Approximations of π (English Wikipedia)

See #Imputed biblical value. Beckmann 1971 "There has been concern over the apparent biblical statement of $π$ ≈ 3 from the early times of rabbinical Judaism, addressed by Rabbi Nehemiah in the 2nd century."^{[page needed]} Beckmann, Petr (1971). A History of $π$ . New York: St. Martin's Press. ISBN 978-0-88029-418-8. MR 0449960.
See also Beckmann 1971, pp. 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. $π$ = 3/0.96 = 25/8]". Beckmann, Petr (1971). A History of $π$ . New York: St. Martin's Press. ISBN 978-0-88029-418-8. MR 0449960.
Lam, Lay Yong; Ang, Tian Se (1986), "Circle measurements in ancient China", Historia Mathematica, 13 (4): 325–340, doi:10.1016/0315-0860(86)90055-8, MR 0875525. Reprinted in Berggren, J. L.; Borwein, Jonathan M.; Borwein, Peter, eds. (2004). Pi: A Source Book. Springer. pp. 20–35. ISBN 978-0387205717.. See in particular pp. 333–334 (pp. 28–29 of the reprint).
Beckmann 1971, p. 66 Beckmann, Petr (1971). A History of $π$ . New York: St. Martin's Press. ISBN 978-0-88029-418-8. MR 0449960.
Beckmann 1971, pp. 94–95 Beckmann, Petr (1971). A History of $π$ . New York: St. Martin's Press. ISBN 978-0-88029-418-8. MR 0449960.
Ramanujan, S. (1914). "Modular equations and approximations to $π$ ". Quarterly Journal of Mathematics. 45: 350–372. Reprinted in Berggren, Lennart; Borwein, Jonathan; Borwein, Peter (2004). "Modular equations and approximations to $π$ ". Pi: A Source Book (3rd ed.). New York: Springer-Verlag. pp. 241–257. doi:10.1007/978-1-4757-4217-6_29. ISBN 0-387-20571-3. MR 2065455.

ams.org

Capra, B. "Digits of Pi" (PDF). Retrieved 13 January 2018.

archive.org (Global: 6^th place; English: 6^th place)

Jones, William (1706). Synopsis Palmariorum Matheseos. London: J. Wale. pp. 243, 263. 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
${\overline {{\tfrac {16}{5}}-{\tfrac {4}{239}}}}-{\tfrac {1}{3}}{\overline {{\tfrac {16}{5^{3}}}-{\tfrac {4}{239^{3}}}}}+{\tfrac {1}{5}}{\overline {{\tfrac {16}{5^{5}}}-{\tfrac {4}{239^{5}}}}}-,\,\&c.=$
$3.14159, & c. = π$ . 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, Van Ceulen's Number, or that in Art. 64.38. may be Examin'd with all desireable Ease and Dispatch.
Reprinted in Smith, David Eugene (1929). "William Jones: The First Use of $π$ for the Circle Ratio". A Source Book in Mathematics. McGraw–Hill. pp. 346–347.
Vega, Géorge (1795) [1789]. "Detérmination de la demi-circonférence d'un cercle dont le diameter est $= 1$ , exprimée en $140$ figures decimals". Supplement. Nova Acta Academiae Scientiarum Petropolitanae. 11: 41–44.
Sandifer, Edward (2006). "Why 140 Digits of Pi Matter" (PDF). Jurij baron Vega in njegov čas: Zbornik ob 250-letnici rojstva [Baron Jurij Vega and His Times: Celebrating 250 Years]. Ljubljana: DMFA. ISBN 978-961-6137-98-0. LCCN 2008467244. OCLC 448882242. Archived from the original (PDF) on 28 August 2006. We should note that Vega's value contains an error in the 127th digit. Vega gives a 4 where there should be an [6], and all digits after that are incorrect.
Shanks, William (1853). Contributions to Mathematics: Comprising Chiefly the Rectification of the Circle to 607 Places of Decimals. Macmillan Publishers. p. viii – via the Internet Archive.
Unpublished work by Newton (1684), later independently discovered by others, and popularized by Euler (1755).
Roy, Ranjan (2021) [1st ed. 2011]. Series and Products in the Development of Mathematics. Vol. 1 (2 ed.). Cambridge University Press. pp. 215–216, 219–220.

