repeatingdigit

A repeating digit is a digit that appears in the infinite, repeating part of the decimal expansion of a rational number. In base 10, numbers are either terminating decimals, which end after a finite number of digits, or repeating decimals, which continue forever with a recurring block of digits.

In a repeating decimal, the digits that form the recurring block are called the repetend, and the length of that block is the period. For example, 0.333... has the repeating digit 3 with period 1; 0.1666... has the repeating digit 6 with period 1; and 0.142857142857... has the repetend 142857 with period 6. Some fractions also have a non-repeating prefix before the repetend, as in 0.083333... where the non-repeating part is 08 and the repeating part is 3.

Mathematical notes: If a fraction p/q is in lowest terms, its decimal expansion terminates if and only if q has no prime factors other than 2 and 5. If q contains other primes, the decimal is repeating. The length of the repetend equals the multiplicative order of 10 modulo q', where q' is q with all factors of 2 and 5 removed. In other bases, similar rules apply with the corresponding base. Repeating digits and their patterns are central to the study of rational numbers, with practical implications for fraction representation, computer arithmetic, and numerical analysis.