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A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to be terminating, and is not considered as repeating.
In mathematics, Midy's theorem, named after French mathematician E. Midy, [1] is a statement about the decimal expansion of fractions a/p where p is a prime and a/p has a repeating decimal expansion with an even period (sequence A028416 in the OEIS). If the period of the decimal representation of a/p is 2n, so that
A conventional way to indicate a repeating decimal is to place a bar (known as a vinculum) over the digits that repeat, for example 0. 789 = 0.789789789... For repeating patterns that begin immediately after the decimal point, the result of the conversion is the fraction with the pattern as a numerator, and the same number of nines as a ...
where the repeating block is indicated by dots over its first and last terms. [2] If the initial non-repeating block is not present – that is, if k = -1, a 0 = a m and = [;,, …, ¯], the regular continued fraction x is said to be purely periodic.
The continued fraction representation for a real number is finite if and only if it is a rational number. In contrast, the decimal representation of a rational number may be finite, for example 137 / 1600 = 0.085625, or infinite with a repeating cycle, for example 4 / 27 = 0.148148148148...
The conversion of a repeating balanced ternary number to a fraction is analogous to converting a repeating decimal. For example (because of 111111 bal3 = ( 3 6 − 1 / 3 − 1 ) dec ):