The Complex Plane
Figure 2.2:
Plotting a complex number as a point in the complex plane.
![\includegraphics[scale=0.5]{eps/ComplexPlane}](http://www.dsprelated.com/josimages_new/mdft/img176.png) |

We can plot any complex number

in a plane as an ordered pair

, as shown in Fig.
2.2. A
complex plane (or
Argand diagram) is any 2D graph in which the horizontal axis is
the
real part and the vertical axis is the
imaginary
part of 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
Cartesian or
rectilinear coordinates. We can
also express complex numbers in terms of
polar coordinates as
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
geometry, it is quick to show that conversion from
rectangular to polar coordinates is accomplished by the formulas
where

denotes the arctangent of

(the angle

in radians whose tangent is

), taking the
quadrant of the vector

into account. We will take

in
the range

to

(although we could choose any interval of
length

radians, such as 0 to

, etc.).
In
Matlab and Octave,
atan2(y,x) performs the
``quadrant-sensitive'' arctangent function. On the other hand,
atan(y/x), like the more traditional mathematical notation

does not ``know'' the quadrant of

, so it maps
the entire real line to the interval

. As a specific
example, the angle of the vector

(in quadrant I) has the
same tangent as the angle of

(in quadrant III).
Similarly,

(quadrant II) yields the same tangent as

(quadrant IV).
The formula

for converting rectangular
coordinates to radius

, follows immediately from the
Pythagorean theorem, while the

follows from the definition of the tangent
function itself.
Similarly, conversion from polar to rectangular coordinates is simply
These follow immediately from the definitions of cosine and sine,
respectively.
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