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).