Pi (π), approximately 3.14159, is a fundamental mathematical constant representing the ratio of a circle's circumference to its diameter. Foundational in mathematics and physics, π is irrational, meaning it cannot be expressed as a simple fraction, resulting in a non-repeating, infinite decimal representation. Also transcendental, π cannot be derived from algebraic equations with integers. This property makes squaring the circle using a compass and straightedge impossible. While its digits seem random, this remains unproven.
1900: Early Written Approximations of Pi
The earliest written approximations of Pi, accurate to within 1% of the true value, are found in Babylon and Egypt around 1900 BC. A Babylonian clay tablet uses a geometric statement that implies Pi as 25/8, or 3.125. The Egyptian Rhind Papyrus, copied from an 1850 BC document, uses a formula for the area of a circle that estimates Pi as (16/9)^2, approximately 3.16.
1914: Ramanujan's Formulae for Pi
Srinivasa Ramanujan published new formulae for Pi in 1914, based on modular equations, which converged much faster than other methods, like Machin's formula, and laid the groundwork for modern algorithms.
1929: Reprinting of William Jones' use of π
A reprint of William Jones' first use of the symbol π for the circle ratio was included in David Eugene Smith's "A Source Book in Mathematics" in 1929.
1946: Discovery of an error in William Shanks' calculation of π
An error in William Shanks' 1873 calculation of π, displayed in the Palais de la Découverte, was detected in 1946.
1946: Machin-like Formulae for Pi Calculation
In 1946, Daniel Ferguson achieved the best approximation of Pi (620 digits) without a calculating device using Machin-like formulae, which remained the most popular method until the computer age.
1949: Computer-Aided Pi Calculations
In 1949, John Wrench and Levi Smith calculated 1,120 digits of Pi using a desk calculator. In the same year, George Reitwiesner and John von Neumann used an inverse tangent infinite series and the ENIAC computer to compute 2,037 digits, taking 70 hours.
1949: Correction of the error in π display at Palais de la Découverte
The error in William Shanks' calculation of π, discovered in 1946, was corrected on the display in the Palais de la Découverte in 1949.
1955: Breaking Records in Pi Calculation
The record for Pi calculations, always relying on an arctan series, was broken repeatedly: in 1955 with 3,089 digits.
1957: Breaking Records in Pi Calculation
In 1957, the record for Pi calculations was broken with 7,480 digits using an arctan series.
1958: Breaking Records in Pi Calculation
The Pi calculation record was broken again in 1958, reaching 10,000 digits using an arctan series.
1961: Surpassing 100,000 Digits of Pi
In 1961, the record for Pi calculation was pushed further, surpassing 100,000 digits using an arctan series.
1971: Publication of Isaac Newton's mathematical papers
In 1971, "The Mathematical Papers of Isaac Newton," edited by Derek Thomas Whiteside, was published, containing Newton's work from 1674–1684, including his contributions to the understanding and calculation of π.
1973: Reaching One Million Digits of Pi
By 1973, the continuous advancements in Pi calculations reached a significant milestone, computing one million digits.
1975: Iterative Algorithms for Pi Calculation
Physicist Eugene Salamin independently published an iterative algorithm for Pi calculation in 1975, moving away from infinite series.
1976: Iterative Algorithms for Pi Calculation
Scientist Richard Brent independently published an iterative algorithm for Pi calculation in 1976, offering an alternative to infinite series methods.
1980: New Algorithms Accelerate Pi Computation
Around 1980, two key developments accelerated Pi computation: faster iterative algorithms and fast multiplication algorithms like Karatsuba, Toom-Cook, and Fourier transform-based methods. These algorithms were crucial as multiplication consumes the majority of computer time during computations.
1980: Howe's observation on the Gaussian integral and Fourier analysis
In 1980, Howe observed the significance of the Gaussian integral in establishing the fundamental theorems of Fourier analysis, highlighting the connection between the Gaussian function, the Heisenberg uncertainty principle, and the value of π.
1980: Widespread Adoption of Iterative Algorithms
Iterative algorithms gained widespread adoption after 1980 due to their faster speeds compared to infinite series algorithms. They multiply the number of correct digits with each step, as opposed to infinite series which add digits in successive terms.
1984: Borwein Brothers' Iterative Algorithm
In 1984, John and Peter Borwein created an iterative algorithm that quadrupled the number of correct digits in every step, significantly advancing Pi calculations.
1985: Carl Sagan's "Contact" and π
Carl Sagan's 1985 novel "Contact" featured the idea of a message from the creator of the universe hidden within the digits of π.
1985: Gosper's Use of Ramanujan's Formula
In 1985, Bill Gosper used Ramanujan's formula to set a new record by calculating 17 million digits of Pi, showcasing the formula's efficacy.
