Complex Numbers, Part 3

Complex numbers cannot be plotted on the traditional Cartesian coordinate system because of the imaginary part. A cartesian system is used to plot a pair of real numbers. Since a pair of complex numbers consists of four real numbers, a pair of complex numbers cannot be plotted. However, we can plot a single complex number on a modified version of a Cartesian system where the x-axis is relabelled as the real axis and the y-axis is the imaginary axis. This modified coordinate system is called an Argand diagram after its inventor Jean-Robert Argand, a Swiss mathematician.

Here is a plot of a generic complex number z = x + yi:

An Argand plot can be used to see the results of adding and subtracting complex numbers. These are interesting and you can see these in any textbook but the real power of an Argand plot is to see the relationship between the rectangular form of a complex number (the form we have been using so far) and the polar form.

Remember that polar coordinates are a different way to identify the location of a point in a two-dimensional plane. Instead of giving x and y coordinates, you can give an angle from the positive x-axis to define a direction, then a distance along that direction to define where a point is located. The same is true for an Argand plot of a complex number, and this polar form is more convenient to use than the rectangular form in certain scenarios. Below is the same plot as above, but I’ve added the polar coordinates:

From your past experience with triangles and trigonometry, you can see that the following relationships are true:

\[\begin{array}{l}
r = |z| = \sqrt{{x}^{2}+{y}^{2}}\\
x = r \text{ cos} πœƒ, y = r \text{ sin} πœƒ\\
πœƒ =\text{tan}^{-1}\frac{y}{x}\end{array}\]

From the above, you can see that in terms of r and πœƒ,

z = r cosπœƒ + ir sinπœƒ

The polar form notation that is used in VCE Australia is

z = r cisπœƒ

where “cis” is shorthand for “cos + i sin”. There are other ways to express a polar complex number as well. In electronics, rβˆ πœƒ is frequently used.

Some definitions: modulus is the length of a complex number, that is, r. Arg(z) is the principal value of πœƒ. This is pronounced as “the argument of z“. As any particular πœƒ can have multiples of 2πœ‹ radians or 360Β° added to it, arg(z) restricts πœƒ to -πœ‹ < arg(z) ≀ πœ‹. Be careful when computing πœƒ with the inverse tangent formula. A calculator will only give values between -πœ‹/2 and πœ‹/2 (-90Β° and 90Β°), that is the first and fourth quadrants. You need to interpret this result to the correct angle by looking at the signs of x and y to know which quadrant your particular complex number falls in. I will do some sample problems in my next post to illustrate this.