ADDITION AND SUBTRACTION OF FRACTIONS. 120. If the fractions to be added together have a common denominator, their sum is found by adding the numerators together for a new numerator and retaining the common denomi nator. a с Thus + in Art. 99. = a + c This follows from the principle laid down 121. If the fractions have not a common denominator they must be transformed to others of the same value, which have a common denominator (Art. 112...114), and then the addition may take place as before. Here a is considered as a fraction whose denominator is 1. of the denominators is Ex. 6. Required the sum of x3 + x2 + x + 1 By Art. 116 or Art. 119 the Least Com. Mult. found to be a-1; therefore the sum required is 122. If two fractions have a common denominator, their difference is found by taking the difference of the numerators for a new numerator and retaining the common denominator. 123. If they have not a common denominator, they must be transformed to others of the same value, which have a common denominator, and then the subtraction may take place as before. The sign of bd in the numerator is negative, because every part of the latter fraction is to be taken from the former. (See Art. 87). MULTIPLICATION AND DIVISION OF FRACTIONS. 124. To multiply a fraction by any quantity multiply the numerator by that quantity and retain the denominator. For if the quantity to be divided be c times as great as before, and the divisor the same, the quotient must be plied by its denominator, the product is the numerator. 126. COR. 2. The result is the same, whether the numerator be multiplied by a given quantity, or the denominator divided by it. arises from the division of its denominator by c. 127. The product of two fractions is found by multiplying the numerators together for a new numerator, and the denominators for a new denominator. α C X = and = y by multiplying the equal quantities, a and b d a b' by b, baa (Art. 81); in the same manner, dy = c; therefore, by the same axiom, bx.dy ac, that is, bday = ac; dividing these equal quantities, bday and ac, by bd (Art. 82), we have xy = ac bd' 128. To divide a fraction by any quantity multiply the denominator by that quantity, and retain the numerator. what it was before, and the divisor the same. 129. COR. The result is the same, whether the denominator is multiplied by the quantity, or the numerator divided by it. 130. To divide one fraction by another invert the numerator and denominator of the divisor, and use it as a multiplier according to the rule for multiplication. also bdx = ad, and bdy = bc; therefore by Art. 82, 131. To prove the Rules for Multiplication and Division of Decimal Fractions. P 10m By the Definition of a Decimal Fraction (Art. 39,) is equivalent to a Decimal having m decimal places, and P being the number which the Q decimal represents excluding the decimal point; similarly is equivalent to a decimal which has n decimal places. Now P Q PQ X = 10m 10" 10m+n 10" = a decimal fraction formed by multiplying P into Q and then marking off m+n decimal places:-which proves the rule for multiplication. Again, P Q P Q 10m Hence the rule for division is, Divide P by Q, as in whole numbers; and then, If m>n, or the number of decimal places in the dividend> the number in the divisor, mark off m―n decimal places in the quotient. If m = n, the quotient is an integer. If m<n, affix n-m ciphers to the right of the quotient. 132. The rule for multiplying the powers of the same quantity (Art. 91), will hold when one or both of the indices are negative. 133. COR. If m = n, a”×a ̄m = am−m = a; also axa-m a am = 1; therefore a = 1, according to the notation adopted in Arts. 63, 65. 134. The rule for dividing any power of a quantity by any other power of the same quantity (Art. 96.) holds, whether those powers are positive or negative. 135. 1 an = an am (Art. 130.) a"-m. = COR. Hence it appears, that a factor may be transferred from the numerator of a fraction to the denominator, and vice versâ, by changing the sign of its index. |