\documentclass[12pt,a4paper,portrait,notitlepage]{report}
\usepackage[utf8]{inputenc}
\usepackage[french]{babel}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{ifthen}
\usepackage{comment}
\usepackage[top=0.5cm,bottom=2.5cm,left=1.5cm,right=1.5cm]{geometry}
\author{Alain Busser\\IREMI 974}
\usepackage{tikz}
\usetikzlibrary{automata,shapes,arrows,shadings}
\tikzset{
v/.style={initial text={\tikz{\node[draw,fill,color=green] {};}},
circle,draw,ultra thick,minimum width=1.4cm},
a/.style={rectangle,draw,ultra thick,minimum width=1.4cm,minimum height=1.4cm},
rouge/.style={->,>=stealth',ultra thick,color=red},
bleu/.style={->,>=stealth',ultra thick,color=blue},
arc/.style={->,>=stealth',ultra thick}
}

\begin{document}

\begin{center}
\begin{LARGE}
\textbf{Dérivée de la fonction carré}
\end{LARGE}
\end{center}

La fonction carré est celle qui, à un nombre (ou autre jeu), associe son produit par lui-même. On la note $x \mapsto x^{2}$.

\section{En des nombres négatifs}

\subsection{En -2}

Le carré de

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v,accepting] (b) at (1,0) {}; 
\draw[rouge] (o) to (a);
\draw[rouge] (a) to (b);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

(c'est-à-dire le produit de ce jeu par lui-même) est, après développement et réduction, égal à

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v] (b) at (1,0) {}; 
\node[v] (c) at (2,0) {}; 
\node[v,accepting] (d) at (3,0) {}; 
\draw[bleu] (o) to (a);
\draw[bleu] (a) to (b);
\draw[bleu] (b) to (c);
\draw[bleu] (c) to (d);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (0,0) {};
\node[v] (b) at (1,1) {};
\node[v] (c) at (2,0)  {};
\node[v] (d) at (1,-1) {};
\node[v] (e) at (-1,-1) {};
\node[v,initial] (a) at (-1,1) {};
\draw[bleu,bend left=15] (d) to (o);
\draw[rouge,bend right=15] (d) to (o);
\draw[rouge] (b) to (o);
\draw[bleu] (b) to (c);
\draw[bleu,bend left=15] (e) to (o);
\draw[rouge,bend right=15] (e) to (o);
\draw[bleu] (a) to (b);
\draw[rouge] (a) to (o);
\draw[bleu] (c) to (d);
\draw[rouge] (c) to (o);
\draw[rouge] (d) to (e);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

C'est la somme de l'entier

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v] (b) at (1,0) {}; 
\node[v] (c) at (2,0) {}; 
\node[v,accepting] (d) at (3,0) {}; 
\draw[bleu] (o) to (a);
\draw[bleu] (a) to (b);
\draw[bleu] (b) to (c);
\draw[bleu] (c) to (d);
\end{tikzpicture}
\end{center}


et de l'infiniment petit négatif

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (0,0) {};
\node[v] (b) at (1,1) {};
\node[v] (c) at (2,0)  {};
\node[v] (d) at (1,-1) {};
\node[v] (e) at (-1,-1) {};
\node[v,initial] (a) at (-1,1) {};
\draw[bleu,bend left=15] (d) to (o);
\draw[rouge,bend right=15] (d) to (o);
\draw[rouge] (b) to (o);
\draw[bleu] (b) to (c);
\draw[bleu,bend left=15] (e) to (o);
\draw[rouge,bend right=15] (e) to (o);
\draw[bleu] (a) to (b);
\draw[rouge] (a) to (o);
\draw[bleu] (c) to (d);
\draw[rouge] (c) to (o);
\draw[rouge] (d) to (e);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

Mais cet infiniment petit négatif est lui-même le produit par

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

de

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v] (b) at (1,0) {}; 
\node[v] (c) at (2,0) {}; 
\node[v,accepting] (d) at (3,0) {}; 
\draw[rouge] (o) to (a);
\draw[rouge] (a) to (b);
\draw[rouge] (b) to (c);
\draw[rouge] (c) to (d);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

qui est la somme de l'entier

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v] (b) at (1,0) {}; 
\node[v] (c) at (2,0) {}; 
\node[v,accepting] (d) at (3,0) {}; 
\draw[rouge] (o) to (a);
\draw[rouge] (a) to (b);
\draw[rouge] (b) to (c);
\draw[rouge] (c) to (d);
\end{tikzpicture}
\end{center}


et de l'infiniment petit positif

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

Conclusion~: le nombre dérivé de la fonction carré en 

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v,accepting] (b) at (1,0) {}; 
\draw[rouge] (o) to (a);
\draw[rouge] (a) to (b);
\end{tikzpicture}
\end{center}

est

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v] (b) at (1,0) {}; 
\node[v] (c) at (2,0) {}; 
\node[v,accepting] (d) at (3,0) {}; 
\draw[rouge] (o) to (a);
\draw[rouge] (a) to (b);
\draw[rouge] (b) to (c);
\draw[rouge] (c) to (d);
\end{tikzpicture}
\end{center}






\subsection{En -1,5}

Le carré de 

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,-1) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[rouge] (o) to (b);
\draw[bleu] (o) to (a);
\draw[rouge] (a) to (b);
\draw[rouge] (b) to (c);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

(c'est-à-dire le produit de ce jeu par lui-même) est, après développement et réduction, égal à

