The Complex Plane

Figure: 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

where $\tan^{-1}(y/x)$ denotes the arctangent of $y/x$ (the angle $\theta$ in radians whose tangent is $\tan(\theta)=y/x$). 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.). The first equation (for $r$) 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

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