## Journey into PI

### Some history

The first estimates for $$\pi$$ resulted from a direct measurement. By this method one can obtain $$\pi$$ with one or two decimal places, which was certainly enough for the practical requirements of antiquity.

However, even then there were those who dedicated themselves to the calculation of $$\pi$$ beyond any practical need.

The first to achieve results in this field was Archimedes, who presented a geometric method for calculating $$\pi$$, known today by his name. The method consists of circumscribing and inscribing a polygon of $$n$$ sides into a given circumference. The perimeter of the circumference would be comprised between the perimeters of the polygons. In this way he deduced that the value of $$\pi$$ is comprised between $3\frac{10}{71}<\pi<3\frac{1}{7},$ that is, $$3.140<\pi<3.142$$

The result of Archimedes was obtained using 96 sided polygons.

This must have been the starting signal for the race initiated by the digit hunters of $$\pi$$.

From this method were deduced numerous formulas that allowed the computation of $$\pi$$ more and more accurately.

Other methods have since been discovered which enabled getting $$\pi$$ faster, until we get to the algorithms used today that allow at each iteration to quadruple, and more, the number of computed digits.

Below is a summary of the most significant steps for computing $$\pi$$ throughout the ages.

François Viéte em 1593:$\frac{2}{\pi}=\sqrt{\frac{1}{2}}\sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}}}\sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}}}}...$ Based on the method of Archimedes.
John Wallis in 1655:$\frac{\pi}{2}=\frac{2}{1}\frac{2}{3}\frac{4}{3}\frac{4}{5}\frac{6}{5}\frac{6}{7}\frac{8}{7}\frac{8}{9}...$ Easy to use but with a slow convergence to $$\pi.$$
William Brouncker in 1658:$\frac{4}{\pi}=1+\frac{1^{2}}{2+\frac{3^{2}}{2+\frac{5^{2}}{2+\frac{7^{2}}{2+\frac{9^{2}}{2+...}}}}}$
James Gregory in 1671:$\arctan(x)=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+...$ Ushered in a new era for the computation of $$\pi$$ since $$\arctan(1)=\frac{\pi}{4}$$.
Very slow convergence to $$\pi$$. It was published by Leibnitz in 1673.
Newton:$\arcsin(x)=x+\frac{1}{2}\frac{x^{3}}{3}+\frac{1}{2}\frac{3}{4}\frac{x^{5}}{5}+...$$$\arcsin(\frac{1}{2})=\frac{\pi}{6}$$. Converges faster than the formula by Gregory/Leibnitz.
John Machin in 1706:$\frac{\pi}{4}=4\arctan\left(\frac{1}{5}\right)-\arctan\left(\frac{1}{239}\right).$This formula converges much faster than $$\arctan(1)$$.
With this formula Machin computed the first 100 significant digit of $$\pi$$.
It marked the beginning of a new era.
Euler:$\arctan(x)=\frac{y}{x}\left(1+\frac{2}{3}y+\frac{2}{3}\frac{4}{5}y^{2}+\frac{2}{3}\frac{4}{5}\frac{6}{7}y^{3}+...\right),$ with $$y=\frac{x^{2}}{1+x^{2}}.$$
Faster formula although it requires a greater effort of computation.
With a set of relationships involving $$\arctan$$ deduced by Euler from the ideas of Machin, it was possible to deduce a number of expressions to compute $$\pi$$ faster than ever. Just a few examples, $\begin{array}{ccl} \frac{\pi}{4} & = & \arctan(1)\\ \frac{\pi}{4} & = & \arctan\left(\frac{1}{2}\right)+\arctan\left(\frac{1}{3}\right)\\ \frac{\pi}{4} & = & 6\arctan\left(\frac{1}{8}\right)+2\arctan\left(\frac{1}{15}\right)+2\arctan\left(\frac{1}{239}\right)\\ \frac{\pi}{4} & = & 8\arctan\left(\frac{1}{10}\right)-\arctan\left(\frac{1}{239}\right)-4\arctan\left(\frac{1}{515}\right)\\ \frac{\pi}{4} & = & 12\arctan\left(\frac{1}{18}\right)+8\arctan\left(\frac{1}{57}\right)-5\arctan\left(\frac{1}{239}\right)\\ & & ... \end{array}$
Salamin in 1972, Brent in 1976:$\begin{array}{ccl} a_{0} & = & 1\\ b_{0} & = & \frac{1}{\sqrt{2}}\\ a_{n+1} & = & \frac{a_{n}+b_{n}}{2}\\ b_{n+1} & = & \sqrt{a_{n}b_{n}}\\ U_{m} & = & \frac{4a_{m}^{2}}{1-2\sum_{j=1}^{m}2^{j}(a_{j}^{2}-b_{j}^{2})}\begin{array}{c} \\ \longrightarrow\\ m\rightarrow\infty \end{array}\pi \end{array}$Beginning of the modern era for the computation of $$\pi$$.
With this algorithm, at each iteration, the number of correctly computed significant digits for $$\pi$$ doubles.
Jonathan and Peter Borwein:$\begin{array}{ccl} y_{0} & = & \sqrt{2}-1\\ a_{0} & = & 6-4\sqrt{2}\\ y_{n+1} & = & \frac{\left(1-y_{n}^{4}\right)^{-\frac{1}{4}}-1}{\left(1-y_{n}^{4}\right)^{-\frac{1}{4}}+1}\\ a_{n+1} & = & a_{n}(1+y_{n+1})^{4}-2^{2n+3}y_{n+1}(1+y_{n+1}+y_{n+1}^{2})\begin{array}{c} \\ \longrightarrow\\ n\rightarrow\infty \end{array}\frac{1}{\pi} \end{array}$Based on the work of Ramanujan. In each iteration, the correct number of computed digits is quadrupled.
Therefore it is said to be a 4th order algorithm.
Chudnovsky brothers:$\frac{1}{\pi}=\frac{12}{\sqrt{6403203^{3}}}\sum_{k=0}^{\infty}(-1)^{k}\frac{(6k)!}{(k!)^{3}(3k)!}\frac{13591409+545140134k}{(640320^{3})^{k}}$ Formula derived with the aid of a mathematical symbolic manipulator.
Bailey, P.Borwein and Plouffe:$\pi=\sum_{n=0}^{\infty}\frac{1}{16^{n}}\left(\frac{4}{8n+1}-\frac{2}{8n+4}-\frac{1}{8n+5}-\frac{1}{8n+6}\right)$This formula was published in 1997 and allows to calculate the $$n$$th hexadecimal digit of $$\pi$$.

On the next page we present some results computed for the value of $$\pi$$ throughout the ages, based on some of the methods described.

How to calculate $$\pi$$ with a billion of significant digits?