The decimal period of a repeating decimal is the number of digits that repeat. For example, 1/3 = 0.3^_ has decimal period one, 1/11 = 0.9^_ has decimal period two, and 1/37 = 0.27^_ has decimal period three. Any nonregular fraction m/n is periodic and has decimal period λ(n) independent of m, which is at most n - 1 digits long. If n is relatively prime to 10, then the period λ(n) of m/n is a divisor of ϕ(n) and has at most ϕ(n) digits, where ϕ is the totient function. It turns out that λ(n) is the multiplicative order of 10 (mod n). 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.

decimal comma | decimal expansion | decimal point | repeating decimal | unique prime