Approximations for the mathematical constant pi () in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era. In Chinese mathematics, this was improved to approximations correct to what corresponds to about seven decimal digits by the 5th century.
Further progress was not made until the 14th century, when Madhava of Sangamagrama developed approximations correct to eleven and then thirteen digits. Jamshīd al-Kāshī achieved sixteen digits next. Early modern mathematicians reached an accuracy of 35 digits by the beginning of the 17th century (Ludolph van Ceulen), and 126 digits by the 19th century (Jurij Vega).
The record of manual approximation of is held by William Shanks, who calculated 527 decimals correctly in 1853. Since the middle of the 20th century, the approximation of has been the task of electronic digital computers (for a comprehensive account, see chronology of computation of pi). On December 11, 2025, the current record was established by StorageReview with Alexander Yee's y-cruncher with 314 trillion (3.14×) digits.
Early history The best known approximations to dating to before the Common Era were accurate to two decimal places; this was improved upon in Chinese mathematics in particular by the mid-first millennium, to an accuracy of seven decimal places. After this, no further progress was made until the late medieval period.
Some Egyptologists have claimed that the ancient Egyptians used an approximation of as = 3.142857 (about 0.04% too high) from as early as the Old Kingdom (c. 2700–2200 BC). This claim has been met with skepticism.
Babylonian mathematics usually approximated to 3, sufficient for the architectural projects of the time (notably also reflected in the description of Solomon's Temple in the Hebrew Bible).See #Imputed biblical value. "There has been concern over the apparent biblical statement of ≈ 3 from the early times of rabbinical Judaism, addressed by Rabbi Nehemiah in the 2nd century." The Babylonians were aware that this was an approximation, and one Old Babylonian mathematical tablet excavated near Susa in 1936 (dated to between the 19th and 17th centuries BCE) gives a better approximation of as = 3.125, about 0.528% below the exact value.See also "in 1936, a tablet was excavated some 200 miles from Babylon. ... The mentioned tablet, whose translation was partially published only in 1950, ... states that the ratio of the perimeter of a regular hexagon to the circumference of the circumscribed circle equals a number which in modern notation is given by 57/60+36/(60)2 [i.e. = 3/0.96 = 25/8]".
At about the same time, the Egyptian Rhind Mathematical Papyrus (dated to the Second Intermediate Period, c. 1600 BCE, although stated to be a copy of an older, Middle Kingdom text) implies an approximation of as ≈ 3.16 (accurate to 0.6 percent) by calculating the area of a circle via approximation with the octagon.
Astronomical calculations in the Shatapatha Brahmana (c. 6th century BCE) use a fractional approximation of .Chaitanya, Krishna. A profile of Indian culture. Indian Book Company (1975). p. 133.
The Mahabharata (500 BCE – 300 CE) offers an approximation of 3, in the ratios offered in Bhishma Parva verses: 6.12.40–45.
In the 3rd century BCE, Archimedes proved the sharp inequalities < < , by means of regular 96-gons (accuracies of 2·10−4 and 4·10−4, respectively).
In the 2nd century CE, Ptolemy used the value , the first known approximation accurate to three decimal places (accuracy 2·10−5). It is equal to which is accurate to two sexagesimal digits.
The Chinese mathematician Liu Hui in 263 CE computed to between and by inscribing a 96-gon and 192-gon; the average of these two values is (accuracy 9·10−5). He also suggested that 3.14 was a good enough approximation for practical purposes. He has also frequently been credited with a later and more accurate result, π ≈ = 3.1416 (accuracy 2·10−6), although some scholars instead believe that this is due to the later (5th-century) Chinese mathematician Zu Chongzhi.. Reprinted in . See in particular pp. 333–334 (pp. 28–29 of the reprint). Zu Chongzhi is known to have computed to be between 3.1415926 and 3.1415927, which was correct to seven decimal places. He also gave two other approximations of : π ≈ and π ≈ , which are not as accurate as his decimal result. The latter fraction is the best possible rational approximation of using fewer than five decimal digits in the numerator and denominator. Zu Chongzhi's results surpass the accuracy reached in Hellenistic mathematics, and would remain without improvement for close to a millennium.
In Gupta-era India (6th century), mathematician Aryabhata, in his astronomical treatise Āryabhaṭīya stated:
Approximating to four decimal places: π ≈ = 3.1416,How Aryabhata got the earth's circumference right Āryabhaṭīya (): . "Add four to one hundred, multiply by eight and then add sixty-two thousand. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given." In other words, (4 + 100) × 8 + 62000 is the circumference of a circle with diameter 20000. This provides a value of π ≈ = 3.1416, Aryabhata stated that his result "approximately" ( "approaching") gave the circumference of a circle. His 15th-century commentator Nilakantha Somayaji (Kerala school of astronomy and mathematics) has argued that the word means not only that this is an approximation, but that the value is incommensurable (irrational).
