Difference between revisions of "Infinite decimal expansion"
From Encyclopedia of Mathematics
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| − | A number written as a decimal fraction, such that there is no last digit. For example, | + | {{TEX|done}} |
| + | 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. $\sqrt2$). | ||
====Comments==== | ====Comments==== | ||
| − | The period length of the decimal expansion of a rational number | + | 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|Euler function]]. |
Latest revision as of 10:22, 27 September 2014
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. $\sqrt2$).
Comments
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: 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