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Conversely the period of the repeating decimal of a fraction c / d will be (at most) the smallest number n such that 10 n − 1 is divisible by d. For example, the fraction 2 / 7 has d = 7, and the smallest k that makes 10 k − 1 divisible by 7 is k = 6, because 999999 = 7 × 142857. The period of the fraction 2 / 7 is ...
Sometimes an infinite repeating decimal is required to reach the same precision. Thus, it is often useful to convert repeating decimals into fractions. 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 ...
The continued fraction representation for a real number is finite if and only if it is a rational number. In contrast, the decimal representation of a rational number may be finite, for example 137 / 1600 = 0.085625, or infinite with a repeating cycle, for example 4 / 27 = 0.148148148148...
In 1802, H. Goodwyn published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain prime numbers. [43] Examples include: = 0. 142857 and 142 + 857 = 999. = 0. 01369863 and 0136 + 9863 = 9999.
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator p and a non-zero denominator q. [1] For example, is a rational number, as is every integer (e.g., ). The set of all rational numbers, also referred to as " the rationals ", [2] the field of rationals[3] or ...
Midy's theorem. 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 2 n, so that.
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