Inverse of a complex number

Assume that z is a non-zero complex number expressed by its algebraic form,
`z = x + i * y`

Then, the inverse of z is written,

`1/z = 1/(x + i * y)`

The numerator and denominator are multiplied by the conjugate of z (in order to get rid of i).

`bar z = x -i*y`

`1/z = bar z/(z*bar z) = (x-i*y)/((x + i * y)(x-i*y)) = (x -i*y)/(x^2-(i*y)^2)`

`1/z = x/(x^2+y^2) -i*y/(x^2+y^2)`

Inverse calculation examples

`z = 1+i , bar z = 1-i`.

`1/z = (1-i)/((1-i)(1+i)) = (1-i)/(1^2-i^2) = (1-i)/2`

`1/z = 1/2 -i/2`

Inverse properties

The inverse conjugate of a complex number is equal to the inverse of its conjugate.

`bar ((1/z)) = 1/bar z`

See also

Algebraic form of a complex number
Conjugate of a complex number