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Real Exponents

The closest we can actually get to most real numbers is to compute a rational number that is as close as we need. It can be shown that rational numbers are dense in 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.3.1An irrational number can be defined as any real number having a non-repeating decimal expansion. For example, $ \sqrt{2}$ is an irrational real number whose decimal expansion starts out as3.2

$\displaystyle \sqrt{2} =

Every truncated, rounded, or repeating expansion is a rational number. That is, it can be rewritten as an integer divided by another integer. For example,

$\displaystyle 1.414 = \frac{1414}{1000}

and, using $ \overline{\mbox{overbar}}$ to denote the repeating part of a decimal expansion, a repeating example is as follows:

x &=& 0.\overline{123} \\ [5pt]
\quad\Rightarrow\quad 1000x &=& 123.\overline{123} = 123 + x\\ [5pt]
\quad\Rightarrow\quad 999x &=& 123\\ [5pt]
\quad\Rightarrow\quad x &=& \frac{123}{999}

Examples of irrational numbers include

\pi &=& 3.1415926535897932384626433832795028841971693993751058209749\dots\\
e &=& 2.7182818284590452353602874713526624977572470936999595749669\dots\,.

Their decimal expansions do not repeat.

Let $ {\hat x}_n$ denote the $ n$ -digit decimal expansion of an arbitrary real number $ x$ . Then $ {\hat x}_n$ is a rational number (some integer over $ 10^n$ ). We can say

$\displaystyle \lim_{n\to\infty} {\hat x}_n = x.

That is, the limit of $ {\hat x}_n$ as $ n$ goes to infinity is $ x$ .

Since $ a^{{\hat x}_n}$ is defined for all $ n$ , we naturally define $ a^x$ as the following mathematical limit:

$\displaystyle \zbox {a^x \isdef \lim_{n\to\infty} a^{{\hat x}_n}}

We have now defined what we mean by real exponents.

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``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 © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University