The number of zeros required may be at once annexed to the numerator of the fraction. Thus, if the approximation is required to tenths, 1 zero must be annexed to the numerator; if to hundredths, 2 zeros; if to thousandths, 3 zeros . . . Since the integral and fractional parts of the quotient are both obtained by the common process of division, the whole calculation may be made continuously, thus, 270)34703-00000(128·52962+ 270 770 540 2303 2160 1430 1350 800 540 2600 2430 1700 1626 800 540 260 When the dividend is less than the divisor, the quotient contains no integral part, and the process of division is resolved into that of reducing a vulgar to a decimal fraction. Let it be required to divide 0037 by 042, and to approximate to the exact quotient to within the part of unity. 0037 and 042, brought to the same denominator, are 0037 and '0420; and taken as whole numbers they are 37 and 420. Since the approximation is to be made to within the 1000000th part of unity, six zeros must be annexed to the numerator, which becomes 37.000000; dividing this number by 420, the quotient is 88095+. Six zeros having been annexed to the numerator of the fraction, the equivalent decimal fraction must contain 6 decimal figures; one, namely, for each zero; to complete this number a zero must be written to the left of the significant figures of the quotient, which, consequently, is '088095+. If the reduction of 37 to a decimal is made as in the case of the 1st Example of this Article, it is found that the place of tenths must be filled by a zero, for 37 420 370 of 1 simple unit = and of 1 tenth = 3700 of 1 hundredth = 8- hundredths, &c. &c. 242. If any decimal, expressed as a vulgar fraction, is to be divided by 10, the division can be effected by multiplying the denominator of the dividend by 10 (Art. 149); the denominator of the quotient, therefore, contains one zero more than that of the dividend, and the quotient, expressed decimally, one decimal figure more than the dividend. Similarly, the divisor being 100, the quotient contains two decimal figures more than the dividend; being 1000, it contains three decimal figures more than the dividend. . . . Whence, to divide a decimal by any power of 10, remove the decimal point of the dividend one figure to the left for each zero contained in the divisor; the result is the quotient required. Example 1. Divide 8497-63 by 100. The divisor containing two zeros, the quotient is obtained by removing the decimal point of the dividend two figures to the left; the quotient, therefore, is 84.9763. Example 2. Divide 320-48 by 100000. The divisor contains five zeros; to obtain the quotient, the decimal point of the dividend must, therefore, be removed five places to the left. As the dividend has only three figures on the left of the decimal point, this cannot be accomplished but by writing two zeros to the left of the number; the quotient obtained in this manner is '0032048. Recurring to the principles on which this rule depends, 100000-100000 1 0032048, the same result as that 243. Rule for the division of decimal fractions : Reduce the dividend and divisor to the same denominator, and divide as in whole numbers; if the division is exact the quotient is a whole number; if the division is not exact, annex to the remainder (or to the dividend) one zero for each decimal place required in the quotient; continue the division till these zeros are exhausted, and point off from the quotient one decimal figure for each zero annexed to the remainder or to the dividend. If the number of zeros annexed to the dividend is greater than the number of figures contained in the quotient, prefix to the significant figures of the quotient zeros equal in number to the difference. To divide a decimal by any power of 10, remove the decimal point of the dividend one figure to the left for each zero contained in the divisor. 244. Exercises in the division of decimal fractions: 6th. 7th. 8th. 9th. 28475÷375 ?.... 15th. 414182 ?........ Ans. 0.35665. 23d. 1000÷001 ?.. 24th. 37.9525÷25000?........ 26th. 015÷75·2 ?.... 27th. 30÷009?...... 28th. 009÷30?. 00110000?. 52d. 53d. 54th. 11?.... 55th. 11?. 56th. 0008136÷67·8?......... 57th. 81-37÷1000?........ 58th. 006453÷6453 ?........... 59th. 128÷37·436?.......... 60th. 004400?........ Ans. 1700.. Ans. 17. ...Ans. 2100. .......Ans. 21.. ...Ans. 0.0021. Ans. 0.0068622. ..Ans. 0·03020408. ...Ans. 0·000033204. Ans. 14400. Ans. 0·13096. .Ans. 7·635. ..Ans. 5425000. ......Ans. 5425. ...........Ans. 0·005425. ...Ans. 0·0002006668. .......Ans. 0·01247. .....Ans. 12470.. .Ans. 0·0000001. Ans. 0-0000000125. Ans. 1. ....Ans. 10. .......Ans. 0.1. .Ans. 0·000012. ......Ans. 0·08137. Ans. 0·000001. ......Ans. 3·41914. ...Ans. 0·00001. 245. If, in division of decimals, the dividend and divisor are made respectively the numerator and denominator of a vulgar fraction (which may be either proper or improper), and this fraction is decomposed into prime factors, and reduced to the lowest terms; 1st. When the denominator, after reduction, is equal to 1, the quotient is a whole number. 2d. When the only prime factors contained in the denominator are 2 and 5, or either of them, the quotient is composed of a limited number of decimal figures. The first is manifest; with respect to the second, if the prime factors 2 and 5 are raised to the same power, since 2×5=10, 22 × 52=100=102, 23 × 53=1000=103, &c. &c., the denominator is a power of 10, marked by the exponent common to the factors 2 and 5. The expression is consequently a decimal fraction, in which it is only necessary to replace the denominator by the decimal point. If 2 is the only prime factor contained in the denominator, let the terms of the fraction be multiplied by the corresponding power of 5; if 5 is the only prime factor contained in the denominator, let the terms be multiplied by the corresponding power of 2; and if both 2 and 5 are contained in the denominator, but the one is raised to a higher power than the other, let the terms of the fraction be multiplied by that power of 2 or of 5 (as the case may require) which makes the exponents of both prime factors the same. Then, as before, the denominator is a power of 10, and the expression a decimal fraction. The exponent of 2 or 5, when the denominator contains but one of these factors, or both, raised to the same power, and the exponent of the higher, when they are raised to different powers, indicates the number of decimal figures contained in the exact quotient. 