{"history":[],"article":{"id":"4e24e392-a467-40e5-a093-e08707ae2db5","title":"Repeating decimal","slug":"repeating-decimal","content":":::summary\n:::\n\n# \nA **repeating decimal** or **recurring decimal** is a [decimal representation](https://bliptext.com/articles/decimal-representation) of a number whose [digits](https://bliptext.com/articles/numerical-digit) are eventually [periodic](https://bliptext.com/articles/periodic-function) (that is, after some place, the same sequence of [digits](https://bliptext.com/articles/numerical-digit) is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero [digits](https://bliptext.com/articles/numerical-digit)), the decimal is said to be terminating, and is not considered as repeating.\nIt can be shown that a number is [rational](https://bliptext.com/articles/rational-number) if and only if its decimal representation is repeating or terminating.\nFor example, the decimal representation of 1⁄3 becomes periodic just after the [decimal point](https://bliptext.com/articles/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](https://bliptext.com/articles/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](https://bliptext.com/articles/decimal-point), repeating the 13-digit pattern \"1886792452830\" forever, i.e. 11.18867924528301886792452830....\nThe infinitely repeated digit sequence is called the **repetend** or **reptend**.\nIf 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.\nEvery terminating decimal representation can be written as a [decimal fraction](https://bliptext.com/articles/decimal-fraction), a fraction whose denominator is a [power](https://bliptext.com/articles/power-math) of 10 (e.g. ); it may also be written as a [ratio](https://bliptext.com/articles/ratio) of the form k⁄2n·5m (e.g. ). 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**.\nThis is obtained by decreasing the final (rightmost) non-zero digit by one and appending a repetend of 9.\nTwo examples of this are [0.999...](https://bliptext.com/articles/0-999) and.\n(This type of repeating decimal can be obtained by long division if one uses a modified form of the usual [division algorithm](https://bliptext.com/articles/division-algorithm). )\nAny number that cannot be expressed as a [ratio](https://bliptext.com/articles/ratio) of two [integers](https://bliptext.com/articles/integer) is said to be ir[ratio](https://bliptext.com/articles/ratio)nal.\nTheir decimal representation neither terminates nor infinitely repeats, but extends forever without repetition (see ).\nExamples of such irrational numbers are [[square root of 2|\n√2\n]] and [π](https://bliptext.com/articles/pi).\n\n# Background\n\n## Notation\nThere are several notational conventions for representing repeating decimals.\nNone of them are accepted universally.\nIn English, there are various ways to read repeating decimals aloud.\nFor 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\".\nLikewise, 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\".\n\n## Decimal expansion and recurrence sequence\nIn order to convert a [rational number](https://bliptext.com/articles/rational-number) represented as a fraction into decimal form, one may use [long division](https://bliptext.com/articles/long-division).\nFor example, consider the rational number 5⁄74:\n0.0675 74 ) 5.00000 4.44 560 518 420 370 500\netc. Observe that at each step we have a remainder; the successive remainders displayed above are 56, 42, 50.\nWhen 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.\nTherefore, the decimal repeats: 0.0675 675 675....\nFor any integer fraction A⁄B, the remainder at step k, for any positive integer k, is A × 10k (modulo B).\n\n## Every rational number is either a terminating or repeating decimal\nFor any given divisor, only finitely many different remainders can occur.\nIn the example above, the 74 possible remainders are 0, 1, 2, ..., 73.\nIf at any point in the division the remainder is 0, the expansion terminates at that point.\nThen the length of the repetend, also called \"period\", is defined to be 0.\nIf 0 never occurs as a remainder, then the division process continues forever, and eventually, a remainder must occur that has occurred before.\nThe 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.\nTherefore, the following division will repeat the same results.\nThe repeating sequence of digits is called \"repetend\" which has a certain length greater than 0, also called \"period\".\nIn base 10, a fraction has a repeating decimal if and only if [in lowest terms](https://bliptext.com/articles/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.