139. The rules which were given in the last chapter for addition, subtraction, multiplication, and division, apply to all fractions, and therefore to decimal fractions among the rest. But the way of writing decimal fractions, which is explained in this chapter, makes the application of these rules more simple. We proceed to the different cases. Suppose it required to add 42·634, 45·2806, 2·001, and 54. By (112) these must be reduced to a common denominator, which is done (138) by writing them as follows: 42·6340, 45·2806, 2·0010, and 54·0000. These are decimal fractions, whose numerators are 426340, 452806, 20010, and 540000, and whose common denominator is 10000. By (112) their sum is 426340 + 452806 + 20010 + 540000 1439156 --------------------------------, which is ------- 10000 10000 or 143·9156. The simplest way of doing this is as follows: write the decimals down under one another, so that the decimal points may fall under one another, thus: 42·634 45·2806 2·001 54 -------- 143·9156 Add the different columns together as in common addition, and place the decimal point under the other decimal points. EXERCISES. What are 1527 + 64·732094 + 2·0013 + ·00001974; 2276·3 + ·107 + ·9 + 26·3172 + 56732·001; and 1·11 + 7·7 + ·0039 + ·00142 + ·8838? _Answer_, 1593·73341374, 59035·6252, 9·69912. 140. Suppose it required to subtract 91·07324 from 137·321. These fractions when reduced to a common denominator are 91·07324 and 137·32100 (138). Their difference is therefore 13732100 - 9107324 4624776 ------------------, which is ------- 100000 100000 or 46·24776. This may be most simply done as follows: write the less number under the greater, so that its decimal point may fall under that of the greater, thus: 137·321 91·07324 --------- 46·24776 Subtract the lower from the upper line, and wherever there is a figure in one line and not in the other, proceed as if there were a cipher in the vacant place. EXERCISES. What is 12362 - 274·22107 + ·5; 9976·2073942 - ·00143976728; and 1·2 + ·03 + ·004 - ·0005? _Answer_, 12088·27893, 9976·20595443272; and 1·2335. 141. The multiplication of a decimal by 10, 100, 1000, &c., is performed by merely moving the decimal point to the right. Suppose, for example, 13·2079 is to be multiplied by 100. The decimal is ¹³²⁰⁷⁹/₁₀₀₀₀, which multiplied by 100 is (117) ¹³²⁰⁷⁹/₁₀₀, or 1320·79. Again, 1·309 × 100000 is ¹³⁰⁹/₁₀₀₀ × 100000, or (116) ¹³⁰⁹⁰⁰⁰⁰⁰/₁₀₀₀ or 130900. From these and other instances we get the following rule: To multiply a decimal fraction by a decimal number (126), move the decimal point as many places to the right as there are ciphers in the decimal number. When this cannot be done, annex ciphers to the right of the decimal (137) until it can.