Editing Pi (section)

== History ==
<!-- [[History of pi]] links here -->
{{Main|Approximations of π{{!}}Approximations of {{pi}}}}
{{See also|Chronology of computation of π|l1=Chronology of computation of {{pi}}}}

Surviving approximations of {{pi}} prior to the 2nd century&nbsp;AD are accurate to one or two decimal places at best. The earliest written approximations are found in [[Babylon]] and Egypt, both within one percent of the true value. In Babylon, a [[clay tablet]] dated 1900–1600&nbsp;BC has a geometrical statement that, by implication, treats {{pi}} as {{sfrac|25|8}}&nbsp;=&nbsp;3.125.{{sfn|Arndt|Haenel|2006|p=167}} In Egypt, the [[Rhind Papyrus]], dated around 1650&nbsp;BC but copied from a document dated to 1850&nbsp;BC, has a formula for the area of a circle that treats {{pi}} as <math display="inline">\bigl(\frac{16}{9}\bigr)^2\approx3.16</math>.{{r|mollin}}{{sfn|Arndt|Haenel|2006|p=167}} Although some [[Pyramidology|pyramidologists]] have theorized that the [[Great Pyramid of Giza]] was built with proportions related to {{pi}}, this theory is not widely accepted by scholars.<ref>{{Cite book |pages=67–77, 165–166 |title=The Shape of the Great Pyramid |first=Roger |last=Herz-Fischler |publisher=Wilfrid Laurier University Press |year=2000 |isbn=978-0-88920-324-2 |url=https://books.google.com/books?id=066T3YLuhA0C&pg=67 |access-date=5 June 2013}}</ref>
In the [[Shulba Sutras]] of [[Indian mathematics]], dating to an oral tradition from the 1st or 2nd millennium&nbsp;BC, approximations are given which have been variously interpreted as approximately 3.08831, 3.08833, 3.004, 3, or 3.125.<ref>{{cite book |page=[https://books.google.com/books?id=DHvThPNp9yMC&pg=PA27 27] |title=Mathematics in India |title-link=Mathematics in India (book) |first=Kim |last=Plofker |author-link=Kim Plofker |date=2009 |publisher=Princeton University Press |isbn=978-0-691-12067-6}}</ref>

=== Polygon approximation era ===
[[File:Domenico-Fetti Archimedes 1620.jpg|alt=A painting of a man studying|thumb|[[Archimedes]] developed the polygonal approach to approximating {{pi}}.]]
[[File:Archimedes pi.svg|thumb|alt=diagram of a hexagon and pentagon circumscribed outside a circle|{{pi}} can be estimated by computing the perimeters of circumscribed and inscribed polygons.]]

The first recorded algorithm for rigorously calculating the value of {{pi}} was a geometrical approach using polygons, devised around 250&nbsp;BC by the Greek mathematician [[Archimedes]], implementing the [[method of exhaustion]].{{sfn|Arndt|Haenel|2006|p=170}} This polygonal algorithm dominated for over 1,000 years, and as a result {{pi}} is sometimes referred to as Archimedes's constant.{{sfn|Arndt|Haenel|2006|pp=175, 205}} Archimedes computed upper and lower bounds of {{pi}} by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that {{math|{{sfrac|223|71}} < {{pi}} < {{sfrac|22|7}}}} (that is, {{math|3.1408 < {{pi}} < 3.1429}}.<ref name=life-of-pi>{{cite book |last=Borwein |first=Jonathan M. |author-link=Jonathan Borwein |editor1-last=Sidoli |editor1-first=Nathan |editor2-last=Van Brummelen |editor2-first=Glen |contribution=The life of {{pi}}: from Archimedes to ENIAC and beyond |doi=10.1007/978-3-642-36736-6_24 |location=Heidelberg |mr=3203895 |pages=531–561 |publisher=Springer |title=From Alexandria, through Baghdad: Surveys and studies in the ancient Greek and medieval Islamic mathematical sciences in honor of J. L. Berggren |year=2014 |isbn=978-3-642-36735-9}}</ref> Archimedes' upper bound of {{math|{{sfrac|22|7}}}} may have led to a widespread popular belief that {{pi}} is equal to {{math|{{sfrac|22|7}}}}.{{sfn|Arndt|Haenel|2006|p=171}} Around 150&nbsp;AD, Greco-Roman scientist [[Ptolemy]], in his ''[[Almagest]]'', gave a value for {{pi}} of 3.1416, which he may have obtained from Archimedes or from [[Apollonius of Perga]].{{sfn|Arndt|Haenel|2006|p=176}}{{sfn|Boyer|Merzbach|1991|p=168}}<!--may be suspect--> Mathematicians using polygonal algorithms reached 39 digits of {{pi}} in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.<ref name="ArPI">{{harvnb|Arndt|Haenel|2006|pp=15–16, 175, 184–186, 205}}. Grienberger achieved 39 digits in 1630; Sharp 71 digits in 1699.</ref>

