period of a decimal expansion

If we take any rational number and write it as a decimal, then its decimal expansion eventually repeats. If the expansion ends in repeating zeros (or nines), we say it is terminating (examples: 1 = 1.000... = 0.999... and 1/20 = 0.0500... = 0.0499...). Otherwise, the length of the (smallest) block of repeating digits is the period. For example:

number	decimal expansion	period
1/3	0.3333333333333...	1
5/7	0.71428571428571...	6
25/13	1.92307692307692...	6
89/26	3.4230769230769...	6

Notice that if x repeats with period n, then (10ⁿ-1) x has a terminating expansion, so there is a non-negative integer m such that 10^m (10ⁿ-1) x is an integer. This shows that x is rational. When x = 1/k for some integer k, it also shows that the period of 1/k is the same as the order of 10 modulo k. In particular the period of 1/k always divides Euler's phi function of k, and the period of 1/p for a prime p always divides p-1.

See Also: PeriodOfAPrime