Complex Conjugates

Definition

The complex conjugate (or conjugate) of a complex number $z$ is the complex number whose real part matches $z$ and whose imaginary part is the additive inverse of $z$. The complex conjugate of a purely real number is the same purely real number. In other words, the complex conjugate of a complex number with no imaginary part is the same complex number.

Notation

We denote the complex conjugate of the complex number $z \in \C $ by $\Cconj{z}$. Other common notation includes $\bar{z}$, read “z bar”. If there exists $a, b \in \R $ so that $z = (a, b)$, then $\Cconj{z} = (a, -b)$.

Geometric interpretation

Taking the conjugate of a complex numbers corresponds to a reflection across the real axis in the plane.

Properties

A complex number $z$ is real if and only if $z = \Cconj{z}$ and it is imaginar if and only if $z = -\Cconj{z}$.