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---
output:
html_document: default
pdf_document: default
---
# À propos du calcul de pi
_Arnaud Legrand_
_25 juin 2018_
## En demandant à la lib maths
Mon ordinateur m’indique que $\pi$ vaut _approximativement_
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<div id="à-propos-du-calcul-de-pi" class="section level1">
<h1>À propos du calcul de pi</h1>
<p><em>Arnaud Legrand</em></p>
<p><em>25 juin 2018</em></p>
<p><em>Arnaud Legrand</em> <em>25 juin 2018</em></p>
<div id="en-demandant-à-la-lib-maths" class="section level2">
<h2>En demandant à la lib maths</h2>
<p>Mon ordinateur m’indique que <span class="math inline">\(\pi\)</span> vaut <em>approximativement</em></p>
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! LaTeX Error: Unicode character π (U+03C0)
not set up for use with LaTeX.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
l.120 Mon ordinateur m'indique que π
vaut \emph{approximativement}
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\begin{document}
\hypertarget{uxe0-propos-du-calcul-de-pi}{%
\section{À propos du calcul de pi}\label{uxe0-propos-du-calcul-de-pi}}
\emph{Arnaud Legrand}
\emph{25 juin 2018}
\hypertarget{en-demandant-uxe0-la-lib-maths}{%
\subsection{En demandant à la lib
maths}\label{en-demandant-uxe0-la-lib-maths}}
Mon ordinateur m'indique que π vaut \emph{approximativement}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{pi}
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\begin{verbatim}
## [1] 3.141593
\end{verbatim}
\hypertarget{en-utilisant-la-muxe9thode-des-aiguilles-de-buffon}{%
\subsection{En utilisant la méthode des aiguilles de
Buffon}\label{en-utilisant-la-muxe9thode-des-aiguilles-de-buffon}}
Mais calculé avec la \textbf{méthode} des
\href{https://fr.wikipedia.org/wiki/Aiguille_de_Buffon}{aiguilles de
Buffon}, on obtiendrait comme \textbf{approximation} :
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\begin{Highlighting}[]
\FunctionTok{set.seed}\NormalTok{(}\DecValTok{42}\NormalTok{)}
\NormalTok{N }\OtherTok{=} \DecValTok{100000}
\NormalTok{x }\OtherTok{=} \FunctionTok{runif}\NormalTok{(N)}
\NormalTok{theta }\OtherTok{=}\NormalTok{ pi}\SpecialCharTok{/}\DecValTok{2}\SpecialCharTok{*}\FunctionTok{runif}\NormalTok{(N)}
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\end{Highlighting}
\end{Shaded}
\begin{verbatim}
## [1] 3.14327
\end{verbatim}
\hypertarget{avec-un-argument-fruxe9quentiel-de-surface}{%
\subsection{Avec un argument ``fréquentiel'' de
surface}\label{avec-un-argument-fruxe9quentiel-de-surface}}
Sinon, une méthode plus simple à comprendre et ne faisant pas intervenir
d'appel à la fonction sinus se base sur le fait que si X∼U(0,1) et
Y∼U(0,1) alors P{[}X2+Y2≤1{]}=π/4 (voir
\href{https://fr.wikipedia.org/wiki/M\%C3\%A9thode_de_Monte-Carlo\#D\%C3\%A9termination_de_la_valeur_de_\%CF\%80}{méthode
de Monte Carlo sur Wikipedia}). Le code suivant illustre ce fait:
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\FunctionTok{library}\NormalTok{(ggplot2)}
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## Warning: le package 'ggplot2' a été compilé avec la version R 4.3.3
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\includegraphics{toy_document_fr_files/figure-latex/unnamed-chunk-3-1.pdf}
Il est alors aisé d'obtenir une approximation (pas terrible) de π en
comptant combien de fois, en moyenne, X2+Y2 est inférieur à 1:
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\begin{Highlighting}[]
\DecValTok{4}\SpecialCharTok{*}\FunctionTok{mean}\NormalTok{(df}\SpecialCharTok{$}\NormalTok{Accept)}
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## [1] 3.156
\end{verbatim}
\end{document}
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