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The Complex Plane

Figure 2.2: Plotting a complex number as a point in the complex plane.
\includegraphics[scale=0.5]{eps/ComplexPlane}

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
\begin{eqnarray*} r &=& \sqrt{x^2 + y^2}\\ \theta &=& \tan^{-1}(y,x). \end{eqnarray*}
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
\begin{eqnarray*} x &=& r\,\cos(\theta)\\ y &=& r\,\sin(\theta). \end{eqnarray*}
These follow immediately from the definitions of cosine and sine, respectively.
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