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

$$\begin{eqnarray*} r &=& \sqrt{x^2 + y^2}\\ \theta &=& \tan^{-1}(y/x), \end{eqnarray*}$$

where $\tan^{-1}(y/x)$ denotes the arctangent of $y/x$ (the angle $\theta$ in radians whose tangent is $\tan(\theta)=y/x$). 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 notation $\tan^{-1}(y,x)$ means ``the angle whose tangent is $y/x$, taking the quadrant of the vector $(x,y)$ into account.'' 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.