# What is Complex conjugate: Definition and 79 Discussions

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.

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