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, is an irrational real number whose decimal expansion starts out as^3.2

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,

and, using to denote the repeating part of a decimal expansion, a repeating example is as follows:

Other examples of irrational numbers include

Their decimal expansions do not repeat.

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

That is, the limit of as goes to infinity is .

Since is defined for all , we naturally define as the following mathematical limit:

We have now defined what we mean by real exponents.

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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.