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 a and b are real, then) the complex conjugate of
a
+
b
i
{\displaystyle a+bi}
is equal to
a
−
b
i
.
{\displaystyle a-bi.}
The complex conjugate of
z
{\displaystyle z}
is often denoted as
z
¯
{\displaystyle {\overline {z}}}
.
In polar form, the conjugate of
r
e
i
φ
{\displaystyle re^{i\varphi }}
is
r
e
−
i
φ
{\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:
a
2
+
b
2
{\displaystyle a^{2}+b^{2}}
(or
r
2
{\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.