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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$ :

\begin{eqnarray*}
r &\isdef & \left\vert z\right\vert = \sqrt{x^2 + y^2}\\
&=& \mbox{\emph{modulus}%
\index{modulus, complex number\vert textbf}\index{complex numbers!modulus, magnitude,
radial coordinate, absolute value, norm\vert textbf},
\emph{magnitude}%
\index{magnitude of a complex number\vert textbf},
\emph{absolute value}\index{absolute value, complex number\vert textbf},
\emph{norm}%
\index{norm of a complex number\vert textbf},
or \emph{radial coordinate} of $z$} \\
\theta &\isdef & \angle{z} = \tan^{-1}(y/x)\\
&=& \mbox{\emph{angle},
\emph{argument}\index{argument of a complex number\vert textbf}\index{complex numbers!argument, angle, or phase\vert textbf},
or \emph{phase} of $z$}
\end{eqnarray*}

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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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA