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 as3.2
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:
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.