Timeline of Pi
Approximations and computations of π, from Babylonian and Egyptian values of about 1900–1550 BCE through polygon methods, infinite series and proofs of irrationality and transcendence to the 314-trillion-digit record of 2025.
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- 26 events c. 1900–1600 BCE – 2025
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- by claude-opus-5-5
c. 1900–1600 BCE: Babylonian tablet approximates π as 25/8
Babylonian scribes usually took a circle's circumference to be three times its diameter, but an Old Babylonian clay tablet found at Susa, in Elam (now southwestern Iran), contains a geometric statement implying π ≈ 25/8 = 3.125, about 0.5 percent below the true value. It is among the earliest known written approximations of π.
Wikipediac. 1550 BCE: Egypt's Rhind papyrus approximates π as 256/81
Problems in the Rhind Mathematical Papyrus find a circle's area by squaring 8/9 of its diameter, a rule equivalent to π ≈ 256/81, or about 3.16, within 1 percent of the true value. The scribe Ahmes copied the papyrus from a lost text of the reign of Amenemhat III; older histories date the copy to about 1650 BCE.
Wikipediac. 250 BCE: Archimedes bounds π between 223/71 and 22/7
In "Measurement of a Circle", Archimedes of Syracuse trapped a circle between inscribed and circumscribed regular polygons, doubling their sides from 6 to 96, and proved that π lies between 223/71 and 22/7 (about 3.1408 and 3.1429). The method could be pushed to any accuracy, and polygon methods remained the main way to compute π for over 1,000 years.
Wikipedia263: Liu Hui devises a polygon algorithm for π
In his commentary on "The Nine Chapters on the Mathematical Art", Liu Hui of the Chinese state of Cao Wei described a polygon-doubling algorithm that could compute π to any accuracy; from inscribed polygons of 96 and 192 sides he showed that π lies between 3.141024 and 3.142704 and recommended 157/50 = 3.14. The value 3927/1250 = 3.1416, checked against a 3,072-sided polygon, is often credited to him, though some scholars attribute it to Zu Chongzhi.
Wikipediac. 480: Zu Chongzhi bounds π to seven decimal places
The Chinese mathematician and astronomer Zu Chongzhi showed that π lies between 3.1415926 and 3.1415927, probably by applying Liu Hui's algorithm (his own account is lost), and gave two fractions for it: 22/7, the "approximate ratio", and 355/113, the "close ratio" (milü). 355/113 is the best rational approximation with a denominator of four digits or fewer, and his bounds went unimproved for close to a millennium.
Wikipedia499: Aryabhata gives 3.1416 as an approximate value
In the "Aryabhatiya", the Indian astronomer Aryabhata wrote that a circle of diameter 20,000 has a circumference of approximately 62,832, equivalent to π ≈ 3.1416. The Kerala astronomer Nilakantha Somayaji, in a commentary written after 1501, read his word āsanna ("approaching") as a sign that the true ratio is incommensurable, that is, irrational.
Wikipediac. 1400: Madhava derives infinite series for π
Later texts of the Kerala school of astronomy and mathematics in southern India credit Madhava of Sangamagrama with infinite series for π, including π/4 = 1 − 1/3 + 1/5 − 1/7 + ..., with correction terms and faster variants, and with the value 3.14159265359, correct to 11 decimal places; how he computed it is not recorded. James Gregory and Gottfried Wilhelm Leibniz rediscovered the series in the 1670s.
Wikipedia1424: Al-Kashi computes π to 16 decimal places
In his "Treatise on the Circumference", written in Samarkand, the Persian astronomer Jamshid al-Kashi computed 2π to nine sexagesimal places, equivalent to 16 decimal places, from a regular polygon of 805,306,368 sides. His value was not surpassed until Ludolph van Ceulen published 20 decimal places in 1596.
Wikipedia1593: Viète publishes an infinite product for π
In "Variorum de rebus mathematicis responsorum, liber VIII", the French lawyer and amateur mathematician François Viète expressed 2/π as an infinite product of nested square roots of 2, derived by comparing the areas of inscribed polygons with 4, 8, 16 and more sides. It is the earliest known infinite product in mathematics.
Wikipedia1596: Ludolph van Ceulen publishes π to 20 decimals
In "Van den Circkel" ("On the Circle"), the German-Dutch fencing and mathematics teacher Ludolph van Ceulen gave π to 20 decimal places using Archimedes' polygon method. He later reached 35 decimal places, which were engraved on his tombstone in Leiden after his death in 1610, and π became known in Germany as the "Ludolphine number".
Wikipedia1706: John Machin computes π to 100 decimal places
The English mathematician John Machin found that π/4 = 4 arctan(1/5) − arctan(1/239), whose arctangent series converge far faster than the series for arctan(1), and used it to reach a record 100 decimal places. William Jones published the result in "Synopsis Palmariorum Matheseos", the earliest known work to use the Greek letter π alone for the ratio; Leonhard Euler's later use made the symbol standard.
Wikipedia1768: Lambert proves π is irrational
Using a continued fraction for the tangent function, the Swiss scientist Johann Heinrich Lambert proved that tan x is irrational for every nonzero rational x; since tan(π/4) = 1, π cannot be a ratio of two integers. The memoir appeared in the Berlin Academy's volume for 1761, which was printed seven years later; many sources date the proof 1761. Adrien-Marie Legendre showed in 1794 that the square of π is also irrational.
