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\title{Exploring Machin's Approximation of $\pi$}
\institute{{\large Helmut Knaust}\medskip\\Department of Mathematical Sciences\\The University of Texas at El Paso\\El Paso TX 79968-0514\medskip\\\tt{hknaust@utep.edu}}
\date{{\small August 8, 2009}}

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\section{Precursors}
\subsection{Method of Exhaustion \`a la Archimedes}
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\textbf{Archimedes of Syracuse} ($\approx$ 287--212 BC) approximated $\pi$ by the Method of Exhaustion:
\begin{figure}
\includegraphics<1>[height=4cm]{nb/ar01.pdf}%
\includegraphics<2>[height=4cm]{nb/ar02.pdf}%
\includegraphics<3>[height=4cm]{nb/ar03.pdf}%
\includegraphics<4>[height=4cm]{nb/ar04.pdf}%
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\visible<4->{The accuracy of the approximation increases by about 1 digit for every 2 iterations.}
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\subsection{Method of Exhaustion: A First Encounter}
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\section{Ingredients}
\subsection{Gregory's Formula}
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\textbf{James Gregory} (1638--1675) found the McLaurin series expansion for the arctangent function:
\[\arctan(x)=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\frac{x^9}{9}-\frac{x^{11}}
   {11}+\ldots\]
The radius of convergence of the series is 1.
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\subsection{Leibniz' Formula}
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As a consequence we obtain \textbf{Leibniz' Formula}:
\[\frac{\pi}{4}=\arctan(1)=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}
   {11}+\ldots\]
(The series converges by the Alternating Series Test.)\bigskip\\
\visible<2->{Convergence of this series is very slow:\smallskip\\
Summing 10,000 terms of the series yields 3.14149 as an approximation of $\pi$.}
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\subsection{Addition Formula for the tangent function}
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This is a nice illustration of the \textbf{General Calculus Philosophy}:
\begin{quote}Convergence behavior is good, if $x$ is close to the center of a Taylor series; convergence is bad if $x$ is close to the ``edge'' of convergence.\end{quote}\medskip\\
\visible<2->{Therefore {\textbf{John Machin} ($\approx$ 1686—-1751) [and others] had the following idea}: 
\begin{quote}Replace the 1 in Leibniz' Formula by one or more numbers closer to 0.\end{quote}\medskip\\}
\visible<3->{The ingredient needed is the \textbf{addition formula} for the tangent function:
\[\hspace{-1cm}\tan(\alpha+\beta)=\frac{\tan \alpha +\tan \beta}{1-\tan \alpha \tan\beta}\]}
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\section{Derivation of Machin's Formula}
\subsection{I}
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Applying the addition formula twice we obtain:
\[\tan(4\alpha)=\frac{4 \tan(\alpha)-4 \tan^3 (\alpha)}{1+\tan^4 (\alpha)-6 \tan^2 (\alpha)}\]
\visible<2->{If we set $x=\tan\alpha$ and $y=\tan\beta$, we can apply the addition formula one more time to get
\[\tan(4\alpha+\beta)=\frac{x^4 y-4 x^3-6 x^2 y+4 x+y}{x^4+4 x^3 y-6 x^2-4 x y+1}\]}
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\section{Derivation of Machin's Formula}
\subsection{II}
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Using this for \[\frac{\pi}{4}=4\alpha+\beta,\] we get the equation:
\[1=\frac{x^4 y-4 x^3-6 x^2 y+4 x+y}{x^4+4 x^3 y-6 x^2-4 x y+1}\]
\visible<2->{At this point, Machin made the decision to choose $x=\frac{1}{5}$, then solved for $y$ to obtain $y=-\frac{1}{239}$.}
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\section{Derivation of Machin's Formula}
\subsection{III}
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Since the angles  $\alpha=\arctan \left(\frac{1}{5}\right)$ and $\beta=\arctan \left(-\frac{1}{239}\right)$ satisfy the equation
\[\frac{\pi}{4}=4\alpha+\beta,\]
we finally arrive at Machin's Formula
\[\frac{\pi}{4}=4\arctan \frac{1}{5}-\arctan \frac{1}{239}\]
\visible<2->{Machin then used this formula in 1706 to compute 100~digits of $\pi$.}
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\section{Current World Record}
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Using similar ideas and identities such as 
\begin{eqnarray*}
\frac{\pi}{4}&=&44 \arctan \frac{1}{57}+7 \arctan\frac{1}{239}-12\arctan\frac{1}{682}
\\&&\mbox{\ \ }+24\arctan\frac{1}{12943},
\end{eqnarray*}
found by \textbf{Carl St\o rmer} in 1896,\medskip\\
\textbf{Yasumasa Kanada} (1949--) and his collaborators computed more than 1.24 trillion digits of $\pi$ in 2002.
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\section{References}
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References:
\begin{itemize}
\item Jack S.~Calcut: \emph{Gaussian Integers and Arctangent Identities for $\pi$.} American Mathematical Monthly 116 (2009), 515--530.
\item Jonathan M.~Borwein: \emph{The Life of Pi.}\\ \small\texttt{http://www.cecm.sfu.ca/\~{}jborwein/pi-slides.pdf}
\item David Blatner: \emph{The Joy of Pi.}\\
\small\texttt{http://www.joyofpi.com/pilinks.html}
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