Complex conjugate

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 and are real, then) the complex conjugate of is equal to The complex conjugate of is often denoted as

Geometric representation (Argand diagram) of and its conjugate in the complex plane. The complex conjugate is found by reflecting across the real axis.

In polar form, the conjugate of is This can be shown using Euler's formula.

The product of a complex number and its conjugate is a real number:  (or  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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