Conjugate

The conjugate of $z = a + bi$ is $\bar{z} = a - bi$ — just flip the sign of the imaginary part. Geometrically it's the mirror image of $z$ across the real axis.

Multiplying a complex number by its own conjugate always gives a real, non-negative result:

$z \bar{z} = (a+bi)(a-bi) = a^2 + b^2$

Example: for $z = 3 + 4i$, $\bar{z} = 3 - 4i$, and $z\bar{z} = 9 + 16 = 25$.

Conjugation also distributes over addition and multiplication: $\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}$ and $\overline{z_1 z_2} = \bar{z_1}\bar{z_2}$. This is the trick behind dividing complex numbers (see the Division topic).