“Approximations of π”的意思、由来-开放百科全书

Approximations for the mathematical constant pi ({{pi}}) in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era (Archimedes). 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 15th century (Jamshīd al-Kāshī). 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), surpassing the accuracy required for any conceivable application outside of pure mathematics.

The record of manual approximation of {{pi}} is held by William Shanks, who calculated 527 digits correctly in the years preceding 1873. Since the middle of the 20th century, the approximation of {{pi}} has been the task of electronic digital computers; {{as of|November 2016|lc=y}}, the record was 22.4{{nbsp}}trillion digits.^[1] (For a comprehensive account, see Chronology of computation of {{pi}}.) In March 2019 Emma Haruka Iwao, a Google employee from Japan calculated to a new world record length of 31 trillion digits with the help of the company's cloud computing service.^[2]

Early history

The best known approximations to {{pi}} 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.

have claimed that the ancient Egyptians used an approximation of {{pi}} as {{frac|22|7}} from as early as the Old Kingdom.^[4]

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 {{pi}} as {{math|1=25/8=3.125}}, about 0.5 percent below the exact value.^[8]^[9]^[10]^[11]

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 {{pi}} as {{frac|256|81}} ≈ 3.16 (accurate to 0.6 percent) by calculating the area of a circle by approximating the circle by an octagon.^[5]^[12]

Astronomical calculations in the Shatapatha Brahmana (c. 6th century BCE) use a fractional approximation of {{math|1= 339/108 ≈ 3.139}}.^[13]

In the 3rd century BCE, Archimedes proved the sharp inequalities {{frac|223|71}} < {{pi}} < {{frac|22|7}}, 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 {{frac|377|120}}, the first known approximation accurate to three decimal places (accuracy 2·10⁻⁵).^[14]

The Chinese mathematician Liu Hui in 263 CE computed {{pi}} to between {{val|3.141024}} and {{val|3.142708}} by inscribing a 96-gon and 192-gon; the average of these two values is {{val|3.141866}} (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 {{math|1=π ≈ 3927/1250 = 3.1416}} (accuracy 2·10⁻⁶), although some scholars instead believe that this is due to the later (5th-century) Chinese mathematician Zu Chongzhi.^[15]

Zu Chongzhi is known to have computed {{pi}} between 3.1415926 and 3.1415927, which was correct to seven decimal places. He gave two other approximations of {{pi}}: {{math|1=π ≈ 22/7}} and {{math|1=π ≈ 355/113}}. The latter fraction is the best possible rational approximation of {{pi}} using fewer than five decimal digits in the numerator and denominator. Zu Chongzhi's result surpasses the accuracy reached in Hellenistic mathematics, and would remain without improvement for close to a millennium.{{citation needed|date=March 2018}}

In Gupta-era India (6th century), mathematician Aryabhata in his astronomical treatise Āryabhaṭīya calculated the value of {{pi}} to five significant figures ({{math|1=π ≈ 62832/20000 = 3.1416}}).^[16]^[17] using it to calculate an approximation of the Earth's circumference.^[18] Aryabhata stated that his result "approximately" ({{IAST|āsanna}} "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).^[19]

Middle Ages

By the 5th century CE, {{pi}} was known to about seven digits in Chinese mathematics, and to about five in Indian mathematics. 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, discovered the infinite series for {{pi}}, now known as the Madhava–Leibniz series,^[20]^[21] and gave two methods for computing the value of {{pi}}. One of these methods is to obtain a rapidly converging series by transforming the original infinite series of {{pi}}. By doing so, he obtained the infinite series

and used the first 21 terms to compute an approximation of {{pi}} correct to 11 decimal places as {{val|3.14159265359}}.

The other method he used was to add a remainder term to the original series of {{pi}}. He used the remainder term

in the infinite series expansion of {{frac|{{pi}}|4}} to improve the approximation of {{pi}} to 13 decimal places of accuracy when {{math|n}} = 75.

Jamshīd al-Kāshī (Kāshānī), a Persian astronomer and mathematician, correctly computed 2{{pi}} to 9 sexagesimal digits in 1424.^[22] This figure is equivalent to 17 decimal digits as

He achieved this level of accuracy by calculating the perimeter of a regular polygon with 3 × 2²⁸ sides.^[23]

16th to 19th centuries

In the second half of the 16th century, the French mathematician François Viète discovered an infinite product that converged on {{pi}} known as Viète's formula.

