Complex conjugate Definition and 79 Threads

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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