Rational Exponents

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Rational Exponents

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 rational number is a real number that can be expressed as a ratio of two integers:

\

Applying property (2) of exponents, we have

\

Thus, the only thing new is $a^{1/M}$. Since

\

we see that $a^{1/M}$ is the $M$th root of $a$. This is sometimes written

\

The $M$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 $M$th roots, there are $M$ distinct values, in general.

How do we come up with $M$ different numbers which when raised to the $M$th power will yield $a$? The answer is to consider complexnumbers in polar form. By Euler's Identity, the real number $a>0$can be expressed, for any integer $k$, as $a \. Using this form for $a$, the number $a^{1/M}$ can be written as

\

We can now see that we get a different complex number for each $k=0,1,2,3,\. When $k=M$, we get the same thing as when $k=0$. When $k=M+1$, we get the same thing as when $k=1$, and so on, so there are only $M$ cases using this construct. The $M$th root for $k=1$ can be called the ''primitive $M$th root of $a$'', since integer powers of it give all of the others.


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