Approximations of π
   HOME

TheInfoList



OR:

Approximations An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ...
for the
mathematical constant A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...
pi () in the
history of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...
reached an accuracy within 0.04% of the true value before the beginning of the
Common Era Common Era (CE) and Before the Common Era (BCE) are year notations for the Gregorian calendar (and its predecessor, the Julian calendar), the world's most widely used calendar era. Common Era and Before the Common Era are alternatives to the or ...
. In
Chinese mathematics Mathematics in China emerged independently by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system ( base 2 and base 10), algebra, geomet ...
, 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 15th century (through the efforts of
Jamshīd al-Kāshī Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) ( fa, غیاث الدین جمشید کاشانی ''Ghiyās-ud-dīn Jamshīd Kāshānī'') (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) was a Persian astronomer a ...
). Early modern mathematicians reached an accuracy of 35 digits by the beginning of the 17th century (
Ludolph van Ceulen Ludolph van Ceulen (, ; 28 January 1540 – 31 December 1610) was a German-Dutch mathematician from Hildesheim. He emigrated to the Netherlands. Biography Van Ceulen moved to Delft most likely in 1576 to teach fencing and mathematics and in 159 ...
), and 126 digits by the 19th century (
Jurij Vega Baron Jurij Bartolomej Vega (also Veha; la, Georgius Bartholomaei Vecha; german: Georg Freiherr von Vega; born ''Vehovec'', March 23, 1754 – September 26, 1802) was a Slovene mathematician, physicist and artillery officer. Early life Bor ...
), surpassing the accuracy required for any conceivable application outside of pure mathematics. The record of manual approximation of is held by
William Shanks William Shanks (25 January 1812 – June 1882) was an English amateur mathematician. He is famous for his calculation of '' '' (pi) to 707 places in 1873, which was correct up to the first 527 places. The error was discovered in 1944 by D. F. Fe ...
, who calculated 527 digits 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 ). On June 8, 2022, the current record was established by Emma Haruka Iwao with Alexander Yee's y-cruncher with 100 trillion digits.


Early history

The best known approximations to dating to
before the Common Era Common Era (CE) and Before the Common Era (BCE) are year notations for the Gregorian calendar (and its predecessor, the Julian calendar), the world's most widely used calendar era. Common Era and Before the Common Era are alternatives to the or ...
were accurate to two decimal places; this was improved upon in
Chinese mathematics Mathematics in China emerged independently by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system ( base 2 and base 10), algebra, geomet ...
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 In ancient Egyptian history, the Old Kingdom is the period spanning c. 2700–2200 BC. It is also known as the "Age of the Pyramids" or the "Age of the Pyramid Builders", as it encompasses the reigns of the great pyramid-builders of the Fourth ...
. This claim has been met with skepticism.
Babylonian mathematics Babylonian mathematics (also known as ''Assyro-Babylonian mathematics'') are the mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the centuries following the fall of Babylon in 539 BC. Babyl ...
usually approximated to 3, sufficient for the architectural projects of the time (notably also reflected in the description of
Solomon's Temple Solomon's Temple, also known as the First Temple (, , ), was the Temple in Jerusalem between the 10th century BC and . According to the Hebrew Bible, it was commissioned by Solomon in the United Kingdom of Israel before being inherited by th ...
in the
Hebrew Bible The Hebrew Bible or Tanakh (;"Tanach"
''Random House Webster's Unabridged Dictionary''.
Hebrew: ''Tān ...
). The Babylonians were aware that this was an approximation, and one Old Babylonian mathematical tablet excavated near
Susa Susa ( ; Middle elx, 𒀸𒋗𒊺𒂗, translit=Šušen; Middle and Neo- elx, 𒋢𒋢𒌦, translit=Šušun; Neo-Elamite and Achaemenid elx, 𒀸𒋗𒐼𒀭, translit=Šušán; Achaemenid elx, 𒀸𒋗𒐼, translit=Šušá; fa, شوش ...
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. At about the same time, the Egyptian Rhind Mathematical Papyrus (dated to the
Second Intermediate Period The Second Intermediate Period marks a period when ancient Egypt fell into disarray for a second time, between the end of the Middle Kingdom and the start of the New Kingdom. The concept of a "Second Intermediate Period" was coined in 1942 by ...
, 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 In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, whi ...
. Astronomical calculations in the ''
Shatapatha Brahmana The Shatapatha Brahmana ( sa, शतपथब्राह्मणम् , Śatapatha Brāhmaṇam, meaning 'Brāhmaṇa of one hundred paths', abbreviated to 'SB') is a commentary on the Śukla (white) Yajurveda. It is attributed to the Vedic ...
'' (c. 6th century BCE) use a fractional approximation of . The
Mahabharata The ''Mahābhārata'' ( ; sa, महाभारतम्, ', ) is one of the two major Sanskrit epics of ancient India in Hinduism, the other being the ''Rāmāyaṇa''. It narrates the struggle between two groups of cousins in the Kuruk ...
(500 BCE - 300 CE) offers an approximation of 3, in the ratios offered in
Bhishma Parva The Bhishma Parva ( sa, भीष्म पर्व), or ''the Book of Bhishma,'' is the sixth of eighteen books of the Indian epic ''Mahabharata''. It is the only Parva in Mahabharata where the main hero is not Arjuna but is rather Bhishma an ...
verses: 6.12.40-45. In the 3rd century BCE,
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
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 Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
used the value , the first known approximation accurate to three decimal places (accuracy 2·10−5). It is equal to 3+8/60+30/60^2, which is accurate to two
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
digits. The Chinese mathematician
Liu Hui Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu (The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state o ...
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 Zu Chongzhi (; 429–500 AD), courtesy name Wenyuan (), was a Chinese astronomer, mathematician, politician, inventor, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3 ...
. 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 Aryabhata (ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the ''Aryabhatiya'' (which ...
, in his astronomical treatise
Āryabhaṭīya ''Aryabhatiya'' (IAST: ') or ''Aryabhatiyam'' ('), a Sanskrit astronomical treatise, is the ''magnum opus'' and only known surviving work of the 5th century Indian mathematician Aryabhata. Philosopher of astronomy Roger Billard estimates that th ...
stated: Approximating to four decimal places: π ≈ = 3.1416,How Aryabhata got the earth's circumference right
Aryabhata stated that his result "approximately" (' "approaching") gave the circumference of a circle. His 15th-century commentator
Nilakantha Somayaji Keļallur Nilakantha Somayaji (14 June 1444 – 1544), also referred to as Keļallur Comatiri, was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehens ...
(
Kerala school of astronomy and mathematics The Kerala school of astronomy and mathematics or the Kerala school was a school of Indian mathematics, mathematics and Indian astronomy, astronomy founded by Madhava of Sangamagrama in Kingdom of Tanur, Tirur, Malappuram district, Malappuram, K ...
) 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 Iriññāttappiḷḷi Mādhavan known as Mādhava of Sangamagrāma () was an Indian mathematician and astronomer from the town believed to be present-day Kallettumkara, Aloor Panchayath, Irinjalakuda in Thrissur District, Kerala, India. He is ...
, founder of the
Kerala school of astronomy and mathematics The Kerala school of astronomy and mathematics or the Kerala school was a school of Indian mathematics, mathematics and Indian astronomy, astronomy founded by Madhava of Sangamagrama in Kingdom of Tanur, Tirur, Malappuram district, Malappuram, K ...
, found the
Maclaurin series Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin ( ...
for arctangent, and then two
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
for . One of them is now known as the Madhava–Leibniz series, based on \pi=4\arctan(1): :\pi=4\left(1-\frac 13+\frac 15-\frac 17 ...\right) The other was based on \pi=6\arctan(1/\sqrt 3): : \pi = \sqrt\sum^\infty_ \frac = \sqrt\sum^\infty_ \frac = \sqrt\left(1-+-+\cdots\right) 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: :\pi/4\approx 1-\frac 13+\frac 15-\frac 17+ ... -\frac\pm\frac 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.
Jamshīd al-Kāshī Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) ( fa, غیاث الدین جمشید کاشانی ''Ghiyās-ud-dīn Jamshīd Kāshānī'') (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) was a Persian astronomer a ...
(Kāshānī), a Persian astronomer and
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
, correctly computed the fractional part of 2 to 9
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
digits in 1424, and translated this into 16 decimal digits after the decimal point: : 2\pi \approx 6.2831853071795864, which gives 16 correct digits for π after the decimal point: : \pi \approx 3.1415926535897932 He achieved this level of accuracy by calculating the perimeter of a
regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
with 3 × 228 sides.


