**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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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## Real Exponents

The closest we can actually get to most real numbers is to compute a

rational numberthat is as close as we need. It can be shown that rational numbers aredensein the real numbers; that is, between every two real numbers there is a rational number, and between every two rational numbers is a real number. Anirrationalnumber can be defined as any real number having a non-repeating decimal expansion. For example, is an irrational real number whose decimal expansion starts out as

(computed viaN[Sqrt[2],80]in Mathematica). Every truncated, rounded, or repeating expansion is arationalnumber. That is, it can be rewritten as an integer divided by another integer. For example,

and, using to denote the repeating part of a decimal expansion,

Other examples of irrational numbers include

Let denote the -digit decimal expansion of an arbitrary real number . Then is a rational number (some integer over ). We can say

Since is defined for all , it is straightforward to define