decimal expansion

Every rational number $\frac{m}{n}$, where $m$ and $n$ are positive integers, has an endless decimal expansion (or decadic expansion — according to Greek).  The decimal expansion of $\frac{m}{n}$ means the series

$\nu .{\nu }_1{\nu }_2{\nu }_3\mathrm{\dots }=\nu +{10}^{-1}{\nu }_1+{10}^{-2}{\nu }_2+{10}^{-3}{\nu }_3+\mathrm{\dots }$ (1)

where  $\nu =\lfloor \frac{m}{n}\rfloor$  is the integer part (http://planetmath.org/Floor) of $\frac{m}{n}$ and the integers ${\nu }_j$ are the remainders of  $\lfloor {10}^j\cdot \frac{m}{n}\rfloor$  when divided by 10; thus  .

We may suppose that $m$ and $n$ are coprime (if necessary, reduce the fraction).  Then the depends only on the denominator $n$.  In the case that  $\gcd (n,\mathrm{\hspace{0.17em}10})=1$,  the is the least positive integer $l$ such that ${10}^l\equiv 1(modn)$ (the does not change if we multiply the fraction by a suitable power of 10 and then reduce all prime factors of 10 from the denominator).  In every case, the is a factor of the number $\phi (n)$, where $\phi$ is Euler’s totient function.

Examples

$\frac{1}{8}=0.125000\mathrm{\dots }=0.124999\mathrm{\dots }$ (one-digit ; N.B. two possibilities),

$\frac{1}{12}=0.08333\mathrm{\dots }$ (one-digit per.),

$\frac{1}{37}=0{.}^{\prime }{027}^{\prime }{027}^{\prime }{027}^{\prime }\mathrm{\dots }$ (three-digit per.),

$\frac{1}{82}={0.0}^{\prime }{12195}^{\prime }{12195}^{\prime }{12195}^{\prime }\mathrm{\dots }$ (five-digit per.),

$\frac{1}{25351}=0.000039446\mathrm{\dots }$ (hundred-digit per.)

The tail of infinitely many 0’s (as in 0.125000…) is of course usually not written out.  Such a tail is possible only when $n$ has no other prime factors except prime factors of the base of the digit system in question.

If the tails of 0’s are not accepted, then the digital expansion of every positive rational is unique (then e.g. 0.124999…  is the only for $\frac{1}{8}$ in the decimal system).

Completely similar results concern the digital expansions in any other positional digit system.  Let the fraction $\frac{1}{31}$ be an example ($\phi (31)=30$); its is

in the decadic (decimal) digit system  $\frac{1}{31}=0{.}^{\prime }{032258064516129}^{\prime }{\mathrm{\dots }}_{\mathrm{ten}}$  (15-digit per.),

in the hexadic (senary) digit system  $\frac{1}{51}=0{.}^{\prime }{010545}^{\prime }{010545}^{\prime }{010545}^{\prime }{\mathrm{\dots }}_{\mathrm{six}}$  (6-digit per.),

in the dyadic () digit system  $\frac{1}{11111}=0.000010000100001{\mathrm{\dots }}_{\mathrm{two}}$  (5-digit per.).

Note.  Also any irrational number has a unique decimal expansion, but it is non-periodic; for example Liouville’s number (http://planetmath.org/ExampleOfTranscendentalNumber)

$$\mathrm{0.110001\hspace{0.17em}000000\hspace{0.17em}000000\hspace{0.17em}000001\hspace{0.17em}000000}\mathrm{\dots }$$

which is transcendental over $\mathbb{Q}$.

Title	decimal expansion
Canonical name	DecimalExpansion
Date of creation	2013-03-22 15:04:01
Last modified on	2013-03-22 15:04:01
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	28
Author	pahio (2872)
Entry type	Result
Classification	msc 11A99
Synonym	decadic expansion
Synonym	digital expansion
Related topic	Base3
Related topic	ExistenceAndUniquenessOfDecimalExpansion
Related topic	0ne1AsRealNumbers
Related topic	Factoradic
Related topic	FactorialBase
Related topic	MatheRealism
Related topic	JosephLiouville
Defines	decadic
Defines	hexadic
Defines	dyadic