16th to 19th centuries

In the second half of the 16th century, the French mathematician
François Viète François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...
discovered an infinite product that converged on known as
Viète's formula In mathematics, Viète's formula is the following infinite product of nested radicals representing twice the reciprocal of the mathematical constant : \frac2\pi = \frac2 \cdot \frac2 \cdot \frac2 \cdots It can also be represented as: \frac2\pi ...
. The German-Dutch mathematician
Ludolph van Ceulen Ludolph van Ceulen (, ; 28 January 1540 – 31 December 1610) was a German-Dutch mathematician from Hildesheim. He emigrated to the Netherlands. Biography Van Ceulen moved to Delft most likely in 1576 to teach fencing and mathematics and in 159 ...
(''circa'' 1600) computed the first 35 decimal places of with a 262-gon. He was so proud of this accomplishment that he had them inscribed on his tombstone. In ''Cyclometricus'' (1621),
Willebrord Snellius Willebrord Snellius (born Willebrord Snel van Royen) (13 June 158030 October 1626) was a Dutch astronomer and mathematician, Snell. His name is usually associated with the law of refraction of light known as Snell's law. The lunar crater Sn ...
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 Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists of ...
in 1654. Snellius was able to obtain seven digits of from a 96-sided polygon. In 1789, the Slovene mathematician
Jurij Vega Baron Jurij Bartolomej Vega (also Veha; la, Georgius Bartholomaei Vecha; german: Georg Freiherr von Vega; born ''Vehovec'', March 23, 1754 – September 26, 1802) was a Slovene mathematician, physicist and artillery officer. Early life Bor ...
calculated the first 140 decimal places for , of which the first 126 were correct, and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places, of which the first 152 were correct. Vega improved John Machin's formula from 1706 and his method is still mentioned today. 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-year A light-year, alternatively spelled light year, is a large unit of length used to express astronomical distances and is equivalent to about 9.46 trillion kilometers (), or 5.88 trillion miles ().One trillion here is taken to be 1012 ...
s) 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 William Shanks (25 January 1812 – June 1882) was an English amateur mathematician. He is famous for his calculation of '' '' (pi) to 707 places in 1873, which was correct up to the first 527 places. The error was discovered in 1944 by D. F. Fe ...
, a man of independent means, 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, all 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 Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis ...
found several rapidly converging infinite series of , including : \frac = \frac \sum^\infty_ \frac 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 . Even using just the first term gives :\pi\approx\frac\approx 3.14159273 See
Ramanujan–Sato series In mathematics, a Ramanujan–Sato series generalizes Ramanujan’s pi formulas such as, :\frac = \frac \sum_^\infty \frac \frac to the form :\frac = \sum_^\infty s(k) \frac by using other well-defined sequences of integers s(k) obeying a cer ...
. From the mid-20th century onwards, all calculations of have been done with the help of
calculators An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first solid-state electronic calculator was created in the early 1960s. Pocket-size ...
or
computers A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These programs ...
. In 1944, 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. In the early years of the computer, an expansion of to decimal places was computed by Maryland mathematician
Daniel Shanks Daniel Shanks (January 17, 1917 – September 6, 1996) was an American mathematician who worked primarily in numerical analysis and number theory. He was the first person to compute π to 100,000 decimal places. Life and education Shanks was b ...
(no relation to the aforementioned William Shanks) and his team at the
United States Naval Research Laboratory The United States Naval Research Laboratory (NRL) is the corporate research laboratory for the United States Navy and the United States Marine Corps. It was founded in 1923 and conducts basic scientific research, applied research, technological ...
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 David Volfovich Chudnovsky (born January 22, 1947 in Kyiv) and Gregory Volfovich Chudnovsky (born April 17, 1952 in Kyiv) are Ukrainian-born American mathematicians and engineers known for their world-record mathematical calculations and developing ...
computed to over 1 billion decimal places on the
supercomputer A supercomputer is a computer with a high level of performance as compared to a general-purpose computer. The performance of a supercomputer is commonly measured in floating-point operations per second ( FLOPS) instead of million instructions ...
IBM 3090 The IBM 3090 family is a family of mainframe computers that was a high-end successor to the IBM System/370 series, and thus indirectly the successor to the IBM System/360 launched 25 years earlier. Announced on 12 February 1985, the press releas ...
using the following variation of Ramanujan's infinite series of : : \frac = 12 \sum^\infty_ \frac. Records since then have all been accomplished using the
Chudnovsky algorithm The Chudnovsky algorithm is a fast method for calculating the digits of , based on Ramanujan’s formulae. It was published by the Chudnovsky brothers in 1988. It was used in the world record calculations of 2.7 trillion digits of in Decembe ...
. In 1999,
Yasumasa Kanada was a Japanese computer scientist most known for his numerous world records over the past three decades for calculating digits of . He set the record 11 of the past 21 times. Kanada was a professor in the Department of Information Science at ...
and his team at the
University of Tokyo , abbreviated as or UTokyo, is a public research university located in Bunkyō, Tokyo, Japan. Established in 1877, the university was the first Imperial University and is currently a Top Type university of the Top Global University Project by ...
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 was a Japanese computer scientist most known for his numerous world records over the past three decades for calculating digits of . He set the record 11 of the past 21 times. Kanada was a professor in the Department of Information Science at ...
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. # In December 2009,
Fabrice Bellard Fabrice Bellard (; born 1972) is a French computer programmer known for writing FFmpeg, QEMU, and the Tiny C Compiler. He developed Bellard's formula for calculating single digits of pi. In 2012, Bellard co-founded Amarisoft, a telecommunication ...
used a home computer to compute 2.7 trillion decimal digits of . Calculations were performed in base 2 (binary), then the result was converted to base 10 (decimal). The calculation, conversion, and verification steps took a total of 131 days. # In August 2010, Shigeru Kondo used Alexander Yee's y-cruncher to calculate 5 trillion digits of . This was the world record for any type of calculation, but significantly it was performed on a home computer built by Kondo. The calculation was done between 4 May and 3 August, with the primary and secondary verifications taking 64 and 66 hours respectively. # In October 2011, Shigeru Kondo broke his own record by computing ten trillion (1013) and fifty digits using the same method but with better hardware. # In December 2013, Kondo broke his own record for a second time when he computed 12.1 trillion digits of . # In October 2014, Sandon Van Ness, going by the pseudonym "houkouonchi" used y-cruncher to calculate 13.3 trillion digits of . # In November 2016, Peter Trueb and his sponsors computed on y-cruncher and fully verified 22.4 trillion digits of (22,459,157,718,361 ( × 1012)). The computation took (with three interruptions) 105 days to complete, the limitation of further expansion being primarily storage space. # In March 2019, Emma Haruka Iwao, an employee at
Google Google LLC () is an American multinational technology company focusing on search engine technology, online advertising, cloud computing, computer software, quantum computing, e-commerce, artificial intelligence, and consumer electronics. ...
, computed 31.4 (approximately ) trillion digits of pi using y-cruncher and
Google Cloud Google Cloud Platform offers numerous integrated cloud-computing services, including compute, network, and storage. Products Past and present products under the Google Cloud platform include: Current * Google Cloud Datastore, a NoSQL databa ...
machines. This took 121 days to complete. # In January 2020, Timothy Mullican announced the computation of 50 trillion digits over 303 days. #On August 14, 2021, a team (DAViS) at the University of Applied Sciences of the Grisons announced completion of the computation of to 62.8 (approximately ) trillion digits. # On June 8th 2022, Emma Haruka Iwao announced on the Google Cloud Blog the computation of 100 trillion (1014) digits of over 158 days using Alexander Yee's y-cruncher.


