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.

The complex conjugate of a complex number ${\displaystyle z}$ is written as ${\displaystyle {\overline {z}}}$ or ${\displaystyle z^{*}.}$ The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred in physics, where dagger (†) is used for the conjugate transpose, as well as electrical engineering and computer engineering, where bar notation can be confused for the logical negation ("NOT") Boolean algebra symbol, while the bar notation is more common in pure mathematics.

If a complex number is represented as a ${\displaystyle 2\times 2}$ matrix, the notations are identical, and the complex conjugate corresponds to the matrix transpose, which is a flip along the diagonal.

The following properties apply for all complex numbers ${\displaystyle z}$ and ${\displaystyle w,}$ unless stated otherwise, and can be proved by writing ${\displaystyle z}$ and ${\displaystyle w}$ in the form ${\displaystyle a+bi.}$

For any two complex numbers, conjugation is distributive over addition, subtraction, multiplication and division: $${\displaystyle {\begin{aligned}{\overline {z+w}}&={\overline {z}}+{\overline {w}},\\{\overline {z-w}}&={\overline {z}}-{\overline {w}},\\{\overline {zw}}&={\overline {z}}\;{\overline {w}},\quad {\text{and}}\\{\overline {\left({\frac {z}{w}}\right)}}&={\frac {\overline {z}}{\overline {w}}},\quad {\text{if }}w\neq 0.\end{aligned}}}$$