%%% % Simplification %%% \makeatletter%by christian Tellechea % Calcul du PGCD de #1 et #2 \newcount\cnt@a\newcount\cnt@b\newcount\pgcd \def\PGCD#1#2{% \ifnum#1>#2\cnt@a#1\cnt@b#2\else\cnt@a#2\cnt@b#1\relax\fi \PGCD@i } \def\PGCD@i{\edef\PGCD@ii##1{##1{\number\cnt@a}{\number\cnt@b}}\PGCD@ii\PGCD@iii} \def\PGCD@iii#1#2{% \cnt@b#1\relax\global\divide\cnt@b#2% \global\cnt@b\numexpr#1-#2*\cnt@b% \global\cnt@a#2\global\pgcd\cnt@a% \ifnum\cnt@b>\z@\expandafter\PGCD@i% \fi}% \makeatother \def\SSimplifie#1#2{% % Simplification d'une \'ecriture #1/#2 \ensuremath{ \newcount\numerateur\newcount\denominateur\newcount\valabsnum\newcount\valabsdeno \numerateur=\number#1 \denominateur=\number#2 \ifnum\number#1<0\relax \valabsnum=\numexpr0-\number#1 \else \valabsnum=\number#1 \fi \ifnum\number#2<0\relax \valabsdeno=\numexpr0-\number#2 \else \valabsdeno=\number#2 \fi \ifnum\the\numerateur<0\relax \ifnum\the\denominateur<0\relax \numerateur=\valabsnum \denominateur=\valabsdeno \fi \else \ifnum\the\denominateur<0\relax \numerateur=-\valabsnum \denominateur=\valabsdeno \fi \fi \ifnum\number#2=0\relax \text{\bfseries(???)} \else \ifnum\number#1=0\relax 0 \else \PGCD{\the\valabsnum}{\the\valabsdeno}% \ifnum\pgcd>1\relax \ifthenelse{\pgcd=\number#2 \OR \pgcd=\the\valabsdeno}{% \divide\numerateur by \denominateur\num{\the\numerateur} }{\divide\numerateur by\pgcd% \divide\denominateur by\pgcd% \frac{\num{\the\numerateur}}{\num{\the\denominateur}} } \else%%%comme on est avec les n\'egatifs, on doit regarder si la valeur absolue est \'egale \`a 1 \ifnum\valabsdeno=1\relax \divide\numerateur by \denominateur\num{\the\numerateur} \else \frac{\num{\the\numerateur}}{\num{\the\denominateur}} \fi \fi% \fi% \fi% }% } \newcommand\SSimpli[2]{% % D\'ecomposition d'une simplification de #1/#2 \newcount\numerateur\newcount\denominateur\newcount\valabsnum\newcount\valabsdeno% \numerateur=\number#1 \denominateur=\number#2 \ifnum\number#1<0 \valabsnum=\numexpr0-\number#1 \else \valabsnum=\number#1 \fi \ifnum\number#2<0 \valabsdeno=\numexpr0-\number#2 \else \valabsdeno=\number#2 \fi \ifnum\the\numerateur<0\relax \ifnum\the\denominateur<0\relax \numerateur=\valabsnum \denominateur=\valabsdeno \fi \else \ifnum\the\denominateur<0\relax \numerateur=-\valabsnum \denominateur=\valabsdeno \fi \fi \ifnum\number#2=0\relax \ensuremath{\text{\bfseries(???)}} \else \ifnum\number#1=0\relax 0 \else \PGCD{\the\valabsnum}{\the\valabsdeno}% \ifnum\pgcd>1\relax \ifthenelse{\pgcd=\number#2 \OR \pgcd=\the\valabsdeno}{% \divide\numerateur by \denominateur\num{\the\numerateur} }{%\divide\numerateur by\pgcd% % \divide\denominateur by\pgcd% \ensuremath{\frac{\num{\the\numerateur}_{\mbox{\tiny$\div\num{\number\pgcd}$}}}{\num{\the\denominateur}_{\mbox{\tiny$\div\num{\number\pgcd}$}}}}% } \else \ifnum\denominateur=1\relax \ensuremath{\frac{\num{\the\numerateur}_{\mbox{\tiny$\div\num{\number\pgcd}$}}}{\num{\the\denominateur}_{\mbox{\tiny$\div\num{\number\pgcd}$}}}} \else \ensuremath{\frac{\num{\the\numerateur}}{\num{\the\denominateur}}} \fi \fi \fi \fi } \newcount\anpdc\newcount\bnpdc\newcount\cnpdc\newcount\dnpdc% \newcount\DivCom \newcommand\DiviseurCommun[2]{% % #1 : le premier nombre entier % #2 : le deuxi\`eme nombre entier \anpdc=#1% \cnpdc=#2% \bnpdc=2\relax% \dnpdc=\numexpr#1+1\relax% \DivCom=1\relax% \whiledo{\bnpdc<\dnpdc}{% \modulo{\the\anpdc}{\the\bnpdc}\relax \ifnum\remainder=0% \modulo{\the\cnpdc}{\the\bnpdc} \ifnum\remainder=0% \DivCom=\bnpdc% \bnpdc=\anpdc% \else% \DivCom=1% \fi \else% \DivCom=1% \fi \bnpdc=\numexpr\bnpdc+1\relax% }% } \newcommand\LongueSimplification[2]{% \xdef\NumerateurDiv{#1}% \xdef\DenominateurDiv{#2}% \ifboolKV[ClesSimplification]{Impose}{% \DivCom=\useKV[ClesSimplification]{Nombre}% }{% \DiviseurCommun{#1}{#2}% }% \ifboolKV[ClesSimplification]{Debut}{% \ensuremath{% \whiledo{\DivCom>1}{% \frac{\mathcolor{PfCSimpliCol}{\num{\the\DivCom}}\times\num{\fpeval{\NumerateurDiv/\the\DivCom}}}{\mathcolor{PfCSimpliCol}{\num{\the\DivCom}}\times\num{\fpeval{\DenominateurDiv/\the\DivCom}}}=\frac{\num{\fpeval{\NumerateurDiv/\DivCom}}}{\num{\fpeval{\DenominateurDiv/\DivCom}}}% \xdef\NumerateurDiv{\fpeval{\NumerateurDiv/\DivCom}}% \xdef\DenominateurDiv{\fpeval{\DenominateurDiv/\DivCom}}% \DiviseurCommun{\NumerateurDiv}{\DenominateurDiv}% \xintifboolexpr{\DivCom>1}{=}{}% }% }% }{% \ensuremath{% \whiledo{\DivCom>1}{% \frac{\num{\fpeval{\NumerateurDiv/\the\DivCom}}\times\mathcolor{PfCSimpliCol}{\num{\the\DivCom}}}{\num{\fpeval{\DenominateurDiv/\the\DivCom}}\times\mathcolor{PfCSimpliCol}{\num{\the\DivCom}}}=\frac{\num{\fpeval{\NumerateurDiv/\DivCom}}}{\num{\fpeval{\DenominateurDiv/\DivCom}}}% \xdef\NumerateurDiv{\fpeval{\NumerateurDiv/\DivCom}}% \xdef\DenominateurDiv{\fpeval{\DenominateurDiv/\DivCom}}% \DiviseurCommun{\NumerateurDiv}{\DenominateurDiv}% \xintifboolexpr{\DivCom>1}{=}{}% }% }% }% }% \setKVdefault[ClesSimplification]{Details=false,All=false,Longue=false,Fleches=false,Contraire=0,Couleur=black,Debut=false,Impose=false} \defKV[ClesSimplification]{Nombre=\setKV[ClesSimplification]{Impose}} \newcounter{NbFrac}% \setcounter{NbFrac}{0}% \newcommand\Simplification[3][]{% \stepcounter{NbFrac}% \useKVdefault[ClesSimplification]% \setKV[ClesSimplification]{#1}% \ifboolKV[ClesSimplification]{Fleches}{% \setsepchar[*]{,*/}%\ignoreemptyitems \readlist*\Listea{#2}% \readlist*\Listeb{#3}% \setbox1=\hbox{\Listea[1,1]{}}% \setbox2=\hbox{\Listeb[1,1]}% \setbox3=\hbox{\Listea[1,3]}% \setbox4=\hbox{\Listeb[1,3]}% \ensuremath{% \frac{\tikzmarknode[anchor=north]{A-\theNbFrac}{\Listea[1,1]}{}}{\tikzmarknode[anchor=south]{B-\theNbFrac}{\Listeb[1,1]}{}}=\frac{\tikzmarknode[anchor=north]{C-\theNbFrac}{\Listea[1,3]}{}}{\tikzmarknode[anchor=south]{D-\theNbFrac}{\Listeb[1,3]}{}}% }% \begin{tikzpicture}[remember picture,overlay]% \draw[out=45,in=135,-stealth,transform canvas={yshift=0.25em}] let \p1=(pic cs:A-\theNbFrac), \p2=(pic cs:C-\theNbFrac) in (pic cs:A-\theNbFrac) to node[midway,above]{\Listea[1,2]}(\x2,\y1); \draw[out=-45,in=-135,-stealth,transform canvas={yshift=-0.25em}] (pic cs:B-\theNbFrac) to node[midway,below]{\Listeb[1,2]}(pic cs:D-\theNbFrac);% \end{tikzpicture}% }{% \xintifboolexpr{\useKV[ClesSimplification]{Contraire}>1}{% \ensuremath{% \frac{\num{#2}}{\num{#3}}=\frac{\num{#2}\times\num{\useKV[ClesSimplification]{Contraire}}}{\num{#3}\times\num{\useKV[ClesSimplification]{Contraire}}}=\frac{\num{\fpeval{\useKV[ClesSimplification]{Contraire}*#2}}}{\num{\fpeval{\useKV[ClesSimplification]{Contraire}*#3}}}% }% }{% \ifboolKV[ClesSimplification]{Longue}{% \colorlet{PfCSimpliCol}{\useKV[ClesSimplification]{Couleur}}% \LongueSimplification{#2}{#3}% }{% \ifboolKV[ClesSimplification]{Details}{\SSimpli{#2}{#3}}{\ifboolKV[ClesSimplification]{All}{\ensuremath{\SSimpli{#2}{#3}=\SSimplifie{#2}{#3}}}{\SSimplifie{#2}{#3}}}% }% }% }% }%