Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if ${\displaystyle a}$ and ${\displaystyle b}$ are real numbers then the complex conjugate of ${\displaystyle a+bi}$ is ${\displaystyle a-bi.}$ The complex conjugate of ${\displaystyle z}$ is often denoted as ${\displaystyle {\overline {z}}}$ or ${\displaystyle z^{*}}$.

In polar form, if ${\displaystyle r}$ and ${\displaystyle \varphi }$ are real numbers then the conjugate of ${\displaystyle re^{i\varphi }}$ is ${\displaystyle re^{-i\varphi }.}$ This can be shown using Euler's formula.

The product of a complex number and its conjugate is a real number: ${\displaystyle a^{2}+b^{2}}$ (or ${\displaystyle r^{2}}$ in polar coordinates).

If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root.