Repeating Decimals Are Rational Numbers

Decimal representation of a number whose digits are periodic

"Repeating fraction" redirects here. Not to exist dislocated with continued fraction.

A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. Information technology can be shown that a number is rational if and only if its decimal representation is repeating or terminating (i.e. all except finitely many digits are cipher). For example, the decimal representation of ane / iii becomes periodic only after the decimal point, repeating the single digit "three" forever, i.e. 0.333.... A more complicated example is 3227 / 555 , whose decimal becomes periodic at the 2nd digit following the decimal point so repeats the sequence "144" forever, i.e. 5.8144144144.... At present, at that place is no single universally accepted notation or phrasing for repeating decimals.

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 exist 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 ten (e.g. ane.585 = 1585 / chiliad ); it may likewise be written as a ratio of the form k / 2 ^{due north} 5 ^m (due east.m. one.585 = 317 / 2³5² ). However, every number with a terminating decimal representation also trivially has a second, culling representation every bit a repeating decimal whose repetend is the digit nine. This is obtained past decreasing the terminal (rightmost) non-nothing digit by one and appending a repetend of nine. i.000... = 0.999... and ane.585000... = i.584999... are ii examples of this. (This type of repeating decimal tin can be obtained by long division if one uses a modified form of the usual partitioning algorithm.^[ii])

Whatever number that cannot be expressed as a ratio of two integers is said to be irrational. Their decimal representation neither terminates nor infinitely repeats just extends forever without repetition (see § #Every rational number is either a terminating or repeating decimal). Examples of such irrational numbers are √2 and π.

Groundwork [edit]

Notation [edit]

In that location are several notational conventions for representing repeating decimals. None of them are accepted universally.

• In the United States, Canada, India, French republic, Germany, Italy, Switzerland, the Czech Republic, Slovakia, and Turkey the convention is to describe a horizontal line (a vinculum) above the repetend. (Meet examples in table beneath, column Vinculum.)
• In the United Kingdom, New Zealand, Australia, India, South korea, and red china, the convention is to place dots above the outermost numerals of the repetend. (See examples in table below, cavalcade Dots.)
• In parts of Europe, Vietnam and Russia, the convention is to enclose the repetend in parentheses. (Come across examples in table below, cavalcade Parentheses.) This can crusade confusion with the note for standard doubtfulness.
• In Spain and some Latin American countries, the arc note over the repetend is also used as an alternative to the vinculum and the dots notation. (Run across examples in tabular array below, column Arc.)
• Informally, repeating decimals are often represented by an ellipsis (iii periods, 0.333...), specially when the previous notational conventions are commencement taught in school. This notation introduces doubt every bit to which digits should be repeated and even whether repetition is occurring at all, since such ellipses are also employed for irrational numbers; π, for instance, tin exist represented equally 3.14159....

Examples

Fraction	Vinculum	Dots	Parentheses	Arc	Ellipsis
1 / 9	0.1	0. . one	0.(1)	0.i	0.111...
1 / iii = 3 / ix	0.iii	0. . 3	0.(three)	0.3	0.333...
2 / 3 = 6 / 9	0.half dozen	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)	0.58three	0.58333 ...
1 / 7 = 142857 / 999999	0.142857	0. . 1 4285 . vii	0.(142857)	0.142857	0.142857142857 ...
1 / 81 = 12345679 / 999999999	0.012345679	0. . 0 1234567 . nine	0.(012345679)	0.012345679	0.012345679012345679 ...
22 / 7 = 3142854 / 999999	3.142857	three. . ane 4285 . vii	three.(142857)	three.142857	3.142857142857 ...

In English, there are various ways to read repeating decimals aloud. For example, 1.234 may be read "i point ii repeating three iv", "1 point two repeated iii 4", "one signal two recurring iii 4", "one point two repetend 3 four" or "one signal 2 into infinity three iv".

Decimal expansion and recurrence sequence [edit]

In order to convert a rational number represented equally a fraction into decimal course, i may apply long partitioning. For instance, consider the rational number 5 / 74 :

                      0.0675                    74 ) five.00000          4.44          560          518          420          370          500        

etc. Observe that at each stride we have a remainder; the successive remainders displayed above are 56, 42, l. When we arrive at 50 every bit the remainder, and bring down the "0", we find ourselves dividing 500 by 74, which is the aforementioned problem we began with. Therefore, the decimal repeats: 0.0675675 675 .....

Every rational number is either a terminating or repeating decimal [edit]

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

If 0 never occurs as a balance, then the division procedure continues forever, and eventually, a remainder must occur that has occurred before. The next step in the division will yield the aforementioned new digit in the quotient, and the same new remainder, as the previous fourth dimension the remainder was the same. Therefore, the following sectionalisation will repeat the same results. The repeating sequence of digits is called "repetend" which has a certain length greater than 0, also chosen "flow".^[3]

Every repeating or terminating decimal is a rational number [edit]

Each repeating decimal number satisfies a linear equation with integer coefficients, and its unique solution is a rational number. To illustrate the latter point, the number α = 5.8144144144... above satisfies the equation 10000α − xα = 58144.144144... − 58.144144... = 58086, whose solution is α = 58086 / 9990 = 3227 / 555 . The process of how to discover these integer coefficients is described below.