The German-Dutch mathematician Ludolph van Ceulen (circa 1600) computed the first 35 decimal places of {{pi}} with a 2⁶²-gon. He was so proud of this accomplishment that he had them inscribed on his tombstone.^[24]

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 {{pi}} from a 96-sided polygon.^[25]

In 1789, the Slovene mathematician Jurij Vega calculated the first 140 decimal places for {{pi}}, of which the first 126 were correct^[26] 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.{{Citation needed|date=December 2017}}

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 (93{{nbsp}}billion light-years) to a precision of less than one Planck length (at {{val|1.6162|e=-35|u=meters}}, the shortest unit of length that has real meaning) using {{pi}} expressed to just 62 decimal places.

The English amateur mathematician William Shanks, a man of independent means, spent over 20 years calculating {{pi}} to 707 decimal places. This was accomplished in 1873, with the first 527 places correct. He would calculate new digits all morning and would then spend all afternoon checking his morning's work. This was the longest expansion of {{pi}} until the advent of the electronic digital computer three-quarters of a century later.{{Citation needed|date=December 2017}}

20th and 21st centuries

In 1910, the Indian mathematician Srinivasa Ramanujan found several rapidly converging infinite series of {{pi}}, including

which computes a further eight decimal places of {{pi}} with each term in the series. His series are now the basis for the fastest algorithms currently used to calculate {{pi}}. See also Ramanujan–Sato series.

From the mid-20th century onwards, all calculations of {{pi}} have been done with the help of calculators or computers.

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 {{pi}} to {{val|100000}} decimal places^[28]{{Rp|78}} was computed by Maryland mathematician Daniel Shanks (no relation to the above-mentioned 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 {{pi}}. 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 {{pi}} were published in 1962.^[27]{{Rp|80–99}} The authors outlined what would be needed to calculate {{pi}} 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.^[27]{{Rp|78}}

In 1989, the Chudnovsky brothers computed {{pi}} to over 1 billion decimal places on the supercomputer IBM 3090 using the following variation of Ramanujan's infinite series of {{pi}}:

In 1999, Yasumasa Kanada and his team at the University of Tokyo computed {{pi}} to over 200 billion decimal places on the supercomputer HITACHI SR8000/MPP (128 nodes) using another variation of Ramanujan's infinite series of {{pi}}. In October 2005, they claimed to have calculated it to 1.24 trillion places.^[28]

Records since then have all been accomplished on personal computers using the Chudnovsky algorithm. In 2009, Fabrice Bellard computed just under 2.7 trillion digits, and from 2010 onward, all records have been set using Alexander Yee's y-cruncher software. {{As of|2016|November}}, the record stands at 22,459,157,718,361 ({{pi}}^e × 10¹²) digits.^[29] The limitation on further expansion is primarily storage space for the computation.^[30]

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 {{pi}} to roughly 1.24 trillion digits in around 600 hours.

In August 2009, a Japanese supercomputer called the T2K Open Supercomputer more than doubled the previous record by calculating {{pi}} to roughly 2.6 trillion digits in approximately 73 hours and 36 minutes.

In December 2009, Fabrice Bellard used a home computer to compute 2.7 trillion decimal digits of {{pi}}. 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.^[31]

In August 2010, Shigeru Kondo used Alexander Yee's y-cruncher to calculate 5 trillion digits of {{pi}}. This was the world record for any type of calculation, but significantly it was performed on a home computer built by Kondo.^[32] The calculation was done between 4 May and 3 August, with the primary and secondary verifications taking 64 and 66 hours respectively.^[33]

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.^[34]^[35]

In December 2013, Kondo broke his own record for a second time when he computed 12.1 trillion digits of {{pi}}.^[36]

In October 2014, Sandon Van Ness, going by the pseudonym "houkouonchi" used y-cruncher to calculate 13.3 trillion digits of {{pi}}.^[37]

In November 2016, Peter Trueb and his sponsors computed on y-cruncher and fully verified 22.4 trillion digits of {{pi}}. The computation took (with three interruptions) 105 days to complete.^[38]

In March 2019, Emma Haruka Iwao, an employee at Google, computed 31.4 trillion digits of pi using y-cruncher and Google Cloud machines. This took 121 days to complete.^[39]

Practical approximations

Depending on the purpose of a calculation, {{pi}} can be approximated by using fractions for ease of calculation. The most notable such approximations are {{frac|22|7}} (relative error of about 4·10⁻⁴) and {{frac|355|113}} (relative error of about 8·10⁻⁸).

