# Complex Conjugate **The [[complex number]] equal in magnitude but with opposite imaginary sign to a complex number.** > [!Example] Complex Conjugate > If $z=x+iy$, then its *complex conjugate* is > $\bar{z}=z^{*}=x-iy=r\angle{-\phi}$ Conjugation is *distributive* over addition, subtraction, multiplication, and division. $\begin{align*} \overline{z+w}&=\bar{z}+\bar{w} \\ \overline{z-w}&=\bar{z}-\bar{w} \\ \overline{zw}&=\bar{z}\;\bar{w} \\ \overline{\frac{z}{w}}&=\frac{\bar{z}}{\bar{w}} \end{align*}$ Conjugation is *commutative* under composition with exponentiation to integer powers, the exponential function, and the natural logarithm. Conjugation is also an [[involution]]. ## 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.