In [[ancient China]], values for {{pi}} included 3.1547 (around 1&nbsp;AD), <math>\sqrt{10}</math> (100&nbsp;AD, approximately 3.1623), and {{math|{{sfrac|142|45}}}} (3rd century, approximately 3.1556).{{sfn|Arndt|Haenel|2006|pp=176–177}} Around 265&nbsp;AD, the [[Cao Wei]] mathematician [[Liu Hui]] created a [[Liu Hui's π algorithm|polygon-based iterative algorithm]], with which he constructed a 3,072-sided polygon to approximate {{pi}} as&nbsp;3.1416.{{sfn|Boyer|Merzbach|1991|p=202}}{{sfn|Arndt|Haenel|2006|p=177}} Liu later invented a faster method of calculating {{pi}} and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of&nbsp;4.{{sfn|Boyer|Merzbach|1991|p=202}} Around 480&nbsp;AD, [[Zu Chongzhi]] calculated that <math>3.1415926 < \pi < 3.1415927</math> and suggested the approximations <math display="inline">\pi \approx \frac{355}{113} = 3.14159292035...</math> and <math display="inline">\pi \approx \frac{22}{7} = 3.142857142857...</math>, which he termed the ''[[milü]]'' ('close ratio') and ''yuelü'' ('approximate ratio') respectively, iterating with Liu Hui's algorithm up to a 12,288-sided polygon. With a correct value for its seven first decimal digits, Zu's result remained the most accurate approximation of {{pi}} for the next 800 years.{{sfn|Arndt|Haenel|2006|p=178}}

The Indian astronomer [[Aryabhata]] used a value of 3.1416 in his ''[[Āryabhaṭīya]]'' (499&nbsp;AD).{{sfn|Arndt|Haenel|2006|p=179}} Around 1220, [[Fibonacci]] computed 3.1418 using a polygonal method devised independently of Archimedes.{{sfn|Arndt|Haenel|2006|p=180}} Italian author [[Dante]] apparently employed the value <math display="inline">3+\frac{\sqrt{2}}{10} \approx 3.14142</math>.{{sfn|Arndt|Haenel|2006|p=180}}

The Persian astronomer [[Jamshīd al-Kāshī]] produced nine [[sexagesimal]] digits, roughly the equivalent of 16 decimal digits, in 1424, using a polygon with <math display="inline">3\times 2^{28}</math> sides,<ref>{{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 |doi=10.35834/mjms/1312233136 |doi-access=free}} {{pb}} {{cite web |last1=O'Connor |first1=John J. |last2=Robertson |first2=Edmund F. |year=1999 |title=Ghiyath al-Din Jamshid Mas'ud al-Kashi |work=[[MacTutor History of Mathematics archive]] |url=http://www-history.mcs.st-and.ac.uk/history/Biographies/Al-Kashi.html |access-date=11 August 2012 |url-status=live |archive-url=https://web.archive.org/web/20110412192025/http://www-history.mcs.st-and.ac.uk/history/Biographies/Al-Kashi.html |archive-date=12 April 2011}}</ref> which stood as the world record for about 180 years.{{sfn|Arndt|Haenel|2006|p=182}} French mathematician [[François Viète]] in 1579 achieved nine digits with a polygon of <math display="inline">3\times 2^{17}</math> sides.{{sfn|Arndt|Haenel|2006|p=182}} Flemish mathematician [[Adriaan van Roomen]] arrived at 15 decimal places in 1593.{{sfn|Arndt|Haenel|2006|p=182}} In 1596, Dutch mathematician [[Ludolph van Ceulen]] reached 20 digits, a record he later increased to 35 digits (as a result, {{pi}} was called the "Ludolphian number" in Germany until the early 20th century).{{sfn|Arndt|Haenel|2006|pp=182–183}} Dutch scientist [[Willebrord Snellius]] reached 34 digits in 1621,{{sfn|Arndt|Haenel|2006|p=183}} and Austrian astronomer [[Christoph Grienberger]] arrived at 38 digits in 1630 using 10<sup>40</sup> sides.<ref>{{cite book |first=Christophorus |last=Grienbergerus |author-link=Christoph Grienberger |language=la |year=1630 |title=Elementa Trigonometrica |url=http://librarsi.comune.palermo.it/gesuiti2/06.04.01.pdf |archive-url=https://web.archive.org/web/20140201234124/http://librarsi.comune.palermo.it/gesuiti2/06.04.01.pdf |archive-date=1 February 2014}} His evaluation was 3.14159 26535 89793 23846 26433 83279 50288 4196 < {{pi}} < 3.14159 26535 89793 23846 26433 83279 50288 4199.</ref> [[Christiaan Huygens]] was able to arrive at 10 decimal places in 1654 using a slightly different method equivalent to [[Richardson extrapolation]].<ref>{{cite book |last=Brezinski |first=C. |contribution=Some pioneers of extrapolation methods |date=2009 |url=https://www.worldscientific.com/doi/10.1142/9789812836267_0001 |title=The Birth of Numerical Analysis |pages=1–22 |publisher=World Scientific |doi=10.1142/9789812836267_0001 |isbn=978-981-283-625-0 |editor1-first=Adhemar |editor1-last=Bultheel |editor1-link=Adhemar Bultheel |editor2-first=Ronald |editor2-last=Cools}} {{pb}} {{Cite journal |last=Yoder |first=Joella G. |author-link=Joella Yoder |date=1996 |title=Following in the footsteps of geometry: The mathematical world of Christiaan Huygens |journal=De Zeventiende Eeuw |volume=12 |pages=83–93 |url=https://www.dbnl.org/tekst/_zev001199601_01/_zev001199601_01_0009.php |via=[[Digital Library for Dutch Literature]]}}</ref>