Table of values [edit]

• fraction	decimal expansion	ℓ 10	binary expansion	ℓ 2
  ane / ii	0.five	0	0.1	0
  ane / 3	0.3	1	0.01	2
  1 / iv	0.25	0	0.01	0
  1 / 5	0.2	0	0.0011	4
  1 / half-dozen	0.ane6	1	0.001	ii
  1 / 7	0.142857	vi	0.001	3
  1 / 8	0.125	0	0.001	0
  1 / nine	0.ane	1	0.000111	6
  1 / ten	0.1	0	0.00011	four
  1 / 11	0.09	2	0.0001011101	ten
  1 / 12	0.083	1	0.0001	2
  i / thirteen	0.076923	6	0.000100111011	12
  1 / 14	0.0714285	vi	0.0001	three
  ane / xv	0.06	1	0.0001	4
  1 / 16	0.0625	0	0.0001	0

• fraction	decimal expansion	ℓ 10
  1 / 17	0.0588235294117647	16
  ane / eighteen	0.05	1
  1 / 19	0.052631578947368421	18
  one / xx	0.05	0
  1 / 21	0.047619	6
  1 / 22	0.045	2
  i / 23	0.0434782608695652173913	22
  one / 24	0.0416	1
  i / 25	0.04	0
  ane / 26	0.0384615	vi
  1 / 27	0.037	iii
  i / 28	0.03571428	half-dozen
  1 / 29	0.0344827586206896551724137931	28
  ane / 30	0.03	1
  1 / 31	0.032258064516129	15

• fraction	decimal expansion	ℓ 10
  1 / 32	0.03125	0
  ane / 33	0.03	2
  1 / 34	0.02941176470588235	sixteen
  1 / 35	0.0285714	half-dozen
  i / 36	0.027	1
  i / 37	0.027	iii
  1 / 38	0.0263157894736842105	18
  one / 39	0.025641	6
  1 / 40	0.025	0
  ane / 41	0.02439	5
  1 / 42	0.0238095	6
  ane / 43	0.023255813953488372093	21
  1 / 44	0.0227	2
  1 / 45	0.02	i
  1 / 46	0.02173913043478260869565	22

Thereby fraction is the unit of measurement fraction 1 / north and ℓ _x is the length of the (decimal) repetend.

The lengths ℓ ₁₀(north) of the decimal repetends of 1 / n , n = 1, 2, three, ..., are:

0, 0, 1, 0, 0, 1, 6, 0, 1, 0, 2, 1, vi, six, one, 0, 16, 1, 18, 0, 6, two, 22, 1, 0, 6, 3, 6, 28, 1, 15, 0, 2, 16, half-dozen, 1, three, 18, 6, 0, 5, 6, 21, 2, 1, 22, 46, i, 42, 0, xvi, 6, 13, 3, 2, 6, 18, 28, 58, 1, lx, xv, 6, 0, half dozen, 2, 33, 16, 22, 6, 35, ane, eight, 3, 1, ... (sequence A051626 in the OEIS).

For comparing, the lengths ℓ ₂(n) of the binary repetends of the fractions 1 / north , northward = 1, 2, 3, ..., are:

0, 0, ii, 0, 4, ii, 3, 0, half dozen, 4, 10, ii, 12, three, iv, 0, viii, vi, 18, 4, six, 10, xi, 2, twenty, 12, 18, 3, 28, 4, 5, 0, x, viii, 12, 6, 36, 18, 12, 4, 20, 6, 14, 10, 12, 11, ... (=A007733[n], if n not a power of 2 else =0).

The decimal repetends of 1 / due north , n = 1, two, 3, ..., are:

0, 0, three, 0, 0, half dozen, 142857, 0, 1, 0, 09, 3, 076923, 714285, half dozen, 0, 0588235294117647, 5, 052631578947368421, 0, 047619, 45, 0434782608695652173913, 6, 0, 384615, 037, 571428, 0344827586206896551724137931, 3, ... (sequence A036275 in the OEIS).

The decimal repetend lengths of ane / p , p = 2, 3, 5, ... (nth prime number), are:

0, 1, 0, half-dozen, 2, vi, xvi, 18, 22, 28, fifteen, three, five, 21, 46, thirteen, 58, lx, 33, 35, 8, 13, 41, 44, 96, 4, 34, 53, 108, 112, 42, 130, 8, 46, 148, 75, 78, 81, 166, 43, 178, 180, 95, 192, 98, 99, 30, 222, 113, 228, 232, 7, 30, fifty, 256, 262, 268, 5, 69, 28, ... (sequence A002371 in the OEIS).

The least primes p for which ane / p has decimal repetend length n, due north = 1, two, 3, ..., are:

three, 11, 37, 101, 41, 7, 239, 73, 333667, 9091, 21649, 9901, 53, 909091, 31, 17, 2071723, 19, 1111111111111111111, 3541, 43, 23, 11111111111111111111111, 99990001, 21401, 859, 757, 29, 3191, 211, ... (sequence A007138 in the OEIS).

The least primes p for which k / p has due north unlike cycles (ane ≤ k ≤ p−1), n = ane, 2, 3, ..., are:

seven, 3, 103, 53, 11, 79, 211, 41, 73, 281, 353, 37, 2393, 449, 3061, 1889, 137, 2467, 16189, 641, 3109, 4973, 11087, 1321, 101, 7151, 7669, 757, 38629, 1231, ... (sequence A054471 in the OEIS).

