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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.
The following list includes a decimal expansion and set containing each number, ordered by year of discovery. The column headings may be clicked to sort the table alphabetically, by decimal value, or by set. Explanations of the symbols in the right hand column can be found by clicking on them.
(also written as 0. 9, 0.., or 0.(9)) is a repeating decimal that is an alternative way of writing the number 1. Following the standard rules for representing numbers in decimal notation, its value is the smallest number greater than or equal to every number in the sequence 0.9, 0.99, 0.999, ... .
For base 10 it is called a repeating decimal or recurring decimal. An irrational number has an infinite non-repeating representation in all integer bases. Whether a rational number has a finite representation or requires an infinite repeating representation depends on the base. For example, one third can be represented by:
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...
Recurring expense, an ongoing (continual) expenditure Repeating decimal , or recurring decimal, a real number in the decimal numeral system in which a sequence of digits repeats infinitely Curiously recurring template pattern (CRTP), a software design pattern
The difference between the cumulative sum and the natural logarithm of n converges to the Euler–Mascheroni constant, commonly denoted as , which is approximately 0.5772. The sum of the reciprocals of the primes diverges. Given coprime positive integers a and b, the sum of the reciprocal of the primes of the form an + b diverges.
In his later years, George Salmon (1819–1904) concerned himself with the repeating periods of these decimal representations of reciprocals of primes. [ 1 ] Contemporaneously, William Shanks (1812–1882) calculated numerous reciprocals of primes and their repeating periods, and published two papers "On Periods in the Reciprocals of Primes" in ...