=== Infinite series ===
{{comparison_pi_infinite_series.svg}}
The calculation of {{pi}} was revolutionized by the development of [[infinite series]] techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite [[sequence (mathematics)|sequence]]. Infinite series allowed mathematicians to compute {{pi}} with much greater precision than [[Archimedes]] and others who used geometrical techniques.{{sfn|Arndt|Haenel|2006|pp=185–191}} Although infinite series were exploited for {{pi}} most notably by European mathematicians such as [[James Gregory (mathematician)|James Gregory]] and [[Gottfried Wilhelm Leibniz]], the approach also appeared in the [[Kerala school of astronomy and mathematics|Kerala school]] sometime in the 14th or 15th century.{{r|Roypp}}{{sfn|Arndt|Haenel|2006|pp=185–186}} Around 1500, an infinite series that could be used to compute {{pi}}, written in the form of [[Sanskrit]] verse, was presented in ''[[Tantrasamgraha]]'' by [[Nilakantha Somayaji]].<ref name="Roypp">{{cite journal |last=Roy |first=Ranjan |author-link=Ranjan Roy |year=1990 |title=The Discovery of the Series Formula for {{mvar|π}} by Leibniz, Gregory and Nilakantha |journal=Mathematics Magazine |volume=63 |number=5 |pages=291–306 |url=https://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1991/0025570x.di021167.02p0073q.pdf |doi=10.1080/0025570X.1990.11977541 |access-date=21 February 2023 |archive-date=14 March 2023 |archive-url=https://web.archive.org/web/20230314224252/https://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1991/0025570x.di021167.02p0073q.pdf |url-status=dead}}</ref> The series are presented without proof, but proofs are presented in the later work ''[[Yuktibhāṣā]]'', published around 1530. Several infinite series are described, including series for sine (which Nilakantha attributes to [[Madhava of Sangamagrama]]), cosine, and arctangent which are now sometimes referred to as [[Madhava series]]. The series for arctangent is sometimes called [[Gregory's series]] or the Gregory–Leibniz series.{{r|Roypp}} Madhava used infinite series to estimate {{pi}} to 11 digits around 1400.<ref>{{cite book |last=Joseph |first=George Gheverghese |author-link=George Gheverghese Joseph |title=The Crest of the Peacock: Non-European Roots of Mathematics |publisher=Princeton University Press |year=1991 |isbn=978-0-691-13526-7 |url=https://books.google.com/books?id=c-xT0KNJp0cC&pg=PA264 |page=264}}<!-- This ISBN is for the third edition from 2011! --></ref>{{sfn|Andrews|Askey|Roy|1999|p=59}}<ref>{{Cite journal |first=R. C. |last=Gupta |author-link=Radha Charan 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}}</ref>

In 1593, [[François Viète]] published what is now known as [[Viète's formula]], an [[infinite product]] (rather than an [[infinite sum]], which is more typically used in {{pi}} calculations):<ref>{{harvnb|Arndt|Haenel|2006|p=187}}. {{pb}} {{cite book |url=https://books.google.com/books?id=7_BCAAAAcAAJ |title=Variorum de rebus mathematicis responsorum |volume=VIII |first=Franciscus |last=Vieta |year=1593}} {{pb}} {{OEIS2C|id=A060294}}</ref>

<math display=block> \frac2\pi = \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdots</math>

In 1655, [[John Wallis]] published what is now known as [[Wallis product]], also an infinite product:{{sfn|Arndt|Haenel|2006|p=187}}

<math display=block>
\frac{\pi}{2} = \Big(\frac{2}{1} \cdot \frac{2}{3}\Big) \cdot \Big(\frac{4}{3} \cdot \frac{4}{5}\Big) \cdot \Big(\frac{6}{5} \cdot \frac{6}{7}\Big) \cdot \Big(\frac{8}{7} \cdot \frac{8}{9}\Big) \cdots
</math>

[[File:GodfreyKneller-IsaacNewton-1689.jpg|thumb|upright=0.8|alt=A formal portrait of a man, with long hair|[[Isaac Newton]]
used [[infinite series]] to compute {{pi}} to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".{{r|Newton}}]]
In the 1660s, the English scientist [[Isaac Newton]] and German mathematician [[Gottfried Wilhelm Leibniz]] discovered [[calculus]], which led to the development of many infinite series for approximating {{pi}}. Newton himself used an arcsine series to compute a 15-digit approximation of {{pi}} in 1665 or 1666, writing, "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."<ref name="Newton">{{harvnb|Arndt|Haenel|2006|p=188}}. Newton quoted by Arndt.</ref>

In 1671, [[James Gregory (mathematician)|James Gregory]], and independently, Leibniz in 1673, discovered the [[Taylor series]] expansion for [[arctangent]]:{{r|Roypp}}<ref>{{cite journal |last=Horvath |first=Miklos |title=On the Leibnizian quadrature of the circle. |journal=Annales Universitatis Scientiarum Budapestiensis (Sectio Computatorica) |volume=4 |year=1983 |pages=75–83 |url=http://ac.inf.elte.hu/Vol_004_1983/075.pdf}}</ref>{{sfn|Eymard|Lafon|2004|pp=53–54}}

<math display=block>
\arctan z = z - \frac {z^3} {3} +\frac {z^5} {5} -\frac {z^7} {7} +\cdots
</math>