\n\n## Every repeating or terminating decimal is a rational number\nEach repeating decimal number satisfies a [linear equation](https://bliptext.com/articles/linear-equation) with integer coefficients, and its unique solution is a rational number.\nIn the example above, satisfies the equation\nThe process of how to find these integer coefficients is described [below](https://bliptext.com/articles/).\n\n### Formal proof\nGiven a repeating decimal where a, b, and c are groups of digits, let, the number of digits of b. Multiplying by 10^n separates the repeating and terminating groups:\nIf the decimals terminate (c=0), the proof is complete.\nFor c\\neq0 with digits, let where is a terminating group of digits.\nThen,\nwhere d_i denotes the i-th digit, and\nSince ,\nSince x is the sum of an integer (y -c) and a rational number, x is also rational.\n\n# Table of values\n

\nThereby fraction is the [unit fraction](https://bliptext.com/articles/unit-fraction) 1⁄n and ℓ10 is the length of the (decimal) repetend.\nThe lengths ℓ10(n) of the decimal repetends of 1⁄n, n = 1, 2, 3, ..., are:\nFor comparison, the lengths ℓ2(n) of the [binary](https://bliptext.com/articles/binary-number) repetends of the fractions 1⁄n, n = 1, 2, 3, ..., are:\nThe decimal repetends of 1⁄n, n = 1, 2, 3, ..., are:\nThe decimal repetend lengths of 1⁄p, p = 2, 3, 5, ...\n(nth prime), are:\nThe least primes p for which 1⁄p has decimal repetend length n, n = 1, 2, 3, ..., are:\nThe least primes p for which k⁄p has n different cycles (1 ≤ k ≤ p−1), n = 1, 2, 3, ..., are:\n\n# Fractions with prime denominators\nA fraction [in lowest terms](https://bliptext.com/articles/in-lowest-terms) with a [prime](https://bliptext.com/articles/prime-number) denominator other than 2 or 5 (i.e. co[prime](https://bliptext.com/articles/prime-number) to 10) always produces a repeating decimal.\nThe length of the repetend (period of the repeating decimal segment) of 1⁄p is equal to the [order](https://bliptext.com/articles/multiplicative-order) of 10 modulo p.\nIf 10 is a [primitive root](https://bliptext.com/articles/primitive-root-modulo-n) modulo p, then the repetend length is equal to p − 1; if not, then the repetend length is a factor of p − 1.\nThis result can be deduced from [Fermat's little theorem](https://bliptext.com/articles/fermat-s-little-theorem), which states that 10p−1 ≡ 1 (mod p).\nThe base-10 [digital root](https://bliptext.com/articles/digital-root) of the repetend of the reciprocal of any prime number greater than 5 is 9.\nIf the repetend length of 1⁄p for prime p is equal to p − 1 then the repetend, expressed as an integer, is called a **cyclic number**.\n\n## Cyclic numbers\nExamples of fractions belonging to this group are:\nThe list can go on to include the fractions 1⁄109, 1⁄113, 1⁄131, 1⁄149, 1⁄167, 1⁄179, 1⁄181, 1⁄193, 1⁄223, 1⁄229, etc..\nEvery proper multiple of a cyclic number (that is, a multiple having the same number of digits) is a rotation:\nThe reason for the cyclic behavior is apparent from an arithmetic exercise of long division of 1⁄7: the sequential remainders are the cyclic sequence {1, 3, 2, 6, 4, 5}.\nSee also the article [142,857](https://bliptext.com/articles/142-857) for more properties of this cyclic number.\nA fraction which is cyclic thus has a recurring decimal of even length that divides into two sequences in [nines' complement](https://bliptext.com/articles/nines-complement) form.\nFor example 1⁄7 starts '142' and is followed by '857' while 6⁄7 (by rotation) starts '857' followed by its nines' complement '142'.\nThe rotation of the repetend of a cyclic number always happens in such a way that each successive repetend is a bigger number than the previous one.\nIn the succession above, for instance, we see that 0.142857... < 0.285714... < 0.428571... < 0.571428... < 0.714285... < 0.857142.... This, for cyclic fractions with long repetends, allows us to easily predict what the result of multiplying the fraction by any natural number n will be, as long as the repetend is known.\nA proper prime is a prime p which ends in the digit 1 in base 10 and whose reciprocal in base 10 has a repetend with length p − 1.\nIn such primes, each digit 0, 1,..., 9 appears in the repeating sequence the same number of times as does each other digit (namely, p − 1⁄10 times).\nThey are:\nA prime is a proper prime if and only if it is a [full reptend prime](https://bliptext.com/articles/full-reptend-prime) and [congruent](https://bliptext.com/articles/modular-arithmetic) to 1 mod 10.