Repeating decimal

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Repeating decimal" – news · newspapers · books · scholar · JSTOR (July 2023) (Learn how and when to remove this message) This article may require cleanup to meet Wikipedia's quality standards. The specific problem is: The article is polluted with multiple non-encyclopedic tables, often placed before the true content that is hidden by them. Section are placed in a random order with the most important facts relegated toward the end of the article. Please help improve this article if you can. (July 2024) (Learn how and when to remove this message) (Learn how and when to remove this message)

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.

It can be shown that a number is rational if and only if its decimal representation is repeating or terminating. For example, the decimal representation of ⁠1/3⁠ becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e. 0.333.... A more complicated example is ⁠3227/555⁠, whose decimal becomes periodic at the second digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144.... Another example of this is ⁠593/53⁠, which becomes periodic after the decimal point, repeating the 13-digit pattern "1886792452830" forever, i.e. 11.18867924528301886792452830....

The infinitely repeated digit sequence is called the repetend or reptend. 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 = ⁠1585/1000⁠); it may also be written as a ratio of the form ⁠k/2ⁿ·5^m⁠ (e.g. 1.585 = ⁠317/2³·5²⁠). However, every number with a terminating decimal representation also trivially has a second, alternative representation as a repeating decimal whose repetend is the digit 9. This is obtained by decreasing the final (rightmost) non-zero digit by one and appending a repetend of 9. Two examples of this are 1.000... = 0.999... and 1.585000... = 1.584999.... (This type of repeating decimal can be obtained by long division if one uses a modified form of the usual division algorithm.^[2])

Any number that cannot be expressed as a ratio of two integers is said to be irrational. Their decimal representation neither terminates nor infinitely repeats, but extends forever without repetition (see § Every rational number is either a terminating or repeating decimal). Examples of such irrational numbers are √2 and π.^[3]