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (n5) at (-1,-1) {};
\node[v] (n4) at (-2,0) {};
\node[v] (n3) at (-1,1) {};
\node[v] (n0) at (0,0) {};
\node[v] (n1) at (1,0) {};
\node[v,accepting] (n2) at (2,0) {};
\draw[bleu] (n5) to (n0);
\draw[rouge] (n5) to (n4);
\draw[bleu] (n4) to (n0);
\draw[rouge] (n4) to (n3);
\draw[bleu] (n3) to (n0);
\draw[bleu] (n0) to (n1);
\draw[bleu] (n1) to (n2);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (0,0) {};
\node[v] (d) at (1,-1) {};
\node[v] (c) at (2,0) {};
\node[v] (b) at (1,1) {};
\node[v,initial] (a) at (-1,1) {};
\draw[bleu,bend left=15] (d) to (o);
\draw[rouge,bend right=15] (d) to (o);
\draw[rouge] (c) to (o);
\draw[bleu] (c) to (d);
\draw[bleu,bend left=15] (b) to (c);
\draw[rouge,bend right=15] (b) to (c);
\draw[bleu] (a) to (b);
\draw[rouge] (a) to (o);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

C'est la somme de la fraction

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (n5) at (-1,-1) {};
\node[v] (n4) at (-2,0) {};
\node[v] (n3) at (-1,1) {};
\node[v] (n0) at (0,0) {};
\node[v] (n1) at (1,0) {};
\node[v,accepting] (n2) at (2,0) {};
\draw[bleu] (n5) to (n0);
\draw[rouge] (n5) to (n4);
\draw[bleu] (n4) to (n0);
\draw[rouge] (n4) to (n3);
\draw[bleu] (n3) to (n0);
\draw[bleu] (n0) to (n1);
\draw[bleu] (n1) to (n2);
\end{tikzpicture}
\end{center}

et de l'infiniment petit négatif

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (0,0) {};
\node[v] (d) at (1,-1) {};
\node[v] (c) at (2,0) {};
\node[v] (b) at (1,1) {};
\node[v,initial] (a) at (-1,1) {};
\draw[bleu,bend left=15] (d) to (o);
\draw[rouge,bend right=15] (d) to (o);
\draw[rouge] (c) to (o);
\draw[bleu] (c) to (d);
\draw[bleu,bend left=15] (b) to (c);
\draw[rouge,bend right=15] (b) to (c);
\draw[bleu] (a) to (b);
\draw[rouge] (a) to (o);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

Mais cet infiniment petit négatif est lui-même le produit par

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

de

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v] (b) at (1,0) {}; 
\node[v,accepting] (c) at (2,0) {}; 
\draw[rouge] (o) to (a);
\draw[rouge] (a) to (b);
\draw[rouge] (b) to (c);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

qui est la somme de l'entier

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v] (b) at (1,0) {}; 
\node[v,accepting] (c) at (2,0) {}; 
\draw[rouge] (o) to (a);
\draw[rouge] (a) to (b);
\draw[rouge] (b) to (c);
\end{tikzpicture}
\end{center}

et de l'infiniment petit positif

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

\newpage

Conclusion~: le nombre dérivé de la fonction carré en

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,-1) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[rouge] (o) to (b);
\draw[bleu] (o) to (a);
\draw[rouge] (a) to (b);
\draw[rouge] (b) to (c);
\end{tikzpicture}
\end{center}

est

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v] (b) at (1,0) {}; 
\node[v,accepting] (c) at (2,0) {}; 
\draw[rouge] (o) to (a);
\draw[rouge] (a) to (b);
\draw[rouge] (b) to (c);
\end{tikzpicture}
\end{center}



\subsection{En -1}

Le carré de 

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v,accepting] (a) at (0,0) {}; 
\draw[rouge] (o) to (a);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

(c'est-à-dire le produit de ce jeu par lui-même) est, après développement et réduction, égal à

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v,accepting] (a) at (0,0) {}; 
\draw[bleu] (o) to (a);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v] (a) at (1,1) {};
\node[v] (b) at (2,0) {};
\node[v,accepting] (c) at (0,0) {};
\node[v,initial] (d) at (-1,1) {};
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (a) to (c);
\draw[rouge,bend right=15] (a) to (c);
\draw[bleu,bend left=15] (b) to (c);
\draw[rouge,bend right=15] (b) to (c);
\draw[rouge] (d) to (c);
\draw[bleu] (d) to (a);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

C'est la somme de l'entier

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v,accepting] (a) at (0,0) {}; 
\draw[bleu] (o) to (a);
\end{tikzpicture}
\end{center}

et de l'infiniment petit négatif

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v] (a) at (1,1) {};
\node[v] (b) at (2,0) {};
\node[v,accepting] (c) at (0,0) {};
\node[v,initial] (d) at (-1,1) {};
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (a) to (c);
\draw[rouge,bend right=15] (a) to (c);
\draw[bleu,bend left=15] (b) to (c);
\draw[rouge,bend right=15] (b) to (c);
\draw[rouge] (d) to (c);
\draw[bleu] (d) to (a);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

Mais cet infiniment petit négatif est lui-même le produit par

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

de

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v,accepting] (b) at (1,0) {}; 
\draw[rouge] (o) to (a);
\draw[rouge] (a) to (b);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

qui est la somme de l'entier

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v,accepting] (b) at (1,0) {}; 
\draw[rouge] (o) to (a);
\draw[rouge] (a) to (b);
\end{tikzpicture}
\end{center}

et de l'infiniment petit

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

Conclusion~: le nombre dérivé de la fonction carré en

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v,accepting] (a) at (0,0) {}; 
\draw[rouge] (o) to (a);
\end{tikzpicture}
\end{center}

est

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v,accepting] (b) at (1,0) {}; 
\draw[rouge] (o) to (a);
\draw[rouge] (a) to (b);
\end{tikzpicture}
\end{center}



\subsection{En -0,75}

Le carré de 

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[v,initial] (n6) at (-1,0) {};
\node[v] (n0) at (0,1) {};
\node[v] (n1) at (0,-1) {};
\node[v,accepting] (n2) at (1,0) {};
\draw[rouge] (n6) to (n0);
\draw[bleu] (n6) to (n1);
\draw[rouge] (n0) to (n2);
\draw[bleu] (n0) to (n1);
\draw[rouge] (n1) to (n2);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