\nIf a prime p is both [full reptend prime](https://bliptext.com/articles/full-reptend-prime) and [safe prime](https://bliptext.com/articles/safe-prime), then 1⁄p will produce a stream of p − 1 [pseudo-random digits](https://bliptext.com/articles/pseudo-random-numbers).\nThose primes are\n\n## Other reciprocals of primes\nSome reciprocals of primes that do not generate cyclic numbers are:\nThe reason is that 3 is a divisor of 9, 11 is a divisor of 99, 41 is a divisor of 99999, etc. To find the period of 1⁄p, we can check whether the prime p divides some number 999...999 in which the number of digits divides p − 1.\nSince the period is never greater than p − 1, we can obtain this by calculating 10p−1 − 1⁄p.\nFor example, for 11 we get\nand then by inspection find the repetend 09 and period of 2.\nThose reciprocals of primes can be associated with several sequences of repeating decimals.\nFor example, the multiples of 1⁄13 can be divided into two sets, with different repetends.\nThe first set is:\nwhere the repetend of each fraction is a cyclic re-arrangement of 076923.\nThe second set is:\nwhere the repetend of each fraction is a cyclic re-arrangement of 153846.\nIn general, the set of proper multiples of reciprocals of a prime p consists of n subsets, each with repetend length k, where nk = p − 1.\n\n## Totient rule\nFor an arbitrary integer n, the length L(n) of the decimal repetend of 1⁄n divides φ(n), where φ is the [totient function](https://bliptext.com/articles/totient-function).\nThe length is equal to φ(n) if and only if 10 is a primitive root modulo n.\nIn particular, it follows that L(p) = p − 1 [if and only if](https://bliptext.com/articles/if-and-only-if) p is a prime and 10 is a primitive root modulo p.\nThen, the decimal expansions of n⁄p for n = 1, 2, ..., p − 1, all have period p − 1 and differ only by a cyclic permutation.\nSuch numbers p are called [full repetend primes](https://bliptext.com/articles/full-repetend-prime).\n\n# Reciprocals of composite integers coprime to 10\nIf p is a prime other than 2 or 5, the decimal representation of the fraction 1⁄p2 repeats:\nThe period (repetend length) L(49) must be a factor of λ(49) = 42, where λ(n) is known as the [Carmichael function](https://bliptext.com/articles/carmichael-function).\nThis follows from [Carmichael's theorem](https://bliptext.com/articles/carmichael-function) which states that if n is a positive integer then λ(n) is the smallest integer m such that\nfor every integer a that is [coprime](https://bliptext.com/articles/coprime) to n.\nThe period of 1⁄p2 is usually pTp, where Tp is the period of 1⁄p.\nThere are three known primes for which this is not true, and for those the period of 1⁄p2 is the same as the period of 1⁄p because p2 divides 10p−1−1.\nThese three primes are 3, 487, and 56598313.\nSimilarly, the period of 1⁄pk is usually pk–1Tp\nIf p and q are primes other than 2 or 5, the decimal representation of the fraction 1⁄pq repeats.\nAn example is 1⁄119:\nwhere LCM denotes the [least common multiple](https://bliptext.com/articles/least-common-multiple).\nThe period T of 1⁄pq is a factor of λ(pq) and it happens to be 48 in this case:\nThe period T of 1⁄pq is LCM(Tp, Tq), where Tp is the period of 1⁄p and Tq is the period of 1⁄q.\nIf p, q, r, etc. are primes other than 2 or 5, and k, ℓ, m, etc. are positive integers, then\nis a repeating decimal with a period of where Tpk, Tqℓ, Trm,... are respectively the period of the repeating decimals 1⁄pk, 1⁄qℓ, 1⁄rm,... as defined above.\n\n# Reciprocals of integers not coprime to 10\nAn integer that is not coprime to 10 but has a prime factor other than 2 or 5 has a reciprocal that is eventually periodic, but with a non-repeating sequence of digits that precede the repeating part.\nThe reciprocal can be expressed as:\nwhere a and b are not both zero.\nThis fraction can also be expressed as:\nif a > b, or as\nif b > a, or as\nif a = b.\nThe decimal has:\nFor example 1⁄28 = 0.03571428:\n\n# Converting repeating decimals to fractions\nGiven a repeating decimal, it is possible to calculate the fraction that produces it.\nFor example:\nAnother example:\n\n## A shortcut\nThe procedure below can be applied in particular if the repetend has n digits, all of which are 0 except the final one which is 1.\nFor instance for n = 7:\nSo this particular repeating decimal corresponds to the fraction 1⁄10n − 1, where the denominator is the number written as n 9s.\nKnowing just that, a general repeating decimal can be expressed as a fraction without having to solve an equation.