# Repeating decimal

A repeating decimal (also called a recurring decimal) is a decimal representation in which, at some point after the decimal point, a digit or block of digits repeats endlessly. The property is exactly equivalent to rationality: a real number is rational if and only if its decimal expansion either terminates or is eventually periodic.<sup>[1](https://people.math.wisc.edu/~jwrobbin/112online/UWCABook.pdf)</sup><sup> • </sup><sup>[2](https://clp.math.uky.edu/clp_2_ic_text.pdf)</sup> Irrational numbers such as π = 3.141592... have expansions that neither terminate nor repeat.<sup>[3](https://archive.lib.msu.edu/crcmath/math/math/r/r223.htm)</sup>

| Key fact | Detail |
|---|---|
| Definition | A decimal in which the last digit or group of digits repeats endlessly, e.g. 1.333... = 1.\overline{3}<sup>[4](https://openstax.org/books/prealgebra/pages/5-3-decimals-and-fractions)</sup> |
| Characterization | A real number is rational if and only if its decimal expansion is eventually periodic<sup>[1](https://people.math.wisc.edu/~jwrobbin/112online/UWCABook.pdf)</sup> |
| Why repetition occurs | Long division of p by q has fewer than q possible remainders, so a remainder must recur within at most q steps<sup>[5](https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter01/exercise04.html)</sup> |
| Conversion example | 0.1363636... = 1/10 + (36/10)·∑(1/100)^k = 3/22<sup>[1](https://people.math.wisc.edu/~jwrobbin/112online/UWCABook.pdf)</sup> |
| Period length | For n/m in lowest terms, the number of repeated digits depends only on m; every fraction with denominator 7 has period length 6<sup>[6](https://digitalcommons.unl.edu/mathfacpub/48/)</sup> |
| Dual representations | 0.5 = 0.5000... = 0.4999..., so terminating decimals can be written as repeating ones<sup>[3](https://archive.lib.msu.edu/crcmath/math/math/r/r223.htm)</sup> |

## Definition and notation

A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly.<sup>[4](https://openstax.org/books/prealgebra/pages/5-3-decimals-and-fractions)</sup> The repeating block is called the period or repetend. The common shorthand places a line (a vinculum) over the digits that repeat: 1.333... is written 1.\overline{3}, 4.161616... as 4.\overline{16}, and 0.271271271... as 0.\overline{271}.<sup>[4](https://openstax.org/books/prealgebra/pages/5-3-decimals-and-fractions)</sup> Other notations exist, including parentheses and an asterisk convention; mathematicians use several marks to indicate which digits belong to the recurring part.<sup>[7](https://fibonacci-numbers.surrey.ac.uk/Fractions/fractions.html)</sup><sup> • </sup><sup>[5](https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter01/exercise04.html)</sup>

The word *eventually* matters. In 43/22 = 1.95454..., the digit 9 after the decimal point is not part of the pattern; only the block 54 repeats, so the expansion is 1.9\overline{54}.<sup>[4](https://openstax.org/books/prealgebra/pages/5-3-decimals-and-fractions)</sup> Formally, an expansion is eventually periodic if there are positive integers k and p such that the digit in place n + p equals the digit in place n for every n beyond k: a non-repeating prefix of k digits followed by a repeating block of p digits.<sup>[2](https://clp.math.uky.edu/clp_2_ic_text.pdf)</sup> A terminating decimal is the special case in which the repeating block is a single zero.<sup>[2](https://clp.math.uky.edu/clp_2_ic_text.pdf)</sup>

## Why rational numbers repeat

The reason lies in the long division algorithm. To expand a fraction p/q, each step computes the next quotient digit, multiplies it by q, subtracts to get a remainder, and brings down the next digit of the dividend. The remainder is always smaller than q, otherwise a larger quotient digit would have been used.<sup>[1](https://people.math.wisc.edu/~jwrobbin/112online/UWCABook.pdf)</sup> Once the division passes the decimal point, the digit brought down is always zero, so the next remainder depends only on the current remainder.<sup>[5](https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter01/exercise04.html)</sup>

Since there are only q possible remainders, after at most q steps a remainder must recur. From that point the sequence of remainders, and therefore the sequence of quotient digits, repeats endlessly.<sup>[5](https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter01/exercise04.html)</sup> If the remainder 0 appears, the expansion terminates, which is the same as repeating zeroes.<sup>[8](https://mathresearch.utsa.edu/wiki/index.php?title=Real_Numbers%3AIrrational)</sup> Examples include 1/11 = 0.090909... and 1/7 = 0.142857142857..., with blocks 09 and 142857 respectively.<sup>[5](https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter01/exercise04.html)</sup>

## Why repeating decimals are rational

The converse direction converts a repeating decimal into a fraction using a geometric series. A number between 0 and 1 has an expansion x = ∑ d_k·10^(−k) with each digit d_k between 0 and 9, and the infinite sum makes this definition precise.<sup>[1](https://people.math.wisc.edu/~jwrobbin/112online/UWCABook.pdf)</sup> For an eventually periodic expansion with prefix α of k digits and repeating block β of p digits, the value is α/10^k plus a geometric tail of terms β/10^(k+p), β/10^(k+2p), and so on.<sup>[2](https://clp.math.uky.edu/clp_2_ic_text.pdf)</sup>

