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# Complex numbers

Complex numbers are written as:

where and are real numbers and . (In this book we'll use capital letters to denote complex numbers and lowercase for real numbers.) 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 real and imaginary parts of , written as:

If is a complex number, its magnitude, written as , is just the distance in the plane from the origin to the point :

and its argument, written as , is the angle from the positive axis to the point :

If we know the magnitude and argument of a complex number (say they are and , for instance) we can reconstruct the real and imaginary parts:

A complex number may be written in terms of its real and imaginary parts and (this is called rectangular form), or alternatively in polar form, in terms of and :

The rectangular and polar formulations are equivalent, and 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 encode sums of angles. 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:

And so, 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 of the result.

Subsections

Next: Sinusoids as geometric series Up: Time shifts Previous: Time shifts   Contents   Index
Miller Puckette 2006-03-03