In Babylon, a clay tablet implies an approximation of π as 3.125.
The Rhind Papyrus from Egypt suggests π as approximately 3.16, calculated as \((\frac{16}{9})^2\).
In the Shulba Sutras of Indian mathematics, various approximations are given, like 3.08831, 3.08833, 3.004, 3, and 3.125.
Archimedes of Greece used a polygonal method to establish that 3.1408 < π < 3.1429, famously approximating π to \(\frac{22}{7}\). This method dominated for over 1,000 years.
Ptolemy documented a π value of 3.1416, possibly deriving from earlier works of Archimedes or Apollonius.
Zu Chongzhi, around 480 AD, estimated π between 3.1415926 and 3.1415927, offering the highly accurate fraction \(\frac{355}{113}\).
Aryabhata, in 499 AD, used a π value of 3.1416 in his Āryabhaṭīya.
Fibonacci approximated π as 3.1418 using a polygonal method, independent of Archimedes.
Itailian poet Dante Alighieri approximated π as \(3 + \frac{\sqrt{2}}{10}\) around 3.14142.
Jamshīd al-Kāshī produced nine sexagesimal digits, roughly the equivalent of 16 decimal digits, in 1424, using a polygon with \(3 \times 2^{28}\) sides.
François Viète achieved nine digits with a polygon of \(3 \times 2^{17}\) sides.
Adriaan van Roomen arrived at 15 decimal places in 1593.
Ludolph van Ceulen reached 20 digits, a record he later increased to 35 digits.
Willebrord Snellius reached 34 digits in 1621.
Christoph Grienberger arrived at 38 digits in 1630 using \(10^{40}\) sides.
François Viète in 1593 published what is now known as Viète's formula, an infinite product. He could only calculate up to 9 digits but he showed that there were other ways to calculate pi.
\( {\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots\)
Newton himself used an arcsine series to compute a 15-digit approximation of π in 1665 or 1666, writing, "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."
Abraham Sharp used the Gregory-Leibniz series to compute π to 71 digits, using \(z = \frac{1}{\sqrt{3}}\).
James Gregory, and independently, Leibniz in 1673, discovered the Taylor series expansion for arctangent:
\(\arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} - \cdots\)
John Machin used the Gregory-Leibniz series to produce an algorithm that converged much faster. Machin reached 100 digits of π with this formula.
\(\frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{239}\)
Zacharias Dase used a Machin-like formula to calculate 200 decimals of π in his head at the behest of German mathematician Carl Friedrich Gauss.
William Shanks calculated π to 707 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect. This is the number he calculated:
John Wrench and Levi Smith reached 1,120 digits in 1949 using a desk calculator.
George Reitwiesner and John von Neumann achieved 2,037 digits with a calculation that took 70 hours of computer time on the ENIAC computer.
Nicholas W. M. Ritchie achieved 3,089 digits in 1955.
George E. Felton achieved 7,480 digits in 1957. He calculated up to 10,021 digits but only the first 7,480 were correct. (I tried to find the output from his attempt but could not find a source of it online so I only included here the first correct digits)
Francois Genuys achieved 10,000 digits in 1958 using an IBM 704 computer.
We keep calculatating more and more pi to this day. The current record was broken this year (2024) by Jordan Ranous, Kevin O'Brien, and Brian Beeler on pi day. They calculated 105 Trillion (105,000,000,000,000) digits of pi using 2 x AMD EPYC 9754 (128 cores, 1.5 TiB RAM) and 1,105 TB of storage. The calculation took 78 days
Over the years, Matt Parker has calculated pi to various degrees of accuracy. Here are some of his attempts:
Calculatating pi with pies. Video
For ultimate pi day, Matt Parker calculated pi by weighing a circle. Video
In 2015 Matt Parker also calculated pi by swinging a pie on a pendulum. Video
For this pi day Matt Parker calculated pi by hand using alternating adding and subtracting odd fractions. Video
\(\pi = 4 \cdot \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + \cdots\right)\)
For this pi day Matt Parker calculated pi by rolling dice and using the probability of the numbers sharing a factor with eachother is \(\frac{6}{\pi^2}\). Video
For this pi day Matt Parker calculated pi by hand again, this time using the chudnovsky algorithm. Video
\(\frac{1}{\pi} = 12 \sum_{k=0}^{\infty} \frac{(-1)^k (6k)! (545140134k + 13591409)}{(3k)!(k!)^3 640320^{3k + 3/2}}\)
For this pi day Matt Parker calculated pi with a balancing beam. Video
\(\frac{\pi^2}{6} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots\)
For this pi day Matt Parker calculated pi the isaac newton way but slightly modified. Video
\(\pi = \frac{3\sqrt{3}}{4} + 2 - 24(\frac{1}{5 \times 2^5} + \frac{1}{7 \times 2^9} + \frac{1}{9 \times 2^{13}} + \frac{1}{11 \times 2^{17}} + \cdots)\)
For this pi day Matt Parker calculated pi with Avogadro's number. Video
\(\pi = \frac{6.2 \times 10^{-3}}{0.004^2}\)
For this pi day Matt Parker calculated pi by hand the William Shanks way, if done correctly it would have been the most accurate pi calculation by hand in a few centuries. Video
For this pi day Matt Parker calculated pi using an out of control car. Video
\(\pi = \frac{15 \times c \times f}{2 \times s^2}\)
For this pi day Matt Parker tried to break the record again set in the 1870s by williams shanks. This year he and his volunteers used a modified arc tan formula to calculate pi. Video
\(\frac{\pi}{4} = 1587 \arctan{\frac{1}{2852}} + 295 \arctan{\frac{1}{4193}} + 593 \arctan{\frac{1}{4246}} + 359 \arctan{\frac{1}{39307}} + 481 \arctan{\frac{1}{55603}} + 625 \arctan{\frac{1}{211050}} - 708 \arctan{\frac{1}{390112}} \) then multiply by 4 to get
\(\pi = 6348 \arctan{\frac{1}{2852}} + 1180 \arctan{\frac{1}{4193}} + 2372 \arctan{\frac{1}{4246}} + 1436 \arctan{\frac{1}{39307}} + 1924 \arctan{\frac{1}{55603}} + 2500 \arctan{\frac{1}{211050}} - 2832 \arctan{\frac{1}{390112}} \)
Combining all of Matt Parker's pi calculations, we can see they're average value of pi. This is a fun way to see how accurate Matt Parker's pi calculations are.