The complex conjugate of a complex number is its reflection on the real axis in the complex plane.

Definition: Complex Conjugate

Let \(z=x + yi\in\mathbb C\) be a complex number. The complex conjugate of \(z\), denoted by \(z^* \), is the complex number with the same real part and the negative imaginary part, i.e.

\[z^* :=\Re(z) - \Im (z) i=x - y i.\]

In the complex plane, complex conjugates can be interpreted as the reflections of complex numbers across the real axis, as shown in the below figure

Complex_conjugate_picture

(from Wikipedia, uploaded by Aflafla1)

Below, you can experiment with complex conjugation of a polygon in the complex plane. Just move around (e.g. using your mouse, mouse-pad or touchscreen) the points in the \(z\)-Plane and see the result in the \(w\)-Plane:

\(z\)-Plane:

\(w\)-Plane with \(w:= z^\ast \):

Corollaries: 1
Definitions: 2 3 4
Examples: 5
Proofs: 6 7 8 9 10 11 12
Propositions: 13 14 15 16


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