This series, sometimes called the [[Gregory's series|Gregory–Leibniz series]], equals <math display="inline">\frac{\pi}{4}</math> when evaluated with <math>z=1</math>.{{sfn|Eymard|Lafon|2004|pp=53–54}} But for <math>z=1</math>, [[Leibniz formula for π#Convergence|it converges impractically slowly]] (that is, approaches the answer very gradually), taking about ten times as many terms to calculate each additional digit.<ref>{{cite journal |last=Cooker |first=M. J. |year=2011 |title=Fast formulas for slowly convergent alternating series |journal=Mathematical Gazette |volume=95 |number=533 |pages=218–226 |doi=10.1017/S0025557200002928 |s2cid=123392772 |url=https://www.cambridge.org/core/services/aop-cambridge-core/content/view/F7C083868DEB95FE049CD44163367592/S0025557200002928a.pdf/div-class-title-fast-formulas-for-slowly-convergent-alternating-series-div.pdf |access-date=23 February 2023 |archive-date=4 May 2019 |archive-url=https://web.archive.org/web/20190504091131/https://www.cambridge.org/core/services/aop-cambridge-core/content/view/F7C083868DEB95FE049CD44163367592/S0025557200002928a.pdf/div-class-title-fast-formulas-for-slowly-convergent-alternating-series-div.pdf |url-status=deviated <!-- now paywalled -->}}</ref>

In 1699, English mathematician [[Abraham Sharp]] used the Gregory–Leibniz series for <math display="inline">z=\frac{1}{\sqrt{3}}</math> to compute {{pi}} to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.{{sfn|Arndt|Haenel|2006|p=189}}

In 1706, [[John Machin]] used the Gregory–Leibniz series to produce an algorithm that converged much faster:<ref name=jones>{{cite book |last=Jones |first=William |author-link=William Jones (mathematician) |year=1706 |title=Synopsis Palmariorum Matheseos |place=London |publisher=J. Wale |url=https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/page/n283/ |pages=[https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/page/n261/ 243], [https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/page/n283/ 263] |quote-page=263 |quote=There are various other ways of finding the ''Lengths'', or ''Areas'' of particular ''Curve Lines'' or ''Planes'', which may very much facilitate the Practice; as for instance, in the ''Circle'', the Diameter is to Circumference as 1 to {{br}}<math>
\overline{\tfrac{16}5 - \tfrac4{239}}
- \tfrac13\overline{\tfrac{16}{5^3} - \tfrac4{239^3}}
+ \tfrac15\overline{\tfrac{16}{5^5} - \tfrac4{239^5}}
-,\, \&c. =</math>{{br}}{{math|1=3.14159, &''c.'' = ''π''}}. This ''Series'' (among others for the same purpose, and drawn from the same Principle) I receiv'd from the Excellent Analyst, and my much Esteem'd Friend Mr. ''[[John Machin]]''; and by means thereof, ''[[Ludolph van Ceulen|Van Ceulen]]''{{'}}s Number, or that in Art. 64.38. may be Examin'd with all desireable Ease and Dispatch.
}} {{pb}} Reprinted in {{cite book |last=Smith |first=David Eugene |author-link=David Eugene Smith |year=1929 |title=A Source Book in Mathematics |publisher=McGraw–Hill |chapter=William Jones: The First Use of {{mvar|π}} for the Circle Ratio |chapter-url=https://archive.org/details/sourcebookinmath1929smit/page/346/ |pages=346–347}}</ref><ref name=tweddle>{{cite journal |last=Tweddle |first=Ian |year=1991 |title=John Machin and Robert Simson on Inverse-tangent Series for {{mvar|π}} |journal=Archive for History of Exact Sciences |volume=42 |number=1 |pages=1–14 |doi=10.1007/BF00384331 |jstor=41133896 |s2cid=121087222}}</ref>{{sfn|Arndt|Haenel|2006|pp=192–193}}

<math display=block> \frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{239}.</math>

Machin reached 100 digits of {{pi}} with this formula.{{sfn|Arndt|Haenel|2006|pp=72–74}} Other mathematicians created variants, now known as [[Machin-like formula]]e, that were used to set several successive records for calculating digits of {{pi}}.<ref>{{cite journal |last=Lehmer |first=D. H. |author-link=D. H. Lehmer |year=1938 |title=On Arccotangent Relations for π |journal=American Mathematical Monthly |volume=45 |number=10 |pages=657–664 Published by: Mathematical Association of America |jstor=2302434 |doi=10.1080/00029890.1938.11990873 |url=https://www.maa.org/sites/default/files/pdf/pubs/amm_supplements/Monthly_Reference_7.pdf |access-date=21 February 2023 |archive-date=7 March 2023 |archive-url=https://web.archive.org/web/20230307164817/https://www.maa.org/sites/default/files/pdf/pubs/amm_supplements/Monthly_Reference_7.pdf |url-status=dead}}</ref>{{sfn|Arndt|Haenel|2006|pp=72–74}}

Isaac Newton [[series acceleration|accelerated the convergence]] of the Gregory–Leibniz series in 1684 (in an unpublished work; others independently discovered the result):<ref>{{cite book |last=Roy |first=Ranjan |author-link=Ranjan Roy |year=2021 |orig-year=1st ed. 2011 |title=Series and Products in the Development of Mathematics |edition=2 |volume=1 |publisher=Cambridge University Press |pages=215–216, 219–220}} {{pb}} {{cite book |last=Newton |first=Isaac |author-link=Isaac Newton |year=1971 |editor-last=Whiteside |editor-first=Derek Thomas |editor-link=Tom Whiteside |title=The Mathematical Papers of Isaac Newton |volume=4, 1674–1684 |publisher=Cambridge University Press |chapter=De computo serierum |trans-chapter=On the computation of series |at="De transmutatione serierum" [On the transformation of series] §{{nbsp}}3.2.2 {{pgs|604–615}} |chapter-url=https://archive.org/details/mathematicalpape0004newt/page/604/mode/2up |chapter-url-access=limited}}</ref>