Fractions with prime denominators [edit]

A fraction in lowest terms with a prime denominator other than 2 or five (i.east. coprime to 10) always produces a repeating decimal. The length of the repetend (period of the repeating decimal segment) of one / p is equal to the society of 10 modulo p. If x is a primitive root modulo p, the repetend length is equal to p − 1; if not, the repetend length is a factor of p − 1. This result can exist deduced from Fermat's little theorem, which states that 10 ^p−1 ≡ 1 (modernistic p).

The base-10 repetend of the reciprocal of any prime number greater than 5 is divisible past nine.^[4]

If the repetend length of 1 / p for prime number p is equal to p − 1 and so the repetend, expressed every bit an integer, is chosen a cyclic number.

Cyclic numbers [edit]

Examples of fractions belonging to this group are:

• ane / seven = 0.142857, 6 repeating digits
• one / 17 = 0.0588235294117647, sixteen repeating digits
• 1 / xix = 0.052631578947368421, 18 repeating digits
• 1 / 23 = 0.0434782608695652173913, 22 repeating digits
• 1 / 29 = 0.0344827586206896551724137931, 28 repeating digits
• 1 / 47 = 0.0212765957446808510638297872340425531914893617, 46 repeating digits
• 1 / 59 = 0.0169491525423728813559322033898305084745762711864406779661, 58 repeating digits
• 1 / 61 = 0.016393442622950819672131147540983606557377049180327868852459, 60 repeating digits
• one / 97 = 0.010309278350515463917525773195876288659793814432989690721649484536082474226804123711340206185567, 96 repeating digits

The list tin proceed to include the fractions one / 109 , ane / 113 , i / 131 , 1 / 149 , 1 / 167 , 1 / 179 , i / 181 , one / 193 , etc. (sequence A001913 in the OEIS).

Every proper multiple of a cyclic number (that is, a multiple having the same number of digits) is a rotation:

• i / 7 = 1 × 0.142857... = 0.142857...
• ii / vii = ii × 0.142857... = 0.285714...
• three / seven = 3 × 0.142857... = 0.428571...
• 4 / 7 = four × 0.142857... = 0.571428...
• 5 / vii = 5 × 0.142857... = 0.714285...
• 6 / 7 = 6 × 0.142857... = 0.857142...

The reason for the cyclic beliefs is apparent from an arithmetic exercise of long segmentation of ane / 7 : the sequential remainders are the cyclic sequence {ane, three, two, half-dozen, 4, 5}. Run across also the commodity 142,857 for more backdrop of this circadian number.

A fraction which is cyclic thus has a recurring decimal of even length that divides into two sequences in nines' complement form. For example ane / 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 mode that each successive repetend is a bigger number than the previous one. In the succession higher up, for case, nosotros see that 0.142857... < 0.285714... < 0.428571... < 0.571428... < 0.714285... < 0.857142.... This, for cyclic fractions with long repetends, allows usa to easily predict what the effect of multiplying the fraction by any natural number n volition be, as long as the repetend is known.

A proper prime number is a prime number p which ends in the digit 1 in base x and whose reciprocal in base ten 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 every bit does each other digit (namely, p − 1 / 10 times). They are:^{[five] : 166}

61, 131, 181, 461, 491, 541, 571, 701, 811, 821, 941, 971, 1021, 1051, 1091, 1171, 1181, 1291, 1301, 1349, 1381, 1531, 1571, 1621, 1741, 1811, 1829, 1861,... (sequence A073761 in the OEIS).

A prime is a proper prime if and only if information technology is a full reptend prime and congruent to 1 mod 10.

If a prime p is both full reptend prime and safe prime, then ane / p will produce a stream of p − ane pseudo-random digits. Those primes are

seven, 23, 47, 59, 167, 179, 263, 383, 503, 863, 887, 983, 1019, 1367, 1487, 1619, 1823,... (sequence A000353 in the OEIS).

Other reciprocals of primes [edit]

Some reciprocals of primes that do not generate cyclic numbers are:

• i / three = 0.3, which has a period (repetend length) of 1.
• ane / xi = 0.09, which has a period of two.
• 1 / 13 = 0.076923, which has a period of half dozen.
• 1 / 31 = 0.032258064516129, which has a menses of xv.
• 1 / 37 = 0.027, which has a menstruum of 3.
• 1 / 41 = 0.02439, which has a period of 5.
• 1 / 43 = 0.023255813953488372093, which has a period of 21.
• 1 / 53 = 0.0188679245283, which has a period of xiii.
• 1 / 67 = 0.014925373134328358208955223880597, which has a period of 33.

(sequence A006559 in the OEIS)

The reason is that iii is a divisor of ix, 11 is a divisor of 99, 41 is a divisor of 99999, etc. To discover the flow of ane / p , nosotros tin cheque whether the prime number p divides some number 999...999 in which the number of digits divides p − one. Since the period is never greater than p − 1, nosotros can obtain this by calculating x ^p−1 − one / p . For example, for 11 we get

${\displaystyle {\frac {10^{eleven-i}-1}{11}}=909090909}$

then past inspection find the repetend 09 and flow of 2.

Those reciprocals of primes can be associated with several sequences of repeating decimals. For example, the multiples of 1 / 13 tin can exist divided into ii sets, with different repetends. The first set is:

• 1 / 13 = 0.076923...
• 10 / thirteen = 0.769230...
• 9 / 13 = 0.692307...
• 12 / 13 = 0.923076...
• 3 / xiii = 0.230769...
• 4 / xiii = 0.307692...,

where the repetend of each fraction is a circadian re-organisation of 076923. The second set is:

• two / xiii = 0.153846...
• 7 / thirteen = 0.538461...
• five / xiii = 0.384615...
• 11 / 13 = 0.846153...
• 6 / 13 = 0.461538...
• 8 / 13 = 0.615384...,

where the repetend of each fraction is a cyclic re-arrangement of 153846.

