Here the dividend contains fewer decimal places than the divisor, by three; wherefore at least three decimal ciphers must be annexed to the dividend. Attending to this, and proceeding as in whole numbers, it will be, •1523)152.3000(1000 1523 000 Now, as in this example the number of decimal places in the dividend has no excess over the number of decimals in the divisor, there will be no decimals in the quotient; hence the divisor is contained in the dividend 1000 times without a remainder, which is easily verified by multiplication. For further illustration, let 6285 be given to be divided by 2.7. The operation will appear thus : 2.7) 6285(-232 quotient. 54 88 81 75 54 21 remainder. In this example the decimal places in the dividend are three more than in the divisor, and consequently three places are pointed off in the quotient. The point being thus settled, the quotient may be carried to any number of places by conceiving ciphers annexed to the dividend. One more example will suffice to set the Rule in the clearest light; and for this purpose let 06 be given to be divided by 94.32. Here it becomes necessary to annex ciphers to the dividend to extend it in order to subtract the figures that must unavoidably be placed under it. Annexing, therefore, five or six ciphers at pleasure, and proceeding as already directed, it will be, 94.32) 06000000(000636 quotient. 56592 34080 28296 57840 56592 1248 remainder. In this dividend there is an excess of six decimals over the number of decimals in the divisor; but in the quotient there are only three significant figures. Now as the number of decimals in the quotient must equal the excess of the decimals in the dividend over those in the divisor, it follows that three ciphers prefixed must make up the deficiency; and thus the quotient becomes 000636, which may be extended (as there is a remainder) to a degree of approximation still nearer, by imagining more ciphers in the dividend, and continuing the division. REDUCTION OF VULGAR TO DECIMAL FRACTIONS. Any vulgar fraction may be reduced to a decimal, by dividing the numerator by the denominator. In division of decimals, it is plain, that any number, however small, may be divided by any other number, however great, if a sufficient quantity of decimal ciphers be annexed to the dividend in many instances, however, the quotient will not be finite, shewing that no decimal fraction whatever can equal exactly the vulgar fraction to be reduced. In some cases the quotient will consist of the same figure in endless succession, and in others, several figures in regular order will arise and circulate for ever. When the quotient, in reducing a vulgar to a decimal fraction, is likely to consist of several places, five or six decimals will in general be found to be accurate enough for common purposes, and where great precision is not requisite, three or four (or even fewer) will suffice. When a vulgar fraction is given to be reduced to a decimal, annex (as already directed) a competent number of decimal ciphers to the numerator, and then divide by the denominator; pointing off the quotient as above instructed in division. The quotient will be the decimal fraction required;-if finite, precisely equal to the vulgar fraction—if not, nearly so. To render these observations more intelligible, and to aid the learner in putting them in practice, we shall subjoin a few examples. And first, let it be proposed to reduce to a decimal fraction. Setting 3, the numerator, with decimal ciphers annexed to it, for a dividend, and then dividing by 4, the denominator, it will be, 4)3:00(-75 decimal required. 28 20 20 Hence it is manifest that of a unit are equivalent to the decimal 75, that is, to 100. 75 Next, let it be required to find the decimal equal to , and the operation will be as follows: 25)1.00(04 decimal sought. 1 00 Again, let a decimal fraction equal to be demanded, and the work will be as under: 8)7.000(-875 decimal required. 64 60 56 40 40 Whereby it is seen that the finite decimal 875 is the equivalent of. If were given to be reduced to a decimal, the quotient would be infinite, as may easily be made apparent; for treating this fraction according to the Rule, we obtain the figure 6 in endless succession by the process following: 3)2·0000000(·6666666 quotient. 18 20 18 20 18 20 18 20 18 20 18 20 18 2 remainder. So that are equivalent to 6 tenths plus 6 hundredths plus 6 thousandth-parts plus &c. to infinity. A decimal of this description is termed a pure repeater, and is generally expressed by the first figure written with a dot over it. A pure repeating decimal is reduced to a vulgar fraction by expunging the point, and setting a 9 under each repeater; therefore the above decimal is equal to 5 or 6, manifestly equivalent to, the fraction from which it was obtained. 1 If it were expedient to convert into a decimal fraction, the result would be ⚫002083, which not being terminate, is only an approximation to the value of the given vulgar fraction, and obtained as follows: |
