In Appendix A, the basic manipulations of complex numbers
are presented. The algebraic rules for combining
complex numbers are reviewed, and then a geometric viewpoint is
taken to explain various operations by drawing vector diagrams. The
following four significant ideas will be pointed out concerning
- Simple Algebraic Rules: operations on complex numbers
\(z=x+j\, y\) follow exactly the same rules as real numbers, with
\(j^2\) replaced everywhere by \(-1\).
- Eliminate Trigonometry: in polar form,
\(z=\cos(\theta) + j\, \sin(\theta)\) appears in formulas,
so many trig identities reduce to simple algebraic operations on a
- Represent Vectors: a vector drawn from the origin to a point
\((x,y)\) in a two-dimensional plane
is equivalent to \(z=x+j\,y\). The algebraic rules
for \(z\) are, in effect, the basic rules for vector operations. More
important, however, is the visualization gained from the vector
- Represent Sinusoids: the magnitude and phase of the sinusoid
are used to define the polar form of a complex number. Then operations
such as adding sine waves are reduced to adding complex numbers.
Examples of how complex numbers and complex exponentials
can be handled by MATLAB.
PhasorRaces began as a speed drill for testing complex addition.
Now it includes many other related operations that can be tested
in a "drill" scenario: adding sinusoids, z-transforms, etc.
A timer starts as soon as the problem is posed, so that a
student can try to solve questions quickly and accurately.
ZDrill is a program that tests the users ability to calculate the result
of simple operations on complex numbers.
The program emphasizes the vectorial view of a complex number.
The following six operations are supported: