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
z
z + \overline z = 2 \mathrm{Re}(z)
\mathrm{Re}(z)
z
z - \overline{z} = 2i \mathrm{Im}(z)
\mathrm{Im}(z)
z$.
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