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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. 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·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 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]

Background

Notation

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There are several notational conventions for representing repeating decimals. None of them are accepted universally.

Different notations with examples
Fraction Vinculum Dots Parentheses Arc Ellipsis
1/9 0.1 0..1 0.(1) 0.1 0.111...
1/3 = 3/9 0.3 0..3 0.(3) 0.3 0.333...
2/3 = 6/9 0.6 0..6 0.(6) 0.6 0.666...
9/11 = 81/99 0.81 0..8.1 0.(81) 0.81 0.8181...
7/12 = 525/900 0.583 0.58.3 0.58(3) 0.583 0.58333...
1/7 = 142857/999999 0.142857 0..14285.7 0.(142857) 0.142857 0.142857142857...
1/81 = 12345679/999999999 0.012345679 0..01234567.9 0.(012345679) 0.012345679 0.012345679012345679...
22/7 = 3142854/999999 3.142857 3..14285.7 3.(142857) 3.142857 3.142857142857...
593/53 = 111886792452819/9999999999999 11.1886792452830 11..188679245283.0 11.(1886792452830) 11.1886792452830 11.18867924528301886792452830...

In English, there are various ways to read repeating decimals aloud. For example, 1.234 may be read "one point two repeating three four", "one point two repeated three four", "one point two recurring three four", "one point two repetend three four" or "one point two into infinity three four". Likewise, 11.1886792452830 may be read "eleven point repeating one double eight six seven nine two four five two eight three zero", "eleven point repeated one double eight six seven nine two four five two eight three zero", "eleven point recurring one double eight six seven nine two four five two eight three zero" "eleven point repetend one double eight six seven nine two four five two eight three zero" or "eleven point into infinity one double eight six seven nine two four five two eight three zero".

Decimal expansion and recurrence sequence

In order to convert a rational number represented as a fraction into decimal form, one may use long division. For example, consider the rational number 5/74:

        0.0675
   74 ) 5.00000
        4.44
          560
          518
           420
           370
            500

etc. Observe that at each step we have a remainder; the successive remainders displayed above are 56, 42, 50. When we arrive at 50 as the remainder, and bring down the "0", we find ourselves dividing 500 by 74, which is the same problem we began with. Therefore, the decimal repeats: 0.0675675675....

For any integer fraction A/B, the remainder at step k, for any positive integer k, is A × 10k (modulo B).

Every rational number is either a terminating or repeating decimal

For any given divisor, only finitely many different remainders can occur. In the example above, the 74 possible remainders are 0, 1, 2, ..., 73. If at any point in the division the remainder is 0, the expansion terminates at that point. Then the length of the repetend, also called "period", is defined to be 0.

If 0 never occurs as a remainder, then the division process continues forever, and eventually, a remainder must occur that has occurred before. The next step in the division will yield the same new digit in the quotient, and the same new remainder, as the previous time the remainder was the same. Therefore, the following division will repeat the same results. The repeating sequence of digits is called "repetend" which has a certain length greater than 0, also called "period".[4]

In base 10, a fraction has a repeating decimal if and only if in lowest terms, its denominator has any prime factors besides 2 or 5, or in other words, cannot be expressed as 2m 5n, where m and n are non-negative integers.

Every repeating or terminating decimal is a rational number

Each repeating decimal number satisfies a linear equation with integer coefficients, and its unique solution is a rational number. In the example above, α = 5.8144144144... satisfies the equation

10000α − 10α = 58144.144144... − 58.144144...
9990α = 58086
Therefore, α = 58086/9990 = 3227/555

The process of how to find these integer coefficients is described below.

Formal proof

Given a repeating decimal where , , and are groups of digits, let , the number of digits of . Multiplying by separates the repeating and terminating groups:

If the decimals terminate (), the proof is complete.[5] For with digits, let where is a terminating group of digits. Then,

where denotes the i-th digit, and

Since ,[6]

Since is the sum of an integer () and a rational number (), is also rational.[7]

Table of values