Appendix B

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Complex numbers arise naturally from an attempt to solve the equation

It's easy enough to write such an equation down, but how would you solve it? The answer is

We let the symbol represent , so that . Then any number of the form

where and are real, is called a complex number. Let's take some other complex number to be , where and are real. Then the two complex numbers are equal,

which is to say

if and only if

We refer to as the real part of the complex number and as the complex part. Sometimes these are written as Re() and Im(), respectively.

We may restate the equivalence condition as if and only if the real part of is equal to the real part of and the imaginary part of is equal to the imaginary part of .

Complex numbers are multiplied like any other binomial expression:

where we have used .

The complex conjugate of the complex number is denoted and is given by

One reason for defining this is that a number times its own complex conjugate is real,

Note that the complex conjugate of the complex conjugate is the original complex number and

Note that the complex conjugate of a product is the product of complex conjugates:


We also call this the modulus squared so that the modulus is


It is often useful to look at a graph for a complex number. The graph consists of an x-axis for the real part, and a y-axis for the complex part. This is shown in Fig.~?. In this figure, it is easily seen that we can think of as a two-dimensional vector and that the magnitude (length) of the vector is the modulus of the complex number, .

Complexgraph1.jpeg

Figure B.1: A complex number in Cartesian coordinates.



Another useful way to represent this is with polar coordinates. We can do this by writing

It turns out that

so we could also write

It is often the case that people will write this as

where as is usual for polar coordinates. Then, everything is just like polar coordinates, with the exception of the inclusion of the factor . (See Fig.~?.)

Complexgraph2.jpeg

Figure B.2: A polar coordinate representation of a complex number.