<math display=block>
\arctan x
= \frac{x}{1 + x^2} + \frac23\frac{x^3}{(1 + x^2)^2}
+ \frac{2\cdot 4}{3 \cdot 5}\frac{x^5}{(1 + x^2)^3} + \cdots
</math>

[[Leonhard Euler]] popularized this series in his 1755 differential calculus textbook, and later used it with Machin-like formulae, including <math display=inline>\tfrac\pi4 = 5\arctan\tfrac17 + 2\arctan\tfrac{3}{79},</math> with which he computed 20 digits of {{pi}} in one hour.<ref>{{cite web |last=Sandifer |first=Ed |year=2009 |title=Estimating π |website=How Euler Did It |url=http://eulerarchive.maa.org/hedi/HEDI-2009-02.pdf}} Reprinted in {{cite book |last=Sandifer |first=Ed |display-authors=0 |year=2014 |title=How Euler Did Even More |pages=109–118 |publisher=Mathematical Association of America}} {{pb}} {{cite book |last=Euler |first=Leonhard |author-link=Leonhard Euler |year=1755 |title=[[Institutiones calculi differentialis|Institutiones Calculi Differentialis]] |chapter=§&nbsp;2.2.30 |page=318 |publisher=Academiae Imperialis Scientiarium Petropolitanae |language=la |chapter-url=https://archive.org/details/institutiones-calculi-differentialis-cum-eius-vsu-in-analysi-finitorum-ac-doctri/page/318 |id=[https://scholarlycommons.pacific.edu/euler-works/212/ E 212]}} {{pb}} {{cite journal |last=Euler |first=Leonhard |author-link=Leonhard Euler |year=1798 |orig-year=written 1779 |title=Investigatio quarundam serierum, quae ad rationem peripheriae circuli ad diametrum vero proxime definiendam maxime sunt accommodatae |journal=Nova Acta Academiae Scientiarum Petropolitinae |volume=11 |pages=133–149, 167–168 |url=https://archive.org/details/novaactaacademia11petr/page/133 |id=[https://scholarlycommons.pacific.edu/euler-works/705/ E 705]}} {{pb}} {{cite journal |first=Chien-Lih |last=Hwang |year=2004 |title=88.38 Some Observations on the Method of Arctangents for the Calculation of {{mvar|π}} |journal=Mathematical Gazette |volume=88 |number=512 |pages=270–278 |doi=10.1017/S0025557200175060|jstor=3620848 |s2cid=123532808}} {{pb}} {{cite journal |first=Chien-Lih |last=Hwang |year=2005 |title=89.67 An elementary derivation of Euler's series for the arctangent function |journal=Mathematical Gazette |volume=89 |number=516 |pages=469–470 |doi=10.1017/S0025557200178404|jstor=3621947 |s2cid=123395287}}
</ref>

Machin-like formulae remained the best-known method for calculating {{pi}} well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson&nbsp;– the best approximation achieved without the aid of a calculating device.{{sfn|Arndt|Haenel|2006|pp=192–196, 205}}

In 1844, a record was set by [[Zacharias Dase]], who employed a Machin-like formula to calculate 200 decimals of {{pi}} in his head at the behest of German mathematician [[Carl Friedrich Gauss]].{{sfn|Arndt|Haenel|2006|pp=194–196}}

In 1853, British mathematician [[William Shanks]] calculated {{pi}} to 607 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect. Though he calculated an additional 100 digits in 1873, bringing the total up to 707, his previous mistake rendered all the new digits incorrect as well.<ref name="hayes-2014">{{cite magazine |last=Hayes |first=Brian |author-link=Brian Hayes (scientist) |url=https://www.americanscientist.org/article/pencil-paper-and-pi |title=Pencil, Paper, and Pi |volume=102 |issue=5 |page=342 |magazine=[[American Scientist]] |date=September 2014 |access-date=22 January 2022 |doi=10.1511/2014.110.342}}</ref>

==== Rate of convergence ====
Some infinite series for {{pi}} [[convergent series|converge]] faster than others. Given the choice of two infinite series for {{pi}}, mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate {{pi}} to any given accuracy.<ref name="Aconverge">{{cite journal |last1=Borwein |first1=J. M. |last2=Borwein |first2=P. B. |title=Ramanujan and Pi |year=1988 |journal=Scientific American |volume=256 |issue=2 |pages=112–117 |bibcode=1988SciAm.258b.112B |doi=10.1038/scientificamerican0288-112}}{{br}} {{harvnb|Arndt|Haenel|2006|pp=15–17, 70–72, 104, 156, 192–197, 201–202}}.</ref> A simple infinite series for {{pi}} is the [[Leibniz formula for π|Gregory–Leibniz series]]:{{sfn|Arndt|Haenel|2006|pp=69–72}}

<math display=block>
\pi = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \frac{4}{11} + \frac{4}{13} - \cdots
</math>

