View source for Infinite decimal expansion

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" />).



====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]].