In general, the fix of proper multiples of reciprocals of a prime number p consists of n subsets, each with repetend lengthk, where nk =p − 1.

Totient rule [edit]

For an arbitrary integer n, the length L(due north) of the decimal repetend of 1 / n divides φ(due north), where φ is the totient function. The length is equal to φ(n) if and only if ten is a archaic root modulo n.^[vi]

In item, information technology follows that L(p) = p − 1 if and only if p is a prime number and x is a primitive root modulo p. Then, the decimal expansions of north / p for due north = 1, ii, ..., p − one, all have period p − ane and differ simply past a cyclic permutation. Such numbers p are called full repetend primes.

Reciprocals of composite integers coprime to 10 [edit]

If p is a prime other than 2 or 5, the decimal representation of the fraction 1 / p ² repeats:

1 / 49 = 0.020408163265306122448979591836734693877551.

The period (repetend length) Fifty(49) must be a factor of λ(49) = 42, where λ(n) is known as the Carmichael function. This follows from Carmichael's theorem which states that if n is a positive integer then λ(n) is the smallest integer thousand such that

${\displaystyle a^{m}\equiv 1{\pmod {n}}}$

for every integer a that is coprime to n.

The period of 1 / p ² is unremarkably pT _p , where T _p is the period of one / p . There are three known primes for which this is non truthful, and for those the period of 1 / p ² is the same as the menstruation of 1 / p because p ⁱⁱ divides 10 ^p−1−1. These iii primes are 3, 487, and 56598313 (sequence A045616 in the OEIS).^[7]

Similarly, the period of 1 / p ^chiliad is usually p ^yard–one T _p

If p and q are primes other than 2 or 5, the decimal representation of the fraction one / pq repeats. An case is ane / 119 :

119 = vii × 17

λ(vii × 17) = LCM(λ(7), λ(17)) = LCM(6, 16) = 48,

where LCM denotes the to the lowest degree mutual multiple.

The menstruum T of 1 / pq is a factor of λ(pq) and it happens to be 48 in this example:

i / 119 = 0.008403361344537815126050420168067226890756302521.

The period T of 1 / pq is LCM(T _p ,T _q ), where T _p is the period of 1 / p and T _q is the menstruum of one / q .

If p, q, r, etc. are primes other than two or 5, and k, ℓ, m, etc. are positive integers, then

${\displaystyle {\frac {ane}{p^{k}q^{\ell }r^{g}\cdots }}}$

is a repeating decimal with a period of

${\displaystyle \operatorname {LCM} (T_{p^{g}},T_{q^{\ell }},T_{r^{yard}},\ldots )}$

where T_p^g , T_q^ℓ , T_r^thou ,... are respectively the period of the repeating decimals ane / p^grand , one / q^ℓ , 1 / r^k ,... as divers higher up.

Reciprocals of integers not coprime to 10 [edit]

An integer that is non coprime to 10 only has a prime number factor other than 2 or five has a reciprocal that is somewhen periodic, but with a non-repeating sequence of digits that precede the repeating part. The reciprocal can be expressed every bit:

${\displaystyle {\frac {one}{2^{a}5^{b}p^{k}q^{\ell }\cdots }}\,,}$

where a and b are non both zero.

This fraction can also be expressed as:

${\displaystyle {\frac {5^{a-b}}{10^{a}p^{one thousand}q^{\ell }\cdots }}\,,}$

if a > b, or as

${\displaystyle {\frac {ii^{b-a}}{10^{b}p^{one thousand}q^{\ell }\cdots }}\,,}$

if b > a, or every bit

${\displaystyle {\frac {1}{x^{a}p^{k}q^{\ell }\cdots }}\,,}$

if a = b.

The decimal has:

• An initial transient of max(a,b) digits afterward the decimal point. Some or all of the digits in the transient can exist zeros.
• A subsequent repetend which is the aforementioned every bit that for the fraction ane / p^k q^ℓ ⋯ .

For example 1 / 28 = 0.03571428:

• a = two, b = 0, and the other factors p^k q^ℓ ⋯ = vii
• in that location are 2 initial non-repeating digits, 03; and
• there are six repeating digits, 571428, the same amount as 1 / 7 has.

Converting repeating decimals to fractions [edit]

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

${\displaystyle x}$	${\displaystyle =0.333333\ldots }$
${\displaystyle 10x}$	${\displaystyle =3.333333\ldots }$	(multiply each side of the higher up line by ten)
${\displaystyle 9x}$	${\displaystyle =iii}$	(subtract the 1st line from the second)
${\displaystyle x}$	${\displaystyle ={\frac {3}{9}}={\frac {ane}{3}}}$	(reduce to lowest terms)

Some other example:

