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\usepackage{times}
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\newcommand{\Z}{\boldmath Z}



\begin{document}

\title{Episode 03 -- Gaussian integers}
\subtitle{European section -- Season 3}
\date{}

\begin{frame}
  \titlepage
\end{frame}


\begin{frame}{Gaussian integers on a lattice}

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\end{frame}

\begin{frame}{Operations in the set of Gaussian integers}

\begin{definition}[Gaussian integers]
A \emph{Gaussian integer} is a number of the form $a+bi$ where $a$ and $b$ are two integers and $i$ is an imaginary number such that $i^2=-1$. The set of all Gaussian integers is usually noted $\Z[i]$.
\end{definition}\pause


\begin{block}{Golden rule}
Addition and multiplication in the set $\Z[i]$ follow the same rules as addition and multiplication of integers but ``real parts'' and ``imaginary parts'' cannot be mixed up in addition.
\end{block}

\end{frame}

\begin{frame}{Addition and multiplication of points ?}


\begin{block}{Question 1}
What is the graphical effect of adding a Gaussian integer to another ?
\end{block}\pause

\begin{block}{Question 2}
What is the graphical effect of multiplying a Gaussian integer by another ?\pause\\
Study the problem for :\pause
\begin{itemize}
\item multiplication by a real number $b$ ;\pause
\item multiplication by the imaginary number $i$ ;\pause
\item multiplication by the imaginary number $bi$.
\end{itemize}

\end{block}


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\begin{frame}{Addition and translation}

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\end{frame}

\begin{frame}{Multiplication, homothety and rotation}%similarity transformation

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\end{document}
