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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a multiplicative inverse or reciprocal for a
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a
fraction A fraction (from la, fractus, "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 ...
''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 Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orient ...
''f''(''x'') that maps ''x'' to 1/''x'', is one of the simplest examples of a function which is its own inverse (an involution). 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 Encyclopædia") is a general knowledge English-language encyclopaedia. It is published by Encyclopædia Britannica, Inc.; the company has existed since the 18th century, although it has changed ownership various ti ...
'' (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; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
's '' Elements''. In the phrase ''multiplicative inverse'', the qualifier ''multiplicative'' is often omitted and then tacitly understood (in contrast to the additive inverse). 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. The notation ''f'' −1 is sometimes also used for the
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X ...
of the function ''f'', which is for most functions not 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 a ...
of x, and not the inverse sine of ''x'' denoted by or . 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
French French (french: français(e), link=no) may refer to: * Something of, from, or related to France ** French language, which originated in France, and its various dialects and accents ** French people, a nation and ethnic group identified with France ...
, the inverse function is preferably called the bijection réciproque).

# Examples and counterexamples

In the real numbers,
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
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, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
are real, reciprocals of every rational number are rational, and reciprocals of every
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
are complex. The property that every element other than zero has a multiplicative inverse is part of the definition of a field, of which these are all examples. On the other hand, no
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
other than 1 and −1 has an integer reciprocal, and so the integers are not a field. In
modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...
, the
modular multiplicative inverse In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer is an integer such that the product is congruent to 1 with respect to the modulus .. In the standard notation of modular arithmetic this congr ...
of ''a'' is also defined: it is the number ''x'' such that . This multiplicative inverse exists
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicond ...
''a'' and ''n'' are
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equival ...
. For example, the inverse of 3 modulo 11 is 4 because . The
extended Euclidean algorithm In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers ''a'' and ''b'', also the coefficients of Bézout's id ...
may be used to compute it. The
sedenion In abstract algebra, the sedenions form a 16- dimensional noncommutative and nonassociative algebra over the real numbers; they are obtained by applying the Cayley–Dickson construction to the octonions, and as such the octonions are isomorphic t ...
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, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...
has an inverse
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicond ...
its determinant has an inverse in the coefficient ring. The linear map that has the matrix ''A''−1 with respect to some base is then the inverse 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, but they still do not coincide, since the multiplicative inverse of ''Ax'' would be (''Ax'')−1, not ''A''−1x. These two notions of an inverse function do sometimes coincide, for example for the function $f\left(x\right)=x^i=e^$ where $\ln$ is the principal branch of the complex logarithm and
trigonometric functions 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 al ...
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 ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
; likewise an algebra in which this holds is a
division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fiel ...
.

# 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 its
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
$\bar z = a - bi$ and using the property that $z\bar z = \, z\, ^2$, the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), a ...
of ''z'' squared, which is the real number : :$\frac = \frac = \frac = \frac = \frac - \fraci.$ The intuition is that :$\frac$ gives us the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
with a magnitude reduced to a value of $1$, so dividing again by $\, z\,$ ensures that the magnitude is now equal to the reciprocal of the original magnitude as well, hence: :$\frac = \frac$ In particular, if , , ''z'', , =1 (''z'' has unit magnitude), then $1/z = \bar z$. Consequently, the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
s, , have additive inverse 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: :$\frac = \frac\left\left(\cos\left(-\varphi\right) + i \sin\left(-\varphi\right)\right\right).$

# Calculus

In real calculus, the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of is given by the power rule with the power −1: :$\frac x^ = \left(-1\right)x^ = -x^ = -\frac.$ The power rule for integrals ( Cavalieri's quadrature formula) cannot be used to compute the integral of 1/''x'', because doing so would result in division by 0: $\int \frac = \frac + C$ Instead the integral is given by: $\int_1^a \frac = \ln a,$ $\int \frac = \ln x + C.$ where ln is the natural logarithm. To show this, note that $\frac e^y = e^y$, so if $x = e^y$ and $y = \ln x$, we have:

# Algorithms

The reciprocal may be computed by hand with the use of long division. 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. Divi ...
s, since the quotient ''a''/''b'' can be computed by first computing 1/''b'' and then multiplying it by ''a''. Noting that $f\left(x\right) = 1/x - b$ has a
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
at ''x'' = 1/''b'',
Newton's method In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real- ...
can find that zero, starting with a guess $x_0$ and iterating using the rule: :$x_ = x_n - \frac = x_n - \frac = 2x_n - bx_n^2 = x_n\left(2 - bx_n\right).$ 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''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 bit shifts to compute its reciprocal. In
constructive mathematics In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove t ...
, 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 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, matrix inverses.

# Reciprocals of irrational numbers

Every real or complex number excluding zero has a reciprocal, and reciprocals of certain
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
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; $f\left(1/e\right)$ is the
global minimum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ra ...
of $f\left(x\right)=x^x$. The second number is the only positive number that is equal to its reciprocal plus one:$\varphi = 1/\varphi + 1$. Its additive inverse is the only negative number that is equal to its reciprocal minus one:$-\varphi = -1/\varphi - 1$. The function $f(n) = n + \sqrt, n \in \N, n>0$ gives an infinite number of irrational numbers that differ with their reciprocal by an integer. For example, $f\left(2\right)$ is the irrational $2+\sqrt 5$. Its reciprocal $1 / \left(2 + \sqrt 5\right)$ is $-2 + \sqrt 5$, exactly $4$ less. Such irrational numbers share an evident property: they have the same fractional part 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 a
zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zer ...
(''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, the sedenions form a 16- dimensional noncommutative and nonassociative algebra over the real numbers; they are obtained by applying the Cayley–Dickson construction to the octonions, and as such the octonions are isomorphic t ...
s provide a counterexample. The converse does not hold: an element which is not a
zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zer ...
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 infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past partici ...
, 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, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
: implies : :$\begin ax &= ay &\quad \rArr & \quad ax-ay = 0 \\ & &\quad \rArr &\quad a\left(x-y\right) = 0 \\ & &\quad \rArr &\quad x-y = 0 \\ & &\quad \rArr &\quad x = y. \end$ 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, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
. 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 published six times per year since January 1977. Its remit is all aspects of cryptography, with a special emphasis on historical aspects of the subject. The founding editors were Brian J. Winkel, Davi ...
'' 17, January 1993, 55–62.
as a source of pseudo-random numbers, if ''q'' is a "suitable" safe prime, 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.

* Division (mathematics) *
Exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
*
Fraction (mathematics) A fraction (from la, fractus, "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 ...
*
Group (mathematics) In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. Th ...
*
Hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
* Inverse distribution * List of sums of reciprocals *
Repeating decimal 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. It can be shown that a number is rational if ...
* Six-sphere coordinates *
Unit fraction A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/''n''. Examples are 1/1, 1/2, 1/3, 1/4, 1/5 ...
s – reciprocals of integers

# 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