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With the exception of zero, reciprocals of every real number are real, reciprocals of every rational number are rational, and reciprocals of every complex number are complex.
The property that every element other than zero has a multiplicative inverse is part of the definition of a field , of which these are all examples.
This multiplicative inverse exists if and only if a and n are coprime. The extended Euclidean algorithm may be used to compute it.
A square matrix has an inverse if and only if its determinant has an inverse in the coefficient ring.
Thus, the two distinct notions of the inverse of a function are strongly related in this case, while they must be carefully distinguished in the general case as noted above.
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.
A ring in which every nonzero element has a multiplicative inverse is a division ring ; likewise an algebra in which this holds is a division algebra.
The reciprocal may be computed by hand with the use of long division. This continues until the desired precision is reached.
A typical initial guess can be found by rounding b to a nearby power of 2, then using bit shifts to compute its reciprocal. In terms of the approximation algorithm described above, this is needed to prove that the change in y will eventually become arbitrarily small.
This iteration can also be generalized to a wider sort of inverses; for example, matrix inverses. Every real or complex number excluding zero has a reciprocal, and reciprocals of certain irrational numbers can have important special properties.
Such irrational numbers share an evident property: they have the same fractional part as their reciprocal, since these numbers differ by an integer.
In the absence of associativity, the sedenions provide a counterexample. The converse does not hold: an element which is not a zero divisor is not guaranteed to have a multiplicative inverse.








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