Infinite decimal expansion

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A number written as a decimal fraction, such that there is no last digit. For example, $1/11=0.090909\dots$, $7/4=1.75000\dots$ or $7/4=1.74999\dots$, $\sqrt2=1.4142\dots$, etc. If the number is rational, the infinite decimal fraction is recurrent: starting from a certain digit, it consists of an infinitely recurring digit or group of digits called a period. In the above examples these are: 09 for $1/11$ and 0 or 9 for $7/4$. If the number is irrational, the infinite decimal fraction cannot be recurrent (e.g. $\sqrt2$).


The period length of the decimal expansion of a rational number $p/q$ with $q$ not divisible by 2 or 5, is precisely the smallest positive integer $n$ such that $q$ divides $10^n-1$. Thus, the period length divides $\phi(q)$, the Euler function.

How to Cite This Entry:
Infinite decimal expansion. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article