
Glossary: Prime Pages: Top 5000: 
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:
Notice that if x repeats with period n, then (10^{n}1) x has a terminating expansion, so there is a nonnegative integer m such that 10^{m} (10^{n}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 p1.
See Also: PeriodOfAPrime
Chris K. Caldwell © 19992018 (all rights reserved)