\nFor example, one could reason:\nor\nIt is possible to get a general formula expressing a repeating decimal with an n-digit period (repetend length), beginning right after the decimal point, as a fraction:\nMore explicitly, one gets the following cases:\nIf 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 the n-digit block divided by the one represented by n 9s.\nFor example,\nIf 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).\nFor example,\nAny 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).\nFor example,\nAn even faster method is to ignore the decimal point completely and go like this\nIt follows that any repeating decimal with [period](https://bliptext.com/articles/periodic-function) 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.\nConversely the period of the repeating decimal of a fraction c⁄d will be (at most) the smallest number n such that 10n − 1 is divisible by d.\nFor example, the fraction 2⁄7 has d = 7, and the smallest k that makes 10k − 1 divisible by 7 is k = 6, because 999999 = 7 × 142857.\nThe period of the fraction 2⁄7 is therefore 6.\n\n### In compressed form\nThe following picture suggests kind of compression of the above shortcut.\nThereby \\mathbf{I} represents the digits of the integer part of the decimal number (to the left of the decimal point), \\mathbf{A} makes up the string of digits of the preperiod and its length, and \\mathbf{P} being the string of repeated digits (the period) with length which is nonzero.\nIn the generated fraction, the digit 9 will be repeated times, and the digit 0 will be repeated times.\nNote that in the absence of an **integer** part in the decimal, \\mathbf{I} will be represented by zero, which being to the left of the other digits, will not affect the final result, and may be omitted in the calculation of the generating function.\nExamples:\nThe symbol \\emptyset in the examples above denotes the absence of digits of part \\mathbf{A} in the decimal, and therefore and a corresponding absence in the generated fraction.\n\n# Repeating decimals as infinite series\nA repeating decimal can also be expressed as an [infinite series](https://bliptext.com/articles/infinite-series).\nThat is, a repeating decimal can be regarded as the sum of an infinite number of rational numbers.\nTo take the simplest example,\nThe above series is a [geometric series](https://bliptext.com/articles/geometric-series) with the first term as 1⁄10 and the common factor 1⁄10.\nBecause the absolute value of the common factor is less than 1, we can say that the geometric series [converges](https://bliptext.com/articles/convergent-series) 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.\nSimilarly,\n\n# Multiplication and cyclic permutation\nThe cyclic behavior of repeating decimals in multiplication also leads to the construction of integers which are [cyclically permuted](https://bliptext.com/articles/cyclic-permutation) when multiplied by certain numbers.\nFor example, 102564 × 4 = 410256.\n102564 is the repetend of 4⁄39 and 410256 the repetend of 16⁄39.\n\n# Other properties of repetend lengths\nVarious properties of repetend lengths (periods) are given by Mitchell and Dickson.\nFor some other properties of repetends, see also.\n\n# Extension to other bases\nVarious features of repeating decimals extend to the representation of numbers in all other integer bases, not just base 10:\nQ\nand\nR\n.\nr :=\n,\ndr := d1 + r − 1\nand\n0 ∈ D\n, then a terminating sequence is obviously equivalent to the same sequence with non-terminating repeating part consisting of the digit 0.\nIf the base is positive, then there exists an [order homomorphism](https://bliptext.com/articles/order-isomorphism) from the [lexicographical order](https://bliptext.com/articles/string-computer-science) of the [right-sided infinite strings](https://bliptext.com/articles/sequence) over the [alphabet](https://bliptext.com/articles/alphabet)\nD\ninto some closed interval of the reals, which maps the strings\n0.A1A2...Andb\nand\n0.A1A2...(An+1)d1\nwith\nAi ∈ D\nand\nAn ≠ db\nto the same real number – and there are no other duplicate images.\nIn the decimal system, for example, there is 0.9 = 1.0 = 1; in the [balanced ternary](https://bliptext.com/articles/balanced-ternary) system there is 0.1 = 1.T = 1⁄2.\nbℓ − 1\n. It is the [multiplicative order](https://bliptext.com/articles/multiplicative-order)\nordq(b)\nof the residue class\nb mod q\nwhich is a divisor of the [Carmichael function](https://bliptext.com/articles/carmichael-function)\nλ(q)\nwhich in turn is smaller than q. The repeating sequence is preceded by a transient of finite length if the reduced fraction also shares a prime factor with the base.