Non-mathematical "definitions" of {{pi}}

Of some notability are legal or historical texts purportedly "defining {{pi}}" to have some rational value, such as the "Indiana Pi Bill" of 1897, which stated "the ratio of the diameter and circumference is as five-fourths to four" (which would imply "{{math|1=π = 3.2}}") and a passage in the Hebrew Bible that implies that {{math|1=π = 3}}.

Indiana bill

The so-called "Indiana Pi Bill" of 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".^[40]

The bill was nearly passed by the Indiana General Assembly in the U.S., and has been claimed to imply a number of different values for {{pi}}, 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 {{math|1=π = 16/5 = 3.2}}, 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 tabling it indefinitely.

Imputed biblical value

It is sometimes claimed that the Hebrew Bible implies that "{{pi}} equals three", based on a passage in {{bibleverse|1 Kings|7:23|NKJV|}} and {{bibleverse|2 Chronicles|4:2|NKJV|}} giving measurements for the round basin located in front of the Temple in Jerusalem as having a diameter of 10 cubits and a circumference of 30 cubits.

The issue is discussed in the Talmud and in Rabbinic literature.^[41] Among the many explanations and comments are these:

There is still some debate on this passage in biblical scholarship.{{failed verification|date=April 2015}}^[43]^[44] 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 {{bibleverse|1|Kings|7:26||NKJV}}^[45] 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" {{bibleverse|2|Chronicles|4:5||NKJV}}, which suggests a shape that can be encompassed with a string shorter than the total length of the brim, e.g., a Lilium flower or a Teacup.

Development of efficient formulae

Polygon approximation to a circle

Archimedes, in his Measurement of a Circle, created the first algorithm for the calculation of {{pi}} 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 {{math|p_k}} and {{math|P_k}} denote the perimeters of regular polygons of {{math|k}} sides that are inscribed and circumscribed about the same circle, respectively. Then,

Archimedes uses this to successively compute {{math|P₁₂, p₁₂, P₂₄, p₂₄, P₄₈, p₄₈, P₉₆}} and {{math|p₉₆}}.^[46] Using these last values he obtains

It is not known why Archimedes stopped at a 96-sided polygon; it only takes patience to extend the computations. Heron 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.^[47]

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 {{pi}} given in the Almagest (circa 150 CE).^[48]

Advances in the approximation of {{pi}} (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 (1621) obtains better bounds from a pair of bounds gotten from the polygon method. Thus, more accurate results were obtained from polygons with fewer sides.^[49] Viète's formula, published by François Viète 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.^[50]

The last major attempt to compute {{pi}} by this method was carried out by Grienberger in 1630 who calculated 39 decimal places of {{pi}} using Snell's refinement.^[49]

Machin-like formula

together with the Taylor series expansion of the function arctan(x). This formula is most easily verified using polar coordinates of complex numbers, producing:

(Note also that {{{mvar|x}},{{mvar|y}}} = {239, 13²} is a solution to the Pell equation {{mvar|x}}²−2{{mvar|y}}² = −1.)

Formulae of this kind are known as Machin-like formulae. Machin's particular formula was used well into the computer era for calculating record numbers of digits of {{pi}},^[27] 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 {{pi}}:^[27]

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:

Other classical formulae

Ramanujan's work is the basis for the Chudnovsky algorithm, the fastest algorithms used, as of the turn of the millennium, to calculate {{pi}}.

Modern algorithms

Extremely long decimal expansions of {{pi}} are typically computed with iterative formulae like the Gauss–Legendre algorithm and Borwein's algorithm. The latter, found in 1985 by Jonathan and Peter Borwein, converges extremely quickly:

where

, the sequence

converges quartically to {{pi}}, giving about 100 digits in three steps and over a trillion digits after 20 steps. However, it is known that using an algorithm such as the Chudnovsky algorithm (which converges linearly) is faster than these iterative formulae.

