Pattern in Period
Date: 01/31/99 at 23:04:50 From: Michael Lau Subject: Pattern in Period I tried to find some periods with 1 as numerator. I found that if the period is even, the first half of the period added to the second half equals a series of 9's. Is there any pattern in odd periods too? e.g.: 1/11 = 0.09090909... 0 + 9 = 9 1/7 = 0.142857142857... 142 + 857 = 999 etc.
Date: 02/02/99 at 08:50:06 From: Doctor Peterson Subject: Re: Pattern in Period I played with your pattern for a while to see if I could prove it to be true. I found that it definitely isn't true for all denominators; for 1/21, the sum of the two halves of the period is 666. But maybe it's true for prime denominators, and also for some composites including 14, 22, and 26. One thing I found is that if 1/a has period 2n, and the sum of the two halves of the period (call them x and y) is 10^n - 1, then the equations 1 10^n x + y --- = ----------- and x + y = 10^n - 1 a 10^(2n) - 1 can be simplified to give a(x + 1) = 10^n + 1 so that a must be a divisor of 10^n + 1. (My equations assume the period starts at the decimal point, restricting the problem somewhat.) This is true, for instance, for small primes such as 7, 11, and 13, since 1001 = 7*11*13 and for 17 (100000001 = 17*5882353) and 19 (1000000001 = 7*11*13*19*52579). I don't know enough number theory to see easily that this is true for all primes. But a little search found Eric Weisstein's page on Midy's theorem, ("If the period of a repeating decimal for a/p has an even number of digits, the sum of the two halves is a string of 9's, where p is prime and a/p is a reduced fraction.") http://mathworld.wolfram.com/MidysTheorem.html This is your pattern, restricted to prime denominators, but extended for any numerator except a multiple of the denominators. There is no mention of any extension to odd periods. A further search showed only that Martin Gardner discussed this theorem in his book _Mathematical Circus_, which you might be able to find. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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