(c'est-à-dire le produit de ce jeu par lui-même) est, après développement et réduction, égal à

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[v,initial] (n8) at (-2,0) {};
\node[v] (n7) at (-1,-1) {};
\node[v] (n6) at (1,-1) {};
\node[v] (n0) at (0,0) {};
\node[v] (n1) at (2,0) {};
\node[v,accepting] (n2) at (1,1) {};
\draw[bleu] (n8) to (n0);
\draw[rouge] (n8) to (n7);
\draw[bleu] (n7) to (n0);
\draw[rouge] (n7) to (n6);
\draw[bleu] (n6) to (n0);
\draw[rouge] (n6) to (n1);
\draw[bleu] (n1) to (n2);
\draw[bleu] (n0) to (n2);
\draw[rouge] (n0) to (n1);
\end{tikzpicture}

\begin{tikzpicture}[scale=1.5]
\node[v] (d) at (-2,2) {};
\node[v] (e) at (0,2) {};
\node[v,accepting] (f) at (2,2) {}; 
\node[v] (b) at (-1,1) {};
\node[v] (c) at (1,1) {};
\node[v] (a) at (-2,0) {};
\node[v,initial] (o) at (2,0) {};
\draw[bleu,bend left=15] (e) to (f);
\draw[rouge,bend right=15] (e) to (f);
\draw[bleu,bend left=15] (c) to (f);
\draw[rouge,bend right=15] (c) to (f);
\draw[rouge] (a) to (b);
\draw[bleu] (a) to (d);
\draw[bleu] (o) to (c);
\draw[rouge,bend left] (o) to (a);
\draw[rouge] (b) to (d);
\draw[bleu] (b) to (c);
\draw[bleu] (d) to (e);
\draw[rouge] (e) to (c);
\draw[rouge,bend left] (d) to (f);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

C'est la somme de la fraction

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[v,initial] (n8) at (-2,0) {};
\node[v] (n7) at (-1,-1) {};
\node[v] (n6) at (1,-1) {};
\node[v] (n0) at (0,0) {};
\node[v] (n1) at (2,0) {};
\node[v,accepting] (n2) at (1,1) {};
\draw[bleu] (n8) to (n0);
\draw[rouge] (n8) to (n7);
\draw[bleu] (n7) to (n0);
\draw[rouge] (n7) to (n6);
\draw[bleu] (n6) to (n0);
\draw[rouge] (n6) to (n1);
\draw[bleu] (n1) to (n2);
\draw[bleu] (n0) to (n2);
\draw[rouge] (n0) to (n1);
\end{tikzpicture}
\end{center}

et de l'infiniment petit négatif

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[v] (d) at (-2,2) {};
\node[v] (e) at (0,2) {};
\node[v,accepting] (f) at (2,2) {}; 
\node[v] (b) at (-1,1) {};
\node[v] (c) at (1,1) {};
\node[v] (a) at (-2,0) {};
\node[v,initial] (o) at (2,0) {};
\draw[bleu,bend left=15] (e) to (f);
\draw[rouge,bend right=15] (e) to (f);
\draw[bleu,bend left=15] (c) to (f);
\draw[rouge,bend right=15] (c) to (f);
\draw[rouge] (a) to (b);
\draw[bleu] (a) to (d);
\draw[bleu] (o) to (c);
\draw[rouge,bend left] (o) to (a);
\draw[rouge] (b) to (d);
\draw[bleu] (b) to (c);
\draw[bleu] (d) to (e);
\draw[rouge] (e) to (c);
\draw[rouge,bend left] (d) to (f);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

Mais cet infiniment petit négatif est lui-même le produit par

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

de

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,-1) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[rouge] (o) to (b);
\draw[bleu] (o) to (a);
\draw[rouge] (a) to (b);
\draw[rouge] (b) to (c);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

qui est la somme de la fraction

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,-1) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[rouge] (o) to (b);
\draw[bleu] (o) to (a);
\draw[rouge] (a) to (b);
\draw[rouge] (b) to (c);
\end{tikzpicture}
\end{center}

et de l'infiniment petit positif

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

Conclusion~: le nombre dérivé de la fonction carré en

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[v,initial] (n6) at (-1,0) {};
\node[v] (n0) at (0,1) {};
\node[v] (n1) at (0,-1) {};
\node[v,accepting] (n2) at (1,0) {};
\draw[rouge] (n6) to (n0);
\draw[bleu] (n6) to (n1);
\draw[rouge] (n0) to (n2);
\draw[bleu] (n0) to (n1);
\draw[rouge] (n1) to (n2);
\end{tikzpicture}
\end{center}

est

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,-1) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[rouge] (o) to (b);
\draw[bleu] (o) to (a);
\draw[rouge] (a) to (b);
\draw[rouge] (b) to (c);
\end{tikzpicture}
\end{center}


 

\subsection{En -0,5}

Le carré de

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v,accepting] (a) at (0.8,0.5) {}; 
\node[v] (b) at (0,0) {}; 
\draw[rouge] (o) to (a);
\draw[bleu] (o) to (b);
\draw[rouge] (b) to (a);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

(c'est-à-dire le produit de ce jeu par lui-même) est, après développement et réduction, égal à

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[v,initial] (z) at (1,1) {};
\node[v,accepting] (y) at (2,0) {};
\node[v] (o) at (0,0) {};
\node[v] (b) at (1,-1) {}; 
\draw[bleu] (o) to (y);
\draw[rouge] (o) to (b);
\draw[rouge] (z) to (o);
\draw[bleu] (z) to (y);
\draw[bleu] (b) to (y);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[rouge] (a) to (o);
\draw[bleu] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