Practical approximations

Depending on the purpose of a calculation, can be approximated by using fractions for ease of calculation. The most notable such approximations are (
relative error The approximation error in a data value is the discrepancy between an exact value and some ''approximation'' to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute er ...
of about 4·10−4) and (relative error of about 8·10−8).


Non-mathematical "definitions" of

Of some notability are legal or historical texts purportedly "defining " to have some rational value, such as the "
Indiana Pi Bill The Indiana Pi Bill is the popular name for bill #246 of the 1897 sitting of the Indiana General Assembly, one of the most notorious attempts to establish mathematical truth by legislative fiat. Despite its name, the main result claimed by the b ...
" of 1897, which stated "the ratio of the diameter and circumference is as five-fourths to four" (which would imply "") and a passage in the
Hebrew Bible The Hebrew Bible or Tanakh (;"Tanach"
''Random House Webster's Unabridged Dictionary''.
Hebrew: ''Tān ...
that implies that .


Indiana bill

The so-called "Indiana Pi Bill" from 1897 has often been characterized as an attempt to "legislate the value of Pi". Rather, the bill dealt with a purported solution to the problem of geometrically "
squaring the circle Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficulty ...
". The bill was nearly passed by the
Indiana General Assembly The Indiana General Assembly is the state legislature, or legislative branch, of the state of Indiana. It is a bicameral legislature that consists of a lower house, the Indiana House of Representatives, and an upper house, the Indiana Senate. ...
in the U.S., and has been claimed to imply a number of different values for , although the closest it comes to explicitly asserting one is the wording "the ratio of the diameter and circumference is as five-fourths to four", which would make , a discrepancy of nearly 2 percent. A mathematics professor who happened to be present the day the bill was brought up for consideration in the Senate, after it had passed in the House, helped to stop the passage of the bill on its second reading, after which the assembly thoroughly ridiculed it before postponing it indefinitely.


