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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.
For any integer coprime to 10, its reciprocal is a repeating decimal without any non-recurring digits. E.g. 1 ⁄ 143 = 0. 006993 006993 006993.... While the expression of a single series with vinculum on top is adequate, the intention of the above expression is to show that the six cyclic permutations of 006993 can be obtained from this repeating decimal if we select six consecutive digits ...
Rules for calculating the periods of repeating decimals from rational fractions were given by James Whitbread Lee Glaisher in 1878. [5] For a prime p, the period of its reciprocal divides p − 1. [6] The sequence of recurrence periods of the reciprocal primes (sequence A002371 in the OEIS) appears in the 1973 Handbook of Integer Sequences.
The repeating decimal commonly written as 0.999... represents exactly the same quantity as the number one. Despite having the appearance of representing a smaller number, 0.999... is a symbol for the number 1 in exactly the same way that 0.333... is an equivalent notation for the number represented by the fraction 1 ⁄ 3. [438]
Some real numbers have decimal expansions that eventually get into loops, endlessly repeating a sequence of one or more digits: 1 ⁄ 3 = 0.33333... 1 ⁄ 7 = 0.142857142857... 1318 ⁄ 185 = 7.1243243243... Every time this happens the number is still a rational number (i.e. can alternatively be represented as a ratio of an integer and a ...
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