# Reciprocal

In mathematics, the**reciprocal**, or

**multiplicative inverse**, of a number

*x*is the number which, when multiplied by

*x*, yields 1.

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. The reciprocal of *x* is denoted 1/*x* or *x*^{-1}.

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*-*x**y*^{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 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.

The multiplicative group of every finite field is cyclic. For all *a* in GF(*n*), *a* = *a ^{n}*. For all nonzero

*a*, it follows that

*a*

^{-1}=

*a*

^{n-2}. The multiplicative inverse of

*a*can thus be found by raising

*a*to a positive exponent, which can be done quickly through exponentiating by squaring.

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. And the cosecant is the reciprocal of the sine.

See also: Additive inverse, Division, Fraction, group (mathematics), ring (mathematics)

In navigation a **reciprocal bearing** is the bearing that will take you in the reverse direction to that of the original bearing.

In the humanities and social sciences, an interaction between actors is said to be **reciprocal** when each action or favour given by one party is matched by another in return.
See also the principle of reciprocity in international negotiations.