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.
31-458: [REDACTED] Look up recurring , recur , or recursion in Wiktionary, the free dictionary. Recurring means occurring repeatedly and can refer to several different things: Mathematics and finance [ edit ] Recurring expense , an ongoing (continual) expenditure Repeating decimal , or recurring decimal, a real number in
62-703: A linear equation with integer coefficients, and its unique solution is a rational number. In the example above, α = 5.8144144144... satisfies the equation The process of how to find these integer coefficients is described below . Given a repeating decimal x = a . b c ¯ {\displaystyle x=a.b{\overline {c}}} where a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} are groups of digits, let n = ⌈ log 10 b ⌉ {\displaystyle n=\lceil {\log _{10}b}\rceil } ,
93-549: A prime denominator other than 2 or 5 (i.e. coprime to 10) always produces a repeating decimal. The length of the repetend (period of the repeating decimal segment) of 1 / p is equal to the order of 10 modulo p . If 10 is a primitive root modulo p , then the repetend length is equal to p − 1; if not, then the repetend length is a factor of p − 1. This result can be deduced from Fermat's little theorem , which states that 10 ≡ 1 (mod p ) . The base-10 digital root of
124-552: A ratio of the form k / 2 ·5 (e.g. 1.585 = 317 / 2 ·5 ). 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
155-431: A rational number represented as a fraction into decimal form, one may use long division . For example, consider the rational number 5 / 74 : 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,
186-515: A character, usually on a television series, that appears from time to time and may grow into a larger role Recurring status , condition whereby a soap opera actor may be used for extended period without being under contract Other uses [ edit ] Recurring (album) , a 1991 album by the British psychedelic-rock group, Spacemen 3 See also [ edit ] All pages with titles beginning with Recurring Topics referred to by
217-465: A character, usually on a television series, that appears from time to time and may grow into a larger role Recurring status , condition whereby a soap opera actor may be used for extended period without being under contract Other uses [ edit ] Recurring (album) , a 1991 album by the British psychedelic-rock group, Spacemen 3 See also [ edit ] All pages with titles beginning with Recurring Topics referred to by
248-1359: A modified form of the usual division algorithm . ) 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 . There are several notational conventions for representing repeating decimals. None of them are accepted universally. In English, there are various ways to read repeating decimals aloud. For example, 1.2 34 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". In order to convert
279-504: A prime p is both full reptend prime and safe prime , then 1 / p will produce a stream of p − 1 pseudo-random digits . Those primes are Some reciprocals of primes that do not generate cyclic numbers are: (sequence A006559 in the OEIS ) The 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
310-430: 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. In 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). They are: A prime is a proper prime if and only if it is a full reptend prime and congruent to 1 mod 10. If
341-422: A real number in the decimal numeral system in which a sequence of digits repeats infinitely Curiously recurring template pattern (CRTP), a software design pattern Processes [ edit ] Recursion , the process of repeating items in a self-similar way Recurring dream , a dream that someone repeatedly experiences over an extended period Television [ edit ] Recurring character ,
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#1733092739811372-430: 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
403-406: 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
434-399: Is a bigger number than the previous one. In 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. A proper prime is
465-447: 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 decimal fraction , a fraction whose denominator is a power of 10 (e.g. 1.585 = 1585 / 1000 ); it may also be written as
496-407: Is cyclic thus has a recurring decimal of even length that divides into two sequences in nines' complement form. For example 1 / 7 starts '142' and is followed by '857' while 6 / 7 (by rotation) starts '857' followed by its nines' complement '142'. The rotation of the repetend of a cyclic number always happens in such a way that each successive repetend
