Repeating decimal

Repeating decimal

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In arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

, a decimal representation of a real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

 is called a repeating decimal (or recurring decimal) if at some point it becomes periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

, that is, if there is some finite sequence of digits that is repeated indefinitely. For example, the decimal representation of or 0.3 (spoken as "0.3 repeating", or "0.3 recurring") becomes periodic just after the decimal point, repeating the single-digit sequence "3" infinitely. A somewhat more complicated example is where the decimal representation becomes periodic at the second digit after the decimal point, repeating the sequence of digits "144" infinitely.

A real number has an ultimately periodic decimal representation if and only if it is a rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

. Rational numbers are numbers that can be expressed in the form a/b where a and b are integers  and b is non-zero. This form is known as a vulgar fraction. On the one hand, the decimal representation of a rational number is ultimately periodic because it can be determined by a long division
Long division
In arithmetic, long division is a standard procedure suitable for dividing simple or complex multidigit numbers. It breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a...

 process, which must ultimately become periodic as there are only finitely many different remainders and so eventually it will find a remainder that has occurred before. On the other hand, each repeating decimal number satisfies a linear equation
Linear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....

 with integral coefficients, and its unique solution is a rational number. To illustrate the latter point, the number above satisfies the equation whose solution is

A decimal representation written with a repeating final 0 is said to terminate before these zeros. Instead of "1.585000…" one simply writes "1.585". The decimal is also called a terminating decimal. Terminating decimals represent rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s of the form k/(2n5m). For example, . A terminating decimal can be written as a decimal fraction: . However, a terminating decimal also has a representation as a repeating decimal, obtained by decreasing the final (nonzero) digit by one and appending an infinitely repeating sequence of nines.
In arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

, a decimal representation of a real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

 is called a repeating decimal (or recurring decimal) if at some point it becomes periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

, that is, if there is some finite sequence of digits that is repeated indefinitely. For example, the decimal representation of {{nowrap|1/3 {{=}} 0.3333333…}} or 0.3 (spoken as "0.3 repeating", or "0.3 recurring") becomes periodic just after the decimal point, repeating the single-digit sequence "3" infinitely. A somewhat more complicated example is {{nowrap|3227/555 {{=}} 5.8144144144…,}} where the decimal representation becomes periodic at the second digit after the decimal point, repeating the sequence of digits "144" infinitely.

A real number has an ultimately periodic decimal representation if and only if it is a rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

. Rational numbers are numbers that can be expressed in the form a/b where a and b are integers  and b is non-zero. This form is known as a vulgar fraction. On the one hand, the decimal representation of a rational number is ultimately periodic because it can be determined by a long division
Long division
In arithmetic, long division is a standard procedure suitable for dividing simple or complex multidigit numbers. It breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a...

 process, which must ultimately become periodic as there are only finitely many different remainders and so eventually it will find a remainder that has occurred before. On the other hand, each repeating decimal number satisfies a linear equation
Linear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....

 with integral coefficients, and its unique solution is a rational number. To illustrate the latter point, the number {{nowrap|α {{=}} 5.8144144144…}} above satisfies the equation {{nowrap|10000α − 10α {{=}} 58144.144144… − 58.144144… {{=}} 58086,}} whose solution is {{nowrap|α {{=}} 58086/9990 {{=}} 3227/555.}}

A decimal representation written with a repeating final 0 is said to terminate before these zeros. Instead of "1.585000…" one simply writes "1.585". The decimal is also called a terminating decimal. Terminating decimals represent rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s of the form k/(2n5m). For example, {{nowrap|1.585 {{=}} 317/200 {{=}} 317/(2352)}}. A terminating decimal can be written as a decimal fraction: {{nowrap|317/200 {{=}} 1585/1000}}. However, a terminating decimal also has a representation as a repeating decimal, obtained by decreasing the final (nonzero) digit by one and appending an infinitely repeating sequence of nines.
In arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

, a decimal representation of a real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

 is called a repeating decimal (or recurring decimal) if at some point it becomes periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

, that is, if there is some finite sequence of digits that is repeated indefinitely. For example, the decimal representation of {{nowrap|1/3 {{=}} 0.3333333…}} or 0.3 (spoken as "0.3 repeating", or "0.3 recurring") becomes periodic just after the decimal point, repeating the single-digit sequence "3" infinitely. A somewhat more complicated example is {{nowrap|3227/555 {{=}} 5.8144144144…,}} where the decimal representation becomes periodic at the second digit after the decimal point, repeating the sequence of digits "144" infinitely.