Sandifer, Ed (2009). "Estimating π" (PDF). How Euler Did It. Reprinted in How Euler Did Even More. Mathematical Association of America. 2014. pp. 109–118.

Newton, Isaac (1971). Whiteside, Derek Thomas (ed.). The Mathematical Papers of Isaac Newton. Vol. 4, 1674–1684. Cambridge University Press. pp. 526–653.

Euler, Leonhard (1755). "§2.30". Institutiones Calculi Differentialis (in Latin). Academiae Imperialis Scientiarium Petropolitanae. p. 318. E 212.

Euler, Leonhard (1798) [written 1779]. "Investigatio quarundam serierum, quae ad rationem peripheriae circuli ad diametrum vero proxime definiendam maxime sunt accommodatae". Nova Acta Academiae Scientiarum Petropolitinae. 11: 133–149, 167–168. E 705.

Hwang Chien-Lih (2005), "An elementary derivation of Euler's series for the arctangent function", The Mathematical Gazette, 89 (516): 469–470, doi:10.1017/S0025557200178404, S2CID 123395287

arxiv.org (Global: 69^th place; English: 59^th place)

Damini, D.B.; Abhishek, Dhar (2020). "How Archimedes showed that π is approximately equal to 22/7". p. 8. arXiv:2008.07995 [math.HO].
Treub, Peter (30 November 2016). "Digit Statistics of the First 22.4 Trillion Decimal Digits of Pi". arXiv:1612.00489 [math.NT].
Trueb, Peter (2020). The Borwein brothers, Pi and the AGM. Springer Proceedings in Mathematics & Statistics. Vol. 313. arXiv:1802.07558. doi:10.1007/978-3-030-36568-4. ISBN 978-3-030-36567-7. S2CID 214742997.
Plouffe, Simon (2009). "On the computation of the n^th decimal digit of various transcendental numbers". arXiv:0912.0303v1 [math.NT].

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Tsaban, Boaz; Garber, David (February 1998). "On the rabbinical approximation of $π$ " (PDF). Historia Mathematica. 25 (1): 75–84. doi:10.1006/hmat.1997.2185. ISSN 0315-0860. Retrieved 14 July 2009.

books.google.com (Global: 3^rd place; English: 3^rd place)

Romano, David Gilman (1993). Athletics and Mathematics in Archaic Corinth: The Origins of the Greek Stadion. American Philosophical Society. p. 78. ISBN 978-0871692061. 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 $π$ was 3 1/8 or 3.125.
Chaitanya, Krishna. A profile of Indian culture. Indian Book Company (1975). p. 133.
Lam, Lay Yong; Ang, Tian Se (1986), "Circle measurements in ancient China", Historia Mathematica, 13 (4): 325–340, doi:10.1016/0315-0860(86)90055-8, MR 0875525. Reprinted in Berggren, J. L.; Borwein, Jonathan M.; Borwein, Peter, eds. (2004). Pi: A Source Book. Springer. pp. 20–35. ISBN 978-0387205717.. See in particular pp. 333–334 (pp. 28–29 of the reprint).

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"Even more pi in the sky: Calculating 100 trillion digits of pi on Google Cloud". Google Cloud Platform. 8 June 2022. Retrieved 10 June 2022.

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doi.org (Global: 2^nd place; English: 2^nd place)

