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

All rational numbers have either finite decimal expansions (i.e., are regular numbers; e.g., ) or repeating decimals (e.g., ). However, transcendental 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 repeating decimals are 3, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, ... (Sloane's A085837).

The repeating portion of a rational number can be found in Mathematica using the command RealDigits[r][[1,-1]]. 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. The periods of the decimal expansions of for n = 1, 2, ... are 0, 0, 1, 0, 0, 1, 6, 0, 1, 0, 2, 1, 6, 6, 1, 0, 16, 1, 18, ... (Sloane's A051626), where 0 indicates that the number is regular.

If is a repeating decimal and is a terminating decimal, them has a nonperiodic part whose length is that of and a repeating part whose length is that of (Wells 1986, p. 60).

 

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