A real number has an ultimately periodic decimal representation if and only if it is a rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

. Rational numbers are numbers that can be expressed in the form a/b where a and b are integers  and b is non-zero. This form is known as a vulgar fraction. On the one hand, the decimal representation of a rational number is ultimately periodic because it can be determined by a long division
Long division
In arithmetic, long division is a standard procedure suitable for dividing simple or complex multidigit numbers. It breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a...

 process, which must ultimately become periodic as there are only finitely many different remainders and so eventually it will find a remainder that has occurred before. On the other hand, each repeating decimal number satisfies a linear equation
Linear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....

 with integral coefficients, and its unique solution is a rational number. To illustrate the latter point, the number {{nowrap|α {{=}} 5.8144144144…}} above satisfies the equation {{nowrap|10000α − 10α {{=}} 58144.144144… − 58.144144… {{=}} 58086,}} whose solution is {{nowrap|α {{=}} 58086/9990 {{=}} 3227/555.}}

A decimal representation written with a repeating final 0 is said to terminate before these zeros. Instead of "1.585000…" one simply writes "1.585". The decimal is also called a terminating decimal. Terminating decimals represent rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s of the form k/(2n5m). For example, {{nowrap|1.585 {{=}} 317/200 {{=}} 317/(2352)}}. A terminating decimal can be written as a decimal fraction: {{nowrap|317/200 {{=}} 1585/1000}}. However, a terminating decimal also has a representation as a repeating decimal, obtained by decreasing the final (nonzero) digit by one and appending an infinitely repeating sequence of nines. {{nowrap and {{nowrap|1.585 {{=}} 1.584999999…}} are two examples of this.

A decimal that is neither terminating nor repeating represents an irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

 (which cannot be expressed as a fraction of two integers), such as the square root of 2
Square root of 2
The square root of 2, often known as root 2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.Geometrically the square root of 2 is the...

 or the number π. Conversely, an irrational number always has a non-terminating non-repeating decimal representation.

Notation


One convention to indicate a repeating decimal is to put a horizontal line (known as a vinculum) above the repeated numerals (). Another convention is to place dots above the outermost numerals of the repeating digits. Where these methods are impossible, the extension may be represented by an ellipsis
Ellipsis
Ellipsis is a series of marks that usually indicate an intentional omission of a word, sentence or whole section from the original text being quoted. An ellipsis can also be used to indicate an unfinished thought or, at the end of a sentence, a trailing off into silence...

 (…), although this may introduce uncertainty as to exactly which digits should be repeated. Another notation, used for example in Europe and China, encloses the repeating digits in brackets.
Fraction Ellipsis Vinculum Dots Brackets
1/9 0.111… 0.1 0.(1)
1/3 0.333… 0.3 0.(3)
2/3 0.666… 0.6 0.(6)
9/11 0.8181… 0.81 0.(81)
7/12 0.58333… 0.583 0.58(3)
1/81 0.012345679… 0.012345679 0.(012345679)
22/7 3.142857142857… 3.142857 3.(142857)

Decimal expansion and recurrence sequence


In order to convert a rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

 represented as a fraction into decimal form, one may use long division
Long division
In arithmetic, long division is a standard procedure suitable for dividing simple or complex multidigit numbers. It breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a...

. For example, consider the rational number 5/74:

0.0675
74 ) 5.00000
4.44
560
518
420
370
500

etc. Observe that at each step we have a remainder; the successive remainders displayed above are 56, 42, 50. When we arrive at 50 as the remainder, and bring down the "0", we find ourselves dividing 500 by 74, which is the same problem we began with. Therefore the decimal repeats: 0.0675 675 675 ….

Every rational number is either a terminating or repeating decimal


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 is 0, the expansion terminates at that point. 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 was the same. Therefore the following division will repeat the same results.

Fractions with prime denominators


A fraction in lowest terms with a prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

 denominator other than 2 or 5 (i.e. coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...

 to 10) always produces a repeating decimal. The period of the repeating decimal of 1/p is equal to the order of 10 modulo p. If 10 is a primitive root
Primitive root modulo n
In modular arithmetic, a branch of number theory, a primitive root modulo n is any number g with the property that any number coprime to n is congruent to a power of g modulo n. In other words, g is a generator of the multiplicative group of integers modulo n...

 modulo p, the period is equal to p − 1; if not, the period is a factor of p − 1. This result can be deduced from Fermat's little theorem
Fermat's little theorem
Fermat's little theorem states that if p is a prime number, then for any integer a, a p − a will be evenly divisible by p...

, which states that 10p−1 = 1 (mod p).

The base-10 repetend (the repeating decimal part) of the reciprocal of any prime number greater than 5 is divisible by 9.

Cyclic numbers


{{Main|Cyclic number}}

If the period of the repeating decimal of 1/p for prime p is equal to p − 1 then the repeating decimal part is called a cyclic number.

