# Complex Conjugate **The [[complex number]] equal in magnitude but with opposite imaginary sign to another complex number.** ![[ComplexConjugate.svg|550]] *A complex number (light blue) and its conjugate (red) on the complex plane* > [!NOTE] Complex Conjugate > If $z=x+iy$, then its *complex conjugate* is > $\bar{z}=z^{*}=x-iy=r\angle{-\phi}$ ## Properties | Property | Description | |:------------------------------------------------------------------------------------------------------------------------------------------------------------:|:-----------------------------------------------------------------------------------------------------------------------------:| | $\begin{gather}\overline{z\pm w}=\bar{z}\pm\bar{w}\\\overline{zw}=\bar{z}\bar{w}\\\overline{\left(\frac{z}{w}\right)}=\frac{\bar{z}}{\bar{w}}\end{gather}$ | Distributive over addition, subtraction,<br>multiplication, and division | | $\overline{\overline{z}}=z$ | [[Involution]] | | $\overline{z^{n}}=(\bar{z})^{n},\quad \forall n\in\mathbb{Z}$ | Commutation under composition<br>with [[Exponentiation#Complex exponents with positive real base\|integer exponentiation]] | | $\exp(\bar{z})=\overline{\exp(z)}$ | Commutation under composition<br>with the [[Exponential Function#Complex exponential function\|complex exponential function]] | | $\ln(\bar{z})=\overline{\ln(z)},\quad z\neq0\vee z\notin\mathbb{R}^{-}$ | Commutation under composition<br> with the [[complex logarithm]] | ## Complex conjugate root theorem > [!Complex conjugate root theorem] > The *complex conjugate root theorem* states that for a single-variable polynomial with real coefficients, if a non-real complex number $z$ is a root then its complex conjugate $\bar{z}$ is also a root. Together with the [[fundamental theorem of algebra]], this implies that a polynomial of odd degree has at least one real root.