the product of the denominators of the other fractions, and we shall have The resulting fractions have the same value as the proposed fractions, and they have the common denominator bdn. Hence we have the following RULE. Multiply each numerator into all the denominators, except its own, for a new numerator, and all the denominators together for the common denominator. |