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50. Whenever the prime factors of the denominator of a vulgar fraction contain neither of the factors 2 and 5, the repetend will be simple. But when they contain one or both of the factors 2 and 5, together with other factors, then the repetend will be compound.
51. Those simple repetends which have as many terms, less one, as there are units in the denominators of their equivalent vulgar fractions, we shall call perfect repetends. The following are all of the perfect repetends, whose denominators are less than 100.
0.61030927835051546391752577319587 wor= 62886597.9381443298.96907216494845 360824.742.268041237] 1340206185567.
The value of Pł may be made to assume the following forms:
* =0.0}}=0.05}}=0.058+3=0.0588,” –0.05882,', = 0.058823# = 0.0588235;} = 0.05882352}} =&c., where each successive value is extended one decimal place further than its preceding value. The numerators of the vulgar fractions connected with the above decimal expressions are the successive remainders found in the operation of converting the vulgar fraction for into a decimal. (Art. 42.)
If this process of decimating be continued, it will be found to give a simple repetend, consisting of 16 places of figures; it is, therefore, a perfect repetend. We will arrange this repetend by placing above each figure its corresponding remainder, as follows;
8 64 7 || 0 58, &c.
10 15 14 4 5 9 5 16 7. 2 3 13 ro-0 0 588 23 529 4 1 1 If we fix our attention upon a particular remainder, as the fifth, for instance, which is 6, it is evident that the decimals which follow, as 3529, &c., continued to infinity, must express the decimal value of or ; for, had we terminated our division after the fifth decimal figure was obtained, we should have had for = 0.05882%, where # stands instead of the decimal figures which follow 0.05882, so that the decimal figures following the remainder, 6, is equal to #. In the same way, the decimals which follow any other remainder is the value of the vulgar fraction whose denominator is 17, and whose numerator is said remainder. We have already said that the decimal figures would commence repeating when a remainder is found like one which has previously occurred, (Art. 47.) A perfect repetend has been defined (Art. 51,) as one whose number of decimal figures is equal to one less than the units in the denominator of the vulgar fraction from which it is derived. Therefore, in converting the vulgar fraction P; into a perfect repetend, every number, from 1 to 16, inclusive, must appear as a remainder. Let us suppose we have reached that point in the process of decimating, which gives 16 for the remainder; then the decimals which follow, being the value of ##, must, when added to the preceding decimals, the value of or, make a succession of 9's, as 99999, &c., since #1-1-##-1, which, expressed in decimals, is 0.99999, &c., continued to infinity. Hence, when we have obtained as many figures beyond the remainder 16, as we had before we found this remainder, the decimal figures will begin to repeat. But or, giving a perfect repetend, must extend to 16 figures before repeating; consequently, 16 will occur as the 8th remainder, or when we have obtained one half the number of decimals in the period, and such must be the case with all perfect repetends. Therefore, the decimal figures of the first half of the period of a perfect repetend, being added to the figures of the second half, must give 99999, &c.; which, if considered as a decimal, is equivalent to a unit. Hence, also, the remainders of the first half of the period, being added to the remainders.of the second half, must make, respectively, 17, since their corresponding decimal values make a unit, which is equivalent to ##. Whenever the number of figures in the period of a simple repetend, arising from decimating a vulgar fraction, whose numerator is 1, and denominator a prime, is even, the remainder which occurs at the middle of the period will be one less than the denominator of the equivalent vulgar fraction, and the figures of the first half of the period, added to those of the second half, will give 99999, &c.; and their corresponding remainders added, must give the denominator of the equivalent vulgar fraction. Such repetends may be called CoMPLEMENTARY REPETENDs. They, of course, include all perfect repetends, as well as many which are not perfect. The following complementary repetends are not perfect.
The following vulgar fractions, when decimated, give
an even number of figures in a period, and still they are not complementary repetends, their denominators not being primes.
*=0.0204081, &c., to 42 places.
52. Perfect REPETENds possess some very remark able properties, which we will explain by means of the following figure:
In this figure, the inner circle of figures, commencing at the 0, directly under the asterisk, and counting towards the right hand, is the circulating period of #. The outer circle of figures, commencing at the same place, and counting in the same direction, are the successive remainders which will occur in the operation of decimating #, (by Rule under Art. 42.) In this circle of remainders, all the numbers from 1 to 28, inclusive, occur, but not in numerical order. From what has been said, we infer the following properties, which are common to all perfect repetends: