Complex conjugate
The complex conjugate of a complex number 
is the complex number 
.
Geometrically, if 
is a point in the complex plane, 
is the reflection of across the real axis.
Properties
Conjugation is its own functional inverse and commutes with the usual operations on complex numbers:
(
is the same as 
(
is the same as 
\mathbb{C}
|\overline{z}| = |z|
\overline{z}\cdot z = |z|^2
z = r\cdot e^{it}
r, t \in \mathbb{R}
\overline z = r\cdot e^{-it}
\overline z![$is the complex number of same [[absolute value]] but opposite [[argument]] of$](//latex.artofproblemsolving.com/e/9/1/e91b25856ffcff7238d2e100654fbaac683af8df.png)
z
z + \overline z = 2 \mathrm{Re}(z)
\mathrm{Re}(z)![$is the [[real part]] of$](//latex.artofproblemsolving.com/1/3/0/130e002ddf8cbf9f4c487be86d057c16dc0a17b2.png)
zz - \overline{z} = 2i \mathrm{Im}(z)\mathrm{Im}(z)![$is the [[imaginary part]] of$](//latex.artofproblemsolving.com/5/6/3/5638ba3a6f76e07c6ccd890d6b0d411d1739590a.png)
z$.
(
is the same as 
(
is the same as 
\mathbb{C}
|\overline{z}| = |z|
\overline{z}\cdot z = |z|^2
z = r\cdot e^{it}
r, t \in \mathbb{R}
\overline z = r\cdot e^{-it}
\overline z![$is the complex number of same [[absolute value]] but opposite [[argument]] of$](http://latex.artofproblemsolving.com/e/9/1/e91b25856ffcff7238d2e100654fbaac683af8df.png)
z
z + \overline z = 2 \mathrm{Re}(z)
\mathrm{Re}(z)![$is the [[real part]] of$](http://latex.artofproblemsolving.com/1/3/0/130e002ddf8cbf9f4c487be86d057c16dc0a17b2.png)
z![$is the [[imaginary part]] of$](http://latex.artofproblemsolving.com/5/6/3/5638ba3a6f76e07c6ccd890d6b0d411d1739590a.png)
z$.
This article is a stub. Help us out by expanding it.
