Repeating decimal

A repeating decimal or recurring decimal is a <a href="/facts/Decimal_representation/HCLBt4UQ">decimal representation</a> of a number whose <a href="/facts/Numerical_digit/JJdLBoGT">digits</a> are eventually <a href="/facts/Periodic_function/Sm8lJQsF">periodic</a> (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 <a href="/facts/Rational_number/VJQmyY05">rational</a> if and only if its decimal representation is repeating or terminating. For example, the decimal representation of ⁠1/3⁠ becomes periodic just after the <a href="/facts/Decimal_point/t6ocVouk">decimal point</a>, 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. Every terminating decimal representation can be written as a <a href="/facts/Decimal_fraction/7omfVNIM">decimal fraction</a>, a fraction whose denominator is a <a href="/facts/Power_(math)/pac6QY2g">power</a> of 10 (e.g. 1.585 = ⁠1585/1000⁠); it may also be written as a <a href="/facts/Ratio/mFmhQt7T">ratio</a> of the form ⁠k/2n·5m⁠ (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 <a href="/facts/0.999.../fdtw8eBL">1.000... = 0.999...</a> 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 <a href="/facts/Division_algorithm/8xc3I1sJ">division algorithm</a>.)
Any number that cannot be expressed as a <a href="/facts/Ratio/mFmhQt7T">ratio</a> of two <a href="/facts/Integer/PUwwrawx">integers</a> is said to be <a href="/facts/Irrational_number/Psv19TET">irrational</a>. 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 <a href="/facts/Square_root_of_2/SD0sag94">√2</a> and <a href="/facts/Pi/vC0UBj1q">π</a>.

Repeating decimal open-in-new

Repeating decimal