The Complex Plane

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). The Complex Plane

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The Complex Plane

Figure 2.2:Plotting a complex number as a point in thecomplex plane.
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We can plot complex numbers $z = x + jy$ in a plane as ordered pairs $(x,y)$, as shown in Fig. 2.2. The complex plane 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,\, where $r$ is the distance from the origin $(0,0)$ to the number being plotted, and $\ 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

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The first equation follows immediately from the Pythagorean theorem, while the second follows immediately from the definition of the tangent function. Similarly, conversion from polar to rectangular coordinates is simply
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These follow immediately from the definitions of cosine and sine, respectively,

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