complex conjugate

Complex conjugate

The complex conjugate of a complex number $z=x+iy$ is $\bar{z}=x-iy$
The sign in front of $i$ (or the angle) is flipped, the magnitude stays the same:

$\overline{z_1+z_1}={\bar{z}}_1+{\bar{z}}_2$
$\overline{z_1\cdot z_1}={\bar{z}}_1\cdot {\bar{z}}_2$
$\frac{z+\bar{z}}{2}=x=\text{Re}(z)\in \mathbb{R}$
$\frac{z-\bar{z}}{2i}=y=\text{Im}(z)\in \mathbb{R}$
$z\cdot \bar{z}=x^2-iyx+iyx-(iy{)}^2=x^2+y^2=|z{|}^2$

Solutions to polynomials with real coefficients come in conjugate pairs: If $a+bi$ is a root, so is $a-bi$.