A worked example from a University of Wisconsin calculus text shows the method for 0.1363636...:

\[ 0.1363636\ldots = \frac{1}{10} + \frac{36}{10}\sum_{k=1}^{\infty}\left(\frac{1}{100}\right)^{k} = \frac{1}{10} + \frac{36}{10}\cdot\frac{0.01}{1-0.01} = \frac{1}{10} + \frac{36}{10}\cdot\frac{1}{99} \]

Evaluating gives (99 + 36)/990 = 135/990 = 3/22.<sup>[1](https://people.math.wisc.edu/~jwrobbin/112online/UWCABook.pdf)</sup> The same algebra underlies the general rule that an n-digit repeating block equals that block's integer value divided by a number written with n nines.<sup>[5](https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter01/exercise04.html)</sup>

## Terminating decimals and the 0.999... case

Every terminating decimal can be regarded as a repeating decimal in two ways. Since 1/2 = 0.5 = 0.5000000..., any terminating expansion can be given a period of zeroes; mathematicians conventionally ignore this and say the decimal terminates.<sup>[7](https://fibonacci-numbers.surrey.ac.uk/Fractions/fractions.html)</sup> Alternatively, the last non-zero digit D can be replaced by D − 1 followed by a period of nines: 1/8 = 0.125 = 0.1249999999....<sup>[7](https://fibonacci-numbers.surrey.ac.uk/Fractions/fractions.html)</sup> The source of the familiar identity 0.5 = 0.4999... is the same phenomenon.<sup>[3](https://archive.lib.msu.edu/crcmath/math/math/r/r223.htm)</sup> Both descriptions are mathematically correct; the choice of the finite form is a convention, made so that everyone writes numbers the same way.<sup>[7](https://fibonacci-numbers.surrey.ac.uk/Fractions/fractions.html)</sup>

## Period length

For a fraction n/m reduced to lowest terms, the number of digits in the repeating period depends only on the denominator m. All fractions with denominator 7, for example, have period length 6.<sup>[6](https://digitalcommons.unl.edu/mathfacpub/48/)</sup> The period length is tied to the multiplicative behavior of 10 modulo the denominator's prime factors, and research continues on computing it: a 2025 arXiv paper details how to calculate the repeating length when the denominator is a power of a prime and examines when the length equals the denominator minus 1.<sup>[9](https://arxiv.org/abs/2507.01295)</sup>

The repeating structure also supports deeper results. Midy's theorem, attributed by the historian L. E. Dickson to E. Midy of Nantes in 1836, concerns primes with even period length and a property of splitting the repetend; the theorem holds for many such primes but is not a criterion for primeness, since [composite numbers](https://www.edgepedia.org/composite-number) such as 77 share the property.<sup>[6](https://digitalcommons.unl.edu/mathfacpub/48/)</sup> Modern treatments generalize periodic decimals to arbitrary bases B.<sup>[10](https://arxiv.org/abs/math/0605182)</sup>

## Base dependence

Terminating versus repeating is a property of the expansion, not of the number. In base 10, fractions whose reduced denominator has no prime factors other than 2 and 5 terminate, while others repeat; in another base the dividing line moves. The same theory transfers to other bases directly: replacing 10 with 3 in the definitions yields the ternary expansion, and the periodicity characterization of rationality holds there as well.<sup>[11](https://webapps.math.uci.edu/~ndonalds/math140a/decimals.html)</sup>

## References

1. [Robbin, J. W. Calculus notes, §9.4 Infinite Sums and Repeating Decimals, University of Wisconsin](https://people.math.wisc.edu/~jwrobbin/112online/UWCABook.pdf)
2. [CLP Calculus text, §3.7 Optional: Rational and Irrational Numbers, University of Kentucky](https://clp.math.uky.edu/clp_2_ic_text.pdf)
3. [Repeating Decimal, CRC concise encyclopedia entry (Michigan State archive)](https://archive.lib.msu.edu/crcmath/math/math/r/r223.htm)
4. [Decimals and Fractions, Prealgebra, OpenStax](https://openstax.org/books/prealgebra/pages/5-3-decimals-and-fractions)
5. [Exercise 1.4, MIT OpenCourseWare 18.013A](https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter01/exercise04.html)
6. [Leavitt, W. G. A Theorem on Repeating Decimals, University of Nebraska–Lincoln](https://digitalcommons.unl.edu/mathfacpub/48/)
7. [Fractions and Decimals, Fibonacci Numbers and the Golden Section, University of Surrey](https://fibonacci-numbers.surrey.ac.uk/Fractions/fractions.html)
8. [Real Numbers: Irrational, UTSA Mathematics Wiki](https://mathresearch.utsa.edu/wiki/index.php?title=Real_Numbers%3AIrrational)
9. [The length of the repeating decimal, arXiv:2507.01295](https://arxiv.org/abs/2507.01295)
10. [Midy's Theorem for Periodic Decimals, arXiv:math/0605182](https://arxiv.org/abs/math/0605182)
11. [Eventually periodic decimals in arbitrary base, UC Irvine Math 140A notes](https://webapps.math.uci.edu/~ndonalds/math140a/decimals.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Number systems › Real and complex number constructions*

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