TABLE OF CONTENTS We established in the previous chapter on Fourier analysis that an arbitrary function could be described in terms of simple sine and cosine functions. The mathematics of the sinusoid was developed using the example of a spot on a bicycle wheel moving in time. We could also consider the amplitude of the sinusoidal wave as if it was a point on a circle. We needed two numbers to describe the point, in a Cartesian coordinate system, the point P( x, y) is the distance in the x direction, and the distance in the y direction, while in a polar coordinate system, was r ncos( ?)and r nsin( ?). But despite the fact that they were real numbers, we could not perform normal algebraic operations of addition and multiplication upon them. Point ( x, y) in essence is a single number and is treated as one entity. For one, we could not perform arithmetic because the algebra does not allow us to have a comma in parentheses, so mathematicians had to invent a different numbering system, just like they did for negative numbers, real numbers, and logarithmic numbers. For representing points in a coordinate system, they invented complex numbers.
In this chapter, you'll see how the new numbering system of complex numbers allows us to apply algebraic rules when such numbers are placed in algebraic equations. We will develop the mathematical foundation and establish the rules of arithmetic...