Imputed biblical value

It is sometimes claimed that the
Hebrew Bible The Hebrew Bible or Tanakh (;"Tanach"
''Random House Webster's Unabridged Dictionary''.
Hebrew: ''Tān ...
implies that " equals three", based on a passage in and giving measurements for the round basin located in front of the
Temple in Jerusalem The Temple in Jerusalem, or alternatively the Holy Temple (; , ), refers to the two now-destroyed religious structures that served as the central places of worship for Israelites and Jews on the modern-day Temple Mount in the Old City of Jerusa ...
as having a diameter of 10
cubit The cubit is an ancient unit of length based on the distance from the elbow to the tip of the middle finger. It was primarily associated with the Sumerians, Egyptians, and Israelites. The term ''cubit'' is found in the Bible regarding No ...
s and a circumference of 30 cubits. The issue is discussed in the
Talmud The Talmud (; he, , Talmūḏ) is the central text of Rabbinic Judaism and the primary source of Jewish religious law (''halakha'') and Jewish theology. Until the advent of modernity, in nearly all Jewish communities, the Talmud was the cente ...
and in
Rabbinic literature Rabbinic literature, in its broadest sense, is the entire spectrum of rabbinic writings throughout Jewish history. However, the term often refers specifically to literature from the Talmudic era, as opposed to medieval and modern rabbinic writ ...
. Among the many explanations and comments are these: *
Rabbi Nehemiah Rabbi Nehemiah was a rabbi who lived circa 150 AD (fourth generation of tannaim). He was one of the great students of Rabbi Akiva, and one of the rabbis who received semicha from R' Judah ben Baba The Talmud equated R' Nechemiah with Rabbi Ne ...
explained this in his ''Mishnat ha-Middot'' (the earliest known
Hebrew Hebrew (; ; ) is a Northwest Semitic language of the Afroasiatic language family. Historically, it is one of the spoken languages of the Israelites and their longest-surviving descendants, the Jews and Samaritans. It was largely preserved ...
text on
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, ca. 150 CE) by saying that the diameter was measured from the ''outside'' rim while the circumference was measured along the ''inner'' rim. This interpretation implies a brim about 0.225 cubit (or, assuming an 18-inch "cubit", some 4 inches), or one and a third " handbreadths," thick (cf. and ). *
Maimonides Musa ibn Maimon (1138–1204), commonly known as Maimonides (); la, Moses Maimonides and also referred to by the acronym Rambam ( he, רמב״ם), was a Sephardic Jewish philosopher who became one of the most prolific and influential Torah ...
states (ca. 1168 CE) that can only be known approximately, so the value 3 was given as accurate enough for religious purposes. This is taken by some as the earliest assertion that is irrational. There is still some debate on this passage in biblical scholarship. Many reconstructions of the basin show a wider brim (or flared lip) extending outward from the bowl itself by several inches to match the description given in In the succeeding verses, the rim is described as "a handbreadth thick; and the brim thereof was wrought like the brim of a cup, like the flower of a lily: it received and held three thousand baths" , which suggests a shape that can be encompassed with a string shorter than the total length of the brim, e.g., a
Lilium ''Lilium'' () is a genus of herbaceous flowering plants growing from bulbs, all with large prominent flowers. They are the true lilies. Lilies are a group of flowering plants which are important in culture and literature in much of the world. M ...
flower or a
Teacup A teacup is a cup for drinking tea. It may be with a handle (grip), handle, generally a small one that may be grasped with the thumb and one or two fingers. It is typically made of a ceramic material. It is usually part of a set, composed of a ...
.


Development of efficient formulae


Polygon approximation to a circle

Archimedes, in his ''Measurement of a Circle'', created the first algorithm for the calculation of based on the idea that the perimeter of any (convex) polygon inscribed in a circle is less than the circumference of the circle, which, in turn, is less than the perimeter of any circumscribed polygon. He started with inscribed and circumscribed regular hexagons, whose perimeters are readily determined. He then shows how to calculate the perimeters of regular polygons of twice as many sides that are inscribed and circumscribed about the same circle. This is a recursive procedure which would be described today as follows: Let and denote the perimeters of regular polygons of sides that are inscribed and circumscribed about the same circle, respectively. Then, :P_ = \frac, \quad \quad p_ = \sqrt. Archimedes uses this to successively compute and . Using these last values he obtains :3 \frac < \pi < 3 \frac. It is not known why Archimedes stopped at a 96-sided polygon; it only takes patience to extend the computations.
Heron The herons are long-legged, long-necked, freshwater and coastal birds in the family Ardeidae, with 72 recognised species, some of which are referred to as egrets or bitterns rather than herons. Members of the genera ''Botaurus'' and ''Ixobrychus ...
reports in his ''Metrica'' (about 60 CE) that Archimedes continued the computation in a now lost book, but then attributes an incorrect value to him. Archimedes uses no trigonometry in this computation and the difficulty in applying the method lies in obtaining good approximations for the square roots that are involved. Trigonometry, in the form of a table of chord lengths in a circle, was probably used by Claudius Ptolemy of Alexandria to obtain the value of given in the ''Almagest'' (circa 150 CE). Advances in the approximation of (when the methods are known) were made by increasing the number of sides of the polygons used in the computation. A trigonometric improvement by
Willebrord Snell Willebrord Snellius (born Willebrord Snel van Royen) (13 June 158030 October 1626) was a Dutch astronomer and mathematician, Snell. His name is usually associated with the law of refraction of light known as Snell's law. The lunar crater Sne ...
(1621) obtains better bounds from a pair of bounds obtained from the polygon method. Thus, more accurate results were obtained from polygons with fewer sides.
Viète's formula In mathematics, Viète's formula is the following infinite product of nested radicals representing twice the reciprocal of the mathematical constant : \frac2\pi = \frac2 \cdot \frac2 \cdot \frac2 \cdots It can also be represented as: \frac2\pi ...
, published by
François Viète François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...
in 1593, was derived by Viète using a closely related polygonal method, but with areas rather than perimeters of polygons whose numbers of sides are powers of two. The last major attempt to compute by this method was carried out by Grienberger in 1630 who calculated 39 decimal places of using Snell's refinement.


Machin-like formula

For fast calculations, one may use formulae such as Machin's: : \frac = 4 \arctan\frac - \arctan\frac together with the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
expansion of the function
arctan In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spec ...
(''x''). This formula is most easily verified using
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, producing: (5+i)^4\cdot(239-i)=2^2 \cdot 13^4(1+i). ( = is a solution to the
Pell equation Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinate ...
2−22 = −1.) Formulae of this kind are known as ''
Machin-like formula In mathematics, Machin-like formulae are a popular technique for computing to a large number of digits. They are generalizations of John Machin's formula from 1706: :\frac = 4 \arctan \frac - \arctan \frac which he used to compute to 100 d ...
e''. Machin's particular formula was used well into the computer era for calculating record numbers of digits of , but more recently other similar formulae have been used as well. For instance, Shanks and his team used the following Machin-like formula in 1961 to compute the first 100,000 digits of : \frac = 6 \arctan\frac + 2 \arctan\frac + \arctan\frac and they used another Machin-like formula, \frac = 12 \arctan\frac + 8 \arctan\frac - 5 \arctan\frac as a check. The record as of December 2002 by Yasumasa Kanada of Tokyo University stood at 1,241,100,000,000 digits. The following Machin-like formulae were used for this: : \frac = 12 \arctan\frac + 32 \arctan\frac - 5 \arctan\frac + 12 \arctan\frac K. Takano (1982). : \frac = 44 \arctan\frac + 7 \arctan\frac - 12 \arctan\frac + 24 \arctan\frac F. C. M. Størmer (1896).


