Infinite decimal expansion
From Encyclopedia of Mathematics
A number written as a decimal fraction, such that there is no last digit. For example, 
, 
or 
, 
, 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 
and 0 or 9 for 
. If the number is irrational, the infinite decimal fraction cannot be recurrent (e.g. 
).
Comments
The period length of the decimal expansion of a rational number 
with 
not divisible by 2 or 5, is precisely the smallest positive integer 
such that 
divides 
. Thus, the period length divides 
, the Euler function.
How to Cite This Entry:
Infinite decimal expansion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Infinite_decimal_expansion&oldid=12068
Infinite decimal expansion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Infinite_decimal_expansion&oldid=12068
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article