### A. Complex Numbers

Overview: 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 complex numbers:
• 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 complex number.
• 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 diagrams.
• 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.
Databases:

#### Demos - MATLAB3

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: