We may suppose that and are coprime (if necessary, reduce the fraction). Then the depends only on the denominator . In the case that , the is the least positive integer such that (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 , where is Euler’s totient function.
(one-digit ; N.B. two possibilities),
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 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 in the decimal system).
Completely similar results concern the digital expansions in any other positional digit system. Let the fraction be an example (); its is
in the decadic (decimal) digit system (15-digit per.),
in the hexadic (senary) digit system (6-digit per.),
in the dyadic () digit system (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)
which is transcendental over .
|Date of creation||2013-03-22 15:04:01|
|Last modified on||2013-03-22 15:04:01|
|Last modified by||pahio (2872)|