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 mathematics, 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.

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.