In ^{−1}, is a number which when by ''x'' yields the ^{−1} is sometimes also used for the

^{−1} with respect to some base is then the reciprocal function of the map having ''A'' as matrix in the same base. Thus, the two distinct notions of the inverse of a function are strongly related in this case, while they must be carefully distinguished in the general case (as noted above).
The

_{0} = 0.1, the following sequence is produced:
:''x''_{1} = 0.1(2 − 17 × 0.1) = 0.03
:''x''_{2} = 0.03(2 − 17 × 0.03) = 0.0447
:''x''_{3} = 0.0447(2 − 17 × 0.0447) ≈ 0.0554
:''x''_{4} = 0.0554(2 − 17 × 0.0554) ≈ 0.0586
:''x''_{5} = 0.0586(2 − 17 × 0.0586) ≈ 0.0588
A typical initial guess can be found by rounding ''b'' to a nearby power of 2, then using

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''multiplicative identity
In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic s ...

, 1. The multiplicative inverse of a fraction
A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths ...

''a''/''b'' is ''b''/''a''. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the function ''f''(''x'') that maps ''x'' to 1/''x'', is one of the simplest examples of a function which is its own inverse (an involution
Involution may refer to:
* Involute, a construction in the differential geometry of curves
* ''Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour input ...

).
Multiplying by a number is the same as dividing by its reciprocal and vice versa. For example, multiplication by 4/5 (or 0.8) will give the same result as division by 5/4 (or 1.25). Therefore, multiplication by a number followed by multiplication by its reciprocal yields the original number (since the product of the number and its reciprocal is 1).
The term ''reciprocal'' was in common use at least as far back as the third edition of ''Encyclopædia Britannica
The (Latin for "British Encyclopaedia") is a general knowledge English-language encyclopaedia which is now published exclusively as an online encyclopedia, online encyclopaedia. It was formerly published by Encyclopædia Britannica, Inc., ...

'' (1797) to describe two numbers whose product is 1; geometrical quantities in inverse proportion are described as in a 1570 translation of Euclid
Euclid (; grc-gre, Εὐκλείδης
Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

's '' Elements''.
In the phrase ''multiplicative inverse'', the qualifier ''multiplicative'' is often omitted and then tacitly understood (in contrast to the additive inverse
In mathematics, the additive inverse of a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be ...

). Multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen that ; then "inverse" typically implies that an element is both a left and right inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when add ...

.
The notation ''f'' inverse function
In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...

of the function ''f'', which is not in general equal to the multiplicative inverse. For example, the multiplicative inverse is the cosecant
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...

of x, and not the inverse sine of ''x'' denoted by or . Only for linear map
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s are they strongly related (see below). The terminology difference ''reciprocal'' versus ''inverse'' is not sufficient to make this distinction, since many authors prefer the opposite naming convention, probably for historical reasons (for example in , the inverse function is preferably called the bijection réciproque).
Examples and counterexamples

In the real numbers,zero
0 (zero) is a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

does not have a reciprocal because no real number multiplied by 0 produces 1 (the product of any number with zero is zero). With the exception of zero, reciprocals of every real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

are real, reciprocals of every rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

are rational, and reciprocals of every complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

are complex. The property that every element other than zero has a multiplicative inverse is part of the definition of a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

, of which these are all examples. On the other hand, no integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

other than 1 and −1 has an integer reciprocal, and so the integers are not a field.
In modular arithmetic #REDIRECT Modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure ( ...

, the modular multiplicative inverseIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

of ''a'' is also defined: it is the number ''x'' such that . This multiplicative inverse exists if and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...

''a'' and ''n'' are coprime
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

. For example, the inverse of 3 modulo 11 is 4 because . The extended Euclidean algorithm
In arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' ...

may be used to compute it.
The sedenion
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

s are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, that is, nonzero elements ''x'', ''y'' such that ''xy'' = 0.
A square matrix
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

has an inverse if and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...

its determinant
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

has an inverse in the coefficient ring. The linear map that has the matrix ''A''trigonometric functions
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

are related by the reciprocal identity: the cotangent is the reciprocal of the tangent; the secant is the reciprocal of the cosine; the cosecant is the reciprocal of the sine.
A ring in which every nonzero element has a multiplicative inverse is a division ringIn algebra, a division ring, also called a skew field, is a ring (mathematics), ring in which division (mathematics), division is possible. Specifically, it is a zero ring, nonzero ring in which every nonzero element has a multiplicative inverse, th ...

; likewise an algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

in which this holds is a division algebra
Division or divider may refer to:
Mathematics
*Division (mathematics)
Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and multi ...

