Rational Exponents

Chapter Contents:

Search Mathematics of the DFT

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

A rational number is a real number that can be expressed as a ratio of two finite integers:

$\displaystyle x = \frac{L}{M}, \quad L\in{\bf Z},\quad M\in{\bf Z}$

Applying property (2) of exponents, we have

$\displaystyle a^x = a^{L/M} = \left(a^{\frac{1}{M}}\right)^L.$

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

$\displaystyle \left(a^{\frac{1}{M}}\right)^M = a^{\frac{M}{M}} = a$

we see that $a^{1/M}$ is the

th root of

. This is sometimes written

$\displaystyle \zbox {a^{\frac{1}{M}} \isdef \sqrt[M]{a}.}$

The

th root of a real (or complex) number is not unique. As we all know, square roots give two values (e.g., $\sqrt{4}=\pm2$ ). In the general case of

th roots, there are

distinct values, in general. After proving Euler's identity, it will be easy to find them all (see §3.11). As an example, $\sqrt[4]{1}=1$ ,

, and

, since

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

Sign in