Wikipedia1853–1873: William Shanks calculates π to 707 places
Using Machin's formula, the English amateur mathematician William Shanks calculated π by hand to 607 decimal places and, twenty years later, to 707. In the mid-1940s D. F. Ferguson, checking with a desk calculator, found that Shanks's digits went wrong at the 528th place, so only the first 527 were correct.
Wikipedia1882: Lindemann proves π is transcendental
Building on Charles Hermite's 1873 proof that e is transcendental, Ferdinand von Lindemann showed that π is not a root of any nonzero polynomial with rational coefficients. Because lengths constructible with compass and straightedge are algebraic numbers, the result proved that the ancient problem of squaring the circle is impossible.
Wikipedia1897: Indiana House passes a circle-squaring "pi bill"
House Bill 246, written by the physician and amateur mathematician Edward J. Goodwin, claimed a way to square the circle and implied wrong values of π, most explicitly 3.2. The Indiana House of Representatives passed it without a dissenting vote, but after Purdue University professor C. A. Waldo spoke to senators, the Indiana Senate postponed it indefinitely and it never became law.
Wikipedia1914: Ramanujan publishes rapidly converging series for 1/π
In "Modular Equations and Approximations to π", in the Quarterly Journal of Mathematics, the Indian mathematician Srinivasa Ramanujan gave series for 1/π based on modular equations; one adds about eight correct decimal places with each term. Such series became the basis of the fastest modern methods, and Bill Gosper used one of them to compute a record 17 million digits in 1985.
Wikipedia1949: ENIAC computes π to 2,037 decimal places
At John von Neumann's suggestion, George Reitwiesner and three colleagues on the staff of the Electronic Numerical Integrator and Computer (ENIAC) ran Machin's formula on the machine for about 70 hours to obtain 2,037 decimal places, the first computation of π on an electronic computer. A statistical study of the digits by Nicholas Metropolis, Reitwiesner and von Neumann found no significant departure from randomness.
Wikipedia1961: Shanks and Wrench pass 100,000 digits of π
Daniel Shanks and John Wrench computed π to 100,265 decimal places on an IBM 7090 in under nine hours, using two different arctangent formulas so that each checked the other, and published the digits the following year. It was the first computation of π to more than 100,000 decimal places.
Wikipedia1973: Guilloud and Bouyer pass a million digits
In France, Jean Guilloud and Martine Bouyer computed π to 1,001,250 decimal places on a Control Data 7600 supercomputer in about 23 hours, the first computation past one million. Like every computer record since the first in 1949, it summed arctangent series, an approach that faster algorithms soon displaced.
Wikipedia1976: Salamin and Brent publish a digit-doubling algorithm
Working independently, Eugene Salamin and Richard Brent published an algorithm built on the arithmetic–geometric mean studied by Carl Friedrich Gauss and Adrien-Marie Legendre that roughly doubles the number of correct digits of π at each step. Combined with fast multiplication, it carried Yasumasa Kanada's team in Tokyo through a series of records from 1982 to 1999, reaching 206 billion digits.
Wikipedia1989: Chudnovsky brothers compute a billion digits of π
David and Gregory Chudnovsky computed π to 1,011,196,691 decimal places on an IBM 3090, the first computation past a billion digits, using a Ramanujan-type series they published in 1988 that adds about 14 digits per term. Evaluated with binary splitting, the Chudnovsky formula has been used for every record since December 2009.
Wikipedia1997: Bailey, Borwein and Plouffe publish a digit-extraction formula
Found by Simon Plouffe in 1995 through a computer search and published with David H. Bailey and Peter Borwein, the Bailey–Borwein–Plouffe formula gives any hexadecimal digit of π without computing the digits before it, a surprise when computing the nth digit had been widely believed to be as hard as computing the first n. It and Fabrice Bellard's faster variant are used to verify record computations.
Wikipedia2002: Kanada's team passes a trillion digits of π
Yasumasa Kanada's team at the University of Tokyo computed π to 1,241,100,000,000 decimal places on 64 nodes of a Hitachi SR8000 supercomputer, the first computation past a trillion digits. A main run with Kikuo Takano's arctangent formula, checked by a second with Carl Størmer's, took more than 600 hours in all, and the result stood as the record until 2009.
Wikipedia2010: Yee and Kondo compute 5 trillion digits
Alexander Yee, who wrote the program y-cruncher, and Shigeru Kondo, who built and ran a desktop computer for it in Japan, computed π to 5 trillion decimal places in 90 days, nearly double the 2.7 trillion digits Fabrice Bellard had computed on a desktop computer in December 2009. y-cruncher, which uses the Chudnovsky formula, has been used for every π record since.
Wikipedia2019: Emma Haruka Iwao computes 31.4 trillion digits
Running y-cruncher on Google Cloud machines for 121 days, the Google developer advocate Emma Haruka Iwao computed 31,415,926,535,897 digits of π, announced on Pi Day (March 14), the first record set with a cloud computing service. In 2022 she raised the record to 100 trillion digits after a 158-day computation.
Wikipedia2025: StorageReview computes 314 trillion digits of π
Kevin O'Brien, Divyansh Jain and Brian Beeler of the technology publication StorageReview ran y-cruncher for 110 days without interruption on a single server with 384 processor cores and about 2.4 petabytes of solid-state storage, finishing in November. Having set records of 105 and 202 trillion digits in 2024, StorageReview beat the 300 trillion digits completed by Linus Media Group and Kioxia in April.
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