3d. When the denominator contains other prime factors than 2 and 5, the quotient is composed of an unlimited number of decimal figures. For the numerator and denominator being, by hypothesis, prime to each other, the former can only be rendered a multiple of the latter by multiplying it by a number which involves all the prime factors contained in the latter (Art. 111 and 115). But the number by which the numerator is multiplied, namely, 10 or some power of 10 (which multiplication is effected by annexing zeros), contains only the prime factors 2 and 5; the division of the product by the denominator must, consequently, end in a remainder. There is no limit to the number of zeros which may be annexed to the numerator, and therefore none to the number of quotient figures, that is, of decimals. 246. When a quotient is not expressible by a limited number of decimal figures, it is partly or altogether composed of periods in which the same figures continually recur in the same order. The numerator being represented by n and the denominator by d, all the remainders which can occur in dividing n× 10, nx 100, n× 1000... by d, are contained in the series 1, 2, 3....d-1. When, therefore, at the utmost d-1 partial divisions have been made, one of the remainders which has been already obtained must be again found. Annexing 0 to this remainder, the result is a partial dividend equal to one of the preceding partial dividends. The quotient and the remainder resulting from the division of this dividend by d are equal to the quotient, and the remainder given by the equal partial dividend at the beginning of the first period. Annexing 0 to the remainder, and dividing by d, a second figure of a second period, corresponding to the second figure of the first period, is found, and similarly a third, &c. (if the period is composed of several figures), till all the figures of the first period are reproduced in their proper order. This process may, it is evident, be continued indefinitely. Such decimal fractions are termed periodic. a. When the last figure of the numerator is significant, the remainders, the partial dividends, and the partial quotients, obtained before the significant figures are exhausted, cannot be equal to the corresponding quantities obtained after the exhaustion of the significant figures. Consequently, the period cannot in such a case begin before the decimal point. b. If the 1, 2, 3 .... last figures of the numerator are zeros, the period may commence 1, 2, 3 .... figures before the decimal point. c. If the denominator contains either or both of the factors, 2, 5, the period cannot commence immediately after the decimal point. For the partial dividend, which gives the first decimal figure of the quotient, is a multiple of 10, being made so by the zero annexed to the preceding remainder. This dividend can, therefore, be divided by 2, 5, 2×5; but such division takes away the zero at the end of the partial dividend, and consequently also the condition by means of which the quotient figure resulting from the division of the reduced partial dividend by the product of the remaining prime factors of the denominator can be the first figure of a period. For the same reason, if the denominator contains the factors 22 or 52, the period cannot commence till after the second decimal figure; if 23 or 53, not till after the third decimal figure.... d. When a period begins at the decimal point the period is termed simple, and the quotient a simple periodic decimal number; when before or after the decimal point, the period is said to be mixed, and the quotient is termed a mixed periodic decimal number. 247. To divide one decimal number by another, and to reduce a vulgar fraction to a number expressed decimally, being in effect the same operation, either form of expression may be employed to describe it. In the following examples, intended to illustrate the observations contained in Articles 245 and 246, the second is used. = 7 239 1st Ex. Let it be required to reduce the vulgar fraction to a decimal: Resolving the terms into simple factors, 2, the fraction containing no common factor is in the lowest terms, and since the denominator is 2 raised to the 3d power, the quotient contains 3 decimal figures; 7x53 875 for= 2d Ex. Let it be required to reduce the vulgar fraction to a decimal: 19 19 Therefore the fraction is in the lowest terms, and the equivalent decimal contains 2 decimal figures; 25 52. for 19 19 x 22 76 100-76. 3d Ex. Let it be required to reduce the vulgar fraction to a decimal : 19 or 80 24x5 57 3 x 19 240 24x3x5 19 is the fraction expressed in the lowest terms; and since the factors of the denominator, 2, 5, are raised to different powers, and the exponent of that which is raised to the higher power is 4, the decimal must contain 4 decimal figures; 3d Ex. Let it be required to reduce the vulgar fraction to a decimal : The terms of the fraction are prime numbers; and since the denominator, which is greater than the numerator, contains a prime factor different from 2 and 5, the result is composed entirely of decimal figures and periodic. 37)13.000000(0·351351+ 190 130 190 50 + In this example the period which begins immediately after the decimal point is composed of three figures. 4th Ex. Let it be required to reduce the vulgar fraction to a decimal: 1300 13 x 102 1300 The numerator and denominator containing no common factor, the fraction is expressed in the lowest terms, the decimal is periodic, and the first period may begin before the decimal point. |