The decimal representation of a real number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic: there is some finite sequence of digits that is repeated indefinitely. This article gives a mathematical definition For a more accessible article see Decimal. In Mathematics, the real numbers may be described informally in several different ways Periodicity is the quality of occurring at regular intervals or periods (in Time or Space) and can occur in different contexts A Clock marks For example, the decimal representation of 13 = 0. 3333333. . . (spoken as "0. 3 repeating") becomes periodic just after the decimal point, repeating the single-digit sequence "3" indefinitely. A somewhat more complicated example is 3227555 = 5. 8144144144. . . , where the decimal representation becomes periodic at the second digit after the decimal point, repeating the sequence of digits "144" forever.

In fact, a real number has an ultimately periodic decimal representation if and only if it is a rational number, that is, a number that can be expressed in the form ab where a and b are integers and b is non-zero, that is, a vulgar fraction. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object On the one hand, the decimal representation of a rational number is ultimately periodic because it can be determined by a long division process, which must ultimately become periodic as there are only finitely many different remainders and so eventually it will find a remainder that has occurred before. Long Division is the second album by the Rustic Overtones, originally released on November 17 1995 On the other hand, each repeating decimal number satisfies a linear equation with integral coefficients, and its unique solution is a rational number. A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable To illustrate the latter point, the number α = 5. 8144144144. . . above satisfies the equation 10000α − 10α = 58144. 144144. . .  − 58. 144144. . .  = 58086, whose solution is α = 580869990 = 3227555.

A decimal representation written with a repeating final 0 is not classified as a repeating decimal, and the decimal is said to terminate before the first final 0 (because it is not necessary to explicitly write that there is a repeating 0; instead of "1. 585000. . . " one simply writes "1. 585"). These terminating decimals represent rational numbers whose fractions in lowest terms are of the form k2n5m. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions An irreducible fraction (or fraction in lowest terms or reduced form) is a Vulgar fraction in which the Numerator and Denominator For example, 1. 585 = 317200 = 3172352. They can be written as a decimal fraction: 317200 = 15851000. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. However, these numbers still also have a representation as a repeating decimal, obtained by decreasing the final (nonzero) digit by one and appending an indefinitely repeating sequence of digits "9" (e. g. 1 = 0. 999999. . . ; 1. 585 = 1. 584999999. . . ). See The case of 0.99999... below.

The remaining type of decimal representations is formed by decimal representations that neither terminate nor repeat. A decimal representation that neither terminates nor repeats represents an irrational number (which cannot be expressed as the ratio of two integers), such as the square root of 2 and the number π. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2}   or   √2 IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems Conversely, an irrational number has a non-terminating non-repeating decimal representation. This is true in other bases than 10 as well.

## Notation

One convention to indicate a repeating decimal is to put a horizontal line (known as a vinculum) above the repeated numerals ($\scriptstyle \frac{1}{3}=\, 0.\overline{3}$). A vinculum is a horizontal line placed over a Mathematical expression, used to indicate that it is to be considered a group Another convention is to place dots above the outermost numerals of the repeating digits. Where these methods are impossible, the extension may be represented by an ellipsis (. . . ), although this may introduce uncertainty as to exactly which digits should be repeated. Another notation, used for example in Europe and China, encloses the repeating digits in brackets.

FractionDecimalOverlineDotsBrackets
190. 111. . . 0. 1$0.\dot{1}$0. (1)
130. 333. . . 0. 3$0.\dot{3}$0. (3)
230. 666. . . 0. 6$0.\dot{6}$0. (6)
170. 142857142857. . . 0. 142857$0.\dot{1}4285\dot{7}$0. (142857)
1810. 012345679. . . 0. 012345679$0.\dot{0}1234567\dot{9}$0. (012345679)
7120. 58333. . . 0. 583$0.58\dot{3}$0. 58(3)

## Fractions with prime denominators

A fraction in lowest terms with a prime denominator other than 2 or 5 (i. An irreducible fraction (or fraction in lowest terms or reduced form) is a Vulgar fraction in which the Numerator and Denominator In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 e. coprime to 10) always produces a repeating decimal. In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than The period of the repeating decimal, 1p, where p is prime, is either p − 1 (the first group) or a divisor of p − 1 (the second group).

Examples of fractions of the first group are:

• 17 = 0. 142857. . . ; 6 repeating digits
• 117 = 0. 0588235294117647. . . ; 16 repeating digits
• 119 = 0. 052631578947368421. . . ; 18 repeating digits
• 123 = 0. 0434782608695652173913. . . ; 22 repeating digits
• 129 = 0. 0344827586206896551724137931. . . ; 28 repeating digits

The list can go on to include the fractions 147, 159, 161, 197, 1109, etc.

The following multiplications exhibit an interesting property:

• 27 = 2 × 0. 142857. . . = 0. 285714. . .
• 37 = 3 × 0. 142857. . . = 0. 428571. . .
• 47 = 4 × 0. 142857. . . = 0. 571428. . .
• 57 = 5 × 0. 142857. . . = 0. 714285. . .
• 67 = 6 × 0. 142857. . . = 0. 857142. . .