${\displaystyle x}$	${\displaystyle =\ \ \ \ 0.836363636\ldots }$
${\displaystyle 10x}$	${\displaystyle =\ \ \ \ 8.36363636\ldots }$	(move decimal to commencement of repetition = move by 1 identify = multiply by 10)
${\displaystyle 1000x}$	${\displaystyle =836.36363636\ldots }$	(collate 2nd repetition here with 1st above = move by ii places = multiply by 100)
${\displaystyle 990x}$	${\displaystyle =828}$	(subtract to clear decimals)
${\displaystyle ten}$	${\displaystyle ={\frac {828}{990}}={\frac {xviii\cdot 46}{eighteen\cdot 55}}={\frac {46}{55}}}$	(reduce to everyman terms)

A shortcut [edit]

The procedure below can be applied in particular if the repetend has n digits, all of which are 0 except the concluding one which is one. For example for n = 7:

${\displaystyle {\begin{aligned}x&=0.000000100000010000001\ldots \\10^{vii}10&=i.000000100000010000001\ldots \\\left(10^{7}-1\right)x=9999999x&=one\\x&={\frac {1}{10^{vii}-1}}={\frac {i}{9999999}}\end{aligned}}}$

And then this particular repeating decimal corresponds to the fraction 1 / 10 ⁿ  − one , where the denominator is the number written as n 9s. Knowing just that, a general repeating decimal tin be expressed every bit a fraction without having to solve an equation. For case, one could reason:

${\displaystyle {\begin{aligned}seven.48181818\ldots &=seven.3+0.18181818\ldots \\[8pt]&={\frac {73}{10}}+{\frac {18}{99}}={\frac {73}{ten}}+{\frac {9\cdot 2}{ix\cdot 11}}={\frac {73}{10}}+{\frac {2}{xi}}\\[12pt]&={\frac {xi\cdot 73+x\cdot 2}{ten\cdot eleven}}={\frac {823}{110}}\end{aligned}}}$

It is possible to get a general formula expressing a repeating decimal with an north-digit period (repetend length), beginning correct after the decimal signal, equally a fraction:

${\displaystyle {\begin{aligned}ten&=0.{\overline {a_{one}a_{2}\cdots a_{n}}}\\10^{n}x&=a_{one}a_{2}\cdots a_{n}.{\overline {a_{i}a_{2}\cdots a_{n}}}\\\left(10^{north}-one\right)10=99\cdots 99x&=a_{1}a_{2}\cdots a_{n}\\x&={\frac {a_{1}a_{ii}\cdots a_{n}}{10^{due north}-1}}={\frac {a_{ane}a_{2}\cdots a_{n}}{99\cdots 99}}\end{aligned}}}$

More explicitly, one gets the following cases:

If the repeating decimal is between 0 and 1, and the repeating cake is n digits long, first occurring right after the decimal indicate, so the fraction (not necessarily reduced) volition be the integer number represented by the northward-digit cake divided by the i represented by north 9s. For example,

• 0.444444... = 4 / 9 since the repeating cake is iv (a 1-digit block),
• 0.565656... = 56 / 99 since the repeating cake is 56 (a 2-digit cake),
• 0.012012... = 12 / 999 since the repeating block is 012 (a 3-digit block); this further reduces to 4 / 333 .
• 0.999999... = 9 / 9 = one, since the repeating block is 9 (too a 1-digit block)

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

• 0.000444... = iv / 9000 since the repeating block is 4 and this block is preceded by iii zeros,
• 0.005656... = 56 / 9900 since the repeating block is 56 and information technology is preceded past 2 zeros,
• 0.00012012... = 12 / 99900 = 1 / 8325 since the repeating block is 012 and it is preceded by ii zeros.

Whatsoever repeating decimal not of the form described above can exist written as a sum of a terminating decimal and a repeating decimal of 1 of the two higher up types (actually the starting time type suffices, only that could require the terminating decimal to be negative). For instance,

• 1.23444... = 1.23 + 0.00444... = 123 / 100 + 4 / 900 = 1107 / 900 + 4 / 900 = 1111 / 900
  • or alternatively 1.23444... = 0.79 + 0.44444... = 79 / 100 + iv / 9 = 711 / 900 + 400 / 900 = 1111 / 900
• 0.3789789... = 0.3 + 0.0789789... = iii / 10 + 789 / 9990 = 2997 / 9990 + 789 / 9990 = 3786 / 9990 = 631 / 1665
  • or alternatively 0.3789789... = −0.6 + 0.9789789... = − half-dozen / x + 978/999 = − 5994 / 9990 + 9780 / 9990 = 3786 / 9990 = 631 / 1665

An even faster method is to ignore the decimal point completely and go similar this

• 1.23444... = 1234 − 123 / 900 = 1111 / 900 (denominator has one 9 and two 0s because ane digit repeats and there are two non-repeating digits later the decimal point)
• 0.3789789... = 3789 − 3 / 9990 = 3786 / 9990 (denominator has three 9s and one 0 because three digits repeat and at that place is one non-repeating digit after the decimal bespeak)

It follows that any repeating decimal with period north, and k digits afterwards the decimal betoken that do not belong to the repeating part, can be written as a (not necessarily reduced) fraction whose denominator is (10 ⁿ  − i)10 ^k .

Conversely the menses of the repeating decimal of a fraction c / d will be (at most) the smallest number north such that 10 ⁿ  − i is divisible past d.

For example, the fraction two / seven has d = seven, and the smallest yard that makes 10 ^k  − 1 divisible past 7 is k = 6, because 999999 = 7 × 142857. The period of the fraction 2 / vii is therefore 6.

