irrationalIrrational2001-11-04 06:49:222008-11-09 18:00:02Definition\usepackage{amssymb}
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An \emph{irrational number} is a real number which cannot be represented as a ratio of two integers. That is, if $x$ is irrational, then
$$ x \ne \frac{a}{b} $$
with $a,b \in \mathbb{Z}$ and $b \ne 0$.
\subsubsection*{Examples}
\begin{enumerate}
\item $\sqrt[p]{2}$ is irrational for $p=2,3,\ldots$,
\item $\pi, e$, and $\sqrt[p]{2}$ for $p=2,3,\ldots$,
are irrational,
\item It is not known whether Euler's constant is rational or irrational.
\end{enumerate}
\subsubsection*{Properties}
\begin{enumerate}
\item It $a$ is a real number and $a^n$ is irrational for some $n=2,3,\ldots$,
then $a$ is irrational (\PMlinkname{proof}{IfAnIsIrrationalThenAIsIrrational}).
\item The sum, difference, product, and quotient (when defined) of two numbers,
one rational and another irrational, is irrational.
(\PMlinkname{proof}{RationalAndIrrational}).
\end{enumerate}