Hayes, Brian (September 2014). "Pencil, Paper, and Pi". American Scientist. Vol. 102, no. 5. p. 342. doi:10.1511/2014.110.342.
Jadhav, Dipak (1 January 2018). "On The Value Implied in the Data Referred To in the Mahābhārata for π". Vidyottama Sanatana: International Journal of Hindu Science and Religious Studies. 2 (1): 18. doi:10.25078/ijhsrs.v2i1.511. ISSN 2550-0651. S2CID 146074061.
Lam, Lay Yong; Ang, Tian Se (1986), "Circle measurements in ancient China", Historia Mathematica, 13 (4): 325–340, doi:10.1016/0315-0860(86)90055-8, MR 0875525. Reprinted in Berggren, J. L.; Borwein, Jonathan M.; Borwein, Peter, eds. (2004). Pi: A Source Book. Springer. pp. 20–35. ISBN 978-0387205717.. See in particular pp. 333–334 (pp. 28–29 of the reprint).
Azarian, Mohammad K. (2010). "al-Risāla al-muhītīyya: A Summary". Missouri Journal of Mathematical Sciences. 22 (2): 64–85. doi:10.35834/mjms/1312233136.
Tweddle, Ian (1991). "John Machin and Robert Simson on Inverse-tangent Series for $π$ ". Archive for History of Exact Sciences. 42 (1): 1–14. doi:10.1007/BF00384331. JSTOR 41133896. S2CID 121087222.
Ferguson, D. F. (16 March 1946). "Value of π". Nature. 157 (3985): 342. Bibcode:1946Natur.157..342F. doi:10.1038/157342c0. ISSN 1476-4687. S2CID 4085398.
Shanks, William (1873). "V. On the extension of the numerical value of π". Proceedings of the Royal Society of London. 21 (139–147). Royal Society Publishing: 318–319. doi:10.1098/rspl.1872.0066. S2CID 120851313.
Ferguson 1946a, doi:10.2307/3608485
Shanks, D.; Wrench, J. W. Jr. (1962). "Calculation of $π$ to 100,000 decimals". Mathematics of Computation. 16 (77): 76–99. doi:10.2307/2003813. JSTOR 2003813.
Hallerberg, Arthur E. (1977). "Indiana's Squared Circle". Mathematics Magazine. 50 (3): 136–140. doi:10.1080/0025570X.1977.11976632.
Tsaban, Boaz; Garber, David (February 1998). "On the rabbinical approximation of $π$ " (PDF). Historia Mathematica. 25 (1): 75–84. doi:10.1006/hmat.1997.2185. ISSN 0315-0860. Retrieved 14 July 2009.
Unpublished work by Newton (1684), later independently discovered by others, and popularized by Euler (1755).
Roy, Ranjan (2021) [1st ed. 2011]. Series and Products in the Development of Mathematics. Vol. 1 (2 ed.). Cambridge University Press. pp. 215–216, 219–220.

Sandifer, Ed (2009). "Estimating π" (PDF). How Euler Did It. Reprinted in How Euler Did Even More. Mathematical Association of America. 2014. pp. 109–118.

Newton, Isaac (1971). Whiteside, Derek Thomas (ed.). The Mathematical Papers of Isaac Newton. Vol. 4, 1674–1684. Cambridge University Press. pp. 526–653.

Euler, Leonhard (1755). "§2.30". Institutiones Calculi Differentialis (in Latin). Academiae Imperialis Scientiarium Petropolitanae. p. 318. E 212.

Euler, Leonhard (1798) [written 1779]. "Investigatio quarundam serierum, quae ad rationem peripheriae circuli ad diametrum vero proxime definiendam maxime sunt accommodatae". Nova Acta Academiae Scientiarum Petropolitinae. 11: 133–149, 167–168. E 705.

Hwang Chien-Lih (2005), "An elementary derivation of Euler's series for the arctangent function", The Mathematical Gazette, 89 (516): 469–470, doi:10.1017/S0025557200178404, S2CID 123395287
Trueb, Peter (2020). The Borwein brothers, Pi and the AGM. Springer Proceedings in Mathematics & Statistics. Vol. 313. arXiv:1802.07558. doi:10.1007/978-3-030-36568-4. ISBN 978-3-030-36567-7. S2CID 214742997.
Popper, K. R. (August 1952). "The nature of philosophical problems and their roots in science". The British Journal for the Philosophy of Science. 3 (10). University of Chicago Press: 124–156. doi:10.1093/bjps/iii.10.124. JSTOR 685553. See p. 150.
Ramanujan, S. (1914). "Modular equations and approximations to $π$ ". Quarterly Journal of Mathematics. 45: 350–372. Reprinted in Berggren, Lennart; Borwein, Jonathan; Borwein, Peter (2004). "Modular equations and approximations to $π$ ". Pi: A Source Book (3rd ed.). New York: Springer-Verlag. pp. 241–257. doi:10.1007/978-1-4757-4217-6_29. ISBN 0-387-20571-3. MR 2065455.
Dutka, J. (1982). "Wallis's product, Brouncker's continued fraction, and Leibniz's series". Archive for History of Exact Sciences. 26 (2): 115–126. doi:10.1007/BF00348349. S2CID 121628039.
Rabinowitz, Stanley; Wagon, Stan (1995). "A Spigot Algorithm for the Digits of π". The American Mathematical Monthly. 102 (3): 195–203. doi:10.2307/2975006. ISSN 0002-9890. JSTOR 2975006.
Hwang Chien-Lih (2005), "An elementary derivation of Euler's series for the arctangent function", The Mathematical Gazette, 89 (516): 469–470, doi:10.1017/S0025557200178404, S2CID 123395287