In compressed course [edit]

The following motion-picture show suggests kind of compression of the higher up shortcut. Thereby ${\displaystyle \mathbf {I} }$ represents the digits of the integer part of the decimal number (to the left of the decimal point), ${\displaystyle \mathbf {A} }$ makes upwardly the string of digits of the preperiod and ${\displaystyle \#\mathbf {A} }$ its length, and ${\displaystyle \mathbf {P} }$ being the cord of repeated digits (the period) with length ${\displaystyle \#\mathbf {P} }$ which is nonzero.

In the generated fraction, the digit ${\displaystyle 9}$ will be repeated ${\displaystyle \#\mathbf {P} }$ times, and the digit ${\displaystyle 0}$ will be repeated ${\displaystyle \#\mathbf {A} }$ times.

Note that in the absence of an integer part in the decimal, ${\displaystyle \mathbf {I} }$ volition exist represented by zero, which being to the left of the other digits, will non touch on the concluding event, and may be omitted in the calculation of the generating function.

Examples:

$${\displaystyle {\brainstorm{array}{lllll}three.254444...&=iii.25{\overline {four}}&={\brainstorm{Bmatrix}\mathbf {I} =3&\mathbf {A} =25&\mathbf {P} =4\\&\#\mathbf {A} =2&\#\mathbf {P} =i\terminate{Bmatrix}}&={\frac {3254-325}{900}}&={\frac {2929}{900}}\\\\0.512512...&=0.{\overline {512}}&={\begin{Bmatrix}\mathbf {I} =0&\mathbf {A} =\emptyset &\mathbf {P} =512\\&\#\mathbf {A} =0&\#\mathbf {P} =3\cease{Bmatrix}}&={\frac {512-0}{999}}&={\frac {512}{999}}\\\\1.09191...&=1.0{\overline {91}}&={\begin{Bmatrix}\mathbf {I} =1&\mathbf {A} =0&\mathbf {P} =91\\&\#\mathbf {A} =1&\#\mathbf {P} =2\end{Bmatrix}}&={\frac {1091-10}{990}}&={\frac {1081}{990}}\\\\1.333...&=i.{\overline {three}}&={\brainstorm{Bmatrix}\mathbf {I} =1&\mathbf {A} =\emptyset &\mathbf {P} =iii\\&\#\mathbf {A} =0&\#\mathbf {P} =1\finish{Bmatrix}}&={\frac {13-1}{9}}&={\frac {12}{9}}&={\frac {4}{3}}\\\\0.3789789...&=0.three{\overline {789}}&={\begin{Bmatrix}\mathbf {I} =0&\mathbf {A} =iii&\mathbf {P} =789\\&\#\mathbf {A} =ane&\#\mathbf {P} =3\end{Bmatrix}}&={\frac {3789-3}{9990}}&={\frac {3786}{9990}}&={\frac {631}{1665}}\end{array}}}$$

The symbol ${\displaystyle \emptyset }$ in the examples to a higher place denotes the absenteeism of digits of office ${\displaystyle \mathbf {A} }$ in the decimal, and therefore ${\displaystyle \#\mathbf {A} =0}$ and a corresponding absenteeism in the generated fraction.

Repeating decimals as space series [edit]

A repeating decimal can also exist expressed as an infinite series. That is, a repeating decimal tin be regarded as the sum of an infinite number of rational numbers. To take the simplest case,

${\displaystyle 0.{\overline {1}}={\frac {1}{x}}+{\frac {1}{100}}+{\frac {ane}{m}}+\cdots =\sum _{due north=1}^{\infty }{\frac {ane}{x^{northward}}}}$

The above series is a geometric serial with the first term as 1 / 10 and the common factor one / 10 . Because the absolute value of the common factor is less than ane, we can say that the geometric series converges and observe the exact value in the form of a fraction past using the following formula where a is the start term of the series and r is the mutual factor.

${\displaystyle {\frac {a}{1-r}}={\frac {\frac {1}{10}}{1-{\frac {1}{10}}}}={\frac {i}{10-i}}={\frac {ane}{nine}}}$

Similarly,

${\displaystyle {\begin{aligned}0.{\overline {142857}}&={\frac {142857}{ten^{6}}}+{\frac {142857}{10^{12}}}+{\frac {142857}{10^{18}}}+\cdots =\sum _{n=1}^{\infty }{\frac {142857}{10^{6n}}}\\[6px]\implies &\quad {\frac {a}{1-r}}={\frac {\frac {142857}{10^{six}}}{one-{\frac {1}{10^{half-dozen}}}}}={\frac {142857}{ten^{6}-1}}={\frac {142857}{999999}}={\frac {1}{seven}}\end{aligned}}}$

Multiplication and cyclic permutation [edit]

The circadian behavior of repeating decimals in multiplication also leads to the construction of integers which are cyclically permuted when multiplied by certain numbers. For example, 102564 × iv = 410256. 102564 is the repetend of 4 / 39 and 410256 the repetend of sixteen / 39 .