Other classical formulae

Other formulae that have been used to compute estimates of include:
Liu Hui Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu (The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state o ...
(see also
Viète's formula In mathematics, Viète's formula is the following infinite product of nested radicals representing twice the reciprocal of the mathematical constant : \frac2\pi = \frac2 \cdot \frac2 \cdot \frac2 \cdots It can also be represented as: \frac2\pi ...
): : \begin \pi &\approx 768 \sqrt\\ &\approx 3.141590463236763. \end
Madhava Mādhava means Lord Krishna an incarnation of Vishnu. It may also refer to: *a Sanskrit patronymic, "descendant of Madhu (a man of the Yadu tribe)". ** especially of Krishna, see Madhava (Vishnu) *** an icon of Krishna ** Madhava of Sangamagrama, ...
: :\pi = \sqrt\sum^\infty_ \frac = \sqrt\sum^\infty_ \frac = \sqrt\left(-+-+\cdots\right)
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
: : = 20 \arctan\frac + 8 \arctan\frac
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * ''Newton'' (film), a 2017 Indian film * Newton ( ...
/ Euler Convergence Transformation: : \frac= \sum_^\infty\frac= \sum_^ \cfrac = 1+\frac\left(1+\frac\left(1+\frac\left(1+\cdots\right)\right)\right) where (2''k'' + 1)!! denotes the product of the odd integers up to 2''k'' + 1. Ramanujan: : \frac = \frac \sum^\infty_ \frac David Chudnovsky and
Gregory Chudnovsky David Volfovich Chudnovsky (born January 22, 1947 in Kyiv) and Gregory Volfovich Chudnovsky (born April 17, 1952 in Kyiv) are Ukrainian-born American mathematicians and engineers known for their world-record mathematical calculations and developing ...
: : \frac = 12 \sum^\infty_ \frac Ramanujan's work is the basis for the
Chudnovsky algorithm The Chudnovsky algorithm is a fast method for calculating the digits of , based on Ramanujan’s formulae. It was published by the Chudnovsky brothers in 1988. It was used in the world record calculations of 2.7 trillion digits of in Decembe ...
, the fastest algorithms used, as of the turn of the millennium, to calculate .


Modern algorithms

Extremely long decimal expansions of are typically computed with iterative formulae like the Gauss–Legendre algorithm and
Borwein's algorithm In mathematics, Borwein's algorithm is an algorithm devised by Jonathan Borwein, Jonathan and Peter Borwein to calculate the value of . They devised several other algorithms. They published the book ''Pi and the AGM – A Study in Analytic Number Th ...
. The latter, found in 1985 by Jonathan and
Peter Borwein Peter Benjamin Borwein (born St. Andrews, Scotland, May 10, 1953 – 23 August 2020) was a Canadian mathematician and a professor at Simon Fraser University. He is known as a co-author of the paper which presented the Bailey–Borwein–Plo ...
, converges extremely quickly: For y_0=\sqrt2-1,\ a_0=6-4\sqrt2 and :y_=(1-f(y_k))/(1+f(y_k)) ~,~ a_ = a_k(1+y_)^4 - 2^ y_(1+y_+y_^2) where f(y)=(1-y^4)^, the sequence 1/a_k converges quartically to , giving about 100 digits in three steps and over a trillion digits after 20 steps. The Gauss–Legendre algorithm (with
time complexity In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...
O(n\log ^2n), using Harvey–Hoeven multiplication algorithm) is asymptotically faster than the Chudnovsky algorithm (with time complexity O(n\log ^3n)) – but which of these algorithms is faster in practice for "small enough" n depends on technological factors such as memory sizes and access times. For breaking world records, the iterative algorithms are used less commonly than the Chudnovsky algorithm since they are memory-intensive. The first one million digits of and are available from
Project Gutenberg Project Gutenberg (PG) is a Virtual volunteering, volunteer effort to digitize and archive cultural works, as well as to "encourage the creation and distribution of eBooks." It was founded in 1971 by American writer Michael S. Hart and is the ...
. A former calculation record (December 2002) by
Yasumasa Kanada was a Japanese computer scientist most known for his numerous world records over the past three decades for calculating digits of . He set the record 11 of the past 21 times. Kanada was a professor in the Department of Information Science at ...
of
Tokyo University , abbreviated as or UTokyo, is a public research university located in Bunkyō, Tokyo, Japan. Established in 1877, the university was the first Imperial University and is currently a Top Type university of the Top Global University Project by ...
stood at 1.24 trillion digits, which were computed in September 2002 on a 64-node
Hitachi () is a Japanese multinational corporation, multinational Conglomerate (company), conglomerate corporation headquartered in Chiyoda, Tokyo, Japan. It is the parent company of the Hitachi Group (''Hitachi Gurūpu'') and had formed part of the Ni ...
supercomputer A supercomputer is a computer with a high level of performance as compared to a general-purpose computer. The performance of a supercomputer is commonly measured in floating-point operations per second ( FLOPS) instead of million instructions ...
with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits). The following Machin-like formulae were used for this: : \frac = 12 \arctan\frac + 32 \arctan\frac - 5 \arctan\frac + 12 \arctan\frac ( Kikuo Takano (1982)) : \frac = 44 \arctan\frac + 7 \arctan\frac - 12 \arctan\frac + 24 \arctan\frac ( F. C. M. Størmer (1896)). These approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers. Properties like the potential normality of will always depend on the infinite string of digits on the end, not on any finite computation.


