Every quantity which is not expressed under a fractional form, is called an entire quantity. An algebraic expression composed partly of an entire quantity and partly of a fraction, is called a mixed quantity. (82.) The proper sign to be prefixed to a fraction may be determined by the rules 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. ab Thus, =+b, for + divided by + gives +. a h is to be subtracted, which is done by changing its sign. -ab a = +b, because the former quotient -b is to be subtracted, whence it becomes +b. ab -a and +b, for the same reason; -ab -a -b, also for the same reason. Hence we have the following equivalent forms: That is, of the three signs belonging to the numerator, denominator, and dividing line of a fraction, any two may be changed from to or from + · value of the fraction. - to, without affecting the In the examples of fractions here employed for illustration, both numerator and denominator have consisted of monomials. The same principles are applicable to polynomials; but it must be remarked, that by the sign of the numerator we understand the entire numerator as distinguished from the sign of any one of its terms taken singly. When no sign is prefixed either to the terms of a fraction or to its dividing line, + is always to be understood. REDUCTION OF FRACTIONS. PROBLEM I. (83.) To reduce a fraction to lower terms. RULE. Divide both numerator and denominator by any quantity which will divide them both without a remainder. According to Remark 3, Art. 81, this will not change the value of the fraction. If the numerator and denominator are both divided by their greatest common divisor, it is evident the fraction will be reduced to its lowest terms. The method of finding the greatest common divisor is considered in Section XV.; but in the following examples the greatest common divisor is easily found, by resolving the quantities into factors according to methods already indicated. (84.) To reduce a fraction to an entire or mixed quantity. RULE. Divide the numerator by the denominator for the entire part, and place the remainder, if any, over the denominator for the fractional part. (85.) To reduce a mixed quantity to the form of a fraction RULE. Multiply the entire part by the denominator of the fraction; to the product add the numerator with its proper sign, and place the result over the denominator. This result may be proved by the preceding Rule. For X- 3 5. Reduce 1+2x+ to the form of a fraction. 5x 3b2-8c2 6. Reduce 7+ a2-ba to the form of a fraction. PROBLEM IV, (86.) To reduce fractions to a common denominator. RULE. Multiply each numerator into all the denominators, except its own, for a new numerator, and all the denominators together for a common denominator. Here it will be seen that the numerator and denominator of the first fraction are both multiplied by d, and in the second fraction they are both multiplied by b. The value of the fractions, therefore, is not changed by this operation. a+b с 2. Reduce and to equivalent fractions having a common denominator. |