Encyclopedia > Reciprocal

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In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1.

Note that zero does not have a reciprocal. Otherwise, every rational number, every real number, and every complex number has a reciprocal of the same type. 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 2y-xy2. 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's 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, you need this to prove that the change in y will eventually get arbitrarily small.

See also: Additive inverse, Division, Fraction, Mathematical group, Mathematical ring

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[?].

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