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Multiplicative inverse

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In mathematics, the '''reciprocal''', or '''multiplicative inverse''', of a number ''x'' is the number which, when multiplied by ''x'', yields {{num|1}}. It is denoted 1/''x'' or ''x''^-1. Number can mean here any element of a unital algebra, but the term ''reciprocal'', as well as the notation 1/''x'', is usually restricted to commutative field (mathematics) fields. In the non-abelian case, "inverse" implies both, left and right inverse element inverse. The qualifier ''multiplicative'' is often omitted and then tacitly self-understood (in contrast to the additive inverse).

Examples and counter-examples
0 (number) Zero does not have a reciprocal. Every complex number except zero has a reciprocal that is a complex number. If it is real number real, then so is its reciprocal, and if it is rational number rational, then so is its reciprocal. To approximate the reciprocal of ''x'', using only multiplication and subtraction, one can guess a number ''y'', and then repeatedly replace ''y'' with 2''y''−''xy''². Once the change in ''y'' becomes (and stays) sufficiently small, ''y'' is an approximation of the reciprocal of ''x''. In constructive mathematics, for a real number x to have a reciprocal, it is not sufficient that it be false (logic) false that ''x'' = 0. Instead, there must be given a ''rational'' number ''r'' such that 0 < ''r'' < |''x''|. In terms of the approximation algorithm in the previous paragraph, this is needed to prove that the change in ''y'' will eventually get arbitrarily small. In modular arithmetic, the multiplicative inverse of ''x'' is also defined: it is the number ''a'' such that (''a·x'') mod ''n'' = 1. However, this multiplicative inverse exists only if ''a'' and ''n'' are coprime relatively prime. For example, the inverse of 3 modulo 11 is 4 because it is the solution to (3''x'') mod 11 = 1. The extended Euclidean algorithm may be used to compute the multiplicative inverse modulo a number. The sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which has nonetheless divisors of zero, i.e. nonzero elements ''x,y'' such that ''xy''=0. A square matrix has an inverse iff its determinant has an inverse in the coefficient ring (mathematics) 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 (see below). The trigonometry 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. It is important to distinguish the reciprocal of a function ''f'' in the multiplicative sense, given by 1/''f'', from the reciprocal or '''inverse function''' w.r.t. composition, rather denoted by ''f''⁻¹, defined by ''f o f''⁻¹ = id. Only for linear maps they are strongly related (see above), while they are completely different for all other cases. The terminology ''reciprocal'' vs ''inverse'' is not sufficient to make this distinction, since many authors prefer the opposite naming convention, probably for historical reasons (for example in French language French, the inverse function is preferably called w:fr:application rÃ©ciproque application rÃ©ciproque). A ring (mathematics) ring or an algebra (ring theory) algebra in which every nonzero element has a multiplicative inverse is called a division ring resp. division algebra.

Practical applications
The multiplicative inverse has innumerable applications in algorithms of computer science, particularly those related to number theory, since many such algorithms rely heavily on the theory of modular arithmetic. As a simple example, consider the ''exact division problem'' where you have a list of odd word-sized numbers each divisible by ''k'' and you wish to divide them all by ''k''. One solution is as follows: # Use the extended Euclidean algorithm to compute ''k''^-1, the multiplicative inverse of ''k'' mod 2^''w'', where ''w'' is the number of bits in a word. This inverse will exist since the numbers are odd and the modulus has no odd factors. # For each number in the list, multiply it by ''k''^-1 and take the least significant word of the result. On many machines, particularly those without hardware support for division, division is a slower operation than multiplication, so this approach can yield a considerable speedup. The first step is relatively slow but only needs to be done once.

Further remarks
An element which has a multiplicative inverse cannot be a zero divisor if the multiplication is associative. To see this, it is sufficient to multiply the equation ''x y'' = 0 by the inverse of ''x'' (on the left), and then simplify using associativity. The sedenions provide a counter example. Conversely, an element which is not a zero divisors needs not to have a multiplicative inverse. The nonzero integers provide an example. (They are not zero divisors but have no inverse in '''Z'''.) If the ring or algebra is finite, however, then all elements ''a'' which are not zero divisors do have a (left and right) inverse. This can be seen by observing that the map ''x→ax'' must be injective (''ax=ay => a(x-y)=0 => x-y=0''), thus surjective, thus there is ''x'' such that ''ax''=1.

See also
* Additive inverse * Division (mathematics) * Fraction (mathematics) * group (mathematics) * ring (mathematics) * division algebra * Exponential decay Category:Abstract algebra Category:Algebra cs:PÅ™evrÃ¡cenÃ¡ hodnota da:Reciprok de:Kehrwert es:Inverso multiplicativo it:Reciproco he:×ž×¡×¤×¨ ×”×•×¤×›×™ hu:Reciprok nl:Omgekeerde ja:é€†æ•° sl:ReciproÄ?na vrednost sv:Reciprok (matematik) zh:å€’æ•°

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