.
Complex numbers

As mentioned above, the reciprocal of every nonzero complex number is complex. It can be found by multiplying both top and bottom of 1/''z'' by itscomplex conjugate
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

$\backslash bar\; z\; =\; a\; -\; bi$ and using the property that $z\backslash bar\; z\; =\; \backslash ,\; z\backslash ,\; ^2$, the absolute value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of ''z'' squared, which is the real number :
:$\backslash frac\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; -\; \backslash fraci.$
The intuition is that
:$\backslash frac$
gives us the complex conjugate
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

with a magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...

reduced to a value of $1$, so dividing again by $\backslash ,\; z\backslash ,$ ensures that the magnitude is now equal to the reciprocal of the original magnitude as well, hence:
:$\backslash frac\; =\; \backslash frac$
In particular, if , , ''z'', , =1 (''z'' has unit magnitude), then $1/z\; =\; \backslash bar\; z$. Consequently, the imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad are ...

s, , have additive inverse
In mathematics, the additive inverse of a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be ...

equal to multiplicative inverse, and are the only complex numbers with this property. For example, additive and multiplicative inverses of are and , respectively.
For a complex number in polar form , the reciprocal simply takes the reciprocal of the magnitude and the negative of the angle:
:$\backslash frac\; =\; \backslash frac\backslash left(\backslash cos(-\backslash varphi)\; +\; i\; \backslash sin(-\backslash varphi)\backslash right).$
Calculus

In realcalculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

, the derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

of is given by the power rule
In calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. ...

with the power −1:
:$\backslash frac\; x^\; =\; (-1)x^\; =\; -x^\; =\; -\backslash frac.$
The power rule for integrals (Cavalieri's quadrature formula
In calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. T ...

) cannot be used to compute the integral of 1/''x'', because doing so would result in division by 0:
$$\backslash int\; \backslash frac\; =\; \backslash frac\; +\; C$$
Instead the integral is given by:
$$\backslash int\_1^a\; \backslash frac\; =\; \backslash ln\; a,$$
$$\backslash int\; \backslash frac\; =\; \backslash ln\; x\; +\; C.$$
where ln is the natural logarithm
The natural logarithm of a number is its logarithm
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

. To show this, note that $\backslash frac\; e^x\; =\; e^x$, so if $y\; =\; e^x$ and $x\; =\; \backslash ln\; y$, we have:
$$\backslash frac\; =\; y\backslash quad\; \backslash Rightarrow\; \backslash quad\; \backslash frac\; =\; dx\; \backslash quad\backslash Rightarrow\backslash quad\; \backslash int\; \backslash frac\; =\; \backslash int\; dx\; \backslash quad\backslash Rightarrow\backslash quad\; \backslash int\; \backslash frac\; =\; x\; +\; C\; =\; \backslash ln\; y\; +\; C.$$
Algorithms

The reciprocal may be computed by hand with the use oflong division
In arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'ar ...

.
Computing the reciprocal is important in many division algorithm
A division algorithm is an algorithm which, given two integers N and D, computes their quotient and/or remainder, the result of Euclidean division. Some are applied by hand, while others are employed by digital circuit designs and software.
Divisi ...

s, since the quotient ''a''/''b'' can be computed by first computing 1/''b'' and then multiplying it by ''a''. Noting that $f(x)\; =\; 1/x\; -\; b$ has a zero
0 (zero) is a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

at ''x'' = 1/''b'', Newton's method
In numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathem ...

can find that zero, starting with a guess $x\_0$ and iterating using the rule:
:$x\_\; =\; x\_n\; -\; \backslash frac\; =\; x\_n\; -\; \backslash frac\; =\; 2x\_n\; -\; bx\_n^2\; =\; x\_n(2\; -\; bx\_n).$
This continues until the desired precision is reached. For example, suppose we wish to compute 1/17 ≈ 0.0588 with 3 digits of precision. Taking ''x''bit shift
In computer programming
Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a particular task. Programming involves tasks such as analysis, gener ...

s to compute its reciprocal.
In constructive mathematics
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. In classical mathematics, one can prove the existence of a mathematical object without "finding ...

, for a real number ''x'' to have a reciprocal, it is not sufficient that ''x'' ≠ 0. There must instead be given a ''rational'' number ''r'' such that 0 < ''r'' < , ''x'', . In terms of the approximation algorithm
In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

described above, this is needed to prove that the change in ''y'' will eventually become arbitrarily small.
This iteration can also be generalized to a wider sort of inverses; for example, .
Reciprocals of irrational numbers

Every real or complex number excluding zero has a reciprocal, and reciprocals of certainirrational number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

s can have important special properties. Examples include the reciprocal of '' e'' (≈ 0.367879) and the golden ratio's reciprocal (≈ 0.618034). The first reciprocal is special because no other positive number can produce a lower number when put to the power of itself; $f(1/e)$ is the global minimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function (mathematics), function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, ei ...

of $f(x)=x^x$. The second number is the only positive number that is equal to its reciprocal plus one:$\backslash varphi\; =\; 1/\backslash varphi\; +\; 1$. Its additive inverse
In mathematics, the additive inverse of a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be ...

