In a monoid or multiplicative group where the operation is a product ·, the multiplicative inverse of any element g is the element g^(-1) such that g·g^(-1) = g^(-1)·g = 1, with 1 the identity element. The multiplicative inverse of a nonzero number z is its reciprocal 1/z (zero is not invertible). For complex z = x + i y!=0, 1/z = 1/(x + i y) = x/(x^2 + y^2) - iy/(x^2 + y^2). The inverse of a nonzero real quaternion h = x + y i + v j + w k (where x, y, v, w are real numbers, and not all of them are zero) is its reciprocal 1/h = x/α - y/α i - v/α j - w/α k, where α = x^2 + y^2 + v^2 + w^2.

additive inverse | invertible element | multiplicative group | multiplicative identity