## 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 $$\phi=\frac{1+\sqrt{5}}{2}$$

As a curiosity, next page lists some examples of expressions involving $$\pi$$ and integer approximations.