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More Notation and Terminology

It's already been mentioned that the rectilinear coordinates of a complex number $ z = x + jy$ in the complex plane are called the real part and imaginary part, respectively.

We also have special notation and various names for the polar coordinates $ (r,\theta)$ of a complex number $ z$:

r &\isdef & \left\vert z\right\vert = \sqrt{x^2 + y^2}\\
...!argument, angle, or phase\vert textbf},
or \emph{phase} of $z$}

The complex conjugate of $ z$ is denoted $ \overline{z}$ (or $ z^\ast$) and is defined by

$\displaystyle \zbox {\overline{z} \isdef x - j y}

where, of course, $ z\isdef x+jy$.

In general, you can always obtain the complex conjugate of any expression by simply replacing $ j$ with $ -j$. In the complex plane, this is a vertical flip about the real axis; i.e., complex conjugation replaces each point in the complex plane by its mirror image on the other side of the $ x$ axis.

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[How to cite this work] [Order a printed hardcopy]

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]