A repeating or recurring decimal is a way of representing rational numbers in base 10 arithmetic. The decimal representation of a number is said to be repeating if it becomes periodic (repeating its values at regular intervals) and the infinitely-repeated portion is not zero. For example, the decimal representation of ⅓ 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 after the second digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144…. At present, there is no single universally accepted notation or phrasing for repeating decimals.
The infinitely-repeated digit sequence is called the repetend or reptend. If the repetend is a zero, this decimal representation is called aterminating 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 divisor is a power of 10 (e.g. 1.585 = 1585/1000); it may also be written as a ratio of the form k/2n5m (e.g. 1.585 = 317/2352). 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 non-zero digit by one and appending a repetend of 9, a fact that some find puzzling. 1.000... = 0.999… and 1.585000... = 1.584999… are two examples of this. (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 for ever without regular repetition. Examples of such irrational numbers are the square root of 2 and pi.