As individual terms of this infinite series are added to the sum, the total gradually gets closer to {{pi}}, and&nbsp;– with a sufficient number of terms&nbsp;– can get as close to {{pi}} as desired. It converges quite slowly, though&nbsp;– after 500,000 terms, it produces only five correct decimal digits of {{pi}}.<ref>{{cite journal |last1=Borwein |first1=J. M. |last2=Borwein |first2=P. B. |last3=Dilcher |first3=K. |year=1989 |title=Pi, Euler Numbers, and Asymptotic Expansions |journal=American Mathematical Monthly |volume=96 |issue=8 |pages=681–687 |doi=10.2307/2324715 |jstor=2324715 |hdl=1959.13/1043679 |hdl-access=free}}</ref>

An infinite series for {{pi}} (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory–Leibniz series is:{{sfn|Arndt|Haenel|2006|loc = Formula 16.10, p. 223}}<ref>{{cite book |last=Wells |first=David |page=35 |title=The Penguin Dictionary of Curious and Interesting Numbers |edition=revised |publisher=Penguin |year=1997 |isbn=978-0-14-026149-3}}</ref>

<math display=block>
\pi = 3 + \frac{4}{2\times3\times4} - \frac{4}{4\times5\times6} + \frac{4}{6\times7\times8} - \frac{4}{8\times9\times10} + \cdots
</math>

The following table compares the convergence rates of these two series:

{|class="wikitable" style="text-align: center; margin: auto;"
|-
! Infinite series for {{pi}} !! After 1st term !! After 2nd term !! After 3rd term !! After 4th term !! After 5th term !! Converges to:
|-
| <math>\pi = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \frac{4}{11} + \frac{4}{13} + \cdots</math>
||4.0000||2.6666 ... ||3.4666 ... ||2.8952 ... ||3.3396 ... ||rowspan=2| {{pi}} = 3.1415 ...
|-
| <math>\pi = {{3}} + \frac{{4}}{2\times3\times4} - \frac{{4}}{4\times5\times6} + \frac{{4}}{6\times7\times8} - \cdots </math>
||3.0000||3.1666 ... ||3.1333 ... ||3.1452 ... ||3.1396 ...
|}

After five terms, the sum of the Gregory–Leibniz series is within 0.2 of the correct value of {{pi}}, whereas the sum of Nilakantha's series is within 0.002 of the correct value. Nilakantha's series converges faster and is more useful for computing digits of {{pi}}. Series that converge even faster include [[Machin-like formula|Machin's series]] and [[Chudnovsky algorithm|Chudnovsky's series]], the latter producing 14 correct decimal digits per term.{{r|Aconverge}}

=== Irrationality and transcendence ===
{{See also|Proof that π is irrational{{!}}Proof that {{pi}} is irrational|Proof that π is transcendental{{!}}Proof that {{pi}} is transcendental}}
Not all mathematical advances relating to {{pi}} were aimed at increasing the accuracy of approximations. When Euler solved the [[Basel problem]] in 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between {{pi}} and the [[prime number]]s that later contributed to the development and study of the [[Riemann zeta function]]:{{sfn|Posamentier|Lehmann|2004|p=284}}

<math display=block> \frac{\pi^2}{6} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots</math>

Swiss scientist [[Johann Heinrich Lambert]] in 1768 proved that {{pi}} is [[irrational number|irrational]], meaning it is not equal to the quotient of any two integers.{{sfn|Arndt|Haenel|2006|p=5}} [[Proof that π is irrational|Lambert's proof]] exploited a continued-fraction representation of the tangent function.<ref>Lambert, Johann, "Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques", reprinted in {{harvnb|Berggren|Borwein|Borwein|1997|pp=129–140}}.</ref> French mathematician [[Adrien-Marie Legendre]] proved in 1794 that {{pi}}<sup>2</sup> is also irrational. In 1882, German mathematician [[Ferdinand von Lindemann]] proved that {{pi}} is [[transcendental number|transcendental]],<ref>{{cite journal |last=Lindemann |first=F. |author-link=Ferdinand Lindemann |year=1882 |title=Über die Ludolph'sche Zahl |journal=Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin |volume=2 |pages=679–682 |url=https://archive.org/details/sitzungsberichte1882deutsch/page/679}}</ref> confirming a conjecture made by both [[Adrien-Marie Legendre|Legendre]] and Euler.{{sfn|Arndt|Haenel|2006|p=196}}<ref>Hardy and Wright 1938 and 2000: 177 footnote §&nbsp;11.13–14 references Lindemann's proof as appearing at ''Math. Ann''. 20 (1882), 213–225.</ref> Hardy and Wright states that "the proofs were afterwards modified and simplified by Hilbert, Hurwitz, and other writers".<ref>cf Hardy and Wright 1938 and 2000:177 footnote §&nbsp;11.13–14. The proofs that e and π are transcendental can be found on pp. 170–176. They cite two sources of the proofs at Landau 1927 or Perron 1910; see the "List of Books" at pp. 417–419 for full citations.</ref>

=== Adoption of the symbol {{pi}} ===
{{Multiple image
| image1            = William Jones, the Mathematician.jpg
| caption1          = The earliest known use of the Greek letter {{pi}} to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician [[William Jones (mathematician)|William Jones]] in 1706
| caption2          = [[Leonhard Euler]] popularized the use of the Greek letter {{pi}} in works he published in 1736 and 1748.
| total_width       = 300
| image2            = Leonhard Euler.jpg
| align             = right
}}