\nA repeating sequence\nFor example, in [duodecimal](https://bliptext.com/articles/duodecimal), 1⁄2 = 0.6, 1⁄3 = 0.4, 1⁄4 = 0.3 and 1⁄6 = 0.2 all terminate; 1⁄5 = 0.2497 repeats with period length 4, in contrast with the equivalent decimal expansion of 0.2; 1⁄7 = 0.186A35 has period 6 in [duodecimal](https://bliptext.com/articles/duodecimal), just as it does in decimal.\nIf b is an integer base and k is an integer, then\nFor example 1/7 in duodecimal:\nwhich is 0.186A35base12.\n10base12 is 12base10, 102base12 is 144base10, 21base12 is 25base10, A5base12 is 125base10.\n\n## Algorithm for positive bases\nFor a rational\n0 < p⁄q < 1\n(and base\nb ∈ N>1\n) there is the following algorithm producing the repetend together with its length: The first highlighted line calculates the digit z.\nThe subsequent line calculates the new remainder p′ of the division [modulo](https://bliptext.com/articles/modular-arithmetic) the denominator q. As a consequence of the [floor function](https://bliptext.com/articles/floor-and-ceiling-functions) we have\nthus\nand\nBecause all these remainders p are non-negative integers less than q, there can be only a finite number of them with the consequence that they must recur in the loop.\nSuch a recurrence is detected by the [associative array](https://bliptext.com/articles/associative-array).\nThe new digit z is formed in the yellow line, where p is the only non-constant.\nThe length L of the repetend equals the number of the remainders (see also section [Every rational number is either a terminating or repeating decimal](https://bliptext.com/articles/)).\n\n# Applications to cryptography\nRepeating decimals (also called decimal sequences) have found cryptographic and error-correction coding applications.\nIn these applications repeating decimals to base 2 are generally used which gives rise to binary sequences.\nThe maximum length binary sequence for 1⁄p (when 2 is a primitive root of p) is given by:\nThese sequences of period p − 1 have an autocorrelation function that has a negative peak of −1 for shift of p − 1⁄2.\nThe randomness of these sequences has been examined by [diehard tests](https://bliptext.com/articles/diehard-tests).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T10:07:08.045Z","updated_at":"2025-01-26T10:07:08.045Z","search_vector":"'decim':2 'repeat':1","search_content":" \n \n\n# \nA **repeating decimal** or **recurring decimal** is a [decimal representation](https://bliptext.com/articles/decimal-representation) of a number whose [digits](https://bliptext.com/articles/numerical-digit) are eventually [periodic](https://bliptext.com/articles/periodic-function) (that is, after some place, the same sequence of [digits](https://bliptext.com/articles/numerical-digit) is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero [digits](https://bliptext.com/articles/numerical-digit)), the decimal is said to be terminating, and is not considered as repeating.\nIt can be shown that a number is [rational](https://bliptext.com/articles/rational-number) if and only if its decimal representation is repeating or terminating.\nFor example, the decimal representation of 1⁄3 becomes periodic just after the [decimal point](https://bliptext.com/articles/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](https://bliptext.com/articles/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](https://bliptext.com/articles/decimal-point), repeating the 13-digit pattern \"1886792452830\" forever, i.e. 11.18867924528301886792452830....\nThe infinitely repeated digit sequence is called the **repetend** or **reptend**.\nIf 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.\nEvery terminating decimal representation can be written as a [decimal fraction](https://bliptext.com/articles/decimal-fraction), a fraction whose denominator is a [power](https://bliptext.com/articles/power-math) of 10 (e.g. ); it may also be written as a [ratio](https://bliptext.com/articles/ratio) of the form k⁄2n·5m (e.g. ). 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**.\nThis is obtained by decreasing the final (rightmost) non-zero digit by one and appending a repetend of 9.\nTwo examples of this are [0.999...](https://bliptext.com/articles/0-999) and.\n(This type of repeating decimal can be obtained by long division if one uses a modified form of the usual [division algorithm](https://bliptext.com/articles/division-algorithm). )\nAny number that cannot be expressed as a [ratio](https://bliptext.com/articles/ratio) of two [integers](https://bliptext.com/articles/integer) is said to be ir[ratio](https://bliptext.com/articles/ratio)nal.