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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 their changes (cal ...
, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when
multiplied
multiplied
by ''x'' yields the
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 ...

Euclid
'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'' −1 is sometimes also used for the
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 ...

cosecant
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 ...

linear map
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
French
French
, 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 ...

zero
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 ...

complex number
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 ...

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 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
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 ...

trigonometric functions
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 its
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 ...
\bar z = a - bi and using the property that z\bar z = \, z\, ^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 ...

absolute value
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 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 \, 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 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: :\frac = \frac\left(\cos(-\varphi) + i \sin(-\varphi)\right).


Calculus

In real
calculus 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 ...

calculus
, 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 ...

derivative
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: : \frac x^ = (-1)x^ = -x^ = -\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: \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 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 \frac e^x = e^x, so if y = e^x and x = \ln y, we have: \frac = y\quad \Rightarrow \quad \frac = dx \quad\Rightarrow\quad \int \frac = \int dx \quad\Rightarrow\quad \int \frac = x + C = \ln y + C.


Algorithms

The reciprocal may be computed by hand with the use of
long 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 ...

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. 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 ...

Newton's method
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(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''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 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 ...

bit shift
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 ...

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
matrix inverses
.


Reciprocals of irrational numbers

Every real or complex number excluding zero has a reciprocal, and reciprocals of certain
irrational 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:\varphi = 1/\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:-\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(2) is the irrational 2+\sqrt 5. Its reciprocal 1 / (2 + \sqrt 5) is -2 + \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 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), ...
(''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 : :\begin ax &= ay &\quad \rArr & \quad ax-ay = 0 \\ & &\quad \rArr &\quad a(x-y) = 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 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 ...

Exponential decay
*
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 ...

Hyperbola
*
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