# 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
(10^{n}-1) *x* has a terminating expansion,
so there is a non-negative 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
*p*-1.

**See Also:** PeriodOfAPrime

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