_a_ _c_ _e_ _g_ --- × --- + --- × --- _b_ _d_ _f_ _h_ _acfh_ + _bdeg_ --------------------- = --------------- _a_ _e_ _c_ _g_ _aedh_ + _bcfg_ --- × --- + --- × --- _b_ _f_ _d_ _h_ 1 _b_ ---------- = --------- 1 _ab_ + 1 _a_ + --- _b_ 1 1 _bc_ + 1 --------------- = -------------- = ----------------- 1 _c_ _abc_ + _a_ + _c_ _a_ + --------- _a_ + -------- 1 _bc_ + 1 _b_ + --- _c_ 1 1 57 Thus, ------------ = --------- = --- 1 8 350 6 + ------- 6 + --- 1 57 7 + --- 8 The rules that have been proved to hold good for all numbers may be applied when the numbers are represented by letters. SECTION VI. DECIMAL FRACTIONS. 126. We have seen (112) (121) the necessity of reducing fractions to a common denominator, in order to compare their magnitudes. We have seen also how much more readily operations are performed upon fractions which have the same, than upon those which have different, denominators. On this account it has long been customary, in all those parts of mathematics where fractions are often required, to use none but such as either have, or can be easily reduced to others having, the same denominators. Now, of all numbers, those which can be most easily managed are such as 10, 100, 1000, &c., where 1 is followed by ciphers. These are called DECIMAL NUMBERS; and a fraction whose denominator is any one of them, is called a DECIMAL FRACTION, or more commonly, a DECIMAL. 127. A whole number may be reduced to a decimal fraction, or one decimal fraction to another, with the greatest ease. For example, 940 9400 94000 94 is ---, or ----, or ----- (106); 10 100 1000 3 30 300 3000 ---- is ----, or ----, or ----- (108). 30 100 1000 10000 The placing of a cipher on the right hand of any number is the same thing as multiplying that number by 10 (57), and this may be done as often as we please in the numerator of a fraction, provided it be done as often in the denominator (108).