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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 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 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 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.
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