The first recorded use of the symbol {{pi}} in circle geometry is in [[William Oughtred|Oughtred's]] ''Clavis Mathematicae'' (1648),<ref>{{cite book |url=https://archive.org/details/bub_gb_ddMxgr27tNkC |title=Clavis Mathematicæ |last=Oughtred |author-link=William Oughtred |first=William |date=1648 |publisher=Thomas Harper |location=London |page=[https://archive.org/details/bub_gb_ddMxgr27tNkC/page/n220 69] |language=la |trans-title=The key to mathematics}} (English translation: {{Cite book |url=https://books.google.com/books?id=S50yAQAAMAAJ&pg=PA99 |title=Key of the Mathematics |last=Oughtred |first=William |author-link=William Oughtred |date=1694 |publisher=J. Salusbury |language=en}})</ref>{{sfn|Arndt|Haenel|2006|p=166}}
where the [[Greek letters]] {{pi}} and ''δ'' were combined into the fraction {{tmath|\tfrac \pi \delta}} for denoting the ratios [[semiperimeter]] to [[semidiameter]] and perimeter to diameter, that is, what is  presently denoted as {{pi}}.<ref name=firstPi>{{Cite book |url=https://books.google.com/books?id=KTgPAAAAQAAJ&pg=PP3 |title=Theorematum in libris Archimedis de sphaera et cylindro declarario |last=Oughtred |first=William |author-link=William Oughtred |date=1652 |publisher=Excudebat L. Lichfield, Veneunt apud T. Robinson |language=la |quote={{math|''δ''.''π''}} :: semidiameter. semiperipheria}}</ref><ref name="Cajori-2007">{{Cite book |url=https://books.google.com/books?id=bT5suOONXlgC&pg=PA9 |title=A History of Mathematical Notations: Vol. II |last=Cajori |first=Florian |author-link=Florian Cajori |date=2007 |publisher=Cosimo, Inc. |isbn=978-1-60206-714-1 |pages=8–13 |language=en |quote=the ratio of the length of a circle to its diameter was represented in the fractional form by the use of two letters ... J.A. Segner ... in 1767, he represented {{math|3.14159...}} by {{math|''δ'':''π''}}, as did Oughtred more than a century earlier}}</ref><ref name="Smith-1958">{{Cite book |url=https://books.google.com/books?id=uTytJGnTf1kC&pg=PA312 |title=History of Mathematics |last=Smith |first=David E. |author-link=David Eugene Smith |date=1958 |publisher=Courier Corporation |isbn=978-0-486-20430-7 |page=312 |language=en}}</ref><ref>{{Cite journal |last=Archibald |first=R. C. |author-link=Raymond C. Archibald |date=1921 |title=Historical Notes on the Relation {{math |1=''e''<sup>−(''π''/2)</sup> = ''i''<sup>''i''</sup>}} |jstor=2972388 |journal=The American Mathematical Monthly |volume=28 |issue=3 |pages=116–121 |doi=10.2307/2972388 |quote=It is noticeable that these letters are ''never'' used separately, that is, {{pi}} is ''not'' used for 'Semiperipheria'}}</ref> (Before then, mathematicians sometimes used letters such as ''c'' or ''p'' instead.{{sfn|Arndt|Haenel|2006|p=166}}) [[Isaac Barrow|Barrow]] likewise used the same notation,<ref>{{Cite book |chapter-url=https://archive.org/stream/mathematicalwor00whewgoog#page/n405/mode/1up |title=The mathematical works of Isaac Barrow |last=Barrow |first=Isaac |author-link=Isaac Barrow |date=1860 |publisher=Cambridge University press |others=Harvard University |editor-last=Whewell |editor-first=William |page=381 |language=la |chapter=Lecture XXIV}}</ref> while [[David Gregory (mathematician)|Gregory]] instead used <math display=inline>\frac \pi \rho</math> to represent {{math|6.28...&thinsp;}}.<ref>{{Cite journal |last=Gregorius |first=David |date=1695 |title=Ad Reverendum Virum D. Henricum Aldrich S.T.T. Decanum Aedis Christi Oxoniae |jstor=102382 |journal=Philosophical Transactions |language=la |volume=19 |issue=231 |pages=637–652 |doi=10.1098/rstl.1695.0114 |bibcode=1695RSPT...19..637G |doi-access=free |url=https://archive.org/download/crossref-pre-1909-scholarly-works/10.1098%252Frstl.1684.0084.zip/10.1098%252Frstl.1695.0114.pdf}}</ref>{{r|Smith-1958}}

The earliest known use of the Greek letter {{pi}} alone to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician [[William Jones (mathematician)|William Jones]] in his 1706 work ''{{lang|la|Synopsis Palmariorum Matheseos|italic=unset}}; or, a New Introduction to the Mathematics''.{{r|jones}}{{sfn|Arndt|Haenel|2006|p=165|ps=: A facsimile of Jones' text is in {{harvnb|Berggren|Borwein|Borwein|1997|pp=108–109}}.}} The Greek letter appears on p.&nbsp;243 in the phrase "<math display=inline>\tfrac12</math> Periphery ({{pi}})", calculated for a circle with radius one. However, Jones writes that his equations for {{pi}} are from the "ready pen of the truly ingenious Mr. [[John Machin]]", leading to speculation that Machin may have employed the Greek letter before Jones.{{sfn|Arndt|Haenel|2006|p=166}} Jones' notation was not immediately adopted by other mathematicians, with the fraction notation still being used as late as 1767.{{r|Cajori-2007}}<ref>{{Cite book |url=https://books.google.com/books?id=NmYVAAAAQAAJ&pg=PA282 |title=Cursus Mathematicus |last=Segner |first=Joannes Andreas |date=1756 |publisher=Halae Magdeburgicae |page=282 |language=la |access-date=15 October 2017 |archive-url=https://web.archive.org/web/20171015150340/https://books.google.co.uk/books?id=NmYVAAAAQAAJ&pg=PA282 |archive-date=15 October 2017 |url-status=live}}</ref>