527-508: Is different from Wikidata All article disambiguation pages All disambiguation pages recurring [REDACTED] Look up recurring , recur , or recursion in Wiktionary, the free dictionary. Recurring means occurring repeatedly and can refer to several different things: Mathematics and finance [ edit ] Recurring expense , an ongoing (continual) expenditure Repeating decimal , or recurring decimal,
558-443: Is different from Wikidata All article disambiguation pages All disambiguation pages Repeating decimal 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
589-782: Is the length of the (decimal) repetend. The lengths ℓ 10 ( n ) of the decimal repetends of 1 / n , n = 1, 2, 3, ..., are: For comparison, the lengths ℓ 2 ( n ) of the binary repetends of the fractions 1 / n , n = 1, 2, 3, ..., are: The decimal repetends of 1 / n , n = 1, 2, 3, ..., are: The decimal repetend lengths of 1 / p , p = 2, 3, 5, ... ( n th prime), are: The least primes p for which 1 / p has decimal repetend length n , n = 1, 2, 3, ..., are: The least primes p for which k / p has n different cycles ( 1 ≤ k ≤ p −1 ), n = 1, 2, 3, ..., are: A fraction in lowest terms with
620-411: Is the sum of an integer ( y − c {\displaystyle y-c} ) and a rational number ( 10 k c 10 k − 1 {\textstyle {\frac {10^{k}c}{10^{k}-1}}} ), x {\displaystyle x} is also rational. Thereby fraction is the unit fraction 1 / n and ℓ 10
651-419: The OEIS ). Every proper multiple of a cyclic number (that is, a multiple having the same number of digits) is a rotation: The 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} . See also the article 142,857 for more properties of this cyclic number. A fraction which
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#1733092739811682-1053: The i- th digit , and x = y + ∑ n = 1 ∞ c ( 10 k ) n = y + ( c ∑ n = 0 ∞ 1 ( 10 k ) n ) − c . {\displaystyle x=y+\sum _{n=1}^{\infty }{\frac {c}{{(10^{k})}^{n}}}=y+\left(c\sum _{n=0}^{\infty }{\frac {1}{{(10^{k})}^{n}}}\right)-c.} Since ∑ n = 0 ∞ 1 ( 10 k ) n = 1 1 − 10 − k {\displaystyle \textstyle \sum _{n=0}^{\infty }{\frac {1}{{(10^{k})}^{n}}}={\frac {1}{1-10^{-k}}}} , x = y − c + 10 k c 10 k − 1 . {\displaystyle x=y-c+{\frac {10^{k}c}{10^{k}-1}}.} Since x {\displaystyle x}
713-405: The decimal numeral system in which a sequence of digits repeats infinitely Curiously recurring template pattern (CRTP), a software design pattern Processes [ edit ] Recursion , the process of repeating items in a self-similar way Recurring dream , a dream that someone repeatedly experiences over an extended period Television [ edit ] Recurring character ,
744-399: The decimal repeats: 0.0675 675 675 .... For any integer fraction A / B , the remainder at step k, for any positive integer k , is A × 10 (modulo B ). 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
775-430: The number of digits of b {\displaystyle b} . Multiplying by 10 n {\displaystyle 10^{n}} separates the repeating and terminating groups: 10 n x = a b . c ¯ . {\displaystyle 10^{n}x=ab.{\bar {c}}.} If the decimals terminate ( c = 0 {\displaystyle c=0} ),
806-427: The prime p divides some number 999...999 in which the number of digits divides p − 1. Since the period is never greater than p − 1, we can obtain this by calculating 10 − 1 / p . For example, for 11 we get and then by inspection find the repetend 09 and period of 2. Those reciprocals of primes can be associated with several sequences of repeating decimals. For example,
837-656: The proof is complete. For c ≠ 0 {\displaystyle c\neq 0} with k ∈ N {\displaystyle k\in \mathbb {N} } digits, let x = y . c ¯ {\displaystyle x=y.{\bar {c}}} where y ∈ Z {\displaystyle y\in \mathbb {Z} } is a terminating group of digits. Then, c = d 1 d 2 . . . d k {\displaystyle c=d_{1}d_{2}\,...d_{k}} where d i {\displaystyle d_{i}} denotes
868-703: The repetend of the reciprocal of any prime number greater than 5 is 9. If 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 . Examples of fractions belonging to this group are: The 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. (sequence A001913 in
899-414: The same term [REDACTED] This disambiguation page lists articles associated with the title Recurring . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Recurring&oldid=936998286 " Category : Disambiguation pages Hidden categories: Short description
930-414: The same term [REDACTED] This disambiguation page lists articles associated with the title Recurring . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Recurring&oldid=936998286 " Category : Disambiguation pages Hidden categories: Short description
961-476: 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". 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 2 5 , where m and n are non-negative integers. Each repeating decimal number satisfies