Reciprocals of Quantities. (79.) The reciprocal of a quantity is a unit divided by the quantity, (80.) The reciprocal of a Fraction is equivalent to the fraction inverted; that is, with its numerator and denominator taken the one for the other. a 1 Thus the reciprocal of, or ax-2 (77), is, which becomes by transferring x-2 from the denominator to the numerator, (78). Under what different forms may the Reciprocal of " α ab pressed? The reciprocal of ? The reciprocal of ? 3c2 Constant Value of a Fraction. (81.) The value of a Fraction remains the same when its numerator and denominator are both multiplied, or both divided, by the same quantity. For the first of these fractions is equal to ax2; and the second is equal to ax-2yy-1, (77). But yy-1 is equal to y°, (41), and by cancelling this factor, (48), we find the second fraction also equal to ax-2. Hence the two fractions are equal to each other. (82.) Since a Fraction represents the quotient of its numerator divided by its denominator, a Fraction is positive when its numerator and denominator have the same sign, and negative when they have contrary signs, (49). . Say whether the Fraction is positive or negative. Say α a b -a whether is positive or negative. Say whether is positive or -b -ax negative. Say whether is positive or negative. by -b (83.) The sign + or - prefixed to a Fraction,-not the sign of either the numerator or denominator,-shows whether the fraction enters additively or subtractively into a calculation. Thus + -a b denotes that the fraction, which is in itself b negative, (82), is to be added, in the calculation into which it enters. When no sign is prefixed to a fraction, + is always understood. (84.) The signs of both the numerator and denominator may be changed, or the sign of either of them with the sign prefixed to the Fraction, without affecting the value of the fraction. its numerator and denominator having contrary signs; this negative fraction becomes positive, when subtracted, as required by the sign-prefixed to it, and it is then equivalent to the first fraction. A Polynomial is changed from positive to negative, or from negative to positive, by changing the sign of each of its terms, (38). For example, if a b is positive, a must be greater than b; then, changing the signs, b-a will be negative. What other changes may be made in the signs, without af ecting the value, of this fraction? FRACTIONS REDUCED TO THEIR LOWEST TERMS. (85.) A Fraction is reduced to lower terms by dividing its numerator and denominator by the same common measure. This simplifies the fraction, without altering its value, (81). A monomial common measure may usually be known by inspection. Thus to reduce the Fraction 4a2b 6ac-8a2x It is obvious that we have only to divide its numerator and denominator by 2a. This gives us the equivalent Fraction, 2ab A binomial common measure may often be discovered from the principles which have been established for the decomposition of Polynomials. Thus to reduce the Fraction a2-b2 By proposition (57) we can divide the numerator by a+b; and by (59), we can divide the denominator by the same quantity Thus dividing we find the equivalent Fraction a-b a+b' In all cases in which a Fraction admits of being reduced, we may apply RULE IX. (86.) To Reduce a Fraction to its Lowest Terms. Divide the numerator and denominator by their greatest common measure, (66): the quotients will be the lowest terms of the given fraction. FRACTIONS REDUCED TO A COMMON DENOMINATOR. (87.) Two or more Fractions are said to have a common denominator, when they have the same quantity for a denominator. Two or more Fractions may often be reduced, very readily, to a common denominator, by multiplying both the numerator and denominator of one or more of them, so as to make the denominator the same for each. For example, to reduce to a common denominator the Fractions We have only to multiply the terms of the first fraction by 3, and those of the second by 2. This gives the equivalent Fractions Observe that this reduction does not alter the values of the given Fractions. When this method cannot be obviously applied, we adopt RULE X. (88.) To Reduce two or more Fractions to a Common Denomi nator. 1. Multiply each numerator by all the denominators except its own, for new numerators; and multiply all the denominators together, for a common denominator. 2. If the Least Common Denominator be required,—Find the least common multiple of the given denominators, for the Common denominator. Divide this Multiple by the denominator of each given Fraction, and multiply the quotient by the numerator, for the new For the new numerators, we have a. 6. (y+2), equal to 6ay+12a; b. 3x. (y+2), 3bxy+6bx; and c. 3x.6 18c. And the common denominator is 3x. 6. (y+2), equal to 18xy+36x. |