Patterns in Repeating DecimalsDate: 08/06/2003 at 19:45:49 From: Patrick J. Igoe Subject: Patterns in repeating decimals 1/7 = .142857... 2/7 = .285714... 3/7 = .428571... 4/7 = .571428... 5/7 = .714285... 6/7= .857142... For each fraction, the repeating decimal is the same basic sequence of six digits [142857], which starts at slightly different starting points [for 1/7, "1" is starting point ... and for 6/7, "8" is starting point] and wraps around. Also, the starting point does not simply move successively down the line of the same basic sequence [from 1 to 4, from 4 to 2, from 2 to 8]. Rather, the starting point always moves to the next highest/largest available number within the same basic sequence [1 to 2, 2 to 4, 4 to 5, 5to 7, 7 to 8]. Why is this? Other fractions are slightly different. For #/13, there appear to two (2) different six-digit sequences. 1/13 = .076923... 2/13 = .153846... 3/13 = .230769... 5/13 = .384615... 4/13 = .307692... 6/13 = .461538... 9/13 = .692307... 7/13 = .538461... 10/13 = .769230... 8/13 = .615384... 12/13 = .923076... 11/13 = .846153... Any thoughts on this? Can it be generalized? Thanks for your help, Patrick Date: 08/07/2003 at 08:34:06 From: Doctor Jacques Subject: Re: Patterns in repeating decimals Hi Patrick, This is no accident. If n is not divisible by 2 or 5, the length of the minimal period of the decimal expansion of 1/n is a divisor of phi(n), where phi(n) is Euler's function (if n is prime, phi(n) = n-1). This is a consequence of Euler's Theorem. If n is prime, and the length of the period is equal to n-1 (the largest possible value), the fractions m/n, where m and n are relatively prime, will have the same period, starting at a different point. This is the case for n = 7, where the expansion of 1/7 has period 6. Another example is n = 19: 1/19 = 0.052631578947368421... 2/19 = 0.105263157894736842... where the period is equal to 18. The length of the period is equal to the smallest exponent k > 0 such that 10^k = 1 (mod n). If the period has length k < phi(n), there may be up to phi(n)/k different patterns. For example, as you noticed, the period of 1/13 has length 6, and there are 12/6 = 2 different patterns. Another example is n = 9, phi(9) = 6, and the period has length 1. In this case, we have 6 different periods (we only consider the numerators relatively prime to 9, since otherwise we are actually dealing with multiples of 1/3): 1/9 = 0.111... 2/9 = 0.222... 4/9 = 0.444.. 5/9 = 0.555.. 7/9 = 0.777.. 8/9 = 0.888... If the denominator is divisible by 2 and / or 5, you must first divide it to remove those factors in order to consider the period length. In that case, the periodic part will not start right after the decimal point. Does this help? Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Jacques, The Math Forum http://mathforum.org/dr.math/ Date: 08/07/2003 at 12:55:12 From: Patrick J. Igoe Subject: Thank you (patterns in repeating decimals) Thank you for your answer, Patrick |
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