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Decimal Period
The decimal period of arepeating decimal is the number of digits that repeat. For example,
has decimal period one,
has decimal period two, and
has decimal period three.
Any nonregular fraction
is periodic and has decimal period
independent of
, which is at most
digits long. If
isrelatively prime to 10, then the period
of
is a divisor of
and has at most
digits, where
is thetotient function. It turns out that
is themultiplicative order of 10 (mod
) (Glaisher 1878, Lehmer 1941). The number of digits in the repeating portion of the decimal expansion of arational number can also be found directly from themultiplicative order of itsdenominator.
A prime
such that
is arepeating decimal with decimal period shared with no other prime is called aunique prime.
See also
Decimal Comma,Decimal Expansion,Decimal Point,Repeating Decimal,Unique PrimeExplore with Wolfram|Alpha
References
Glaisher, J. W. L. "Periods of Reciprocals of Integers Prime to 10."Proc. Cambridge Philos. Soc.3, 185-206, 1878.Lehmer, D. H. "Guide to Tables in the Theory of Numbers." Bulletin No. 105. Washington, DC: National Research Council, pp. 7-12, 1941.Referenced on Wolfram|Alpha
Decimal PeriodCite this as:
Weisstein, Eric W. "Decimal Period." FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/DecimalPeriod.html