C'est la somme de la fraction

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[v,initial] (z) at (1,1) {};
\node[v,accepting] (y) at (2,0) {};
\node[v] (o) at (0,0) {};
\node[v] (b) at (1,-1) {}; 
\draw[bleu] (o) to (y);
\draw[rouge] (o) to (b);
\draw[rouge] (z) to (o);
\draw[bleu] (z) to (y);
\draw[bleu] (b) to (y);
\end{tikzpicture}
\end{center}

et de l'infiniment petit négatif

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[rouge] (a) to (o);
\draw[bleu] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}


Mais cet infiniment petit négatif est lui-même le produit par

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

de

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v,accepting] (a) at (0,0) {}; 
\draw[rouge] (o) to (a);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

qui est la somme de l'entier

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v,accepting] (a) at (0,0) {}; 
\draw[rouge] (o) to (a);
\end{tikzpicture}
\end{center}

et de l'infiniment petit positif

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

Conclusion~: le nombre dérivé de la fonction carré en
\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v,accepting] (a) at (0.8,0.5) {}; 
\node[v] (b) at (0,0) {}; 
\draw[rouge] (o) to (a);
\draw[bleu] (o) to (b);
\draw[rouge] (b) to (a);
\end{tikzpicture}
\end{center}

est

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v,accepting] (a) at (0,0) {}; 
\draw[rouge] (o) to (a);
\end{tikzpicture}
\end{center}


\subsection{En -0,25}

Le carré de

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (z) at (1,1) {};
\node[v,accepting] (y) at (2,0) {};
\node[v] (o) at (0,0) {};
\node[v] (b) at (1,-1) {}; 
\draw[rouge] (o) to (y);
\draw[bleu] (o) to (b);
\draw[bleu] (z) to (o);
\draw[rouge] (z) to (y);
\draw[rouge] (b) to (y);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

(c'est-à-dire le produit de ce jeu par lui-même) est, après développement et réduction, égal à

\begin{center}
\begin{tikzpicture}[scale=1.4]
\node[v,initial] (n12) at (-1,1) {};
\node[v] (n11) at (1,1) {};
\node[v] (n10) at (2,0) {};
\node[v,accepting] (n2) at (0,0) {};
\node[v] (n1) at (-1,-1) {};
\node[v] (n0) at (1,-1) {};
\draw[bleu] (n12) to (n2);
\draw[rouge] (n12) to (n11);
\draw[bleu] (n11) to (n2);
\draw[rouge] (n11) to (n10);
\draw[bleu] (n10) to (n2);
\draw[rouge] (n10) to (n0);
\draw[bleu] (n0) to (n2);
\draw[rouge] (n0) to (n1);
\draw[bleu] (n1) to (n2);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,initial] (a) at (-0.5,1) {};
\node[v] (b) at (0.5,1) {};
\node[v,accepting] (o) at (0,0) {};
\node[v] (d) at (1,0) {};
\node[v] (e) at (0,-1) {};
\node[v] (c) at (-1,0) {};
\draw[bleu,bend left=15] (c) to (o);
\draw[rouge,bend right=15] (c) to (o);
\draw[rouge] (d) to (o);
\draw[bleu] (d) to (e);
\draw[bleu,bend left=15] (e) to (o);
\draw[rouge,bend right=15] (e) to (o);
\draw[bleu] (c) to (e);
\draw[bleu] (b) to (d);
\draw[rouge] (b) to (o);
\draw[rouge] (a) to (b);
\draw[bleu] (a) to (c);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

C'est la somme de la fraction

\begin{center}
\begin{tikzpicture}[scale=1.4]
\node[v,initial] (n12) at (-1,1) {};
\node[v] (n11) at (1,1) {};
\node[v] (n10) at (2,0) {};
\node[v,accepting] (n2) at (0,0) {};
\node[v] (n1) at (-1,-1) {};
\node[v] (n0) at (1,-1) {};
\draw[bleu] (n12) to (n2);
\draw[rouge] (n12) to (n11);
\draw[bleu] (n11) to (n2);
\draw[rouge] (n11) to (n10);
\draw[bleu] (n10) to (n2);
\draw[rouge] (n10) to (n0);
\draw[bleu] (n0) to (n2);
\draw[rouge] (n0) to (n1);
\draw[bleu] (n1) to (n2);
\end{tikzpicture}
\end{center}

et de l'infiniment petit

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (a) at (-0.5,1) {};
\node[v] (b) at (0.5,1) {};
\node[v,accepting] (o) at (0,0) {};
\node[v] (d) at (1,0) {};
\node[v] (e) at (0,-1) {};
\node[v] (c) at (-1,0) {};
\draw[bleu,bend left=15] (c) to (o);
\draw[rouge,bend right=15] (c) to (o);
\draw[rouge] (d) to (o);
\draw[bleu] (d) to (e);
\draw[bleu,bend left=15] (e) to (o);
\draw[rouge,bend right=15] (e) to (o);
\draw[bleu] (c) to (e);
\draw[bleu] (b) to (d);
\draw[rouge] (b) to (o);
\draw[rouge] (a) to (b);
\draw[bleu] (a) to (c);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

Mais cet infiniment petit est lui-même le produit par

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

de

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v,accepting] (a) at (0.8,0.5) {}; 
\node[v] (b) at (0,0) {}; 
\draw[rouge] (o) to (a);
\draw[bleu] (o) to (b);
\draw[rouge] (b) to (a);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

qui est la somme de la fraction

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v,accepting] (a) at (0.8,0.5) {}; 
\node[v] (b) at (0,0) {}; 
\draw[rouge] (o) to (a);
\draw[bleu] (o) to (b);
\draw[rouge] (b) to (a);
\end{tikzpicture}
\end{center}


et de l'infiniment petit positif

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

Conclusion~: le nombre dérivé de la fonction carré en

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (z) at (1,1) {};
\node[v,accepting] (y) at (2,0) {};
\node[v] (o) at (0,0) {};
\node[v] (b) at (1,-1) {}; 
\draw[rouge] (o) to (y);
\draw[bleu] (o) to (b);
\draw[bleu] (z) to (o);
\draw[rouge] (z) to (y);
\draw[rouge] (b) to (y);
\end{tikzpicture}
\end{center}

est

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v,accepting] (a) at (0.8,0.5) {}; 
\node[v] (b) at (0,0) {}; 
\draw[rouge] (o) to (a);
\draw[bleu] (o) to (b);
\draw[rouge] (b) to (a);
\end{tikzpicture}
\end{center}



