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Repeating Decimal

A number whose decimal representation eventually becomes periodic (i.e., the same sequence of digits repeats indefinitely) is called a repeating decimal. Numbers such as 0.5 can be regarded as repeating decimals since $0.5=0.5000\ldots=0.4999\ldots$. All Rational Numbers have repeating decimals, e.g., $1/11=0.\overline{09}$. However, Transcendental Numbers, such as $\pi=3.141592\ldots$ do not.

See also Cyclic Number, Decimal Expansion, Full Reptend Prime, Irrational Number, Midy's Theorem, Rational Number, Regular Number


Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 53-54, 1987.

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.

© 1996-9 Eric W. Weisstein