Miscellaneous approximations

Historically, base 60 was used for calculations. In this base, can be approximated to eight (decimal) significant figures with the number 3;8,29,44, which is : 3 + \frac + \frac + \frac = 3.14159\ 259^+ (The next
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
digit is 0, causing truncation here to yield a relatively good approximation.) In addition, the following expressions can be used to estimate : * accurate to three digits: ::\frac = 3.143^+ * accurate to three digits: :: \sqrt + \sqrt = 3.146^+ :
Karl Popper Sir Karl Raimund Popper (28 July 1902 – 17 September 1994) was an Austrian-British philosopher, academic and social commentator. One of the 20th century's most influential philosophers of science, Popper is known for his rejection of the cl ...
conjectured that
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
knew this expression, that he believed it to be exactly , and that this is responsible for some of Plato's confidence in the omnicompetence of mathematical geometry—and Plato's repeated discussion of special
right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
s that are either
isosceles In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
or halves of
equilateral In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
triangles. :: \sqrt - \sqrt + 1 = 3.140^+ * accurate to four digits: ::\sqrt = 3.1413^+ ::\sqrt - \sqrt = 3.14142\ 2^+ ::\sqrt - \sqrt = 3.14142\ 8^+ * accurate to four digits (or five significant figures): ::\sqrt = 3.1416^+ * an approximation by Ramanujan, accurate to 4 digits (or five significant figures): ::\frac+\sqrt = 3.1416^+ * accurate to five digits: ::\frac = 3.14156^+ ::\sqrt 3.14155^+ * accurate to six digits: :: \left(2 - \frac\right)^2 = 3.14159\ 6^+
::\sqrt + \sqrt + \frac = 3.14159\ 0^+ * accurate to seven digits: ::\frac = 3.14159\ 29^+ ::\frac = 3.14159\ 25^+ ::\frac - inverse of first term of Ramanujan series. * accurate to eight digits: ::\frac = 3.14159\ 269^+ ::\frac = 3.14159\ 260^+ ::\frac=3.14159\ 264^+ *accurate to nine digits: :: \sqrt =\sqrt = 3.14159\ 2652^+ : This is from Ramanujan, who claimed the Goddess of Namagiri appeared to him in a dream and told him the true value of . * accurate to ten digits: ::\sqrt 5/math> * accurate to ten digits: ::\frac \times \frac = 3.14159\ 26538^+ * accurate to ten digits (or eleven significant figures): ::\sqrt 93= 3.14159\ 26536^+ :This curious approximation follows the observation that the 193rd power of 1/ yields the sequence 1122211125... Replacing 5 by 2 completes the symmetry without reducing the correct digits of , while inserting a central decimal point remarkably fixes the accompanying magnitude at 10100. * accurate to eleven digits: ::\sqrt 8/math> * accurate to twelve digits: ::\sqrt 0/math> * accurate to 16 digits: ::\frac - inverse of sum of first two terms of Ramanujan series. ::\frac=3.1415926535897 \ 934^+ * accurate to 18 digits: ::\frac :This is based on the
fundamental discriminant In mathematics, a fundamental discriminant ''D'' is an integer invariant in the theory of integral binary quadratic forms. If is a quadratic form with integer coefficients, then is the discriminant of ''Q''(''x'', ''y''). Conversely, every integer ...
= 3(89) = 267 which has class number (-) = 2 explaining the
algebraic numbers An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
of degree 2. The core radical \scriptstyle 5^4+53\sqrt is 53 more than the fundamental unit \scriptstyle U_ = 500+53\sqrt which gives the smallest solution = to the
Pell equation Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinate ...
2 − 892 = −1. * accurate to 24 digits: ::\frac - inverse of sum of first three terms of Ramanujan series. * accurate to 30 decimal places: ::\frac = 3.14159\ 26535\ 89793\ 23846\ 26433\ 83279^+ : Derived from the closeness of
Ramanujan constant In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, its ring of integers has unique factoriz ...
to the integer 6403203+744. This does not admit obvious generalizations in the integers, because there are only finitely many
Heegner number In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, its ring of integers has unique factoriza ...
s and negative
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
s ''d'' with class number ''h''(−''d'') = 1, and d = 163 is the largest one in
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
. * accurate to 52 decimal places: ::\frac :Like the one above, a consequence of the
j-invariant In mathematics, Felix Klein's -invariant or function, regarded as a function of a Complex analysis, complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such funct ...
. Among negative discriminants with class number 2, this ''d'' the largest in absolute value. * accurate to 161 decimal places: ::\frac :where ''u'' is a product of four simple quartic units, ::u = (a+\sqrt)^2(b+\sqrt)^2(c+\sqrt)(d+\sqrt) :and, :: \begin a &= \tfrac(23+4\sqrt)\\ b &= \tfrac(19\sqrt+7\sqrt)\\ c &= (429+304\sqrt)\\ d &= \tfrac(627+442\sqrt) \end :Based on one found by
Daniel Shanks Daniel Shanks (January 17, 1917 – September 6, 1996) was an American mathematician who worked primarily in numerical analysis and number theory. He was the first person to compute π to 100,000 decimal places. Life and education Shanks was b ...
. Similar to the previous two, but this time is a quotient of a
modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
, namely the
Dedekind eta function In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string t ...
, and where the argument involves \tau = \sqrt. The discriminant ''d'' = 3502 has ''h''(−''d'') = 16. * The
continued fraction In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...
representation of can be used to generate successive
best rational approximation In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer p ...
s. These approximations are the best possible rational approximations of relative to the size of their denominators. Here is a list of the first thirteen of these: :: \frac, \frac, \frac, \frac, \frac, \frac, \frac, \frac, \frac, \frac, \frac, \frac :Of these, \frac is the only fraction in this sequence that gives more exact digits of (i.e. 7) than the number of digits needed to approximate it (i.e. 6). The accuracy can be improved by using other fractions with larger numerators and denominators, but, for most such fractions, more digits are required in the approximation than correct significant figures achieved in the result.


Summing a circle's area

Pi can be obtained from a circle if its radius and area are known using the relationship: : A = \pi r^2. If a circle with radius ' is drawn with its center at the point (0, 0), any point whose distance from the origin is less than ' will fall inside the circle. The
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
gives the distance from any point (, ) to the center: :d=\sqrt. Mathematical "graph paper" is formed by imagining a 1×1 square centered around each cell (, ), where and are
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s between − and . Squares whose center resides inside or exactly on the border of the circle can then be counted by testing whether, for each cell (, ), :\sqrt \le r. The total number of cells satisfying that condition thus approximates the area of the circle, which then can be used to calculate an approximation of . Closer approximations can be produced by using larger values of . Mathematically, this formula can be written: :\pi = \lim_ \frac \sum_^ \; \sum_^ \begin 1 & \text \sqrt \le r \\ 0 & \text \sqrt > r. \end In other words, begin by choosing a value for . Consider all cells (, ) in which both and are integers between − and . Starting at 0, add 1 for each cell whose distance to the origin (0,0) is less than or equal to '. When finished, divide the sum, representing the area of a circle of radius , by 2 to find the approximation of . For example, if is 5, then the cells considered are: : The 12 cells (0, ±5), (±5, 0), (±3, ±4), (±4, ±3) are ''exactly on'' the circle, and 69 cells are ''completely inside'', so the approximate area is 81, and is calculated to be approximately 3.24 because = 3.24. Results for some values of are shown in the table below: For related results se
The circle problem: number of points (x,y) in square lattice with x^2 + y^2 <= n
Similarly, the more complex approximations of given below involve repeated calculations of some sort, yielding closer and closer approximations with increasing numbers of calculations.