[[Euler]] started using the single-letter form beginning with his 1727 ''Essay Explaining the Properties of Air'', though he used {{math|1=''π'' = 6.28...}}, the ratio of periphery to radius, in this and some later writing.<ref>{{Cite journal |last=Euler |first=Leonhard |author-link=Leonhard Euler |date=1727 |title=Tentamen explicationis phaenomenorum aeris |url=http://eulerarchive.maa.org/docs/originals/E007.pdf#page=5 |journal=Commentarii Academiae Scientiarum Imperialis Petropolitana |language=la |volume=2 |page=351 |id=[http://eulerarchive.maa.org/pages/E007.html E007] |quote=Sumatur pro ratione radii ad peripheriem, {{math|I : π}} |access-date=15 October 2017 |archive-url=https://web.archive.org/web/20160401072718/http://eulerarchive.maa.org/docs/originals/E007.pdf#page=5 |archive-date=1 April 2016 |url-status=live}} [http://www.17centurymaths.com/contents/euler/e007tr.pdf#page=3 English translation by Ian Bruce] {{Webarchive|url=https://web.archive.org/web/20160610172054/http://www.17centurymaths.com/contents/euler/e007tr.pdf#page=3 |date=10 June 2016 }}: "{{mvar|π}} is taken for the ratio of the radius to the periphery [note that in this work, Euler's {{pi}} is double our {{pi}}.]" {{pb}} {{Cite book |url=https://books.google.com/books?id=3C1iHFBXVEcC&pg=PA139 |title=Lettres inédites d'Euler à d'Alembert |last=Euler |first=Leonhard |author-link=Leonhard Euler |series=Bullettino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche |year=1747 |editor-last=Henry |editor-first=Charles |volume=19 |publication-date=1886 |page=139 |language=fr |id=[http://eulerarchive.maa.org/pages/E858.html E858] |quote=Car, soit π la circonference d'un cercle, dout le rayon est {{math|{{=}} 1}}}} English translation in {{Cite journal |last=Cajori |first=Florian |author-link=Florian Cajori |date=1913 |title=History of the Exponential and Logarithmic Concepts |jstor=2973441 |journal=The American Mathematical Monthly |volume=20 |issue=3 |pages=75–84 |doi=10.2307/2973441 |quote=Letting {{pi}} be the circumference (!) of a circle of unit radius}}</ref>   Euler first used {{nowrap|1={{pi}} = 3.14...}} in his 1736 work ''[[Mechanica]]'',<ref>{{Cite book |last=Euler |first=Leonhard |author-link=Leonhard Euler |title=Mechanica sive motus scientia analytice exposita. (cum tabulis) |date=1736 |publisher=Academiae scientiarum Petropoli |volume=1 |page=113 |language=la |chapter=Ch. 3 Prop. 34 Cor. 1 |id=[http://eulerarchive.maa.org/pages/E015.html E015] |quote=Denotet {{math|1 : ''π''}} rationem diametri ad peripheriam |chapter-url=https://books.google.com/books?id=jgdTAAAAcAAJ&pg=PA113}} [http://www.17centurymaths.com/contents/euler/mechvol1/ch3a.pdf#page=26 English translation by Ian Bruce] {{Webarchive|url=https://web.archive.org/web/20160610183753/http://www.17centurymaths.com/contents/euler/mechvol1/ch3a.pdf#page=26|date=10 June 2016}} : "Let {{math|1 : ''π''}} denote the ratio of the diameter to the circumference"</ref> and continued in his widely read 1748 work {{lang|la|[[Introductio in analysin infinitorum]]|italic=yes}} (he wrote: "for the sake of brevity we will write this number as {{pi}}; thus {{pi}} is equal to half the circumference of a circle of radius {{math|1}}").<ref>{{Cite book |url=http://gallica.bnf.fr/ark:/12148/bpt6k69587/f155 |title=Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus / ediderunt Adolf Krazer et Ferdinand Rudio |last=Euler |first=Leonhard |date=1922 |author-link=Leonhard Euler |publisher=B. G. Teubneri |location=Lipsae |pages=133–134 |language=la |id=[http://eulerarchive.maa.org/pages/E101.html E101] |access-date=15 October 2017 |archive-url=https://web.archive.org/web/20171016022758/http://gallica.bnf.fr/ark:/12148/bpt6k69587/f155 |archive-date=16 October 2017 |url-status=live}}</ref> Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly, and the practice was universally adopted thereafter in the [[Western world]],{{sfn|Arndt|Haenel|2006|p=166}} though the definition still varied between {{math|3.14...}} and {{math|6.28...}} as late as 1761.<ref>{{Cite book |url=https://books.google.com/books?id=P-hEAAAAcAAJ&pg=PA374 |title=Cursus Mathematicus: Elementorum Analyseos Infinitorum Elementorum Analyseos Infinitorvm |last=Segner |first=Johann Andreas von |date=1761 |publisher=Renger |page=374 |language=la |quote=Si autem {{pi}} notet peripheriam circuli, cuius diameter eſt {{math|2}}}}</ref>