other (Art. 219), and the annexing of one or more ciphers to the numerator makes the same a multiple of 10, but does not render it divisible by any factor, except 2 and 5, the factors of 10. Therefore, when the denominator of a fraction, in its lowest terms, contains other factors than those of 10, the decimal resulting from dividing the numerator with ciphers annexed, will not terminate, but will contain one or more figures that constantly repeat. 293. A pure repetend is always equivalent to a common fraction whose numerator is the repeating figure or figures, and whose denominator as many places of nines as there are repeating figures. For, by reducing to a decimal, we obtain as its equivalent the repetend .i ; and since .i is equivalent to š, 2 will be equivalent to g,.3 to g, and so on, till .9 is equal to or 1. Again, y's, and ggg, being reduced, give .bi, and voi ; that is, g = .0i, and ggg .001; therefore, .02, and go 002, and so on ; the same principle holding true in all like cases. 294. A mixed repetend is equivalent to a complex decimal (Art. 278), or to a complex fraction. Thus, the mixed repetend .2412 is equivalent to the mixed decimal .247%, which is 243 equal to the complex fraction 100 295. Repetends are of the same denomination only when they are similar and conterminous. For then alone, by having a common denominator, do they express fractional parts of the same unit. REDUCTION OF REPETENDS. 296. To reduce a repetend to an equivalent common fraction. Ex. 1. Reduce .123 to an equivalent common fraction. Ans. 343. We write the figures of the given .i23 13 Ans. repetend with the decimal point omitted for the numerator, and as many nines as places in the repetend for the denominator, of a common fraction (Art. 293), and obtain 1%, which, reduced to its lowest terms, , the answer required. OPERATION. 99 2. Reduce .138 to an equivalent common fraction. Ans. 36 OPERATION. 1 2 5 100: The mixed repetend .138 138 is equivalent to the mixed .1 3 5 6 Ans. decimal .13% (Art. 294), 100 which we readily change to the form of a complex fraction by erasing the decimal point and writing the denominator, 100, 13 which is understood; and thus obtain which, reduced to its simplest form, gives go, the answer required. Hence, If the given repetend be SIMPLE, make the repeating figure or figures the numerator, and take as many nines as the repetend has figures for the denominator. If the given repetend be MIXED, change it to an equivalent complex fraction, and that fraction to its simplest form. NOTE. — Any circulating decimal may be transformed into another decimal, having a repetend of the same number of figures; as, .78 = .787, and .534 .5345 =.53453. Thus, when such expressions as 12.5 or 17.56 occur, they may be also transformed; as 12 5 12.5ż, and 17.56 = 17.567 = 17.5676, &c. EXAMPLES. 3. Required the common fraction equal to ... Ans. & 4. Reduce 1.62 to its equivalent mixed number. Ans. 133. 5. Change .53 to an equivalent common fraction. 6. What common fraction is equivalent to .769230 ? Ans. 18. 7. What common fraction is equivalent to .5925 ? 8. Change 31.62 to an equivalent mixed number. Ans. 313. 9. Reduce .008497133 to an equivalent common fraction. Ans. 96 297. To determine the kind of decimal to which a given common fraction can be reduced. Ex. 1. Required to find whether the decimal equal to 1 be finite or circulating; and if finite, of how many places the decimal will consist. Ans. Finite, of 4 places. OPERATION. We reduce 3 the given frac% = 16 .18 7 5. tion to its lowest 2 X 2 X 2 X 2 terms, and then resolve the denominator, 16, of the fraction obtained, s, into its prime factors, which we find to be 2 X 2 X 2 X 2. Now, since the denominator contains no prime factor other than 2 or 5, it is evident that, by annexing ciphers to the numerator, 3, and dividing by the denominator, 16, the decimal arising will terminate, and thus be finite (Art. 292) Since, in reducing a common fraction to its equivalent decimal, we annex ciphers to the numerator and divide by the denominator (Art. 278), every 10, or 2 and 5, that enter into the denominator as factors must produce one decimal place, and no more, and therefore every other factor 2 or 5 must give one, and only one, decimal place. The denominator, 16, contains only the factor 2 taken 4 times, or 24 ; and the exponent of the 2 indicates that the decimal equivalent to must contain exactly 4 decimal places, which we verify by reducing the id to its equivalent decimal, .1875. 