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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.
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.
It was found very early on that for any prime p greater than 5, the period of the decimal expansion of 1/p is equal to the length of the smallest repunit number that is divisible by p. Tables of the period of reciprocal of primes up to 60,000 had been published by 1860 and permitted the factorization by such mathematicians as Reuschle of all ...
φ c is the intercorrelation of two discrete variables [2] and may be used with variables having two or more levels. φ c is a symmetrical measure: it does not matter which variable we place in the columns and which in the rows.
In this case, in the first decimal representation, all are zero for >, and, in the second representation, all 9. (see 0.999... for details). In summary, there is a bijection between the real numbers and the decimal representations that do not end with infinitely many trailing 9.
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.
The number π (/ p aɪ / ⓘ; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter.It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.
Example: In the case where decimal digits n = 23, since n is an odd number that is not a multiple of 3, the number of equations that can be solved is limited to the following three, and if the operation (denoted by f) defined above is applied once to the numbers corresponding to the solutions of these equations, seven Kaprekar numbers can be ...