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
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
Since
is defined for all
, we naturally define
as the following mathematical limit:
We have now defined what we mean by real exponents.