Other properties of repetend lengths [edit]

Diverse properties of repetend lengths (periods) are given by Mitchell^[8] and Dickson.^[9]

• The period of 1 / k for integer k is always ≤k − i.
• If p is prime, the catamenia of 1 / p divides evenly into p − 1.
• If k is composite, the period of 1 / k is strictly less than m − 1.
• The period of c / 1000 , for c coprime to k, equals the catamenia of ane / k .
• If k = 2 ^a 5 ^b due north where north > i and northward is not divisible by ii or 5, so the length of the transient of 1 / k is max(a,b), and the flow equals r, where r is the smallest integer such that ten ^r ≡ one (mod n).
• If p, p′, p″,... are distinct primes, then the flow of one / p p′ p″ ⋯ equals the lowest mutual multiple of the periods of ane / p , 1 / p′ , i / p″ ,....
• If thousand and 1000′ have no common prime factors other than 2 or 5, then the catamenia of 1 / g grand′ equals the to the lowest degree common multiple of the periods of 1 / k and 1 / chiliad′ .
• For prime p, if

${\displaystyle {\text{period}}\left({\frac {i}{p}}\right)={\text{period}}\left({\frac {1}{p^{2}}}\right)=\cdots ={\text{period}}\left({\frac {1}{p^{thou}}}\correct)}$

for some m, but

${\displaystyle {\text{period}}\left({\frac {1}{p^{yard}}}\right)\neq {\text{flow}}\left({\frac {ane}{p^{m+ane}}}\right),}$

so for c ≥ 0 we have

${\displaystyle {\text{period}}\left({\frac {1}{p^{m+c}}}\right)=p^{c}\cdot {\text{period}}\left({\frac {1}{p}}\right).}$

• If p is a proper prime number ending in a 1, that is, if the repetend of i / p is a cyclic number of length p − ane and p = 10h + 1 for some h, then each digit 0, i, ..., 9 appears in the repetend exactly h = p − one / ten times.

For some other properties of repetends, see also.^[ten]

Extension to other bases [edit]

Various features of repeating decimals extend to the representation of numbers in all other integer bases, not just base 10:

• Whatever existent number can be represented as an integer part followed by a radix point (the generalization of a decimal betoken to non-decimal systems) followed by a finite or infinite number of digits.
• If the base is an integer, a terminating sequence obviously represents a rational number.
• A rational number has a terminating sequence if all the prime number factors of the denominator of the fully reduced partial form are also factors of the base. These numbers make upward a dense set up in Q and R .
• If the positional numeral system is a standard one, that is it has base

${\displaystyle b\in \mathbb {Z} \smallsetminus \{-ane,0,1\}}$

combined with a sequent prepare of digits

${\displaystyle D:=\{d_{1},d_{1}+1,\dots ,d_{r}\}}$

with r := |b|, d_r  := d_one + r − i and 0 ∈ D , then a terminating sequence is obviously equivalent to the same sequence with non-terminating repeating part consisting of the digit 0. If the base of operations is positive, then there exists an order homomorphism from the lexicographical lodge of the right-sided space strings over the alphabet D into some closed interval of the reals, which maps the strings 0.A ₁ A _two...A _n d_b and 0.A ₁ A ₂...(A_n +ane) d _ane with A_i ∈ D and A_n ≠ d_b to the same real number – and there are no other duplicate images. In the decimal organisation, for instance, there is 0.ix = ane.0 = ane; in the balanced ternary system at that place is 0.1 = ane.T = ane / 2 .

• A rational number has an indefinitely repeating sequence of finite length l, if the reduced fraction's denominator contains a prime factor that is non a factor of the base. If q is the maximal factor of the reduced denominator which is coprime to the base of operations, l is the smallest exponent such that q divides b ^l − 1. It is the multiplicative order ord _q (b) of the residue class b mod q which is a divisor of the Carmichael function λ(q) which 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 gene with the base. A repeating sequence

${\displaystyle \left(0.{\overline {A_{1}A_{2}\ldots A_{\ell }}}\correct)_{b}}$

represents the fraction

${\displaystyle {\frac {(A_{ane}A_{2}\ldots A_{\ell })_{b}}{b^{l}-1}}.}$

• An irrational number has a representation of infinite length that is non, from whatever indicate, an indefinitely repeating sequence of finite length.

For example, in duodecimal, 1 / 2 = 0.6, one / 3 = 0.iv, 1 / four = 0.3 and 1 / 6 = 0.2 all cease; ane / v = 0.2497 repeats with period length iv, in contrast with the equivalent decimal expansion of 0.2; 1 / seven = 0.186A35 has flow 6 in duodecimal, just as it does in decimal.

If b is an integer base and k is an integer, so

${\displaystyle {\frac {one}{yard}}={\frac {1}{b}}+{\frac {(b-chiliad)^{i}}{b^{two}}}+{\frac {(b-k)^{2}}{b^{3}}}+{\frac {(b-yard)^{3}}{b^{4}}}+\cdots +{\frac {(b-k)^{Due north-1}}{b^{N}}}+\cdots ={\frac {1}{b}}{\frac {1}{1-{\frac {b-grand}{b}}}}.}$

For example i / vii in duodecimal:

1 / 7 = ( 1 / 10 + v / 10² + 21 / ten³ + A5 / 10⁴ + 441 / 10⁵ + 1985 / 10⁶ + ...)_base12

which is 0.186A35 _base12. 10_base12 is 12_base10, ten² _base12 is 144_base10, 21_base12 is 25_base10, A5_base12 is 125_base10.

