Complex conjugate

Two complex numbers of the form $z = x + iy$ and $\bar z = x - iy$: in polar form, $r e^{i\theta}$ and $r e^{-i\theta}$. In terms of the Argand diagram, they are symmetric in the $x$-axis. A complex number is its own complex conjugate if and only if it is a real number.

Complex conjugation is the map $z \mapsto \bar z$. We have $\overline{z+w} = \bar z + \bar w$, $\overline{z\cdot w} = \bar z \cdot \bar w$, $\overline{z^{-1}} = \bar z^{-1}$, $\overline{\bar z} = z$. Complex conjugation is an automorphism of the field of complex numbers of order two. The absolute value $|z|$ is the positive square root of $z \bar z$.