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\rhead{\sc{Sample Extended Glossary II}}
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\noindent {\bf Extended Glossary.}
Please give a definition of a {\bf rational number}.  
Then give an example of a rational
number, an example of a number that is not rational, and state and prove a theorem 
about rational numbers.


\medskip


\noindent You may work in groups, but please write up 
your solutions {\bf yourself}. If you do work together,
your group should come up with at least 
two examples, two non-examples, and two theorems.
Each one (example/non-example/theorem) should be included in some group member's 
extended glossary.
Your solutions should be written formally,
so that we could cut and paste them into a textbook.


\medskip



\begin{center}

{\bf Your grade:}

\medskip

\begin{tabular}{|c|c|c|}
\hline
Component & Correct? & Well-written? \\ \hline
Definition & $\phantom{5}/6$ & $\phantom{5}/6$ \\ \hline
Example & $\phantom{5}/4$ & $\phantom{5}/4$ \\ \hline
Non-example & $\phantom{5}/4$  & $\phantom{5}/4$ \\ \hline
Theorem & $\phantom{5}/5$  & $\phantom{5}/5$  \\ \hline
Proof & $\phantom{5}/6$  & $\phantom{5}/6$  \\ \hline\hline
Total & $\phantom{5}/25$  & $\phantom{5}/25$  \\ \hline
\end{tabular}
\end{center}

\vfill \pagebreak



\begin{Definition}
A {\bf rational number} is a fraction $\frac{a}{b}$.
\end{Definition}

\begin{Example}
$\frac{1}{2}$.
\hfill $\diamondsuit$
\end{Example}


\begin{Example}
A number that isn't rational is the number $\pi=3.14159$.  
\end{Example}

We now prove that the rational numbers are indeed a field.

\begin{Theorem}
The set of rational numbers $\Q$ is countable.
\end{Theorem}


\begin{proof}
We can see this pretty easily by looking at the following diagram.
\begin{figure}[h]
\begin{center}
 \includegraphics[width=2.5in]{rationals-countable.png}
 \caption{We show the rationals by counting the black numbers in the
 order shown.  The red numbers are the redundant ones.}
\end{center}
\end{figure}
That diagram shows us that the positive rational numbers
are countable.  We can then count all the rational numbers by
starting with zero, and then using this counting but overlaying
the negative rationals.  In other words, we enumerate the rationals
$$
0, 1, -1, 2, -2, \frac{1}{2}, -\frac{1}{2}, \frac{1}{3}, -\frac{1}{3},\dots.
$$
\end{proof}




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