\section{En 0}

Le carré de

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial,accepting] (a) at (0,0) {};
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

est le produit de 

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

par lui-même, autrement dit, par un infinitésimal. Donc le nombre dérivé de la fonction carré en

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial,accepting] (a) at (0,0) {};
\end{tikzpicture}
\end{center} 

est

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial,accepting] (a) at (0,0) {};
\end{tikzpicture}
\end{center} 

\section{En des nombres positifs}

\subsection{En 0,25}

Le carré de

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (z) at (1,1) {};
\node[v,accepting] (y) at (2,0) {};
\node[v] (o) at (0,0) {};
\node[v] (b) at (1,-1) {}; 
\draw[bleu] (o) to (y);
\draw[rouge] (o) to (b);
\draw[rouge] (z) to (o);
\draw[bleu] (z) to (y);
\draw[bleu] (b) to (y);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

(c'est-à-dire le produit de ce jeu par lui-même) est, après développement et réduction, égal à

\begin{center}
\begin{tikzpicture}[scale=1.4]
\node[v,initial] (n12) at (-1,1) {};
\node[v] (n11) at (1,1) {};
\node[v] (n10) at (2,0) {};
\node[v,accepting] (n2) at (0,0) {};
\node[v] (n1) at (-1,-1) {};
\node[v] (n0) at (1,-1) {};
\draw[bleu] (n12) to (n2);
\draw[rouge] (n12) to (n11);
\draw[bleu] (n11) to (n2);
\draw[rouge] (n11) to (n10);
\draw[bleu] (n10) to (n2);
\draw[rouge] (n10) to (n0);
\draw[bleu] (n0) to (n2);
\draw[rouge] (n0) to (n1);
\draw[bleu] (n1) to (n2);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,initial] (a) at (-0.5,1) {};
\node[v] (b) at (0.5,1) {};
\node[v,accepting] (o) at (0,0) {};
\node[v] (d) at (1,0) {};
\node[v] (e) at (0,-1) {};
\node[v] (c) at (-1,0) {};
\draw[bleu,bend left=15] (c) to (o);
\draw[rouge,bend right=15] (c) to (o);
\draw[bleu] (d) to (o);
\draw[rouge] (d) to (e);
\draw[bleu,bend left=15] (e) to (o);
\draw[rouge,bend right=15] (e) to (o);
\draw[rouge] (c) to (e);
\draw[rouge] (b) to (d);
\draw[bleu] (b) to (o);
\draw[bleu] (a) to (b);
\draw[rouge] (a) to (c);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

C'est la somme de la fraction

\begin{center}
\begin{tikzpicture}[scale=1.4]
\node[v,initial] (n12) at (-1,1) {};
\node[v] (n11) at (1,1) {};
\node[v] (n10) at (2,0) {};
\node[v,accepting] (n2) at (0,0) {};
\node[v] (n1) at (-1,-1) {};
\node[v] (n0) at (1,-1) {};
\draw[bleu] (n12) to (n2);
\draw[rouge] (n12) to (n11);
\draw[bleu] (n11) to (n2);
\draw[rouge] (n11) to (n10);
\draw[bleu] (n10) to (n2);
\draw[rouge] (n10) to (n0);
\draw[bleu] (n0) to (n2);
\draw[rouge] (n0) to (n1);
\draw[bleu] (n1) to (n2);
\end{tikzpicture}
\end{center}

et de l'infiniment petit positif

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (a) at (-0.5,1) {};
\node[v] (b) at (0.5,1) {};
\node[v,accepting] (o) at (0,0) {};
\node[v] (d) at (1,0) {};
\node[v] (e) at (0,-1) {};
\node[v] (c) at (-1,0) {};
\draw[bleu,bend left=15] (c) to (o);
\draw[rouge,bend right=15] (c) to (o);
\draw[bleu] (d) to (o);
\draw[rouge] (d) to (e);
\draw[bleu,bend left=15] (e) to (o);
\draw[rouge,bend right=15] (e) to (o);
\draw[rouge] (c) to (e);
\draw[rouge] (b) to (d);
\draw[bleu] (b) to (o);
\draw[bleu] (a) to (b);
\draw[rouge] (a) to (c);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

Mais cet infiniment petit positif est lui-même le produit par

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}


de

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v,accepting] (a) at (0.8,0.5) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (o) to (a);
\draw[rouge] (o) to (b);
\draw[bleu] (b) to (a);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

qui est la somme de la fraction

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v,accepting] (a) at (0.8,0.5) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (o) to (a);
\draw[rouge] (o) to (b);
\draw[bleu] (b) to (a);
\end{tikzpicture}
\end{center}

et de l'infiniment petit positif

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

Conclusion~: le nombre dérivé de la fonction carré en

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (z) at (1,1) {};
\node[v,accepting] (y) at (2,0) {};
\node[v] (o) at (0,0) {};
\node[v] (b) at (1,-1) {}; 
\draw[bleu] (o) to (y);
\draw[rouge] (o) to (b);
\draw[rouge] (z) to (o);
\draw[bleu] (z) to (y);
\draw[bleu] (b) to (y);
\end{tikzpicture}
\end{center}

est

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v,accepting] (a) at (0.8,0.5) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (o) to (a);
\draw[rouge] (o) to (b);
\draw[bleu] (b) to (a);
\end{tikzpicture}
\end{center}



