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.^{3.1}An *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

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

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

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

We have now defined what we mean by

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University