SECTION X. PERIODICAL OR CIRCULATING DECIMALS.* ART. 349. Decimals which consist of the same figures or set of figures repeated, are called PERIODICAL, OR CIRCULATING DECIMALS. (Art. 339.) 350. The repeating figures are called periods, or repetends. If one figure only repeats, it is called a single period, or repetend; as .11111, &c.; .33333, &c. When the same set of figures recurs at equal intervals, it is called a compound period, or repetend; as .01010101, &c.; .123123123, &c. 351. If other figures arise before the period commences, the decimal is said to be a mixed periodical; all others are called pure, or simple periodicals. Thus .42631631, &c., is a mixed periodical; and .33333, &c., is a pure periodical decimal. OBS. 1. When a pure circulating decimal contains as many figures as there are units in the denominator less one, it is sometimes called a perfect period, or repetend. (Art. 344.) Thus, .142857, &c., and is a perfect period. 2. The decimal figures which precede the period, are often called the terminate part of the fraction. 352. Circulating decimals are expressed by writing the period once with a dot over its first and last figure when compound; and when single by writing the repeating figure only once with a dot over it. Thus .46135135, &c., is written .46135 and .33, &c., .3. 353. Similar periods are such as begin at the same place before or after the decimal point; as .i and .3, or 2.34 and 3.76, &c. Dissimilar periods are such as begin at different places; as 123 and .42325. Similar and conterminous periods are such as begin and end in the same places; as .2321 and 1634. Should Periodical Decimals be deemed too intricate for younger classes, they can be omitted till review. REDUCTION OF CIRCULATING DECIMALS. CASE I.-To reduce pure circulating decimals to common fractions. 354. To investigate this problem, let us recur to the origin of circulating decimals, or the manner of obtaining them. For example,=.11111, &c., or .i; therefore the true value of .11111, &c., or .i, must be from which it arose. For the same reason .22222, &c., or .2, must be twice as much or ; (Art. 186;) .33333, &c., or .33; .44; .5=4, &c. Again,=.010101, &c., or .01; consequently .010101, &c., or .01; .020202, &c., or .02; .030303, &c., or .034; .070707, &c., or .07=4, &c. So also.001001001, &c., or .001; therefore .001001, &c., or .001; .002; &c. In like manner =.142857; (Art. 337;) and 142857142857; for, multiplying the numerator and denominator of by 142857, we have. (Art. 191.) So is twice as much as;, three times as much, &c. Thus it will be seen that the value of a pure periodical decimal is expressed by the common fraction whose numerator is the given period, and whose denominator is as many 9s as there are figures in the period. Hence, 355. To reduce a pure circulating decimal to a common fraction. Make the given period the numerator, and the denominator will be as many 9s as there are figures in the period. Ex. 1. Reduce .3 to a common fraction. Ans. &, or . CASE II. To reduce mixed circulating decimals to common fractions. 356. 11. Reduce .16 to a common fraction. Analysis. Separating the mixed decimal into its terminate and periodical part, we have .16 .1+.06. (Art. 320.) Now .1=; (Art. 312;) and .06; for, the pure period .6, (Art. 351,) and since the mixed period .06, begins in hundredths' place, its value is evidently only as much; but ÷10=. (Art. 227.) Therefore .16+%. Now and, reduced to a common denominator and added together, make, or . Ans. OBS. In mixed circulating decimals, if the period begins in hundredths' place it is evident from the preceding analysis that the value of the periodical part is only as much as it would be, if the period were pure or begun in tenths' place; when the period begins in thousandths' place, its value is only as much, &c. Thus .6=g; .06=§÷10-£; .006—§÷100=5&T, &c. part 357. Hence, the denominator of the periodical part of a mixed circulating decimal, is always as many 9s as there are figures in the period with as many ciphers annexed as there are decimals in the terminate part. 12. Reduce .8567923 to a common fraction. Solution. Reasoning as before .8567923=15+5933336. Reducing these two fractions to the least common denominator, (Art. 261.) 9999999915 whose denominator is the same as that of the other. Now $49315+, 978336=3597338. Ans.. Contraction. 8500000 85 8499915 1st Nu. 67923 2d Nu. 8567838 9999900 Ans. 8499915. -8 9999900 9999900 9999900. To multiply by 99999, annex as many ciphers to the multiplicand as there are 9s in the multiplier, &c. (Art. 105.) This gives the numerator of the first fraction or terminate part, to which add the numerator of the second or periodical part, and the sum will be the numerator of the The denominator is the same as answer. that of the second or periodical part. Second Method. 8567923 the given circulating decimal. 85 the terminate part which is subtracted. 8567838 the numerator of the answer. 8567838 Ans. 9999900 Note.-1. The reason of this operation may be shown thus: 8567923= 850000067923. Now 8500000-85+-67923 is equal to 8567923—85. 2. It is evident that the required denominator is the same as that of the periodical part; (Art. 357;) for, the denominator of the periodical part is the least common multiple of the two denominators. Hence, 358. To reduce a mixed circulating decimal to a common fraction. Change both the terminate and periodical part to common fractions separately, and their sum will be the answer required. Or, from the given mixed periodical, subtract the terminate part, and the remainder will be the numerator required. The denominator is always as many 9s as there are figures in the period with as many ciphers annexed as there are decimals in the terminate part. Ans. 15, or 5. PROOF.-Change the common fraction back to a decimal, and if the result is the same as the given circulating decimal, the work is right. 13. Reduce .138 to a common fraction. 14. Reduce .53 to a common fraction. 15. Reduce .5925 to a common fraction. 16. Reduce .583 to a common fraction. 17. Reduce .0227 to a common fraction. 18. Reduce .4745 to a common fraction. 19. Reduce .5925 to a common fraction. CASE III.-Dissimilar periodicals reduced to similar and conter minous ones. 359. In changing dissimilar periods, or repetends, to simila and conterminous ones, the following particulars require attention. 1. Any terminate decimal may be considered as interminato by annexing ciphers continually to the numerator. Thus .46.460000, &c.=.460. 2. Any pure periodical may be considered as mixed, by taking the given period for the terminate part, and making the given period the interminate part. Thus .46 .46+.0046, &c. 3. A single period may be regarded as a compound periodical. Thus .3 may become .33, or .333; so .63 may be made .6333, or .63333, &c. 4. A single period may also be made to begin at a lower order, regarding its higher orders as terminate decimals. Thus .3 may be made .33, or .3333, &c. 5. Compound periods may also be made to begin at a lower order. Thus .36 may be changed to .363, or .36363, &c.; or by extending the number of places .479 may be made .47979, or 4797979, &c.; or making both changes at once .532 may be changed to .5325325, &c. Hence, 360. To make any number of dissimilar periodical decimals similar. Move the points, so that each period shall begin at the same order as the period which has the most figures in its terminate part. 21. Change 6.814, 3.26, and .083 to similar`and conterminous periods. Operation. 6.814 6.81481481 3.26 3.26262626 .083=0.08333333 Having made the given periods similar, the next step is to make them conterminous. Now as one of the given periods contains 3 figures, another 2, and the other 1, it is evident the new periodical must contain a number of figures which is some multiple of the number of figures in the different periods; viz: 3, 2, and 1. But the least common multiple of 3, 2, and 1 is 6; therefore the new periods must at least contain 6 figures. Hence, 361. To make any number of dissimilar periodical decimals, similar and conterminous. First make the periods similar; (Art. 360;) then extend the figures of each to as many places, as there are units in the least common multiple of the NUMBER of periodical figures contained in each of the given decimals. (Art. 176.) |