Continued fractions

Besides its simple
continued fraction In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...
representation ; 7, 15, 1, 292, 1, 1,... which displays no discernible pattern, has many
generalized continued fraction In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values. A ge ...
representations generated by a simple rule, including these two. : \pi= : \pi = \cfrac The well-known values and are respectively the second and fourth continued fraction approximations to π. (Other representations are available a
The Wolfram Functions Site
)


Trigonometry


Gregory–Leibniz series

The Gregory–Leibniz series : \pi = 4\sum_^ \cfrac = 4\left( \frac - \frac + \frac - \frac +- \cdots\right) is the power series for
arctan In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spec ...
(x) specialized to  = 1. It converges too slowly to be of practical interest. However, the power series converges much faster for smaller values of x, which leads to formulae where \pi arises as the sum of small angles with rational tangents, known as
Machin-like formula In mathematics, Machin-like formulae are a popular technique for computing to a large number of digits. They are generalizations of John Machin's formula from 1706: :\frac = 4 \arctan \frac - \arctan \frac which he used to compute to 100 d ...
e.


Arctangent

Knowing that 4 arctan 1 = , the formula can be simplified to get: : \begin \pi &= 2\left( 1 + \cfrac + \cfrac + \cfrac + \cfrac + \cfrac + \cdots\right) \\ &= 2\sum_^ \cfrac = \sum_^ \cfrac = \sum_^ \cfrac \\ &= 2 + \frac + \frac + \frac + \frac + \frac + \frac + \frac + \frac + \frac + \cdots \end with a convergence such that each additional 10 terms yields at least three more digits. Another formula for \pi involving arctangent function is given by : \frac=\arctan \frac, \qquad\qquad k\geq 2, where a_k=\sqrt such that a_1=\sqrt . Approximations can be made by using, for example, the rapidly convergent
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
formula :\arctan(x) = \sum_^\infty \frac \; \frac. Alternatively, the following simple expansion series of the arctangent function can be used : \arctan(x)=2\sum_^, where : \begin & a_1(x)=2/x,\\ & b_1(x)=1,\\ & a_n(x)=a_(x)\,\left(1-4/x^2\right)+4b_(x)/x,\\ & b_n(x)=b_(x)\,\left(1-4/x^2\right)-4a_(x)/x, \end to approximate \pi with even more rapid convergence. Convergence in this arctangent formula for \pi improves as integer k increases. The constant \pi can also be expressed by infinite sum of arctangent functions as :\frac = \sum_^\infty \arctan\frac = \arctan\frac + \arctan\frac + \arctan\frac + \arctan\frac + \cdots and : \frac=\sum_ \arctan \frac, where F_n is the ''n''-th
Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
. However, these two formulae for \pi are much slower in convergence because of set of arctangent functions that are involved in computation.


Arcsine

Observing an equilateral triangle and noting that : \sin\left (\frac\right )=\frac yields : \begin \pi &= 6 \sin^ \left( \frac \right) = 6 \left( \frac + \frac + \frac + \frac + \cdots\! \right) \\ &= \frac + \frac + \frac + \frac + \cdots\! = \sum_^\infty \frac \\ &= 3 + \frac + \frac + \frac + \frac + \frac + \frac + \frac + \cdots \end with a convergence such that each additional five terms yields at least three more digits.


Digit extraction methods

The
Bailey–Borwein–Plouffe formula The Bailey–Borwein–Plouffe formula (BBP formula) is a formula for . It was discovered in 1995 by Simon Plouffe and is named after the authors of the article in which it was published, David H. Bailey, Peter Borwein, and Plouffe. Before that, ...
(BBP) for calculating was discovered in 1995 by Simon Plouffe. Using math, the formula can compute any particular digit of —returning the hexadecimal value of the digit—without having to compute the intervening digits (digit extraction). : \pi=\sum_^\infty \left(\frac-\frac-\frac-\frac\right)\left(\frac\right)^n In 1996, Simon Plouffe derived an algorithm to extract the th decimal digit of (using base10 math to extract a base10 digit), and which can do so with an improved speed of time. The algorithm requires virtually no memory for the storage of an array or matrix so the one-millionth digit of can be computed using a pocket calculator. However, it would be quite tedious and impractical to do so. : \pi+3=\sum_^\infty \frac The calculation speed of Plouffe's formula was improved to by
Fabrice Bellard Fabrice Bellard (; born 1972) is a French computer programmer known for writing FFmpeg, QEMU, and the Tiny C Compiler. He developed Bellard's formula for calculating single digits of pi. In 2012, Bellard co-founded Amarisoft, a telecommunication ...
, who derived an alternative formula (albeit only in base2 math) for computing . : \pi=\frac\sum_^\infty \frac \left (-\frac-\frac+\frac-\frac-\frac-\frac+\frac\right )


