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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 vinculum can indicate a line segment where A and B are the endpoints: ¯. A vinculum can indicate the repetend of a repeating decimal value: . 1 ⁄ 7 = 0. 142857 = 0.1428571428571428571...
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 ...
When multiplied by their generating prime, the result is a sequence of b − 1 digits, where b is the base (e.g. 10 in decimal). For example, in decimal, 142857 × 7 = 999999. When split into groups of equal length (of two, three, four, etc... digits), and the groups are added, the result is a sequence of b - 1 digits. For example, 14 + 28 + 57 ...
The Archimedean property: any point x before the finish line lies between two of the points P n (inclusive).. It is possible to prove the equation 0.999... = 1 using just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series and limits.
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In recreational mathematics, a repdigit or sometimes monodigit [1] is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of "repeated" and "digit". Examples are 11, 666, 4444, and 999999. All repdigits are palindromic numbers and are multiples of ...
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