That is, these multiples can be obtained from rotating the digits of the original decimal of 17. The reason for the rotating behaviour of the digits is apparent from an arithmetics exercise of finding the decimal of 17.

Of course 142857 × 7 = 999999, and 142 + 857 = 999.

Decimals of other prime fractions, such as 117, 119, 123, 129, 147, 159, 161, 197, and 1109, each exhibit the same property.

Fractions of the second group are:

• 13 = 0. 333. . . which has 1 repeating digit.
• 111 = 0. 090909. . . which has 2 repeating digits.
• 113 = 0. 076923. . . which has 6 repeating digits.

Note that the following multiples of 113 exhibit the discussed property of rotating digits:

• 113 = 0. 076923. . .
• 313 = 0. 230769. . .
• 413 = 0. 307692. . .
• 913 = 0. 692307. . .
• 1013 = 0. 769230. . .
• 1213 = 0. 923076. . .

And similarly these multiples:

• 213 = 0. 153846. . .
• 513 = 0. 384615. . .
• 613 = 0. 461538. . .
• 713 = 0. 538461. . .
• 813 = 0. 615384. . .
• 1113 = 0. 846153. . .

Again, 076923 × 13 = 999999, and 076 + 923 = 999.

The period of the repeating decimal of 1p is equal to the order of 10 modulo p. In Number theory, the order of an element a \pmod{n} is the smallest integer k such that a^k \equiv 1\pmod{n} The period is equal to p-1 if 10 is a primitive root modulo p. In Modular arithmetic, a branch of Number theory, a primitive root modulo n is any number g with the property that any number Coprime

## Fraction from repeating decimal

Given a repeating decimal, it is possible to calculate the fraction that produced it. For example:

\begin{alignat}2 x &= 0.333333\ldots\\ 10x &= 3.333333\ldots&\quad&\mbox{(multiplying each side of the above line by 10)}\\ 9x &= 3 &&\mbox{(subtracting the 1st line from the 2nd)}\\ x &= 3/9 = 1/3 &&\mbox{(simplifying)}\\\end{alignat}

Another example:

\begin{align} x &= 7.48181818\ldots\\ 100x &= 748.18181818\ldots\\ 99x &= 740.7\\ x &= 740.7/99 = 7407/990 = 823/110\end{align}

### A shortcut

The above argument can be applied in particular if the repeating sequence has n digits, all of which are 0 except the final one which is 1. For instance for n = 7:

\begin{align} x &= 0.000000100000010000001\ldots\\ 10^7x &= 1.000000100000010000001\ldots\\ (10^7-1)x=9999999x &= 1\\ x &= 1/(10^7-1) = 1/9999999\end{align}

So this particular repeating decimal corresponds to the fraction 1 / (10n − 1), where the numerator is the number written as n digits 9. Knowing just that, a general repeating decimal can be expressed as a fraction without having to solve an equation. For the example above one could reason:

$7.48181818\ldots=7.3+0.18181818\ldots=\frac{73}{10}+\frac{18}{99}=\frac{99\times73+10\times18}{990} = \frac{7407}{990} = \frac{823}{110}$

More explicitly one gets the following cases.

If the repeating decimal is between 0 and 1, and the repeating block is n digits long, first occurring right after the decimal point, then the fraction (not necessarily reduced) will be the integer number represented by then-digit block divided by the one represented by n digits 9. For example,

• 0. 444444. . . = 49 since the repeating block is 4 (a 1-digit block),
• 0. 565656. . . = 5699 since the repeating block is 56 (a 2-digit block),
• 0. 012012. . . = 12999 since the repeating block is 012 (a 3-digit block), and this further reduces to 4333.

If the repeating decimal is as above, except that there are k (extra) digits 0 between the decimal point and the repeating n-digit block, then one can simply add k digits 0 after the n digits 9 of the denominator (and as before the fraction may subsequently be simplified). For example,

• 0. 000444. . . = 49000 since the repeating block is 4 and this block is preceded by 3 zeros,
• 0. 005656. . . = 569900 since the repeating block is 56 and it is preceded by 2 zeros,
• 0. 00012012. . . = 1299900= 216650 since the repeating block is 012 and it is preceded by 2 (!) zeros.