Algorithm for positive bases [edit]

For a rational 0 < p / q < 1 (and base b ∈ N _>one ) there is the post-obit algorithm producing the repetend together with its length:

                        function            b_adic            (            b            ,            p            ,            q            )            // b ≥ ii; 0 < p < q            static            digits            =            "0123..."            ;            // upwardly to the digit with value b–1            begin            southward            =            ""            ;            // the string of digits            pos            =            0            ;            // all places are right to the radix bespeak            while            not            defined            (            occurs            [            p            ])            practice            occurs            [            p            ]            =            pos            ;            // the position of the place with remainder p            bp            =            b            *            p            ;                          z              =              floor              (              bp              /              q              )              ;              // alphabetize z of digit within: 0 ≤ z ≤ b-1                                      p              =              b              *              p              −              z              *              q              ;              // 0 ≤ p < q                        if            p            =            0            then            L            =            0            ;            if            not            z            =            0            then            due south            =            s            .            substring            (            digits            ,            z            ,            i            )            end            if            return            (            s            )            ;            end            if            south            =            s            .            substring            (            digits            ,            z            ,            1            )            ;            // append the graphic symbol of the digit            pos            +=            one            ;            end            while            L            =            pos            -            occurs            [            p            ]            ;            // the length of the repetend (being < q)            // marking the digits of the repetend past a vinculum:            for            i            from            occurs            [            p            ]            to            pos            -            i            practise            substring            (            south            ,            i            ,            one            )            =            overline            (            substring            (            s            ,            i            ,            1            ))            ;            end            for            return            (            s            )            ;            end            function          

The first highlighted line calculates the digit z.

The subsequent line calculates the new residual p′ of the division modulo the denominator q. Equally a outcome of the floor function flooring we have

${\displaystyle {\frac {bp}{q}}-1\;\;<\;\;z=\left\lfloor {\frac {bp}{q}}\right\rfloor \;\;\leq \;\;{\frac {bp}{q}},}$

thus

${\displaystyle bp-q<zq\quad \implies \quad p':=bp-zq<q}$

and

${\displaystyle zq\leq bp\quad \implies \quad 0\leq bp-zq=:p'\,.}$

Because all these remainders p are non-negative integers less than q, in that location can exist only a finite number of them with the consequence that they must recur in the while loop. Such a recurrence is detected past the associative array occurs. The new digit z is formed in the yellow line, where p is the just non-constant. The length Fifty of the repetend equals the number of the remainders (see too department Every rational number is either a terminating or repeating decimal).

Applications to cryptography [edit]

Repeating decimals (also called decimal sequences) take found cryptographic and error-correction coding applications.^[xi] In these applications repeating decimals to base 2 are generally used which gives rise to binary sequences. The maximum length binary sequence for i / p (when ii is a primitive root of p) is given past:^[12]

${\displaystyle a(i)=2^{i}{\bmod {p}}{\bmod {2}}}$

These sequences of period p − one have an autocorrelation function that has a negative peak of −1 for shift of p − 1 / 2 . The randomness of these sequences has been examined past diehard tests.^[xiii]

See likewise [edit]

• Decimal representation
• Full reptend prime
• Midy's theorem
• Parasitic number
• Trailing zero
• Unique prime
• 0.999..., a repeating decimal equal to one
• Pigeonhole principle

References and remarks [edit]

1. ^ Courant, R. and Robbins, H. What Is Mathematics?: An Uncomplicated Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, 1996: p. 67.
2. ^ Beswick, Kim (2004), "Why Does 0.999... = 1?: A Perennial Question and Number Sense", Australian Mathematics Teacher, 60 (4): 7–9
3. ^ For a base b and a divisor n, in terms of group theory this length divides

   ${\displaystyle \operatorname {ord} _{n}(b):=\min\{50\in \mathbb {North} \,\mid \,b^{50}\equiv 1{\text{ mod }}n\}}$

   (with modular arithmetic ≡ one mod n ) which divides the Carmichael function

   ${\displaystyle \lambda (n):=\max\{\operatorname {ord} _{n}(b)\,\mid \,\gcd(b,north)=1\}}$

   which over again divides Euler'south totient function φ(n).
4. ^ Gray, Alexander J., "Digital roots and reciprocals of primes", Mathematical Gazette 84.09, March 2000, p. 86.
5. ^ Dickson, L. East., History of the Theory of Numbers, Book one, Chelsea Publishing Co., 1952.
6. ^ William Eastward. Heal. Some Properties of Repetends. Annals of Mathematics, Vol. iii, No. iv (Aug., 1887), pp. 97–103
7. ^ Albert H. Beiler, Recreations in the Theory of Numbers, p. 79
8. ^ Mitchell, Douglas W., "A nonlinear random number generator with known, long cycle length", Cryptologia 17, January 1993, pp. 55–62.
9. ^ Dickson, Leonard Eastward., History of the Theory of Numbers, Vol. I, Chelsea Publ. Co., 1952 (orig. 1918), pp. 164–173.
10. ^ Armstrong, Due north. J., and Armstrong, R. J., "Some properties of repetends", Mathematical Gazette 87, November 2003, pp. 437–443.
11. ^ Kak, Subhash, Chatterjee, A. "On decimal sequences". IEEE Transactions on Information Theory, vol. IT-27, pp. 647–652, September 1981.
12. ^ Kak, Subhash, "Encryption and error-correction using d-sequences". IEEE Transactios on Computers, vol. C-34, pp. 803–809, 1985.
13. ^ Bellamy, J. "Randomness of D sequences via diehard testing". 2013. arXiv:1312.3618

External links [edit]

• Weisstein, Eric Due west. "Repeating Decimal". MathWorld.

Repeating Decimals Are Rational Numbers,

Source: https://en.wikipedia.org/wiki/Repeating_decimal

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