# Rational Exponents

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

It appears that you are using AdBlocking software. The cost of running this website is covered by advertisements. If you like it please feel free to a small amount of money to secure the future of this website. NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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

<< Previous page  TOC  INDEX  Next page >>

## 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 . 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 different numbers which when raised to the th power will yield ? The answer is to consider complexnumbers in polar 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 a different complex number for 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.

Subsections

<< Previous page  TOC  INDEX  Next page >>