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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 ...
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 fixed-point representation of a fractional number is essentially an integer that is to be implicitly multiplied by a fixed scaling factor. For example, the value 1.23 can be stored in a variable as the integer value 1230 with implicit scaling factor of 1/1000 (meaning that the last 3 decimal digits are implicitly assumed to be a decimal fraction), and the value 1 230 000 can be represented ...
Also the converse is true: The decimal expansion of a rational number is either finite, or endlessly repeating. Finite decimal representations can also be seen as a special case of infinite repeating decimal representations. For example, 36 ⁄ 25 = 1.44 = 1.4400000...; the endlessly repeated sequence is the one-digit sequence "0".
A repeating decimal is an infinite decimal that, after some place, repeats indefinitely the same sequence of digits (e.g., 5.123144144144144... = 5.123 144). [4] An infinite decimal represents a rational number, the quotient of two integers, if and only if it is a repeating decimal or has a finite number of non-zero 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.
Most decimal fractions (or most fractions in general) cannot be represented exactly as a fraction with a denominator that is a power of two. For example, the simple decimal fraction 0.3 (3 ⁄ 10) might be represented as 5404319552844595 ⁄ 18014398509481984 (0.299999999999999988897769…). This inexactness causes many problems that are ...