The first one million digits of {{pi}} and {{frac|1|{{pi}}}} are available from Project Gutenberg (see external links below). A former calculation record (December 2002) by Yasumasa Kanada of Tokyo University stood at 1.24 trillion digits, which were computed in September 2002 on a 64-node Hitachi supercomputer 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 formulæ were used for this:

These approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers.^[52] Properties like the potential normality of {{pi}} 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, {{pi}} can be approximated to eight (decimal) significant figures with the number 3:8:29:44{{sub|60}}, which is

(The next sexagesimal digit is 0, causing truncation here to yield a relatively good approximation.)

Notes

1. ^¹{{cite web|url=http://www.numberworld.org/y-cruncher/|title=y-cruncher: A Multi-Threaded Pi Program|first=Alexander J.|last=Yee|date=2016|access-date=28 March 2017}}
2. ^{{cite web|url=https://www.bbc.co.uk/news/technology-47524760|title=Emma Haruka Iwao smashes pi world record with Google help|first=Zoe |last=Kleinman|date=2019|access-date=14 March 2019}}
3. ^{{cite book |author-last= Petrie |author-first= W.M.F. |title= Wisdom of the Egyptians |year=1940}}
4. ^{{cite book |author-first= Miroslav |author-last= Verner |author-link=Miroslav Verner |title= The Pyramids: The Mystery, Culture, and Science of Egypt's Great Monuments |year= 2001 |orig-year= 1997 |publisher= Grove Press |isbn= 978-0-8021-3935-1 |quote= Based on the Great Pyramid of Giza, supposedly built so that the circle whose radius is equal to the height of the pyramid has a circumference equal to the perimeter of the base (it is 1760 cubits around and 280 cubits in height).}}
5. ^¹{{cite book |author-last= Rossi |title= Corinna Architecture and Mathematics in Ancient Egypt |year= 2007 |orig-year= |publisher= Cambridge University Press |isbn= 978-0-521-69053-9 |quote= }}
6. ^{{cite book |author-first= J. A. R. |author-last= Legon |title= On Pyramid Dimensions and Proportions |series= Discussions in Egyptology |volume= 20 |pages=25–34 |year= 1991 |orig-year= |publisher= |isbn= |url= http://www.legon.demon.co.uk/pyrprop/propde.htm }}
7. ^See #Imputed biblical value. {{cite book |author-first= Petr |author-last= Beckmann |author-link= Petr Beckmann |title= A History of Pi |series= |volume= |page= |year= 1971 |orig-year= |publisher= St. Martin's |isbn= |url= |quote= There has been concern over the apparent biblical statement of {{pi}} ≈ 3 from the early times of rabbinical Judaism, addressed by Rabbi Nehemiah in the 2nd century. |title-link= A History of Pi }}{{page needed|date=April 2015}}
8. ^{{cite book |author-first= David Gilman |author-last= Romano |author-link= David Gilman Romano |title= Athletics and Mathematics in Archaic Corinth: The Origins of the Greek Stadion |series= |volume= |page= 78 |year= 1993 |orig-year= |publisher= American Philosophical Society |isbn=9780871692061 |url= https://books.google.com/books?id=q0gyy5JOZzIC&pg=PA78&lpg=PA78 |quote= A group of mathematical clay tablets from the Old Babylonian Period, excavated at Susa in 1936, and published by E.M. Bruins in 1950, provide the information that the Babylonian approximation of {{pi}} was 3 1/8 or 3.125. }}
9. ^{{cite web |author-first= E. M. |author-last= Bruins |author-link= |title= Quelques textes mathématiques de la Mission de Suse |year= 1950 |orig-year= |publisher= |isbn= |url= http://www.dwc.knaw.nl/DL/publications/PU00018846.pdf }}
