OPERATION. We reduce 3 the given frac 1%°? = Tv = o v « v a v o = -1875. tion to its lowest & A « A * A * terms, and then resolve the denominator, 16, of the fraction obtained, ^8j, 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 Jiore, 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 21; and the exponent of the 2 indicates that the decimal equivalent to J8j must contain exactly 4 decimal places, which we verify by reducing the -fg to its equivalent decimal, .1875. 2. Find whether the decimal equal to T2T35^, 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. Ofrration. We reduce 4 7 . the given Twty = Uvt = 9 v 5 Y i y 7 =-13428571. fraction to i X 0 X O A' its lowest terms, and obtain The denominator, 350, = 2x5x5X7, contains a prime factor, 7, other than 2 and 5; therefore the decimal equivalent to -ffif 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 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 olaces. 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 34^j, 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 ^ must consist of eight places in all. Examples. 3. To what kind of a decimal can T\ be reduced? Ans. A pure repetend, of 2 places. 4. How many places of decimals, finite and repeating, will be required to express TlfiT? Ans. 5 places; 3 finite and 2 repeating. 5. To what kind of a decimal can if, J be reduced? 6. Reduce 13}i to a mixed repetend. Ans. 13.37. 7. Change T£f fB 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, 12.432, 8.181, and 1.307. 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 3.671, 1.0071, 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 887.1 similar and conterminous. 5. Make .312.3, 3.27, and 5.02 similar and conterminous. 6. Make 17.0884, 1563.0929, and 15.i2345 similar and conterminous. ADDITION OF CIRCULATING DECIMALS. 303.° Ex. 1. Add 2.765, 7.16674, 3.671, .7, and .i728 together. Ans. 14.55436. Opcration. Having made the given Dissimilar. Similar and Conterminous. repetends similar and eon-" 2.7 6o = 2.7 6 5 6 5 tcrminous (Art. 302), we 7.1 6 0 7 4 = 7.1 6 6 7 4 add as in addition of whole numbers, and obtain 3.6 7 1 = 3.6 7 1 3 6 14.55433. The right-hand .7 = .7 7 7 7 7 figure of this result we in i 7 2 8 =- 1 7 2 8 i crease by such a number .-—. as would have been car Ans. 1 4.5 5 4 3 6 rfed, if the repetends had been continued farther to the right. In that case we should have had to carry 3 after finding the amount of the first left-hand column of the repetends continued. We therefore increase the sum as first found, and thus have the true amount as in the operation, 14.55436. Rule. — Make the given repetends, when dissimilar, similar and conterminous. Add as in addition of finite decimals, observing to increase the repetend of the amount by the number, if any, to be carried from the left-hand column of the repetends. Examples. 2. Add 3.5, 7.65i, 1.765, 6.173, 51.7, 3.7, 27.63i,and 1.003 together. Ans. 103.2591227. 3. Reduce ^, \, and £ to decimals, and find their sum. 4. Find the sum of 27.56, 5.632, 6.7, 16.356, .7i, and 6.1234. Ans. 63.1690670868888. 5. Add together .165002, 31.64, 1.06, .34634, and 13. 6. Add together .87, .8, and .870. Ans. 2.644553. 7. Required the value of .3 + .45 + .45 + .351 -(-.6468 ,f .0408 + .0408, and -0408. Ans. 4.1760345618. 8. Find the value of 1.25 -f- 3.4 -f- .637 + 7.885 -{- 7.875 f 7.875 + ll.i. Ans. 40.079360724. 9. Add together 131.613, 15.001, 07.134, and 1000.63. 10. Find the value of 5.i6345 -f- 8.6381 +' 3.75. Ans. 17.55919120847374090302. SUBTRACTION OF CIRCULATING DECIMALS. 304° Ex. 1. From 87.1645 take 19.479107. Ans. 67.685377. Operation. Having made the repe Dissimilar^ Similar and Conterminous. tends similar and COnter 8 7.1 6 4 5 = 8 7.1 6 4 5 4 5 ruinous, we subtract as in 1 9.47 9 1 67 = 1 9.47 9 1 67 whole numters, regarding however, the right-hand Ans. 6 7.6 8 5 3 7 7 %ure of the subtrahend as increased by 1, since 1 would have been carried to it in subtracting, if the repetends had been continued farther to the right, as is evident from the circulating part of the subtrahend being greater than that of the minuend. Rule. — Make the repetends, when dissimilar, similar and conterminous. Subtract as in subtraction of finite decimals; observing to regard the repetend of the subtrahend as increased by 1, when it exceeds that of the minuend. Examples. 2. From 7.1 take 5.02. Ans. 2.08. 3. From 315.87 take 78.0378. Ans. 237.838072095497. 4. Subtract 4 from §. Ans. .079365. 5. From 16.1347 take 11.0884. Ans. 5.0462. 6. From 18.1678 take 3.27. Ans. 14.8951. 7. From 3.123 take 0.7 i. Ans. 2.405951. 8. From ? take ft. Ans. .246753. 9. From | take f Ans. .i58730. 10. From T9T_ take fT. Ans. .i764705882352941. 11. From 5.i2345 take 2.3523456. Ans. 2.7711055821666927777988888599994. MULTIPLICATION OF CIRCULATING DECIMALS. 305.° Ex. 1. Multiply .36 by 25. Ans. 0929. Operation. We change the 36 4 • 2J 23 given numbers to •36 = = —; .25 = — = their equivalent com . . mon fractions, and, r^X|5 = sVo- — TV65 — -0929 Ans" multiplying, obtain = T49^ which, reduced to its equivalent decimal, .srives .0925, the answer required. |