**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION:** "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 ©
*2017-09-28* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

**<<
Previous page TOC INDEX Next
page >>**

## The Complex Plane

We can plot complex numbers in a plane as ordered pairs , as shown in Fig. 2.2. The

complex planeis any 2D graph in which the horizontal axis is thereal partand the vertical axis is theimaginary partof a complex number or function. As an example, the number has coordinates in the complex plane while the number has coordinates .Plotting as the point in the complex plane can be viewed as a plot in

Cartesianorrectilinearcoordinates. We can also express complex numbers in terms ofpolar coordinatesas an ordered pair , where is the distance from the origin to the number being plotted, and is the angle of the number relative to the positive real coordinate axis (the line defined by and ). (See Fig. 2.2.)Using elementary

The first equation follows immediately from thegeometry, it is quick to show that conversion from rectangular to polar coordinates is accomplished by the formulasPythagorean theorem, while the second follows immediately from the definition of thetangentfunction. Similarly, conversion from polar to rectangular coordinates is simply These follow immediately from the definitions of cosine and sine, respectively,