**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION:** "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
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Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
*2017-09-28* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

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## Complex Numbers

This section introduces various notation and terms associated with complex numbers. As discussed above, complex numbers are devised by introducing the square-root of as a primitive new algebraic object among real numbers and manipulating it symbolically as if it were a real number itself:

Mathemeticians and physicists often use instead of as . The use of is common in engineering where is more often used for electrical current.As mentioned above, for any negative number , we have , where denotes the absolute value of . Thus, every square root of a negative number can be expressed as times the square root of a positive number.

By definition, we have

and so on. Thus, the sequence , is a periodic sequence with period , since . (We’ll learn later that the sequence is a sampled complex sinusoid havingfrequency equal to one fourth the sampling rate.)Every

complex numbercan be written as

where and are real numbers. We call thereal partand theimaginary part. We may also use the notation

Note that the real numbers are the subset of the complex numbers having a zero imaginary part ().The rule for

complex multiplicationfollows directly from the definition of the imaginary unit :

In some mathematics texts, complex numbers are defined as ordered pairs of real numbers , and algebraic operations such as multiplication are defined more formally as operations on ordered pairs, e.g., . However, such formality tends to obscure the underlying simplicity of complex numbers as a straightforward extension of real numbers to include .It is important to realize that complex numbers can be treated algebraically just like real numbers. That is, they can be added, subtracted, multiplied, divided, etc., using exactly the same rules of algebra (since both real and complex numbers are mathematical

fields). It is often preferable to think of complex numbers as being the true and proper setting for algebraic operations, with real numbers being the limited subset for which .To explore further the magical world of complex variables, see any textbook such as [4,5].

Subsections