Any repeating decimal not of the form described above can be written as a sum of a terminating decimal and a repeating decimal of one of the two above types (actually the first type suffices, but that could require the terminating decimal to be negative). For example,

• 1. 23444. . . = 1. 23 + 0. 00444. . . = 123100 + 4900 = 1107900 + 4900 = 1111900 or alternatively 1. 23444. . . = 0. 79 + 0. 44444. . . = 79100 + 49 = 711900 + 400900 = 1111900
• 0. 3789789. . . = 0. 3 + 0. 0789789. . . = 310 + 7899990 = 29979990 + 7899990 = 37869990 = 6311665 or alternatively 0. 3789789. . . = −0. 6 + 0. 9789789. . . = −610 + 978999 = −59949990 + 97809990 = 37864995 = 6311665

It follows that any repeating decimal with period n, and k digits after the decimal point that do not belong to the repeating part, can be written as a (not necessarily reduced) fraction whose denominator is (10n − 1)10k. Periodicity is the quality of occurring at regular intervals or periods (in Time or Space) and can occur in different contexts A Clock marks

Conversely the period of the repeating decimal of a fraction cd will be (at most) the smallest number n such that 10n − 1 is divisible by d. Periodicity is the quality of occurring at regular intervals or periods (in Time or Space) and can occur in different contexts A Clock marks

For example, the fraction 27 has d = 7, and the smallest k that makes 10k − 1 divisible by 7 is k = 6, because 999999 = 7 × 142857. The period of the fraction 27 is therefore 6.

## Repeating decimals as an infinite series

Repeating decimals can also be expressed as an infinite series. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with That is, repeating decimals can be shown to be a sum of a sequence of numbers. In Mathematics, a sequence is an ordered list of objects (or events To take the simplest example,

$\sum_{n=1}^\infty\frac{1}{10^n} = {1 \over 10} + {1 \over 100} + {1 \over 1000} + \cdots = 0.\overline{1}$

The above series is a geometric series with the first term as 1/10 and the common factor 1/10. In Mathematics, a geometric series is a series with a constant ratio between successive terms. Because the absolute value of the common factor is less than 1, we can say that the geometric series converges and find the exact value in the form of a fraction by using the following formula where "a" is the first term of the series and "r" is the common factor.

$\ \frac{a}{1-r} = \frac{\frac{1}{10}}{1-\frac{1}{10}} = \frac{1}{9} = 0.\overline{1}$

## How a repeating or terminating decimal expansion is found

In order to convert a rational number represented as a fraction into decimal form, one may use long division. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions Long Division is the second album by the Rustic Overtones, originally released on November 17 1995 For example, consider the rational number 574:

        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. . . .

## Why every rational number has a repeating or terminating decimal expansion

Only finitely many different remainders — in the example above, 74 possible remainders: 0, 1, 2, . . . , 73 — can occur. If the remainder is 0, then the expansion terminates. If 0 never occurs as a remainder, then only finitely many other possible remainders exist — in the example above they are 1, 2, ,3, . . . , 73. Therefore eventually a remainder must occur that has occurred before. The same remainder implies the same new digit in the result and the same new remainder. Therefore the whole sequence repeats itself.

## The case of 0. 99999. . .

Main article: 0.999...

A proof that 1 = 0. 99999. . . , using the method of calculating fractions from repeating decimals, follows these steps.

\begin{alignat}2 x &= 0.99999\ldots\\ 10x &= 9.99999\ldots\\ 10x - x &= 9.99999\ldots - 0.99999\ldots\\ 9x &= 9\\ x &= 1\\\end{alignat}

The second step is not 10x = 9. 999. . . 0, because the right-hand side does not terminate (it is repeating) and so there is no end to which a zero can be appended. In Mathematics, LHS is informal shorthand for the left-hand side of an Equation. [1]

One can also think of this as the sum of a geometric series. In Mathematics, a geometric series is a series with a constant ratio between successive terms.

$S_a = \sum_{n=0}^{a} \frac{0.9}{10^n}$
$S_a = 0.9 \sum_{n=0}^{a} \frac{1}{10^n}$

By a standard result,

$S_a = 0.9 \frac{10^{-a-1} - 1}{10^{-1}-1}.$

From the definition,

$\lim_{a \rightarrow \infty} S_a = 0.99999 \ldots$

So applying this on the sum of the geometric series:

$\lim_{a \rightarrow \infty} 0.9 \frac{10^{-a-1} - 1}{10^{-1}-1} = 0.9 \frac{-1}{-0.9}$
$0.9 \frac{-1}{-0.9} = 1.$

Therefore

$0.99999 \ldots = 1.\,$

For a less persuasive but more formal-looking proof, consider the formula

$x = {10^n-1 \over 10^n},$
$n = 1: x = {9 \over 10} = 0.9,$
$n = 2: x = {99\over 100} = 0.99.$

It follows that

$\lim_{n \to \infty}{10^n-1 \over 10^n} = 0.9999\dots\,.$

On the other hand we can evaluate this limit easily as 1, also, by dividing top and bottom by 10n.

The above exposition using formal mathematical notation looks more impressive than the arithmetic proof but it is not persuasive as the crucial step, the division by 10n, is not actually performed. But even were the proof using limits properly completed the arithmetic proof is adequate and simpler and can be followed by those without the proper understanding of limits.

Generalising this, any nonzero number with a finite decimal expression (a decimal fraction) can be written in a second way as a repeating decimal. The decimal ( base ten or occasionally denary) Numeral system has ten as its base.

For example, 34 = 0. 75 = 0. 750000000. . . = 0. 74999999. . . .