Efficient methods

Many other expressions for were developed and published by Indian mathematician
Srinivasa Ramanujan Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis ...
. He worked with mathematician
Godfrey Harold Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
in England for a number of years. Extremely long decimal expansions of are typically computed with the Gauss–Legendre algorithm and
Borwein's algorithm In mathematics, Borwein's algorithm is an algorithm devised by Jonathan Borwein, Jonathan and Peter Borwein to calculate the value of . They devised several other algorithms. They published the book ''Pi and the AGM – A Study in Analytic Number Th ...
; the Salamin–Brent algorithm, which was invented in 1976, has also been used. In 1997, David H. Bailey,
Peter Borwein Peter Benjamin Borwein (born St. Andrews, Scotland, May 10, 1953 – 23 August 2020) was a Canadian mathematician and a professor at Simon Fraser University. He is known as a co-author of the paper which presented the Bailey–Borwein–Plo ...
and
Simon Plouffe Simon Plouffe (born June 11, 1956) is a mathematician who discovered the Bailey–Borwein–Plouffe formula (BBP algorithm) which permits the computation of the ''n''th binary digit of π, in 1995. His other 2022 formula allows extracting the '' ...
published a paper (Bailey, 1997) on a new formula for as an
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
: : \pi = \sum_^\infty \frac \left( \frac - \frac - \frac - \frac\right). This formula permits one to fairly readily compute the ''k''th
binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that t ...
or
hexadecimal In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, hexa ...
digit of , without having to compute the preceding ''k'' − 1 digits. Bailey's website contains the derivation as well as implementations in various
programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming ...
s. The
PiHex PiHex was a distributed computing project organized by Colin Percival to calculate specific bits of . 1,246 contributors used idle time slices on almost two thousand computers to make its calculations. The software used for the project made use of ...
project computed 64 bits around the
quadrillion Two naming scales for large numbers have been used in English and other European languages since the early modern era: the long and short scales. Most English variants use the short scale today, but the long scale remains dominant in many non-Eng ...
th bit of (which turns out to be 0).
Fabrice Bellard Fabrice Bellard (; born 1972) is a French computer programmer known for writing FFmpeg, QEMU, and the Tiny C Compiler. He developed Bellard's formula for calculating single digits of pi. In 2012, Bellard co-founded Amarisoft, a telecommunication ...
further improved on BBP with his formula: :\pi = \frac \sum_^ \frac \left( - \frac - \frac + \frac - \frac - \frac - \frac + \frac \right) Other formulae that have been used to compute estimates of include: : \frac=\sum_^\infty\frac=\sum_^\frac =1+\frac\left(1+\frac\left(1+\frac\left(1+\cdots\right)\right)\right) :
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * ''Newton'' (film), a 2017 Indian film * Newton ( ...
. : \frac = \frac \sum^\infty_ \frac :
Srinivasa Ramanujan Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis ...
. This converges extraordinarily rapidly. Ramanujan's work is the basis for the fastest algorithms used, as of the turn of the millennium, to calculate . In 1988, David Chudnovsky and
Gregory Chudnovsky David Volfovich Chudnovsky (born January 22, 1947 in Kyiv) and Gregory Volfovich Chudnovsky (born April 17, 1952 in Kyiv) are Ukrainian-born American mathematicians and engineers known for their world-record mathematical calculations and developing ...
found an even faster-converging series (the
Chudnovsky algorithm The Chudnovsky algorithm is a fast method for calculating the digits of , based on Ramanujan’s formulae. It was published by the Chudnovsky brothers in 1988. It was used in the world record calculations of 2.7 trillion digits of in Decembe ...
): : \frac = \frac \sum^\infty_ \frac. The speed of various algorithms for computing pi to n correct digits is shown below in descending order of asymptotic complexity. M(n) is the complexity of the multiplication algorithm employed.


Projects


Pi Hex

Pi Hex was a project to compute three specific binary digits of using a distributed network of several hundred computers. In 2000, after two years, the project finished computing the five trillionth (5*1012), the forty trillionth, and the quadrillionth (1015) bits. All three of them turned out to be 0.


Software for calculating

Over the years, several programs have been written for calculating to many digits on
personal computer A personal computer (PC) is a multi-purpose microcomputer whose size, capabilities, and price make it feasible for individual use. Personal computers are intended to be operated directly by an end user, rather than by a computer expert or tec ...
s.


General purpose

Most
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The de ...
s can calculate and other common
mathematical constant A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...
s to any desired precision. Functions for calculating are also included in many general
libraries A library is a collection of materials, books or media that are accessible for use and not just for display purposes. A library provides physical (hard copies) or digital access (soft copies) materials, and may be a physical location or a vir ...
for
arbitrary-precision arithmetic In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are li ...
, for instance
Class Library for Numbers Class Library for Numbers (CLN) is a free library for arbitrary precision arithmetic. It operates on signed integers, rational numbers, floating point numbers, complex numbers, modular numbers, and univariate polynomials. Its implementation pr ...
,
MPFR The GNU Multiple Precision Floating-Point Reliable Library (GNU MPFR) is a GNU portable C library for arbitrary-precision binary floating-point computation with correct rounding, based on GNU Multi-Precision Library. Library MPFR's computation ...
and SymPy.


Special purpose

Programs designed for calculating may have better performance than general-purpose mathematical software. They typically implement
checkpointing Checkpointing is a technique that provides fault tolerance for computing systems. It basically consists of saving a snapshot of the application's state, so that applications can restart from that point in case of failure. This is particularly imp ...
and efficient disk swapping to facilitate extremely long-running and memory-expensive computations. * TachusPi by Fabrice Bellard is the program used by himself to compute world record number of digits of pi in 2009. * -cruncher by Alexander Yee is the program which every world record holder since Shigeru Kondo in 2010 has used to compute world record numbers of digits. -cruncher can also be used to calculate other constants and holds world records for several of them. * PiFast by Xavier Gourdon was the fastest program for
Microsoft Windows Windows is a group of several proprietary graphical operating system families developed and marketed by Microsoft. Each family caters to a certain sector of the computing industry. For example, Windows NT for consumers, Windows Server for serv ...
in 2003. According to its author, it can compute one million digits in 3.5 seconds on a 2.4 GHz
Pentium 4 Pentium 4 is a series of single-core CPUs for desktops, laptops and entry-level servers manufactured by Intel. The processors were shipped from November 20, 2000 until August 8, 2008. The production of Netburst processors was active from 2000 ...
. PiFast can also compute other irrational numbers like and . It can also work at lesser efficiency with very little memory (down to a few tens of megabytes to compute well over a billion (109) digits). This tool is a popular benchmark in the
overclocking In computing, overclocking is the practice of increasing the clock rate of a computer to exceed that certified by the manufacturer. Commonly, operating voltage is also increased to maintain a component's operational stability at accelerated spe ...
community. PiFast 4.4 is available fro
Stu's Pi page
PiFast 4.3 is available from Gourdon's page. * QuickPi by Steve Pagliarulo for Windows is faster than PiFast for runs of under 400 million digits. Version 4.5 is available on Stu's Pi Page below. Like PiFast, QuickPi can also compute other irrational numbers like , , and . The software may be obtained from the Pi-Hacks Yahoo! forum, or fro

* Super PI by Kanada Laboratory in the University of Tokyo is the program for Microsoft Windows for runs from 16,000 to 33,550,000 digits. It can compute one million digits in 40 minutes, two million digits in 90 minutes and four million digits in 220 minutes on a Pentium 90 MHz. Super PI version 1.9 is available fro
Super PI 1.9 page


See also

*
Milü Milü (; "close ratio"), also known as Zulü ( Zu's ratio), is the name given to an approximation to (pi) found by Chinese mathematician and astronomer Zu Chongzhi in the 5th century. Using Liu Hui's algorithm (which is based on the areas of r ...


Notes


References

* * * * * * {{DEFAULTSORT:Approximations of Pi Approximations History of mathematics Pi Pi algorithms Real transcendental numbers