10. ^{{cite book |author-first= E. M. |author-last= Bruins |author-link= |author2-first= M. |author2-last= Rutten |author2-link= |title= Textes mathématiques de Suse |series= Mémoires de la Mission archéologique en Iran |volume= XXXIV |page= |pages= |year= 1961 |orig-year= |publisher= |isbn= |url= |quote= }}
11. ^See also {{harvnb|Beckmann|1971|pages=12, 21–22}} "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. {{pi}} = 3/0.96 = 25/8]".
12. ^{{cite book |editor-last= Katz |editor-first=Victor J. |author-last= Imhausen |author-first= Annette |author-link= Annette Imhausen |title= The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook |publisher= Princeton University Press |year= 2007 |isbn= 978-0-691-11485-9}}
13. ^Chaitanya, Krishna. [https://books.google.com/books?ei=9h35T9PdIoHQrQeO4ZXhBg&id=hDc8AAAAMAAJ&dq=Satapatha+Brahmana+value+of+pi&q=pi#search_anchor A profile of Indian culture.] Indian Book Company (1975). P. 133.
14. ^http://uzweb.uz.ac.zw/science/maths/zimaths/pi.htm{{dead link|date=July 2017 |bot=InternetArchiveBot |fix-attempted=yes }}
15. ^{{citation | last1 = Lam | first1 = Lay Yong | last2 = Ang | first2 = Tian Se | doi = 10.1016/0315-0860(86)90055-8 | issue = 4 | journal = Historia Mathematica | mr = 875525 | pages = 325–340 | title = Circle measurements in ancient China | volume = 13 | year = 1986}}. Reprinted in {{cite book |title=Pi: A Source Book |editor1-first=J. L.|editor1-last=Berggren |editor2-first=Jonathan M.|editor2-last=Borwein |editor3-first=Peter|editor3-last=Borwein |publisher=Springer |year= 2004 |isbn= 9780387205717 |pages= 20–35 |url=https://books.google.com/books?id=QlbzjN_5pDoC&pg=PA20}}. See in particular pp. 333–334 (pp. 28–29 of the reprint).
16. ^How Aryabhata got the earth's circumference right {{webarchive|url=https://web.archive.org/web/20170115063654/http://www.livemint.com/Sundayapp/8wRiLexg1N2IOXjeK2BKcL/How-Aryabhata-got-the-earths-circumference-right-millenia-a.html |date=15 January 2017 }}
17. ^Āryabhaṭīya ({{IAST|gaṇitapāda 10}})::{{IAST|chaturadhikam śatamaṣṭaguṇam dvāśaṣṭistathā sahasrāṇām ayutadvayaviṣkambhasyāsanno vr̥ttapariṇahaḥ}}.:"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 {{math|1=π ≈ 62832/20000 = 3.1416}},{{cite book |title= Geometry: Seeing, Doing, Understanding (Third Edition) |last= Jacobs |first= Harold R. |year= 2003 |publisher= W.H. Freeman and Company |location= New York|isbn= |page= 70}}
18. ^{{cite web|url=http://www-history.mcs.st-and.ac.uk/Biographies/Aryabhata_I.html|title=Aryabhata the Elder|publisher=University of St Andrews, School of Mathematics and Statistics|accessdate=20 July 2011}}
19. ^{{cite book | author = S. Balachandra Rao | title = Indian Mathematics and Astronomy: Some Landmarks | publisher = Jnana Deep Publications | year = 1998 | place = Bangalore | isbn = 978-81-7371-205-0}}
20. ^{{Cite book|title=Special Functions|author1=George E. Andrews |author2=Richard Askey |first=Ranjan Roy|publisher=Cambridge University Press|year=1999|isbn=978-0-521-78988-2|page=58|postscript=}}
21. ^{{Cite journal|first=R. C.|last=Gupta|title=On the remainder term in the Madhava–Leibniz's series|journal=Ganita Bharati|volume=14|issue=1–4|year=1992|pages=68–71|postscript=}}
22. ^{{cite book |author1=Boris A. Rosenfeld |last-author-amp=yes |author2=Adolf P. Youschkevitch |chapter=Ghiyath al-din Jamshid Masud al-Kashi (or al-Kashani) |title=Dictionary of Scientiﬁc Biography |volume=Vol. 7 |date=1981 |page=256}}