\nTheir decimal representation neither terminates nor infinitely repeats, but extends forever without repetition (see ).\nExamples of such irrational numbers are [[square root of 2|\n√2\n]] and [π](https://bliptext.com/articles/pi).\n\n# Background\n\n## Notation\nThere are several notational conventions for representing repeating decimals.\nNone of them are accepted universally.\nIn English, there are various ways to read repeating decimals aloud.\nFor 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\".\nLikewise, 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\".\n\n## Decimal expansion and recurrence sequence\nIn order to convert a [rational number](https://bliptext.com/articles/rational-number) represented as a fraction into decimal form, one may use [long division](https://bliptext.com/articles/long-division).\nFor example, consider the rational number 5⁄74:\n0.0675 74 ) 5.00000 4.44 560 518 420 370 500\netc. Observe that at each step we have a remainder; the successive remainders displayed above are 56, 42, 50.\nWhen 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.\nTherefore, the decimal repeats: 0.0675 675 675....\nFor any integer fraction A⁄B, the remainder at step k, for any positive integer k, is A × 10k (modulo B).\n\n## Every rational number is either a terminating or repeating decimal\nFor any given divisor, only finitely many different remainders can occur.\nIn the example above, the 74 possible remainders are 0, 1, 2, ..., 73.\nIf at any point in the division the remainder is 0, the expansion terminates at that point.\nThen the length of the repetend, also called \"period\", is defined to be 0.\nIf 0 never occurs as a remainder, then the division process continues forever, and eventually, a remainder must occur that has occurred before.\nThe 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.\nTherefore, the following division will repeat the same results.\nThe repeating sequence of digits is called \"repetend\" which has a certain length greater than 0, also called \"period\".\nIn base 10, a fraction has a repeating decimal if and only if [in lowest terms](https://bliptext.com/articles/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.\n\n## Every repeating or terminating decimal is a rational number\nEach repeating decimal number satisfies a [linear equation](https://bliptext.com/articles/linear-equation) with integer coefficients, and its unique solution is a rational number.\nIn the example above, satisfies the equation\nThe process of how to find these integer coefficients is described [below](https://bliptext.com/articles/).\n\n### Formal proof\nGiven a repeating decimal where a, b, and c are groups of digits, let, the number of digits of b. Multiplying by 10^n separates the repeating and terminating groups:\nIf the decimals terminate (c=0), the proof is complete.\nFor c\\neq0 with digits, let where is a terminating group of digits.\nThen,\nwhere d_i denotes the i-th digit, and\nSince ,\nSince x is the sum of an integer (y -c) and a rational number, x is also rational.\n\n# Table of values\n \n \n \n \n \nThereby fraction is the [unit fraction](https://bliptext.com/articles/unit-fraction) 1⁄n and ℓ10 is the length of the (decimal) repetend.\nThe lengths ℓ10(n) of the decimal repetends of 1⁄n, n = 1, 2, 3, ..., are:\nFor comparison, the lengths ℓ2(n) of the [binary](https://bliptext.com/articles/binary-number) repetends of the fractions 1⁄n, n = 1, 2, 3, ..., are:\nThe decimal repetends of 1⁄n, n = 1, 2, 3, ..., are:\nThe decimal repetend lengths of 1⁄p, p = 2, 3, 5, ...\n(nth prime), are:\nThe least primes p for which 1⁄p has decimal repetend length n, n = 1, 2, 3, ..., are:\nThe least primes p for which k⁄p has n different cycles (1 ≤ k ≤ p−1), n = 1, 2, 3, ..., are:\n\n# Fractions with prime denominators\nA fraction [in lowest terms](https://bliptext.com/articles/in-lowest-terms) with a [prime](https://bliptext.com/articles/prime-number) denominator other than 2 or 5 (i.e. co[prime](https://bliptext.com/articles/prime-number) to 10) always produces a repeating decimal.\nThe length of the repetend (period of the repeating decimal segment) of 1⁄p is equal to the [order](https://bliptext.com/articles/multiplicative-order) of 10 modulo p.\nIf 10 is a [primitive root](https://bliptext.com/articles/primitive-root-modulo-n) modulo p, then the repetend length is equal to p − 1; if not, then the repetend length is a factor of p − 1.