Editing Multiplicative inverse (section)

==Examples and counterexamples==
In the real numbers, [[0 (number)|zero]] does not have a reciprocal ([[division by zero]] is [[undefined (mathematics)|undefined]]) because no real number multiplied by 0 produces 1 (the product of any number with zero is zero). 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 (mathematics)|field]], of which these are all examples. On the other hand, no [[integer]] other than 1 and −1 has an integer reciprocal, and so the integers are not a field.

In [[modular arithmetic]], the [[modular multiplicative inverse]] of ''a'' is also defined: it is the number ''x'' such that {{nowrap|''ax'' ≡ 1 (mod ''n'')}}. This multiplicative inverse exists [[if and only if]] ''a'' and ''n'' are [[coprime]]. For example, the inverse of 3 modulo 11 is 4 because {{nowrap|4 ⋅ 3 ≡ 1 (mod 11)}}. The [[extended Euclidean algorithm]] may be used to compute it.

The [[sedenion]]s are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, that is, nonzero elements ''x'', ''y'' such that ''xy''&nbsp;= 0.

A [[square matrix]] has an inverse [[if and only if]] its [[determinant]] has an inverse in the coefficient [[Ring (mathematics)|ring]]. The linear map that has the matrix ''A''<sup>−1</sup> with respect to some base is then the inverse function of the map having ''A'' as matrix in the same base. Thus, the two distinct notions of the inverse of a function are strongly related in this case, but they still do not coincide, since the multiplicative inverse of ''Ax'' would be (''Ax'')<sup>−1</sup>, not ''A''<sup>−1</sup>x.

These two notions of an inverse function do sometimes coincide, for example for the function <math>f(x)=x^i=e^{i\ln(x)}</math> where <math>\ln</math> is the [[Complex logarithm#Definition|principal branch of the complex logarithm]] and <math>e^{-\pi}<|x|<e^{\pi}</math>:
:<math>((1/f)\circ f)(x)=(1/f)(f(x))=1/(f(f(x)))=1/e^{i\ln(e^{i\ln(x)})}=1/e^{ii\ln(x)}=1/e^{-\ln(x)}=x</math>.

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 (ring theory)|algebra]] in which this holds is a [[division algebra]].