## The Complex Plane

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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