23. ^{{cite journal| first1=Mohammad K. | last1=Azarian | title=al-Risāla al-muhītīyya: A Summary | journal=Missouri Journal of Mathematical Sciences | volume=22 | issue=2 | year=2010 | pages=64–85 | url=http://projecteuclid.org/euclid.mjms/1312233136}}
24. ^Capra, B. Digits of Pi Retrieved 13 January 2018
25. ^[https://docs.google.com/viewer?a=v&q=cache:NTSdP7wNFA8J:www.ijpam.eu/contents/2003-7-2/4/4.pdf+&hl=en&gl=uk&pid=bl&srcid=ADGEESjTAd8gQzDqGaN7fo99joDgBNmLm4PCsT69_vWR13A0nR6yT0T-RZFFSpqN-djir-w4lBOV2Juacul9apQNCW2KMOxf0csRinFDa-1DOSRpxTk83Cg4i8qAxvylfWoLRM04qjE8&sig=AHIEtbSw0AiTGzbt4uLFRLrbOXiXNknJCg Google Docs]
26. ^{{cite book |first=Edward |last=Sandifer |year=2007 |title=Jurij baron Vega in njegov čas: Zbornik ob 250-letnici rojstva |page=17 |chapter=Why 140 Digits of Pi Matter |chapter-url=https://web.archive.org/web/20160303184631/http://people.wcsu.edu/sandifere/History/Preprints/Talks/Jurij%20Vega/Vega%20math%20script.pdf#page=17 |trans-title=Baron Jurij Vega and His Times: Celebrating 250 Years |publisher=DMFA |location=Ljubljana |isbn=978-961-6137-98-0 |lccn=2008467244 |oclc=448882242 |quote=We should note that Vega's value contains an error in the 127th digit. Vega gives a 4 where there should be an {{bracket|6}}, and all digits after that are incorrect.}}
27. ^¹²³⁴{{Cite journal|first1=D.|last1=Shanks|author1-link=Daniel Shanks|first2=J. W.|last2=Wrench, Jr.|author2-link=John Wrench|title=Calculation of {{pi}} to 100,000 decimals|journal=Mathematics of Computation|volume=16|year=1962|pages=76–99|doi=10.2307/2003813|jstor=2003813|issue=77}}
28. ^{{cite web|url=http://www.super-computing.org/pi_current.html|title=Announcement at the Kanada lab web site.|website=Super-computing.org|access-date=11 December 2017}}
29. ^{{Cite arxiv |title=Digit Statistics of the First 22.4 Trillion Decimal Digits of Pi |eprint=1612.00489 |class=math.NT |date=30 November 2016 |first=Peter |last=Treub}}
30. ^{{cite web |title=12.1 Trillion Digits of Pi, And we're out of disk space... |first1=Alexander J. |last1=Yee |first2=Shigeru |last2=Kondo |url=http://www.numberworld.org/misc_runs/pi-12t/ |date=December 2013}}
31. ^{{cite web|url=http://bellard.org/pi/pi2700e9/|title=Pi Computation Record|publisher=}}
32. ^McCormick Grad Sets New Pi Record {{webarchive |url=https://web.archive.org/web/20110928084418/http://www.mccormick.northwestern.edu/news/articles/article_743.html |date=28 September 2011 }}
33. ^{{cite web|url=http://www.numberworld.org/misc_runs/pi-5t/announce_en.html|title=Pi - 5 Trillion Digits|publisher=}}
34. ^{{cite web|author=By Glenn |url=https://www.newscientist.com/blogs/shortsharpscience/2011/10/pi-10-trillion.html |title=Short Sharp Science: Epic pi quest sets 10 trillion digit record |publisher=Newscientist.com |date=2011-10-19 |accessdate=2016-04-18}}
35. ^{{cite web |url=http://www.numberworld.org/misc_runs/pi-10t/details.html |title=Round 2... 10 Trillion Digits of Pi |author=Alexander J. Yee |author2=Shigeru Kondo |date=22 October 2011}}
36. ^{{cite web |url=http://www.numberworld.org/misc_runs/pi-12t/ |title=12.1 Trillion Digits of Pi |author=Alexander J. Yee |author2=Shigeru Kondo |date=28 December 2013}}
37. ^{{cite web|url=http://www.numberworld.org/y-cruncher/|title=y-cruncher: A Multi-Threaded Pi Program|first=Alexander J.|last=Yee|date=2018|access-date=14 March 2018}}
38. ^{{Cite web|url=http://www.numberworld.org/y-cruncher/|title=y-cruncher - A Multi-Threaded Pi Program|website=www.numberworld.org|access-date=2016-11-29}}
39. ^{{Cite web|url=http://www.numberworld.org/blogs/2019_3_14_pi_record/|title=Google Cloud Topples the Pi Record|website=www.numberworld.org|access-date=14 March 2019}}
