divide; to the remainder annex another cipher, and again divide by the denominator; and so continue to do until there is no remainder, or until as many decimal figures have been obtained as may be desired. The quotient will be the decimal fraction required. NOTE. It will be seen that this rule bears a close analogy to rule under ART. 91, as it ought; since the values of the successive figures in a decimal fraction decrease in a tenfold ratio. EXAMPLES. 1. What decimal fraction is equivalent to? 16)100(0.0625 40 32 80 80 0 2. Wha decimal is equivalent to ? Ans. 0·05555, &c. 3. What decimal is equivalent to? 4. What decimal is equivalent to? 5. What decimal is equivalent to ? Ans. 0.05. Ans. 0·04. Ans. 0.3333, &c. Ans. 0.142857, &c. 7. What decimal is equivalent to ? Ans. 0.0909, &c. 8. What decimal is equivalent to? Ans. 0·076923, &c. 9. What decimal is equivalent to ? Ans. 0.0588235, &c. 10. Change into an equivalent decimal. Ans: 0.75. into an equivalent decimal. 11. Change 16. Change into an equivalent decimal. Ans. 0.875. into an equivalent decimal. Ans. 0·95. into an equivalent decimal. Ans. 0.98. into an equivalent decimal. Ans. 0.928571428, &c. In the foregoing process of converting a vulgar fraction into an equivalent decimal fraction, we continue to annex ciphers to the remainders, and to divide by the denominator of the vulgar fraction; hence, whenever we obtain a remainder like one that has previously occurred, then the decimal figures will commence a repetition. And as no remainder can exceed or equal the divisor or denominator of the vulgar fraction, the whole number of different remainders cannot exceed the number of units in the denominator less one; consequently, when the decimal figures do not terminate, they must fecur in periods whose number of places cannot exceed the number of units less one in the denominator of the equivalent vulgar fraction. Decimals which recur in this way, are called repetends. When the period begins with the first decimal figure, i is called a simple repetend. But when other decimal figures occur before the period commences, it is called a compound repetend. A repetend is distinguished from ordinary decimals by a period or dot placed over the first and last figure of the circulating period. 96. The following vulgar fractions give simple repetends: 98. Those simple repetends, which have as many terms, less one, as there are units in the denominator, we shall call perfect repetends. The following are some of the perfect repetends: +=0·142857. 0.0588235294117647. 0.052631578947368421. 00434782608695652173913. =0.034482758620689655172413793i. NOTE. For some interesting properties of repetends, see Higher Arithmetic. REDUCTION OF DENOMINATE DECIMALS. 99. A denominate decimal is a decimal fraction of a unit of a particular kind. Thus, 0·45 of a £, is a denominate decimal, since the unit is £1; for the same reason, 0.25 of a foot is a denominate decimal, the unit being 1 foot. What is a denominate decimal? Give some examples. CASE I. To reduce denominate numbers of different denominations to a decimal of a given denomination. Let it be required to reduce 15s. 6d. 3far. to the decimal of a £. I. 3far. d. 0·75d. II. 6d. 3far. is therefore the same as 6-75d.; if we divide this by 12, it will become 6.75 =0.5625s. 12 III. 15s. 6d. 3far.-15.5625s.; this divided by 20, gives for the decimal sought. The work may be more concisely done, as in the following OPERATION. 413far. 126-75d. 20 15.5625s. 0.778125 of a £. EXPLANATION We placed the different denominations above each other, so that the smallest denomination stood at the top; we then supposed ciphers annexed to the 3 farthings, and divided by 4, since 4 farthings make one penny, and the quotient, which must be a decimal. we placed at the right of the 6d.; we next divided 6·75d. with ciphers annexed, by 12, because 12 pence make one shilling, and the quotient, which is also a decimal, we placed at the right of the 15s. ; finally, we divided the 15.5625s. by 20, because 20 shillings make one pound. In dividing by 20, we cut off the cipher, and then divided by 2, observing to remove the decimal point one place to the left. We therefore have this RULE. Place the different denominations above each other, so that the lowest denomination may stund at the top; commencing at |