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

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

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.