Examples of fractions belonging to this group are:
  • 1/7 = 0.142857 ; 6 repeating digits
  • 1/17 = 0.05882352 94117647 ; 16 repeating digits
  • 1/19 = 0.052631578 947368421 ; 18 repeating digits
  • 1/23 = 0.04347826086 95652173913 ; 22 repeating digits
  • 1/29 = 0.0344827 5862068 9655172 4137931 ; 28 repeating digits
  • 1/97 = 0.01030927 83505154 63917525 77319587 62886597 93814432 98969072 16494845 36082474 22680412 37113402 06185567 ; 96 repeating digits


The list can go on to include the fractions 1/47, 1/59, 1/61, 1/97, 1/109, etc.

Every proper multiple of a cyclic number (that is, a multiple having the same number of digits) is a rotation.
  • 1/7 = 1 × 0.142857… = 0.142857…
  • 3/7 = 3 × 0.142857… = 0.428571…
  • 2/7 = 2 × 0.142857… = 0.285714…
  • 6/7 = 6 × 0.142857… = 0.857142…
  • 4/7 = 4 × 0.142857… = 0.571428…
  • 5/7 = 5 × 0.142857… = 0.714285…


See also the article 142857 for more properties.

Other reciprocals of primes


Some reciprocals of primes that do not generate cyclic numbers are:
  • 1/3 = 0.333… which has a period of 1.
  • 1/11 = 0.090909… which has a period of 2.
  • 1/13 = 0.076923… which has a period of 6.


The multiples of 1/13 can be divided into two sets, with different repeating decimal parts. The first set is:
  • 1/13 = 0.076923…
  • 10/13 = 0.769230…
  • 9/13 = 0.692307…
  • 12/13 = 0.923076…
  • 3/13 = 0.230769…
  • 4/13 = 0.307692…


where the repeating decimal part of each fraction is a cyclic re-arrangement of 076923. The second set is:
  • 2/13 = 0.153846…
  • 7/13 = 0.538461…
  • 5/13 = 0.384615…
  • 11/13 = 0.846153…
  • 6/13 = 0.461538…
  • 8/13 = 0.615384…


where the repeating decimal part of each fraction is a cyclic re-arrangement of 153846.

In general, the set of reciprocals of a prime p will consist of n sets each with period k, where nk = p − 1.

Reciprocals of composite integers coprime to 10


If p is a prime other than 2 or 5, the decimal representation of the fraction has a specific period e.g.:
1/49 = 0.0204081 6326530 6122448 9795918 3673469 3877551


The period of the repeating decimal 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 m such that


for every integer a that is coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...

 to n.

The period of the repeating decimal of is usually pTp where Tp is the period of the repeating decimal of . There are three known primes for which this is not true, and for which the period of is the same as the period of because p2 divides 10p−1−1; they are 3, 487 and 56598313 {{OEIS|id=A045616}}.

Similarly, the period of the repeating decimal of is usually pk−1Tp

If p and q are primes other than 2 or 5, the decimal representation of the fraction has a specific period. An example is 1/119:
119 = 7 × 17
λ(7 × 17) = LCM(λ(7), λ(17))
= LCM(6, 16)
= 48


where LCM denotes the least common multiple
Least common multiple
In arithmetic and number theory, the least common multiple of two integers a and b, usually denoted by LCM, is the smallest positive integer that is a multiple of both a and b...

.

The period T of is a factor of λ(pq) and it happens to be 48 in this case:
1/119 = 0.00840336 13445378 15126050 42016806 72268907 56302521


The period T of the repeating decimal of is LCM(Tp, Tq) where Tp is the period of the repeating decimal of and Tq is the period of the repeating decimal of .

If p , q, r etc. are primes other than 2 or 5, and k , ℓ, m etc. are positive integers then is a repeating decimal with a period of where , etc. are respectively the periods of the repeating decimals etc. as defined above.

Reciprocals of integers not co-prime to 10


An integer that is not co-prime 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. The reciprocal can be expressed as:


where a and b are not both zero.

This fraction can also be expressed as:


if a > b, or as


if b > a, or as


if a = b.

The decimal has:
  • An initial transient of max(a, b) digits after the decimal point. Some or all of the digits in the transient can be zeros.
  • A subsequent repetend which is the same as that for the fraction .


For example 1/28 = 0.03571428571428…:
  • the initial non-repeating digits are 03; and
  • the subsequent repeating digits are 571428.

Converting repeating decimals to fractions


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


Another example:

A shortcut


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


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


It is possible to get a general formula expressing a repeating decimal with an n digit period, beginning right after the decimal point, as a fraction:

x = 0.(A1A2…An)

10nx = A1A2…An.(A1A2…An)

(10n - 1)x = 99…99x = A1A2…An

x = A1A2…An/(10n - 1)
A1A2…An/99…99

More explicitly one gets the following cases.