2. Find whether the decimal equal to 1359 be finite or circulating ; and if circulating, of how many places the finite part, if any, and the circulating part, will each consist. Ans. Circulating: the finite part, 2 places; the repetend, 6 places. We reduce 47 the given 138=245 .1 3 4 2 8 5 7 i. fraction to 2 X 5 X 5 X 7 its lowest terms, and obtain 40. The denominator, 350, 2 X 5 X 5 X 7, contains a prime factor, 7, other than 2 and 5; therefore the decimal equivalent to will contain a repetend; and as, of the factors 2 and 5, the higher exponent of either, that of 5, is 2, the decimal will have 2 finite places before the repetend commences. This we verify by reducing 34% to its equivalent decimal, .1342857i. Hence, to determine whether the decimal to which a given common fraction can be reduced is finite or circulating, and the number of finite decimal places, if any, Having reduced the given common fraction to its lowest terms, resolve the denominator into its prime factors. If these factors be not other than 2 or 5, the decimal will be FINITE; if other prime factors occur with 2 or 5, the decimal will be a MIXED REPETEND; and if neither 2 nor 5 occurs as factor, the decimal will be a PURE REPETEND. Whichever factor 2 or 5 occurs in the denominator with the higher exponent will by its exponent denote the number of finite decimal places. OPERATION. NOTE. — The number of figures of which a repetend will consist may be discovered by dividing 1 with ciphers annexed by the factors other than 2 or 5 of the denominator, until there is a remainder 1. Thus, if it be required to discover the number of figures in the repeating part of the decimal equivalent to 397, we divide 1 with ciphers annexed by 7, the only prime factor in the denominator other than 2 or 5, until there is a remainder of 1, which occurs after the sixth division, thereby indicating that the repeating part will consist of six figures. We have seen that these must be preceded by two places of finite decimals, so that the mixed repetend equal to 356 must consist of eight places in all. EXAMPLES. 3. To what kind of a decimal can it be reduced ? Ans. A pure repetend, of 2 places. 4. How many places of decimals, finite and repeating, will be required to express ? Ans. 5 places ; 3 finite and 2 repeating. 5. To what kind of a decimal can } be reduced ? 6. Reduce 1317 to a mixed repetend. Ans. 13.37. 7. Change 13936 to a mixed repetend. Ans. .008497133. 8. Of how many figures will the repetend consist that corresponds to ? Ans. 28 figures. TRANSFORMATION OF REPETENDS. 298. Any finite decimal may be considered as a mixed repetend by making ciphers continually recur; thus, .42 = .420 .4200 = .42000, &c. 299. Any circulating decimal may be transformed into another having the same number of repeating figures; thus, .127 = .1272 = .12727, &c. 300. Any circulating decimal having as repetend any number of figures may be transformed to another having twice or thrice that number of figures, or any multiple thereof; thus, .5925, having a repetend of three figures, may be transformed to one having 6, 9, 12, &c. places; therefore .5925 = .5925925 5925925925 = 5925925925925925, &c. 301. The value of a decimal is not changed by any of the above transformations, as may be seen by reducing the given repetends to their equivalent common fractions (Art. 296) and comparing them together. Hence, they can be used in making dissimilar repetends similar and conterminous. 302. To make any number of dissimilar repetends similar and conterminous. Ex. 1. Make similar and conterminous 9.167, 14.6, 3.165, 12.132, 8.187, and 1.307. OPERATION. Dissimilar. Similar Similar and Conterminous. 9.1 6 7 9.1 6 7 6 9.1 6 7 6 7 6 7 6 1 4.6 0 0 0 0 0 0 0 3.1 6 5 5 5 5 5 5 1 2.4 3 2 1 2.4 3 2 4 3 1 2.4 3 2 4 3 2 4 3 Ans 8.1 8 i 8.1 8 is 8.1 8 1 81818 1.3 0 7 1.3 0 1 3 0 1.3 0 7 3 0 7 30 We make the finite mixed decimal, 14.6, a mixed repetend by annexing recurring ciphers, and make it and all the given repetends similar, by extending the figures to the right, so that the circulating part of each may begin at the same distance from the decimal point as does that repetend which is preceded by the most finite decimal places. Then, to make conterminous the repetends that have thus been rendered similar, as some of them consist of 1, some of 2, and the others of 3 places, we extend the repeating figures of each repetend till those of each occupy as many places as there are units in the least common multiple of 1, 2, and 3, which is 6. Hence, to make dissimilar repetends similar and conterminous, Transform the given repetends so that the circulating parts shall commence at the same distance from the decimal point, and shall consist of as many circulating places as there are units in the least common multiple of the number of repeating figures found in the given decimals. EXAMPLES. 2. Make 3.67i, 1.007i, 8.52, and 7.616325 similar and conterminous. 3. Make 1.52, 8.7156, 3.567, and 1.378 similar and conterminous. 4. Make .0007,.141414, and 887.i similar and conterminous. 5. Make .3i23, 3.27, and 5.02 similar and conterminous. 6. Make 17.0884, 1563.0929, and 15.12345 similar and conterminous. |