decimal expansion

Every rational numberPlanetmathPlanetmathPlanetmath mn, where m and n are positive integers, has an endless decimal expansion (or decadic expansion — according to Greek).  The decimal expansion of mn means the series

ν.ν1ν2ν3=ν+10-1ν1+10-2ν2+10-3ν3+ (1)

where  ν=mn  is the integer part ( of mn and the integers νj are the remainders of  10jmn  when divided by 10; thus  0νj<10.

We may suppose that m and n are coprimeMathworldPlanetmathPlanetmath (if necessary, reduce the fraction).  Then the depends only on the denominator n.  In the case that  gcd(n, 10)=1,  the is the least positive integer l such that 10l1(modn) (the does not change if we multiply the fraction by a suitable power of 10 and then reduce all prime factorsMathworldPlanetmathPlanetmath of 10 from the denominator).  In every case, the is a factor of the number φ(n), where φ is Euler’s totient function.


18=0.125000=0.124999 (one-digit ; N.B. two possibilities),

112=0.08333 (one-digit per.),

137=0.027027027 (three-digit per.),

182=0.0121951219512195 (five-digit per.),

125351=0.000039446 (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 18 in the decimal system).

Completely similar results concern the digital expansions in any other positional digit system.  Let the fraction 131 be an example (φ(31)=30); its is

in the decadic (decimal) digit system  131=0.032258064516129ten  (15-digit per.),

in the hexadic (senary) digit system  151=0.010545010545010545six  (6-digit per.),

in the dyadic () digit system  111111=0.000010000100001two  (5-digit per.).

Note.  Also any irrational number has a unique decimal expansion, but it is non-periodic; for example Liouville’s number (

0.110001 000000 000000 000001 000000

which is transcendental over .

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