\nThis result can be deduced from [Fermat's little theorem](https://bliptext.com/articles/fermat-s-little-theorem), which states that 10p−1 ≡ 1 (mod p).\nThe base-10 [digital root](https://bliptext.com/articles/digital-root) of the repetend of the reciprocal of any prime number greater than 5 is 9.\nIf the repetend length of 1⁄p for prime p is equal to p − 1 then the repetend, expressed as an integer, is called a **cyclic number**.\n\n## Cyclic numbers\nExamples of fractions belonging to this group are:\nThe list can go on to include the fractions 1⁄109, 1⁄113, 1⁄131, 1⁄149, 1⁄167, 1⁄179, 1⁄181, 1⁄193, 1⁄223, 1⁄229, etc..\nEvery proper multiple of a cyclic number (that is, a multiple having the same number of digits) is a rotation:\nThe reason for the cyclic behavior is apparent from an arithmetic exercise of long division of 1⁄7: the sequential remainders are the cyclic sequence {1, 3, 2, 6, 4, 5}.\nSee also the article [142,857](https://bliptext.com/articles/142-857) for more properties of this cyclic number.\nA fraction which is cyclic thus has a recurring decimal of even length that divides into two sequences in [nines' complement](https://bliptext.com/articles/nines-complement) form.\nFor example 1⁄7 starts '142' and is followed by '857' while 6⁄7 (by rotation) starts '857' followed by its nines' complement '142'.\nThe rotation of the repetend of a cyclic number always happens in such a way that each successive repetend is a bigger number than the previous one.\nIn the succession above, for instance, we see that 0.142857... b, or as\nif b > a, or as\nif a = b.\nThe decimal has:\nFor example 1⁄28 = 0.03571428:\n\n# Converting repeating decimals to fractions\nGiven a repeating decimal, it is possible to calculate the fraction that produces it.\nFor example:\nAnother example:\n\n## A shortcut\nThe procedure below can be applied in particular if the repetend has n digits, all of which are 0 except the final one which is 1.\nFor instance for n = 7:\nSo this particular repeating decimal corresponds to the fraction 1⁄10n − 1, where the denominator is the number written as n 9s.\nKnowing just that, a general repeating decimal can be expressed as a fraction without having to solve an equation.\nFor example, one could reason:\nor\nIt is possible to get a general formula expressing a repeating decimal with an n-digit period (repetend length), beginning right after the decimal point, as a fraction:\nMore explicitly, one gets the following cases:\nIf 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 the n-digit block divided by the one represented by n 9s.\nFor example,\nIf 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).\nFor example,\nAny 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).\nFor example,\nAn even faster method is to ignore the decimal point completely and go like this\nIt follows that any repeating decimal with [period](https://bliptext.com/articles/periodic-function) 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.\nConversely the period of the repeating decimal of a fraction c⁄d will be (at most) the smallest number n such that 10n − 1 is divisible by d.\nFor example, the fraction 2⁄7 has d = 7, and the smallest k that makes 10k − 1 divisible by 7 is k = 6, because 999999 = 7 × 142857.\nThe period of the fraction 2⁄7 is therefore 6.\n\n### In compressed form\nThe following picture suggests kind of compression of the above shortcut.\nThereby \\mathbf{I} represents the digits of the integer part of the decimal number (to the left of the decimal point), \\mathbf{A} makes up the string of digits of the preperiod and its length, and \\mathbf{P} being the string of repeated digits (the period) with length which is nonzero.\nIn the generated fraction, the digit 9 will be repeated times, and the digit 0 will be repeated times.\nNote that in the absence of an **integer** part in the decimal, \\mathbf{I} will be represented by zero, which being to the left of the other digits, will not affect the final result, and may be omitted in the calculation of the generating function.\nExamples:\nThe symbol \\emptyset in the examples above denotes the absence of digits of part \\mathbf{A} in the decimal, and therefore and a corresponding absence in the generated fraction.\n\n# Repeating decimals as infinite series\nA repeating decimal can also be expressed as an [infinite series](https://bliptext.com/articles/infinite-series).\nThat is, a repeating decimal can be regarded as the sum of an infinite number of rational numbers.\nTo take the simplest example,\nThe above series is a [geometric series](https://bliptext.com/articles/geometric-series) with the first term as 1⁄10 and the common factor 1⁄10.