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If the repetend is a zero, this decimal representation is called a terminating decimal rather than a repeating decimal, since the zeros can be omitted and the decimal terminates before these zeros. [1] Every terminating decimal representation can be written as a decimal fraction, a fraction whose denominator is a power of 10 (e.g. 1.585 ...
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
The Egyptians produced tables of Egyptian fractions for unit fractions multiplied by two, the numbers that in modern notation would be written , such as the Rhind Mathematical Papyrus table; in these tables, most of these expansions use either two or three terms. [1] These tables were needed, because the obvious expansion = + was not allowed ...
), all finite decimal expansions are unique. However, even finite β-expansions are not necessarily unique, for example φ + 1 = φ 2 for β = φ, the golden ratio. A canonical choice for the β-expansion of a given real number can be determined by the following greedy algorithm, essentially due to Rényi (1957) and formulated as given here by ...
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".
Any such decimal fraction, i.e.: d n = 0 for n > N, may be converted to its equivalent infinite decimal expansion by replacing d N by d N − 1 and replacing all subsequent 0s by 9s (see 0.999... In summary, every real number that is not a decimal fraction has a unique infinite decimal expansion.
The definition of the Champernowne constant immediately gives rise to an infinite series representation involving a double sum, = = = (+), where () = = is the number of digits between the decimal point and the first contribution from an n-digit base-10 number; these expressions generalize to an arbitrary base b by replacing 10 and 9 with b and b − 1 respectively.
The value of n is then the period of the decimal expansion of 1/p. [10] At present, more than fifty decimal unique primes or probable primes are known. However, there are only twenty-three unique primes below 10 100. The decimal unique primes are 3, 11, 37, 101, 9091, 9901, 333667, 909091, ... (sequence A040017 in the OEIS).
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