**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION:** "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
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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## Rational Exponents

A

rationalnumber is a real number that can be expressed as a ratio of twointegers:

Applying property (2) of exponents, we have

Thus, the only thing new is . Since

we see that is the th root of . This is sometimes written

The th root of a real (or complex) number is not unique. As we all know, square roots give two values (e.g., ). In the general case of th roots, there are distinct values, in general.How do we come up with

differentnumbers which when raised to the th power will yield ? The answer is to considercomplexnumbers inpolar form. By Euler's Identity, the real number can be expressed, for any integer , as . Using this form for , the number can be written as

We can now see that we get adifferent complex numberfor each . When , we get the same thing as when . When , we get the same thing as when , and so on, so there are only cases using this construct. The th root for can be called the ''primitive th root of '', since integer powers of it give all of the others.

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