Namespaces
Variants
Actions

Difference between revisions of "Infinite decimal expansion"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
A number written as a decimal fraction, such that there is no last digit. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050830/i0508301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050830/i0508302.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050830/i0508303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050830/i0508304.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050830/i0508305.png" /> and 0 or 9 for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050830/i0508306.png" />. If the number is irrational, the infinite decimal fraction cannot be recurrent (e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050830/i0508307.png" />).
+
{{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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050830/i0508308.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050830/i0508309.png" /> not divisible by 2 or 5, is precisely the smallest positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050830/i05083010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050830/i05083011.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050830/i05083012.png" />. Thus, the period length divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050830/i05083013.png" />, the [[Euler function|Euler function]].
+
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
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article