\nBecause the absolute value of the common factor is less than 1, we can say that the geometric series [converges](https://bliptext.com/articles/convergent-series) 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.\nSimilarly,\n\n# Multiplication and cyclic permutation\nThe cyclic behavior of repeating decimals in multiplication also leads to the construction of integers which are [cyclically permuted](https://bliptext.com/articles/cyclic-permutation) when multiplied by certain numbers.\nFor example, 102564 × 4 = 410256.\n102564 is the repetend of 4⁄39 and 410256 the repetend of 16⁄39.\n\n# Other properties of repetend lengths\nVarious properties of repetend lengths (periods) are given by Mitchell and Dickson.\nFor some other properties of repetends, see also.\n\n# Extension to other bases\nVarious features of repeating decimals extend to the representation of numbers in all other integer bases, not just base 10:\nQ\nand\nR\n.\nr :=\n,\ndr := d1 + r − 1\nand\n0 ∈ D\n, then a terminating sequence is obviously equivalent to the same sequence with non-terminating repeating part consisting of the digit 0.\nIf the base is positive, then there exists an [order homomorphism](https://bliptext.com/articles/order-isomorphism) from the [lexicographical order](https://bliptext.com/articles/string-computer-science) of the [right-sided infinite strings](https://bliptext.com/articles/sequence) over the [alphabet](https://bliptext.com/articles/alphabet)\nD\ninto some closed interval of the reals, which maps the strings\n0.A1A2...Andb\nand\n0.A1A2...(An+1)d1\nwith\nAi ∈ D\nand\nAn ≠ db\nto the same real number – and there are no other duplicate images.\nIn the decimal system, for example, there is 0.9 = 1.0 = 1; in the [balanced ternary](https://bliptext.com/articles/balanced-ternary) system there is 0.1 = 1.T = 1⁄2.\nbℓ − 1\n. It is the [multiplicative order](https://bliptext.com/articles/multiplicative-order)\nordq(b)\nof the residue class\nb mod q\nwhich is a divisor of the [Carmichael function](https://bliptext.com/articles/carmichael-function)\nλ(q)\nwhich in turn is smaller than q. The repeating sequence is preceded by a transient of finite length if the reduced fraction also shares a prime factor with the base.\nA repeating sequence\nFor example, in [duodecimal](https://bliptext.com/articles/duodecimal), 1⁄2 = 0.6, 1⁄3 = 0.4, 1⁄4 = 0.3 and 1⁄6 = 0.2 all terminate; 1⁄5 = 0.2497 repeats with period length 4, in contrast with the equivalent decimal expansion of 0.2; 1⁄7 = 0.186A35 has period 6 in [duodecimal](https://bliptext.com/articles/duodecimal), just as it does in decimal.\nIf b is an integer base and k is an integer, then\nFor example 1/7 in duodecimal:\nwhich is 0.186A35base12.\n10base12 is 12base10, 102base12 is 144base10, 21base12 is 25base10, A5base12 is 125base10.\n\n## Algorithm for positive bases\nFor a rational\n0 1\n) there is the following algorithm producing the repetend together with its length: The first highlighted line calculates the digit z.\nThe subsequent line calculates the new remainder p′ of the division [modulo](https://bliptext.com/articles/modular-arithmetic) the denominator q. As a consequence of the [floor function](https://bliptext.com/articles/floor-and-ceiling-functions) we have\nthus\nand\nBecause all these remainders p are non-negative integers less than q, there can be only a finite number of them with the consequence that they must recur in the loop.\nSuch a recurrence is detected by the [associative array](https://bliptext.com/articles/associative-array).\nThe new digit z is formed in the yellow line, where p is the only non-constant.\nThe length L of the repetend equals the number of the remainders (see also section [Every rational number is either a terminating or repeating decimal](https://bliptext.com/articles/)).\n\n# Applications to cryptography\nRepeating decimals (also called decimal sequences) have found cryptographic and error-correction coding applications.\nIn these applications repeating decimals to base 2 are generally used which gives rise to binary sequences.\nThe maximum length binary sequence for 1⁄p (when 2 is a primitive root of p) is given by:\nThese sequences of period p − 1 have an autocorrelation function that has a negative peak of −1 for shift of p − 1⁄2.\nThe randomness of these sequences has been examined by [diehard tests](https://bliptext.com/articles/diehard-tests).","revisionCount":0},"hasMore":false,"nextCursor":null}

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