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?