40. ^"Indiana's squared circle" by Arthur E. Hallerberg (Mathematics Magazine, vol. 50 (1977), pp. 136–140)
41. ^{{Cite journal|first1=Boaz|last1=Tsaban|first2=David|last2=Garber|date=February 1998|title=On the rabbinical approximation of {{pi}}|journal=Historia Mathematica|volume=25|issue=1|pages=75–84|issn=0315-0860|doi=10.1006/hmat.1997.2185|url=http://u.cs.biu.ac.il/~tsaban/Pdf/latexpi.pdf|accessdate=14 July 2009}}
42. ^Wilbur Richard Knorr, The Ancient Tradition of Geometric Problems, New York: Dover Publications, 1993.
43. ^{{cite web|first=H. Peter|last=Aleff|url=http://www.recoveredscience.com/const303solomonpi.htm|title=Ancient Creation Stories told by the Numbers: Solomon's Pi|publisher=recoveredscience.com|accessdate=30 October 2007| archiveurl= https://web.archive.org/web/20071014201334/http://recoveredscience.com/const303solomonpi.htm| archivedate= 14 October 2007 | deadurl= no}}
44. ^{{cite web|first=J J|last=O'Connor|author2=E F Robertson|url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.html|title=A history of Pi|date=August 2001|accessdate=30 October 2007| archiveurl= https://web.archive.org/web/20071030063811/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.html| archivedate= 30 October 2007 | deadurl= no}}
45. ^Math Forum – Ask Dr. Math
46. ^{{harvnb|Eves|1992|page=131}}
47. ^{{harvnb|Beckmann|1971|page=66}}
48. ^{{harvnb|Eves|1992|page=118}}
49. ^¹{{harvnb|Eves|1992|page=119}}
50. ^{{harvnb|Beckmann|1971|pages=94–95}}
51. ^{{cite web|url=http://mathworld.wolfram.com/PiFormulas.html |title=Pi Formulas - from Wolfram MathWorld |publisher=Mathworld.wolfram.com |date=2016-04-13 |accessdate=2016-04-18}}
52. ^{{cite web|url=http://www.extremetech.com/extreme/122159-what-can-you-do-with-a-supercomputer|title=What can you do with a supercomputer? - ExtremeTech|publisher=}}
53. ^{{Cite journal|first=Martin|last=Gardner|authorlink=Martin Gardner|title=New Mathematical Diversions|publisher=Mathematical Association of America|year=1995|page=92}}
54. ^A nested radical approximation for {{pi}} {{webarchive |url=https://web.archive.org/web/20110706215615/http://www.mschneider.cc/papers/pi.pdf |date=6 July 2011 }}
55. ^"Lost notebook page 16" ,Ramanujan
56. ^Hoffman, D.W. College Mathematics Journal, 40 (2009) 399
57. ^{{cite web|url=http://iamned.com/math|title=Mathematics|publisher=}}
58. ^{{Cite OEIS|sequencenumber=A002485 |name=Numerators of convergents to Pi}}
59. ^{{Cite OEIS|sequencenumber=A002486 |name=Denominators of convergents to Pi}}
60. ^{{cite web|url=http://qin.laya.com/tech_projects_approxpi.html|title=Fractional Approximations of Pi|publisher=}}
61. ^MathWorld: BBP Formula Wolfram.com
62. ^{{cite arXiv|eprint=0912.0303v1 |last1=Plouffe |first1=Simon |title=On the computation of the n^th decimal digit of various transcendental numbers |class=math.NT |year=2009 }}
63. ^Bellard's Website: Bellard.org
64. ^{{cite web|url=http://crd.lbl.gov/~dhbailey/|title=David H Bailey|website=crd.LBL.gov|access-date=11 December 2017}}
65. ^{{cite web|url=http://www.pi314.net/eng/bellard.php |title=The world of Pi - Bellard |publisher=Pi314.net |date=2013-04-13 |accessdate=2016-04-18}}
66. ^"PiFast timings"
67. ^{{cite web|url=http://pi2.cc.u-tokyo.ac.jp/index.html |title=Kanada Laboratory home page |date=10 August 2010 |author1=Takahashi, Daisuke |author2=Kanada, Yasumasa |publisher=University of Tokyo |accessdate=1 May 2011 |deadurl=yes |archiveurl=https://web.archive.org/web/20110824110203/http://pi2.cc.u-tokyo.ac.jp/index.html |archivedate=24 August 2011 |df=dmy }}