\subsection{En 0,5}

Le carré de

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v,accepting] (a) at (0.8,0.5) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (o) to (a);
\draw[rouge] (o) to (b);
\draw[bleu] (b) to (a);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

(c'est-à-dire le produit de ce jeu par lui-même) est, après développement et réduction, égal à

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (z) at (1,1) {};
\node[v,accepting] (y) at (2,0) {};
\node[v] (o) at (0,0) {};
\node[v] (b) at (1,-1) {}; 
\draw[bleu] (o) to (y);
\draw[rouge] (o) to (b);
\draw[rouge] (z) to (o);
\draw[bleu] (z) to (y);
\draw[bleu] (b) to (y);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

C'est la somme de la fraction

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (z) at (1,1) {};
\node[v,accepting] (y) at (2,0) {};
\node[v] (o) at (0,0) {};
\node[v] (b) at (1,-1) {}; 
\draw[bleu] (o) to (y);
\draw[rouge] (o) to (b);
\draw[rouge] (z) to (o);
\draw[bleu] (z) to (y);
\draw[bleu] (b) to (y);
\end{tikzpicture}
\end{center}


et de l'infiniment petit positif

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

Mais cet infiniment petit positif est lui-même le produit par

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

de

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v,accepting] (a) at (0,0) {}; 
\draw[bleu] (o) to (a);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

qui est la somme de l'entier

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v,accepting] (a) at (0,0) {}; 
\draw[bleu] (o) to (a);
\end{tikzpicture}
\end{center}

et de l'infiniment petit positif

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}


Conclusion~: le nombre dérivé de la fonction carré en

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v,accepting] (a) at (0.8,0.5) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (o) to (a);
\draw[rouge] (o) to (b);
\draw[bleu] (b) to (a);
\end{tikzpicture}
\end{center}

est

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v,accepting] (a) at (0,0) {}; 
\draw[bleu] (o) to (a);
\end{tikzpicture}
\end{center}


\subsection{En 0,75}

Le carré de 

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[v,initial] (n6) at (-1,0) {};
\node[v] (n0) at (0,1) {};
\node[v] (n1) at (0,-1) {};
\node[v,accepting] (n2) at (1,0) {};
\draw[bleu] (n6) to (n0);
\draw[rouge] (n6) to (n1);
\draw[bleu] (n0) to (n2);
\draw[rouge] (n0) to (n1);
\draw[bleu] (n1) to (n2);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

(c'est-à-dire le produit de ce jeu par lui-même) est, après développement et réduction, égal à

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[v,initial] (n8) at (-2,0) {};
\node[v] (n7) at (-1,-1) {};
\node[v] (n6) at (1,-1) {};
\node[v] (n0) at (0,0) {};
\node[v] (n1) at (2,0) {};
\node[v,accepting] (n2) at (1,1) {};
\draw[bleu] (n8) to (n0);
\draw[rouge] (n8) to (n7);
\draw[bleu] (n7) to (n0);
\draw[rouge] (n7) to (n6);
\draw[bleu] (n6) to (n0);
\draw[rouge] (n6) to (n1);
\draw[bleu] (n1) to (n2);
\draw[bleu] (n0) to (n2);
\draw[rouge] (n0) to (n1);
\end{tikzpicture}

\begin{tikzpicture}[scale=1.5]
\node[v] (d) at (-2,2) {};
\node[v] (e) at (0,2) {};
\node[v,accepting] (f) at (2,2) {}; 
\node[v] (b) at (-1,1) {};
\node[v] (c) at (1,1) {};
\node[v] (a) at (-2,0) {};
\node[v,initial] (o) at (2,0) {};
\draw[bleu,bend left=15] (e) to (f);
\draw[rouge,bend right=15] (e) to (f);
\draw[bleu,bend left=15] (c) to (f);
\draw[rouge,bend right=15] (c) to (f);
\draw[bleu] (a) to (b);
\draw[rouge] (a) to (d);
\draw[rouge] (o) to (c);
\draw[bleu,bend left] (o) to (a);
\draw[bleu] (b) to (d);
\draw[rouge] (b) to (c);
\draw[rouge] (d) to (e);
\draw[bleu] (e) to (c);
\draw[bleu,bend left] (d) to (f);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

C'est la somme de la fraction

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[v,initial] (n8) at (-2,0) {};
\node[v] (n7) at (-1,-1) {};
\node[v] (n6) at (1,-1) {};
\node[v] (n0) at (0,0) {};
\node[v] (n1) at (2,0) {};
\node[v,accepting] (n2) at (1,1) {};
\draw[bleu] (n8) to (n0);
\draw[rouge] (n8) to (n7);
\draw[bleu] (n7) to (n0);
\draw[rouge] (n7) to (n6);
\draw[bleu] (n6) to (n0);
\draw[rouge] (n6) to (n1);
\draw[bleu] (n1) to (n2);
\draw[bleu] (n0) to (n2);
\draw[rouge] (n0) to (n1);
\end{tikzpicture}
\end{center}

et de l'infiniment petit positif

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[v] (d) at (-2,2) {};
\node[v] (e) at (0,2) {};
\node[v,accepting] (f) at (2,2) {}; 
\node[v] (b) at (-1,1) {};
\node[v] (c) at (1,1) {};
\node[v] (a) at (-2,0) {};
\node[v,initial] (o) at (2,0) {};
\draw[bleu,bend left=15] (e) to (f);
\draw[rouge,bend right=15] (e) to (f);
\draw[bleu,bend left=15] (c) to (f);
\draw[rouge,bend right=15] (c) to (f);
\draw[bleu] (a) to (b);
\draw[rouge] (a) to (d);
\draw[rouge] (o) to (c);
\draw[bleu,bend left] (o) to (a);
\draw[bleu] (b) to (d);
\draw[rouge] (b) to (c);
\draw[rouge] (d) to (e);
\draw[bleu] (e) to (c);
\draw[bleu,bend left] (d) to (f);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

