Journey into PI 
Approximations of \(\pi\)
Mathematicians have always tried to calculate the value of \(\pi\).
In the pages of the history of mathematics about this constant, were recorded some extraordinary expressions, whose sole reason for existence was to determine a value for \(\pi\).
Ramanujan:
\[
\begin{split}
\frac{4}{\sqrt{522}}\ln\left[\left(\frac{5+\sqrt{29}}{\sqrt{2}}\right)^{3}\left(5\sqrt{29}+11\sqrt{6}\right)\left(\sqrt{\frac{9+3\sqrt{6}}{4}}+\sqrt{\frac{5+3\sqrt{6}}{4}}\right)^{6}\right]=\\
~\\
~\\
3.141592653589793238462643383279\;432
\end{split}
\]
Castellanos:\[\left(100-\frac{2125^{3}+214^{3}+30^{3}+37^{2}}{82^{5}}\right)^{\frac{1}{4}}=\\
~\\
~\\
3.1415926535897\mbox{ }80\]
Plouffe:\[(43)^{\frac{7}{23}}=3.1415\mbox{ }39\]\[\frac{\ln\left(262537412640768744\right)}{\sqrt{163}}=\\
~\\
~\\
3.141592653589793238462643383279\mbox{ }726\]
Euler:\[\frac{103993}{33102}=3.141592653\mbox{ }011\]
Dixon
Kochansky approximation:\[\sqrt{\frac{40}{3}-2\sqrt{3}}=3.1415\mbox{ }33\]\[\frac{6}{5}\left(1+\phi\right)=3.141\mbox{ }64,\] with \(\phi=\frac{1+\sqrt{5}}{2}\)
Kochansky approximation:\[\sqrt{\frac{40}{3}-2\sqrt{3}}=3.1415\mbox{ }33\]\[\frac{6}{5}\left(1+\phi\right)=3.141\mbox{ }64,\] with \(\phi=\frac{1+\sqrt{5}}{2}\)
As a curiosity, next page lists some examples of expressions involving \(\pi\) and integer approximations.