If the repeating decimal is between 0 and 1, and the repeating block is n digits long, first occurring right after the decimal point, then the fraction (not necessarily reduced) will be the integer number represented by the n-digit block divided by the one represented by n digits 9. For example,
  • 0.444444… = 4/9 since the repeating block is 4 (a 1-digit block),
  • 0.565656… = 56/99 since the repeating block is 56 (a 2-digit block),
  • 0.012012… = 12/999 since the repeating block is 012 (a 3-digit block), and this further reduces to 4/333.
  • 0.9999999… = 9/9 = 1, since the repeating block is 9 (also a 1-digit block)


If the repeating decimal is as above, except that there are k (extra) digits 0 between the decimal point and the repeating n-digit block, then one can simply add k digits 0 after the n digits 9 of the denominator (and as before the fraction may subsequently be simplified). For example,
  • 0.000444… = 4/9000 since the repeating block is 4 and this block is preceded by 3 zeros,
  • 0.005656… = 56/9900 since the repeating block is 56 and it is preceded by 2 zeros,
  • 0.00012012… = 12/99900 = 2/16650 since the repeating block is 012 and it is preceded by 2 (!) zeros.


Any repeating decimal not of the form described above can be written as a sum of a terminating decimal and a repeating decimal of one of the two above types (actually the first type suffices, but that could require the terminating decimal to be negative). For example,
  • 1.23444… = 1.23 + 0.00444… = 123/100 + 4/900 = 1107/900 + 4/900 = 1111/900 or alternatively 1.23444… = 0.79 + 0.44444… = 79/100 + 4/9 = 711/900 + 400/900 = 1111/900
  • 0.3789789… = 0.3 + 0.0789789… = 3/10 + 789/9990 = 2997/9990 + 789/9990 = 3786/9990 = 631/1665 or alternatively 0.3789789… = −0.6 + 0.9789789… = −6/10 + 978/999 = −5994/9990 + 9780/9990 = 3786/9990 = 631/1665


It follows that any repeating decimal with period
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

 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.

Conversely 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.

For 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. The period of the fraction 2/7 is therefore 6.

Repeating decimals as an infinite series


Repeating decimals can also be expressed as an infinite series. That is, repeating decimals can be shown to be a sum of a sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

 of numbers. To take the simplest example,


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

Multiplication and cyclic permutation


{{Main|Cyclic permutation of integer}}

The cyclic behavior of repeating decimals in multiplication also leads to the construction of integers which are cyclically permuted
Cyclic permutation
A cyclic permutation or circular permutation is a permutation built from one or more sets of elements in cyclic order.The notion "cyclic permutation" is used in different, but related ways:- Definition 1 :right|mapping of permutation...

 when multiplied by a number n. For example, 102564 x 4 = 410256. Note that 102564 is the repeating digits of 4/39 and 410256 the repeating digits of 16/39.

Other properties of repetend lengths


Various properties of repetend lengths (periods) are given in and :

The period of 1/k for integer k is always ≤ k − 1.

If p is prime, the period of 1/p divides evenly into p − 1.

If k is composite, the period of 1/k is strictly less than k − 1.

The period of c/k, for c coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...

 to k, equals the period of 1/k.

If where n > 1 and n is not divisible by 2 or 5, then the length of the transient of 1/k is max(a, b), and the period equals r, where r is the smallest integer such that .

If p, p', p", … are distinct primes, then the period of 1/(pp'p"…) equals the lowest common multiple of the periods of 1/p, 1/p' ,1/p" , ….

If k and k' have no common prime factors other than 2 and/or 5, then the period of equals the least common multiple of the periods of and .

For prime p, if but , then for we have .

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

For some other properties of repetends, see also.

See also

  • 142857
    142857 (number)
    142857 is the six repeating digits of 1/7, 0., and is the best-known cyclic number in base 10. If it is multiplied by 2, 3, 4, 5, or 6, the answer will be a cyclic permutation of itself, and will correspond to the repeating digits of 2/7, 3/7, 4/7, 5/7, or 6/7, respectively.- Calculations :- 22/7...

  • Cyclic number
    Cyclic number
    A cyclic number is an integer in which cyclic permutations of the digits are successive multiples of the number. The most widely known is 142857:For example:Multiples of these fractions exhibit cyclic permutation:...

  • Parasitic number
    Parasitic number
    An n-parasitic number is a positive natural number which can be multiplied by n by moving the rightmost digit of its decimal representation to the front. Here n is itself a single-digit positive natural number. In other words, the decimal representation undergoes a right circular shift by one...

  • Midy's theorem
    Midy's theorem
    In mathematics, Midy's theorem, named after French mathematician E. Midy, is a statement about the decimal expansion of fractions a/p where p is a prime and a/p has a repeating decimal expansion with an even period...

  • 0.999...
    0.999...
    In mathematics, the repeating decimal 0.999... denotes a real number that can be shown to be the number one. In other words, the symbols 0.999... and 1 represent the same number...