is the only negative number that is equal to its reciprocal minus one:$-\backslash varphi\; =\; -1/\backslash varphi\; -\; 1$.
The function $f(n)\; =\; n\; +\; \backslash sqrt,\; n\; \backslash in\; \backslash N,\; n>0$ gives an infinite number of irrational numbers that differ with their reciprocal by an integer. For example, $f(2)$ is the irrational $2+\backslash sqrt\; 5$. Its reciprocal $1\; /\; (2\; +\; \backslash sqrt\; 5)$ is $-2\; +\; \backslash sqrt\; 5$, exactly $4$ less. Such irrational numbers share an evident property: they have the same fractional part
The fractional part or decimal part of a non‐negative real number
Real may refer to:
* Reality, the state of things as they exist, rather than as they may appear or may be thought to be
Currencies
* Brazilian real (R$)
* Central American Republi ...

as their reciprocal, since these numbers differ by an integer.
Further remarks

If the multiplication is associative, an element ''x'' with a multiplicative inverse cannot be azero divisor
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

(''x'' is a zero divisor if some nonzero ''y'', ). To see this, it is sufficient to multiply the equation by the inverse of ''x'' (on the left), and then simplify using associativity. In the absence of associativity, the sedenion
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

s provide a counterexample.
The converse does not hold: an element which is not a zero divisor
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

is not guaranteed to have a multiplicative inverse.
Within Z, all integers except −1, 0, 1 provide examples; they are not zero divisors nor do they have inverses in Z.
If the ring or algebra is finite
Finite is the opposite of Infinity, infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected ...

, however, then all elements ''a'' which are not zero divisors do have a (left and right) inverse. For, first observe that the map must be injective
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

: implies :
:$\backslash begin\; ax\; \&=\; ay\; \&\backslash quad\; \backslash rArr\; \&\; \backslash quad\; ax-ay\; =\; 0\; \backslash \backslash \; \&\; \&\backslash quad\; \backslash rArr\; \&\backslash quad\; a(x-y)\; =\; 0\; \backslash \backslash \; \&\; \&\backslash quad\; \backslash rArr\; \&\backslash quad\; x-y\; =\; 0\; \backslash \backslash \; \&\; \&\backslash quad\; \backslash rArr\; \&\backslash quad\; x\; =\; y.\; \backslash end$
Distinct elements map to distinct elements, so the image consists of the same finite number of elements, and the map is necessarily surjective
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. Specifically, ƒ (namely multiplication by ''a'') must map some element ''x'' to 1, {{nowrap, 1=''ax'' = 1, so that ''x'' is an inverse for ''a''.
Applications

The expansion of the reciprocal 1/''q'' in any base can also act Mitchell, Douglas W., "A nonlinear random number generator with known, long cycle length," ''Cryptologia
''Cryptologia'' is a journal in cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia
''-logy'' is a suffix in the English language, used with words origin ...

'' 17, January 1993, 55–62. as a source of pseudo-random numbers, if ''q'' is a "suitable" safe prime
In number theory, a prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composit ...

, a prime of the form 2''p'' + 1 where ''p'' is also a prime. A sequence of pseudo-random numbers of length ''q'' − 1 will be produced by the expansion.
See also

*Division (mathematics)
Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and multiplication. The division sign , a symbol consisting of a short horizontal ...

* Exponential decay
A quantity is subject to exponential decay if it decreases at a rate proportional
Proportionality, proportion or proportional may refer to:
Mathematics
* Proportionality (mathematics), the property of two variables being in a multiplicative rela ...

* Fraction (mathematics)
A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifth ...

* Group (mathematics)
In mathematics, a group is a set (mathematics), set equipped with an operation that combines any two element (mathematics), elements to form a third element while being associativity, associative as well as having an identity element and inverse ...

* Hyperbola
In mathematics, a hyperbola () (adjective form hyperbolic, ) (plural ''hyperbolas'', or ''hyperbolae'' ()) is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it ...

* List of sums of reciprocals
In mathematics and especially number theory, the sum of reciprocals generally is computed for the multiplicative inverse, reciprocals of some or all of the positive number, positive integers (counting numbers)—that is, it is generally the sum ...

* Repeating decimal
A repeating decimal or recurring decimal is decimal representation of a number whose Numerical digit, digits are periodic function, periodic (repeating its values at regular intervals) and the infinity, infinitely repeated portion is not zero. It c ...

* Six-sphere coordinates
* Unit fractionA unit fraction is a rational number written as a fraction where the numerator
A fraction (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was original ...

s – reciprocals of integers
Notes

References

*Maximally Periodic Reciprocals, Matthews R.A.J. ''Bulletin of the Institute of Mathematics and its Applications'' vol 28 pp 147–148 1992 Elementary special functions Abstract algebra Elementary algebra Multiplication Unary operations