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 Template:Num. 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 fields. In the non-abelian case, "inverse" implies both, left and right inverse.
The qualifier multiplicative is often omitted and then tacitly self-understood (in contrast to the additive inverse).
Examples and counter-examples
Zero does not have a reciprocal. Every complex number except zero has a reciprocal that is a complex number. If it is real, then so is its reciprocal, and if it is 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 2y-x.y2. 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 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 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.
A square matrix has an inverse iff its determinant has an inverse in the coefficient ring. The linear map that has the matrix A-1 w.r.t. some base is then the reciprocal function of the map having A as matrix in the same base. Thus, the two 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 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, probaly for historical reasons (for example in French, the inverse function is preferredly called application réciproque).
See also
- Additive inverse
- Division (mathematics)
- Fraction (mathematics)
- group (mathematics)
- ring (mathematics)
- Exponential decay
cs:Převrácená hodnota de:Kehrwert fr:inverse es:Inverso multiplicativo ja:逆数 sl:Recipročna vrednost sv:Reciprok