       Re: Cyclical Decimal Expansion

• To: mathgroup at smc.vnet.net
• Subject: [mg114495] Re: Cyclical Decimal Expansion
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sun, 5 Dec 2010 21:56:22 -0500 (EST)

```The answer is classical and is (or rather used to be) taught in elementary
number theory classes. Here is how to compute it with Mathematica:

Consider a fraction of the form p/q in reduced form. Use FactorInteger to factorize it into the form 2^a 5^b Q ,where we have GCD[Q,10]====1. Let m== Max[a,b]. Then in the decimal expansion of p/q will get m non-recurring digits, while the number of recurring ones depends only on Q and is given by the Mathematica function MultiplicativeOrder[10,Q]. For example. in the case 1/7 there are no non-recurring digits and the number of recurring ones is:

In:== MultiplicativeOrder[10, 7]

Out== 6

Andrzej Kozlowski

On 3 Dec 2010, at 11:21, Harvey P. Dale wrote:

> What's the easiest way to determine the length of the repeating
> cycle for decimal expansions of fractions?  For example, 1/7 ==
> 0.14285714285714285714 . . . so the length of its repeating cycle
> (142857) is 6.  For 1/3 the length of the cycle is obviously 1.  For
> some fractions, e.g., 1/4, the decimal expansion is not cyclical (in
> base 10).
>
> Thanks.
>
> Harvey
>
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