Mais cet infiniment petit positif est lui-même le produit par

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}


de

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,-1) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu] (o) to (b);
\draw[rouge] (o) to (a);
\draw[bleu] (a) to (b);
\draw[bleu] (b) to (c);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

qui est la somme de la fraction

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,-1) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu] (o) to (b);
\draw[rouge] (o) to (a);
\draw[bleu] (a) to (b);
\draw[bleu] (b) to (c);
\end{tikzpicture}
\end{center}

et de l'infiniment petit positif

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

Conclusion~: le nombre dérivé de la fonction carré en

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[v,initial] (n6) at (-1,0) {};
\node[v] (n0) at (0,1) {};
\node[v] (n1) at (0,-1) {};
\node[v,accepting] (n2) at (1,0) {};
\draw[bleu] (n6) to (n0);
\draw[rouge] (n6) to (n1);
\draw[bleu] (n0) to (n2);
\draw[rouge] (n0) to (n1);
\draw[bleu] (n1) to (n2);
\end{tikzpicture}
\end{center}

est

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,-1) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu] (o) to (b);
\draw[rouge] (o) to (a);
\draw[bleu] (a) to (b);
\draw[bleu] (b) to (c);
\end{tikzpicture}
\end{center}



\subsection{En 1}

Le carré de 

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v,accepting] (a) at (0,0) {}; 
\draw[bleu] (o) to (a);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

(c'est-à-dire le produit de ce jeu par lui-même) est, après développement et réduction, égal à

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v,accepting] (a) at (0,0) {}; 
\draw[bleu] (o) to (a);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v] (a) at (1,1) {};
\node[v] (b) at (2,0) {};
\node[v,accepting] (c) at (0,0) {};
\node[v,initial] (d) at (-1,1) {};
\draw[bleu] (a) to (b);
\draw[bleu,bend left=15] (a) to (c);
\draw[rouge,bend right=15] (a) to (c);
\draw[bleu,bend left=15] (b) to (c);
\draw[rouge,bend right=15] (b) to (c);
\draw[bleu] (d) to (c);
\draw[rouge] (d) to (a);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

C'est la somme de l'entier

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v,accepting] (a) at (0,0) {}; 
\draw[bleu] (o) to (a);
\end{tikzpicture}
\end{center}

et de l'infiniment petit positif

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v] (a) at (1,1) {};
\node[v] (b) at (2,0) {};
\node[v,accepting] (c) at (0,0) {};
\node[v,initial] (d) at (-1,1) {};
\draw[bleu] (a) to (b);
\draw[bleu,bend left=15] (a) to (c);
\draw[rouge,bend right=15] (a) to (c);
\draw[bleu,bend left=15] (b) to (c);
\draw[rouge,bend right=15] (b) to (c);
\draw[bleu] (d) to (c);
\draw[rouge] (d) to (a);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

Mais cet infiniment petit positif est lui-même le produit par

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

de

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v,accepting] (b) at (1,0) {}; 
\draw[bleu] (o) to (a);
\draw[bleu] (a) to (b);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

qui est la somme de l'entier

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v,accepting] (b) at (1,0) {}; 
\draw[bleu] (o) to (a);
\draw[bleu] (a) to (b);
\end{tikzpicture}
\end{center}

et de l'infiniment petit positif

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

Conclusion~: le nombre dérivé de la fonction carré en

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v,accepting] (a) at (0,0) {}; 
\draw[bleu] (o) to (a);
\end{tikzpicture}

\end{center}

est

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v,accepting] (b) at (1,0) {}; 
\draw[bleu] (o) to (a);
\draw[bleu] (a) to (b);
\end{tikzpicture}
\end{center}


\subsection{En 1,5}

Le carré de 

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,-1) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu] (o) to (b);
\draw[rouge] (o) to (a);
\draw[bleu] (a) to (b);
\draw[bleu] (b) to (c);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

(c'est-à-dire le produit de ce jeu par lui-même) est, après développement et réduction, égal à

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (n5) at (-1,-1) {};
\node[v] (n4) at (-2,0) {};
\node[v] (n3) at (-1,1) {};
\node[v] (n0) at (0,0) {};
\node[v] (n1) at (1,0) {};
\node[v,accepting] (n2) at (2,0) {};
\draw[bleu] (n5) to (n0);
\draw[rouge] (n5) to (n4);
\draw[bleu] (n4) to (n0);
\draw[rouge] (n4) to (n3);
\draw[bleu] (n3) to (n0);
\draw[bleu] (n0) to (n1);
\draw[bleu] (n1) to (n2);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (0,0) {};
\node[v] (d) at (1,-1) {};
\node[v] (c) at (2,0) {};
\node[v] (b) at (1,1) {};
\node[v,initial] (a) at (-1,1) {};
\draw[bleu,bend left=15] (d) to (o);
\draw[rouge,bend right=15] (d) to (o);
\draw[bleu] (c) to (o);
\draw[rouge] (c) to (d);
\draw[bleu,bend left=15] (b) to (c);
\draw[rouge,bend right=15] (b) to (c);
\draw[rouge] (a) to (b);
\draw[bleu] (a) to (o);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}


