Geometrical Representation of Conjugate of a Complex Number

The complex conjugate of $z$ is denoted $\overline z$. To find the conjugate of $z$, simply replace $i$ by $-i$ throughout $z$. For instance, $2-5i$ is the conjugate of $2+5i$.

Key fact. The product of a complex number with its own conjugate is always a real number:

For instance: $(1+3i)(1-3i)=1^2+3^2=1+9=10$.

Geometric picture. Since conjugation flips only the sign of $y$, $\overline z$ is obtained from $z$ by reflecting $z$ across the real axis in the Argand plane — the point $(x,y)$ and its conjugate $(x,-y)$ are mirror images through the $x$-axis. …