# inverse

The inverse of a number, or a reciprocal, is 1 divided by the number; for example, the inverse of 8 is 1/8 and the inverse of 3/5 is 5/3.

The inverse of a function or a transformation is the function or transformation that the reverses the effect of the
function or transformation. For example, the inverse of addition is
subtraction, and of clockwise rotation is anticlockwise rotation. For
a function *f(x)*, the function *g(x)* such that *f(a)
= b* implies *g(b) = a* is described as the inverse of *f(x)*.
In practice, the inverse function of *f(x)* is written *f *^{-1}*(x)*. For example, the inverse of *f(x)* = *ax* + *b* is *f *^{-1}*(x)* = *(x - b)/a*, since *f *^{-1}*(ax* + *b)* = *(ax* + *b - b)/a = ax/a = x*.

The inverse of an element of a set, or
a number, with respect to a particular operation, is what has to be
combined with the element or number in order to obtain that operation's **identity element**. In other words, the inverse of element *a* is the element *b* such that *a***b* = *e*, where * is an algebraic operation and *e* is the
identity element relative to the operation * of the set of which *a* and *b* are members. For example, a is the identity element of real numbers relative to multiplication:
hence if *a.b* = -1, *a* is the inverse of *b*, *b* the inverse of *a*, relative to the multiplication
of real numbers. (Moreover, *a* and *b* are reciprocals
in this case.)

## inverse of a proposition

For a proposition *h* → *c* (read *h* implies *c*), the proposition not-*h* → *c* is described as its inverse.

## inverse trigonometric function

For a function of the form *y* = sin *x*, the function of
the form *x* = sin^{-1} *y* (read as "*x* is
the angle whose sine is *y*") is described as its inverse. This may
also be written as *x* = arc sin *y*.