Middle Ages Further progress was not made for nearly a millennium, until the 14th century, when Indian mathematician and astronomer Madhava of Sangamagrama, founder of the Kerala school of astronomy and mathematics, found the Maclaurin series for arctangent, and then two infinite series for . One of them is now known as the Madhava–Leibniz series, based on
The other was based on
He used the first 21 terms to compute an approximation of correct to 11 decimal places as .
He also improved the formula based on arctan(1) by including a correction:
It is not known how he came up with this correction. Using this he found an approximation of to 13 decimal places of accuracy when = 75.
Indian mathematician Bhaskara II (12th century) used regular polygons with up to 384 sides to obtain another approximation of π, calculating it as 3.141666.
Jamshīd al-Kāshī (Kāshānī) (15th century), a Persian astronomer and mathematician, correctly computed the fractional part of 2 to 9 sexagesimal digits in 1424, and translated this into 16 decimal digits after the decimal point:
which gives 16 correct digits for π after the decimal point:
He achieved this level of accuracy by calculating the perimeter of a regular polygon with 3 × 228 sides.
16th to 19th centuries In the second half of the 16th century, the French mathematician François Viète discovered an infinite product that converged on known as Viète's formula.
The German-Dutch mathematician Ludolph van Ceulen (circa 1600) computed the first 35 decimal places of with a 262-gon. He was so proud of this accomplishment that he had the digits inscribed on his tombstone.
In Cyclometricus (1621), Willebrord Snellius demonstrated that the perimeter of the inscribed polygon converges on the circumference twice as fast as does the perimeter of the corresponding circumscribed polygon. This was proved by Christiaan Huygens in 1654. Snellius was able to obtain seven digits of from a 96-sided polygon.
In 1656, John Wallis published the Wallis product:
In 1706, John Machin used Gregory's series (the Taylor series for arctangent) and the identity to calculate 100 digits of (see below).Reprinted in In 1719, Thomas de Lagny used a similar identity to calculate 127 digits (of which 112 were correct). In 1789, the Slovene mathematician Jurij Vega improved John Machin's formula to calculate the first 140 digits, of which the first 126 were correct. In 1841, William Rutherford calculated 208 digits, of which the first 152 were correct.
The magnitude of such precision (152 decimal places) can be put into context by the fact that the circumference of the largest known object, the observable universe, can be calculated from its diameter (93billion light-years) to a precision of less than one Planck length (at , the shortest unit of length expected to be directly measurable) using expressed to just 62 decimal places.
The English amateur mathematician William Shanks calculated to 530 decimal places in January 1853, of which the first 527 were correct (the last few likely being incorrect due to round-off errors). He subsequently expanded his calculation to 607 decimal places in April 1853, but an error introduced right at the 530th decimal place rendered the rest of his calculation erroneous; due to the nature of Machin's formula, the error propagated back to the 528th decimal place, leaving only the first 527 digits correct once again. Twenty years later, Shanks expanded his calculation to 707 decimal places in April 1873. Due to this being an expansion of his previous calculation, most of the new digits were incorrect as well. Shanks was said to have calculated new digits all morning and would then spend all afternoon checking his morning's work. This was the longest expansion of until the advent of the electronic digital computer three-quarters of a century later.
20th and 21st centuries
In 1910, the Indian mathematician Srinivasa Ramanujan found several rapidly converging infinite series of , including
which computes a further eight decimal places of with each term in the series. His series are now the basis for the fastest algorithms currently used to calculate . Evaluating the first term alone yields a value correct to seven decimal places:
See Ramanujan–Sato series.
From the mid-20th century onwards, all improvements in calculation of have been done with the help of calculators or computers.
In 1944−45, D. F. Ferguson, with the aid of a mechanical desk calculator, found that William Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were incorrect.Ferguson 1946a,
In the early years of the computer, an expansion of to decimal places was computed by Maryland mathematician Daniel Shanks (no relation to the aforementioned William Shanks) and his team at the United States Naval Research Laboratory in Washington, D.C. In 1961, Shanks and his team used two different power series for calculating the digits of . For one, it was known that any error would produce a value slightly high, and for the other, it was known that any error would produce a value slightly low. And hence, as long as the two series produced the same digits, there was a very high confidence that they were correct. The first 100,265 digits of were published in 1962. The authors outlined what would be needed to calculate to 1 million decimal places and concluded that the task was beyond that day's technology, but would be possible in five to seven years.
In 1989, the Chudnovsky brothers computed to over 1 billion decimal places on the supercomputer IBM 3090 using the following variation of Ramanujan's infinite series of :
Records since then have all been accomplished using the Chudnovsky algorithm. In 1999, Yasumasa Kanada and his team at the University of Tokyo computed to over 200 billion decimal places on the supercomputer HITACHI SR8000/MPP (128 nodes) using another variation of Ramanujan's infinite series of . In November 2002, Yasumasa Kanada and a team of 9 others used the Hitachi SR8000, a 64-node supercomputer with 1 terabyte of main memory, to calculate to roughly 1.24 trillion digits in around 600 hours (25days).
Recent records
In August 2009, a Japanese supercomputer called the T2K Open Supercomputer more than doubled the previous record by calculating to roughly 2.6 trillion digits in approximately 73 hours and 36 minutes.