Decimal Period

The decimal period of a repeating decimal is the number of digits that repeat. For example, 1/3=0.3^_ has decimal period one, 1/11=0.09^_ has decimal period two, and 1/37=0.027^_ has decimal period three.

Any nonregular fraction m/n is periodic and has decimal period lambda(n) independent of m, which is at most n-1 digits long. If n is relatively prime to 10, then the period lambda(n) of m/n is a divisor of phi(n) and has at most phi(n) digits, where phi is the totient function. It turns out that lambda(n) is the multiplicative order of 10 (mod n) (Glaisher 1878, Lehmer 1941). The number of digits in the repeating portion of the decimal expansion of a rational number can also be found directly from the multiplicative order of its denominator.

A prime p such that 1/p is a repeating decimal with decimal period shared with no other prime is called a unique prime.

See also

Decimal Comma, Decimal Expansion, Decimal Point, Repeating Decimal, Unique Prime

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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.

Cite this as:

Weisstein, Eric W. "Decimal Period." From MathWorld--A Wolfram Web Resource.

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