1987: Chudnovsky Formula
The Chudnovsky brothers developed a formula for Pi calculation in 1987, capable of generating about 14 digits per term, and used it to achieve several record-breaking calculations.
1987: Borwein Brothers' Improved Algorithm
Furthering their work, the Borwein brothers developed an algorithm in 1987 that increased the number of digits fivefold per step. This marked a substantial improvement in computational efficiency.
1989: First to Surpass One Billion Digits
The Chudnovsky brothers, utilizing their formula, became the first to surpass one billion digits of Pi in 1989, marking a significant milestone.
1991: Discovery of π in the Mandelbrot Set
In 1991, David Boll discovered the presence of π within the fractal known as the Mandelbrot set, observing its emergence in specific calculations related to the set's behavior near certain points.
1995: Yasumasa Kanada's Pi Calculation Records
Japanese mathematician Yasumasa Kanada used iterative methods to establish numerous records for calculating Pi between 1995 and 2002, demonstrating the power and efficiency of these algorithms.
1995: Wagon and Rabinowitz's Spigot Algorithm
Mathematicians Stan Wagon and Stanley Rabinowitz developed a simple spigot algorithm in 1995, comparable in speed to arctan algorithms, but not as fast as iterative algorithms.
1995: BBP Digit Extraction Algorithm
Simon Plouffe discovered another spigot algorithm, the BBP digit extraction algorithm, in 1995, offering a new way to calculate Pi digits.
1995: Discovery of Spigot Algorithms
Two spigot algorithms, a new approach to Pi calculation, were discovered in 1995. Unlike previous methods, they produce single digits of Pi without reusing them after calculation.
1998: PiHex Project
The PiHex project used Bellard's formula, a modification of the BBP algorithm, to calculate the quadrillionth bit of Pi between 1998 and 2000.
2000: PiHex Project
The PiHex project, which ran from 1998 to 2000, successfully computed the quadrillionth bit of Pi.
2002: Yasumasa Kanada's Pi Calculation Records
Yasumasa Kanada continued to set records for calculating Pi until 2002 using iterative methods, pushing the boundaries of Pi computation.
2004: Hwang's observations on the arctangent method for calculating π
In 2004, Chien-Lih Hwang published observations on the method of arctangents for calculating π in the Mathematical Gazette.
2005: Kate Bush's song "Pi"
Kate Bush included digits of π in the lyrics of her song "Pi" on her 2005 album "Aerial."
2005: Hwang's elementary derivation of Euler's series
Chien-Lih Hwang presented an elementary derivation of Euler's series for the arctangent function in the Mathematical Gazette in 2005.
2006: Akira Haraguchi's Unverified Claim
In 2006, Akira Haraguchi claimed to have memorized 100,000 digits of π, but this claim was not officially verified by Guinness World Records.
2006: Plouffe's New Formulae for Pi
Mathematician Simon Plouffe used the PSLQ integer relation algorithm in 2006 to develop several new formulae for calculating Pi, expanding the toolkit for Pi computation.
September 2010: Computing Pi with Hadoop
In September 2010, a Yahoo employee used the company's Hadoop application on 1,000 computers for 23 days to calculate 256 bits of Pi at the two-quadrillionth bit.
2010: Proposal to replace π with τ and celebration of Tau Day
In 2010, a movement emerged advocating for the replacement of π with τ (2π), arguing for its more natural representation of a circle's properties, leading to the celebration of Tau Day on June 28.
2011: Ten Trillion Digits of Pi
Alexander Yee and Shigeru Kondo computed ten trillion digits of Pi in 2011, demonstrating the continued progress in computational power and algorithmic efficiency.
2012: New Definition of Pi
In 2012, Remmert explained that a definition of Pi that doesn't rely on integral calculus is desirable since differential calculus is typically taught before integral calculus. Richard Baltzer, popularized by Edmund Landau, defined Pi as twice the smallest positive number at which the cosine function is 0.
March 2015: Guinness World Record for memorizing digits of π
On March 21, 2015, Rajveer Meena set a Guinness World Record by reciting 70,000 digits of π in 9 hours and 27 minutes.
2015: Unique Pi Day
Pi Day in 2015 (3/14/15) held special significance because the date and time 9:26:53 aligned with the first 10 digits of π (3.141592653).
May 2019: Redefinition of vacuum permeability constant
Before May 2019, the vacuum permeability constant μ0, which appears in Maxwell's equations in electromagnetics, was defined as an exact value.
2022: Emma Haruka Iwao's 100 Trillion Digits
Emma Haruka Iwao achieved a new record in 2022 by calculating 100 trillion digits of Pi, pushing the boundaries of Pi computation even further.
2022: Plouffe's Base-10 Algorithm
Simon Plouffe introduced a new base-10 algorithm for calculating the digits of Pi in 2022, providing an alternative method for Pi computation.