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The Complex Plane

Figure 2.2: Plotting a complex number as a point in the complex plane.

We can plot any complex number $ z = x + jy$ in a plane as an ordered pair $ (x,y)$, 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 $ j$ has coordinates $ (0,1)$ in the complex plane while the number $ 1$ has coordinates $ (1,0)$.

Plotting $ z = x + jy$ as the point $ (x,y)$ 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 $ (r,\theta)$, where $ r$ is the distance from the origin $ (0,0)$ to the number being plotted, and $ \theta$ is the angle of the number relative to the positive real coordinate axis (the line defined by $ y=0$ and $ x>0$). (See Fig.2.2.)

Using elementary geometry, it is quick to show that conversion from rectangular to polar coordinates is accomplished by the formulas

r &=& \sqrt{x^2 + y^2}\\
\theta &=& \tan^{-1}(y,x).

where $ \tan^{-1}(y,x)$ denotes the arctangent of $ y/x$ (the angle $ \theta$ in radians whose tangent is $ \tan(\theta)=y/x$), taking the quadrant of the vector $ (x,y)$ into account. We will take $ \theta$ in the range $ -\pi$ to $ \pi $ (although we could choose any interval of length $ 2\pi $ radians, such as 0 to $ 2\pi $, 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 $ \tan^{-1}(y/x)$ does not ``know'' the quadrant of $ (x,y)$, so it maps the entire real line to the interval $ (-\pi/2,\pi/2)$. As a specific example, the angle of the vector $ (x,y)=(1,1)$ (in quadrant I) has the same tangent as the angle of $ (x,y)=(-1,-1)$ (in quadrant III). Similarly, $ (x,y)=(-1,1)$ (quadrant II) yields the same tangent as $ (x,y)=(1,-1)$ (quadrant IV).

The formula $ r = \sqrt{x^2 + y^2}$ for converting rectangular coordinates to radius $ r$, follows immediately from the Pythagorean theorem, while the $ \theta = \tan^{-1}(y,x)$ follows from the definition of the tangent function itself.

Similarly, conversion from polar to rectangular coordinates is simply

x &=& r\,\cos(\theta)\\
y &=& r\,\sin(\theta).

These follow immediately from the definitions of cosine and sine, respectively.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See for details.


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