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A repeating decimal, also called a recurring decimal, is a number whose decimal expansion eventually becomes periodic (i.e., the same sequence of digits repeats indefinitely). The repeating portion of a decimal expansion is conventionally denoted with a vinculum so, for example,

The minimum number of digits that repeats in such a number is known as the decimal period.

Repeating decimal notation was implemented in versions of the Wolfram Language prior to 6 as PeriodicForm[RealDigits[r]] after loading the add-on package NumberTheory`ContinuedFractions`.

All rational numbers have either finite decimal expansions (e.g., ) or repeating decimals (e.g., ). However, irrational numbers, such as , neither terminate nor become periodic.

Numbers such as 0.5 are sometimes regarded as repeating decimals since .

The denominators of the first few unit fractions having nonterminating repeating decimals are 3, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, ... (OEIS A085837).

For an exact rational number with a nonterminating decimal expansion, the repeating portion can be found in the Wolfram Language using the command RealDigits[r][[1,-1]]. The decimal period of a rational number in lowest terms is the multiplicative order of 10 modulo for , where is obtained by removing all prime factors 2 and 5 from its denominator. If , the decimal expansion terminates and its period is recorded as 0. The periods of the decimal expansions of for , 2, ... are 0, 0, 1, 0, 0, 1, 6, 0, 1, 0, 2, 1, 6, 6, 1, 0, 16, 1, 18, ... (OEIS A051626), where 0 indicates a terminating decimal expansion.

For positive integers and , if is relatively prime to 10, is a nonterminating repeating decimal, and is a terminating decimal, then has a nonperiodic part whose length is that of and a repeating part whose length is that of (Wells 1986, p. 60).

See also

Cyclic Number, Decimal Expansion, Decimal Period, Euler's Totient Rule, Full Reptend Prime, Irrational Number, Midy's Theorem, Multiplicative Order, Rational Number, Regular Number

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References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 53-54, 1987.Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 167-168, 1996.Courant, R. and Robbins, H. "Rational Numbers and Periodic Decimals." §2.2.4 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 66-68, 1996.Numberphile. "998,001 and Its Mysterious Recurring Decimals." Mar. 6, 2012. https://www.youtube.com/watch?v=daro6K6mym8.Numberphile. "The Reciprocals of Primes." Featuring M. Parker. Mar. 14, 2022. https://www.youtube.com/watch?v=DmfxIhmGPP4.Sloane, N. J. A. Sequences A051626 and A085837 in "The On-Line Encyclopedia of Integer Sequences."Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 60, 1986.

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Repeating Decimal

Cite this as:

Weisstein, Eric W. "Repeating Decimal." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RepeatingDecimal.html

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