ARITHMETIC 531 20. To multiply fractions, multiply the numerators together for the numerator of the product, and the de nominators together for the denominator. Thus, in multi plying | by I, if we multiply f by 7, we have 2 ( 13); but our multiplier, ^. is the ninth part of 7 ; we must therefore divide ^- by 9, which gives -ff- ( 13). When there are several fractions to be multiplied con tinuously together, we proceed in the same way. Mixed numbers are reduced to improper fractions, and common factors may be struck out, precisely as in 16. "When an integer has to be multiplied by a fraction, we may convert the integer into a fraction by putting 1 as the denominator; or we may multiply the integer by the numerator of the fraction and divide by the denominator, since to multiply by is, as has just been shown, to multiply by 7 and divide by 9. So, to multiply an integer by a mixed number, the common method is to multiply by the integer and the fraction of the latter separately, and add the results. In multiplying mixed numbers like 46|- and 14^ together, instead of using improper fractions, we may take the four products 46x1 4, ^ x 14, 46 x ^, and -5- x |-, and add them. The amount is 644 + 3 + 15 + ^ = 662^. 21. To divide fractions, invert the terms of the divisor (i.e., interchange the positions of the numerator and deno minator), and multiply the dividend by the inverted fraction; thus, f - = f x = ||-. If we divide -| by 7, we have -/ v ( 13). But since our divisor, , is the ninth part of 7, in dividing by 7 we divide by a number nine times too large. The true quotient must therefore be nine times -/ v , i.e., f. The common method of dividing an integer by a mixed number is a modification of this division. When, in dividing by 37, for instance, we multiply both divisor and dividend by 5, and then divide, we really multiply the dividend by y|-g-. Or the method may be explained on the principle (identical with that of 12) that the multi plication of the divisor and dividend by the same number does not affect the quotient. As an instance of the divi sion of a mixed number by an integer, let 3982|-| be divided by 54. The quotient is 73, with remainder 40g-| ; and to obtain the complete quotient, this remainder must be divided by 54 (see 11), giving -^^^<-^f = j^g- ; i.e., 73^5- is the result of the division. A complex fraction is reduced to a simple one by divid ing the numerator by the denominator ; thus, _6rT_77 _9 _33 10J = 12 X 91 = 52 When one term only is fractional, it will be found conve nient to multiply both terms of the complex fraction by the denominator that occurs in the fractional term ; thus, 13| 13jx4 55 5 22 ~ 22 x 4 = 22 x 4 = 8 imals. Decimals. 22. In the ordinary denary notation, a figure in combination with others has only the tenth part of the value it would have if removed a place towards the left ( 1); thus, in 374, the 3 signifies 3 times 100; the 7, 7 times 10 ; the 4, simply 4. By an extension of this notation we obtain a species of fractions that are often of very great use, especially for purposes of comparison. If we mark the place of units by a point put after it, and write other figures after the point, we can denote by the first of these figures one-tenth of the value it would have in the units place ; by the second, one-hundredth part, and so on. In 374-691, then, the 6 is 6 times ^, i.e., . . the 9, 9 times ^ or ^ ; and the 1, T ^. Whence, by giving these fractions a common denominator and adding them, we have 374 691 = 374^^- These decimal frac tions or decimals, therefore, are fractions of which the numerator only is written, the denominator being the continued product of as many tens as there are decimal figures. In addition, subtraction, multiplication, and division of decimals, the operations are, and from the structure of decimals must be, the same as the like operations with integers. The position of the decimal point in the results is the only thing that needs particular explanation. Results, it may be mentioned here, are often expressed by decimals that are not exact, but approximate only, it being held sufficient to give the correct value to some assigned number of decimal places. An amount correct to four decimal places differs from the true amount by less than the ten-thousandth part of unity. See above, p. 524. To reduce a decimal to a vulgar fraction, write the deci mal as the numerator of the fraction, and set under it for the denominator 1 followed by as many ciphers as there are decimal places. This follows at once from the definition of a decimal. To reduce a vulgar fraction to a decimal, annex ciphers to both terms of the fraction the same number in both cases ; divide both terms by the significant figures of the denominator, and then write the numerator as a decimal, pointing off as many decimal places (prefixing ciphers, if necessary) as there are ciphers in the denominator. (See alr> X 9^ I nfrn Tlina 3 . . 300 . 75 _ -007 -"i aiso zo, injia.) inus, XTTO" ~~ TTTGITO~ ~ i o o o o ~ Here the vulgar fractions retain their values unchanged ( 12), and from the last the decimal is set down accord ing to the definition. It very often happens that in dividing as above we find there must always be a remainder. In this case, however, a remainder we had before may soon recur, giving the same figure or group of figures over and over again in the quotient; thus | is found to be 47222 ; -^ is 05729729 The recurring figures are distinguished by points placed over them, the above results being written 472 and 05729. We shall return to the consideration of these recurring decimals at 26. 23. To add or subtract decimals, write the numbers under each other, placing units under units, &c., add or subtract as with integers, and place the decimal point under the points in the given numbers. This follows directly from the definition of decimals. 24. To multiply decimals, multiply as with integers, and point off as many decimal places as there are in both factors taken together. Thus, in 37 64 x 082 = 3-OS648, the 4 and 2 in the factors are -^^ and ^ 2 ; their pro duct is therefore 100 8 000 , i.e., 00008 ; and so with the others. If an approximate product be sufficient, the multipli cation may be shortened thus. Let it be required to find the product, say, of 4 273 and 6 859 correct to three places of decimals. Write the figures of the multi plier in reverse order, with the unit figure under the third decimal place of the multipli cand. Begin each multiplication with the figure above the multiplier, adding what has to be earned from the right hand figure, and observing to carry the nearest ten (i.e., for 36, for instance, not 3, but 4) ; place the first figures of each multiplication under each other; then add and point off three decimal places. It will be seen that every figure of the product falls into its proper place, e.g., the 8 and 7, the 5 and 2, &c., each give thousandth parts. 25. To divide decimals, divide as with integers, and point off as many decimal places in the quotient as the dividend has more than the divisor. This follows from the dividend being the product of the divisor and quotient Exs. 228-956^3-64 = 62-9; 2-28956 ~ 36-4 = "0629 ; 22895-6^-364 = 62900. In the second exair.ple. the 4273 9586 25633 3418 214 38
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