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 as3.2
![$\displaystyle \sqrt{2} =
1.414213562373095048801688724209698078569671875376948073176679\dots
$](http://www.dsprelated.com/josimages_new/mdft/img258.png)
![$\displaystyle 1.414 = \frac{1414}{1000}
$](http://www.dsprelated.com/josimages_new/mdft/img259.png)
![$ \overline{\mbox{overbar}}$](http://www.dsprelated.com/josimages_new/mdft/img260.png)
![\begin{eqnarray*}
x &=& 0.\overline{123} \\ [5pt]
\quad\Rightarrow\quad 1000x &=...
...999x &=& 123\\ [5pt]
\quad\Rightarrow\quad x &=& \frac{123}{999}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img261.png)
Other examples of irrational numbers include
![\begin{eqnarray*}
\pi &=& 3.1415926535897932384626433832795028841971693993751058...
...82818284590452353602874713526624977572470936999595749669\dots\,.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img262.png)
Their decimal expansions do not repeat.
Let
denote the
-digit decimal expansion of an arbitrary real
number
. Then
is a rational number (some integer over
).
We can say
![$\displaystyle \lim_{n\to\infty} {\hat x}_n = x.
$](http://www.dsprelated.com/josimages_new/mdft/img265.png)
![$ {\hat x}_n$](http://www.dsprelated.com/josimages_new/mdft/img263.png)
![$ n$](http://www.dsprelated.com/josimages_new/mdft/img80.png)
![$ x$](http://www.dsprelated.com/josimages_new/mdft/img25.png)
Since
is defined for all
, we naturally define
as the following mathematical limit:
![$\displaystyle \zbox {a^x \isdef \lim_{n\to\infty} a^{{\hat x}_n}}
$](http://www.dsprelated.com/josimages_new/mdft/img267.png)
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A First Look at Taylor Series
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Rational Exponents