C'est la somme de la fraction

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (n5) at (-1,-1) {};
\node[v] (n4) at (-2,0) {};
\node[v] (n3) at (-1,1) {};
\node[v] (n0) at (0,0) {};
\node[v] (n1) at (1,0) {};
\node[v,accepting] (n2) at (2,0) {};
\draw[bleu] (n5) to (n0);
\draw[rouge] (n5) to (n4);
\draw[bleu] (n4) to (n0);
\draw[rouge] (n4) to (n3);
\draw[bleu] (n3) to (n0);
\draw[bleu] (n0) to (n1);
\draw[bleu] (n1) to (n2);
\end{tikzpicture}
\end{center}

et de l'infiniment petit positif

\begin{center}

\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (0,0) {};
\node[v] (d) at (1,-1) {};
\node[v] (c) at (2,0) {};
\node[v] (b) at (1,1) {};
\node[v,initial] (a) at (-1,1) {};
\draw[bleu,bend left=15] (d) to (o);
\draw[rouge,bend right=15] (d) to (o);
\draw[bleu] (c) to (o);
\draw[rouge] (c) to (d);
\draw[bleu,bend left=15] (b) to (c);
\draw[rouge,bend right=15] (b) to (c);
\draw[rouge] (a) to (b);
\draw[bleu] (a) to (o);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

Mais cet infiniment petit positif est lui-même le produit par

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

de

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v] (b) at (1,0) {}; 
\node[v,accepting] (c) at (2,0) {}; 
\draw[bleu] (o) to (a);
\draw[bleu] (a) to (b);
\draw[bleu] (b) to (c);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

qui est la somme de l'entier 

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v] (b) at (1,0) {}; 
\node[v,accepting] (c) at (2,0) {}; 
\draw[bleu] (o) to (a);
\draw[bleu] (a) to (b);
\draw[bleu] (b) to (c);
\end{tikzpicture}
\end{center}

et de l'infiniment petit

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

Conclusion~: le nombre dérivé de la fonction carré en 

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,-1) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu] (o) to (b);
\draw[rouge] (o) to (a);
\draw[bleu] (a) to (b);
\draw[bleu] (b) to (c);
\end{tikzpicture}
\end{center}

est

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v] (b) at (1,0) {}; 
\node[v,accepting] (c) at (2,0) {}; 
\draw[bleu] (o) to (a);
\draw[bleu] (a) to (b);
\draw[bleu] (b) to (c);
\end{tikzpicture}
\end{center}

\subsection{En 2}

Le carré de

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v,accepting] (b) at (1,0) {}; 
\draw[bleu] (o) to (a);
\draw[bleu] (a) to (b);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

(c'est-à-dire le produit de ce jeu par lui-même) est, après développement et réduction, égal à

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v] (b) at (1,0) {}; 
\node[v] (c) at (2,0) {}; 
\node[v,accepting] (d) at (3,0) {}; 
\draw[bleu] (o) to (a);
\draw[bleu] (a) to (b);
\draw[bleu] (b) to (c);
\draw[bleu] (c) to (d);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (0,0) {};
\node[v] (b) at (1,1) {};
\node[v] (c) at (2,0)  {};
\node[v] (d) at (1,-1) {};
\node[v] (e) at (-1,-1) {};
\node[v,initial] (a) at (-1,1) {};
\draw[bleu,bend left=15] (d) to (o);
\draw[rouge,bend right=15] (d) to (o);
\draw[bleu] (b) to (o);
\draw[rouge] (b) to (c);
\draw[bleu,bend left=15] (e) to (o);
\draw[rouge,bend right=15] (e) to (o);
\draw[rouge] (a) to (b);
\draw[bleu] (a) to (o);
\draw[rouge] (c) to (d);
\draw[bleu] (c) to (o);
\draw[bleu] (d) to (e);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

C'est la somme de l'entier

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v] (b) at (1,0) {}; 
\node[v] (c) at (2,0) {}; 
\node[v,accepting] (d) at (3,0) {}; 
\draw[bleu] (o) to (a);
\draw[bleu] (a) to (b);
\draw[bleu] (b) to (c);
\draw[bleu] (c) to (d);
\end{tikzpicture}

\end{center}

et de l'infiniment petit positif

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (0,0) {};
\node[v] (b) at (1,1) {};
\node[v] (c) at (2,0)  {};
\node[v] (d) at (1,-1) {};
\node[v] (e) at (-1,-1) {};
\node[v,initial] (a) at (-1,1) {};
\draw[bleu,bend left=15] (d) to (o);
\draw[rouge,bend right=15] (d) to (o);
\draw[bleu] (b) to (o);
\draw[rouge] (b) to (c);
\draw[bleu,bend left=15] (e) to (o);
\draw[rouge,bend right=15] (e) to (o);
\draw[rouge] (a) to (b);
\draw[bleu] (a) to (o);
\draw[rouge] (c) to (d);
\draw[bleu] (c) to (o);
\draw[bleu] (d) to (e);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

Mais cet infiniment petit positif est le produit par

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

de

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v] (b) at (1,0) {}; 
\node[v] (c) at (2,0) {}; 
\node[v,accepting] (d) at (3,0) {}; 
\draw[bleu] (o) to (a);
\draw[bleu] (a) to (b);
\draw[bleu] (b) to (c);
\draw[bleu] (c) to (d);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

qui est lui-même la somme de l'entier 

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v] (b) at (1,0) {}; 
\node[v] (c) at (2,0) {}; 
\node[v,accepting] (d) at (3,0) {}; 
\draw[bleu] (o) to (a);
\draw[bleu] (a) to (b);
\draw[bleu] (b) to (c);
\draw[bleu] (c) to (d);
\end{tikzpicture}
\end{center}

et de l'infiniment petit

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

Conclusion~: le nombre dérivé de la fonction carré en

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v,accepting] (b) at (1,0) {}; 
\draw[bleu] (o) to (a);
\draw[bleu] (a) to (b);
\end{tikzpicture}
\end{center}


est

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (-1,0) {};
\node[v] (a) at (0,0) {}; 
\node[v] (b) at (1,0) {}; 
\node[v] (c) at (2,0) {}; 
\node[v,accepting] (d) at (3,0) {}; 
\draw[bleu] (o) to (a);
\draw[bleu] (a) to (b);
\draw[bleu] (b) to (c);
\draw[bleu] (c) to (d);
\end{tikzpicture}
\end{center}

\end{document}


