# Repeating decimal

## Decimal representation of a number whose digits are periodic / From Wikipedia, the free encyclopedia

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....

This article needs additional citations for verification. (July 2023) |

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*/2^{n}·5^{m} (e.g. 1.585 = 317/2^{3}·5^{2}). 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]}