Complex numbers

Complex
numbers are written as:

where and are real numbers and . (In this book we'll use the upper case Roman letters such as to denote complex numbers. Real numbers appear as lower case Roman or Greek letters, except for integer bounds, usually written as or .) Since a complex number has two real components, we use a Cartesian plane (in place of a number line) to graph it, as shown in Figure 7.1. The quantities and are called the

If is a complex number, its
*magnitude* (or *absolute value*),
written as , is just the distance in the plane from the origin to the
point :

and its

If we know the magnitude and argument of a complex number (call them and ) we can reconstruct the real and imaginary parts:

A complex number may be written in terms of its real and imaginary parts and , as (this is called

The rectangular and polar formulations are interchangeable; the equations above show how to compute and from and and vice versa.

The main reason we use complex numbers in electronic music is because they
magically automate trigonometric calculations. We frequently have to add
angles together in order to talk about the changing phase of an audio signal as
time progresses (or as it is shifted in time, as in this chapter). It turns
out that, if you multiply two complex numbers, the argument of the product is
the sum of the arguments of the two factors. To see how this happens, we'll
multiply two numbers and , written in polar form:

giving:

Here the minus sign in front of the term comes from multiplying by itself, which gives . We can spot the cosine and sine summation formulas in the above expression, and so it simplifies to:

By inspection, it follows that the product has magnitude and argument .

We can use this property of complex numbers to add and subtract angles (by multiplying and dividing complex numbers with the appropriate arguments) and then to take the cosine and sine of the result by extracting the real and imaginary parts.