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"Die FH Graubünden kennt Pi am genauesten – Weltrekord!". Retrieved 31 August 2021.

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"Most accurate value of pi". Guinness World Records. Archived from the original on 8 May 2025. Retrieved 16 May 2025.

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Ferguson, D. F. (16 March 1946). "Value of π". Nature. 157 (3985): 342. Bibcode:1946Natur.157..342F. doi:10.1038/157342c0. ISSN 1476-4687. S2CID 4085398.

hathitrust.org (Global: 441^st place; English: 311^th place)

babel.hathitrust.org

Ramanujan, S. (1914). "Modular equations and approximations to $π$ ". Quarterly Journal of Mathematics. 45: 350–372. Reprinted in Berggren, Lennart; Borwein, Jonathan; Borwein, Peter (2004). "Modular equations and approximations to $π$ ". Pi: A Source Book (3rd ed.). New York: Springer-Verlag. pp. 241–257. doi:10.1007/978-1-4757-4217-6_29. ISBN 0-387-20571-3. MR 2065455.

ijpam.eu (Global: low place; English: low place)

Chakrabarti, Gopal; Hudson, Richard (2003). "An Improvement of Archimedes Method of Approximating π" (PDF). International Journal of Pure and Applied Mathematics. 7 (2): 207–212.

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jstor.org (Global: 26^th place; English: 20^th place)

Tweddle, Ian (1991). "John Machin and Robert Simson on Inverse-tangent Series for $π$ ". Archive for History of Exact Sciences. 42 (1): 1–14. doi:10.1007/BF00384331. JSTOR 41133896. S2CID 121087222.
Shanks, D.; Wrench, J. W. Jr. (1962). "Calculation of $π$ to 100,000 decimals". Mathematics of Computation. 16 (77): 76–99. doi:10.2307/2003813. JSTOR 2003813.
Popper, K. R. (August 1952). "The nature of philosophical problems and their roots in science". The British Journal for the Philosophy of Science. 3 (10). University of Chicago Press: 124–156. doi:10.1093/bjps/iii.10.124. JSTOR 685553. See p. 150.
Hoffman, David W. (November 2009). "A pi curiosity". College Mathematics Journal. 40 (5). JSTOR 25653799.
Rabinowitz, Stanley; Wagon, Stan (1995). "A Spigot Algorithm for the Digits of π". The American Mathematical Monthly. 102 (3): 195–203. doi:10.2307/2975006. ISSN 0002-9890. JSTOR 2975006.

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How Aryabhata got the earth's circumference right Archived 15 January 2017 at the Wayback Machine

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lccn.loc.gov

Vega, Géorge (1795) [1789]. "Detérmination de la demi-circonférence d'un cercle dont le diameter est $= 1$ , exprimée en $140$ figures decimals". Supplement. Nova Acta Academiae Scientiarum Petropolitanae. 11: 41–44.
Sandifer, Edward (2006). "Why 140 Digits of Pi Matter" (PDF). Jurij baron Vega in njegov čas: Zbornik ob 250-letnici rojstva [Baron Jurij Vega and His Times: Celebrating 250 Years]. Ljubljana: DMFA. ISBN 978-961-6137-98-0. LCCN 2008467244. OCLC 448882242. Archived from the original (PDF) on 28 August 2006. We should note that Vega's value contains an error in the 127th digit. Vega gives a 4 where there should be an [6], and all digits after that are incorrect.

maa.org (Global: 3,479^th place; English: 2,444^th place)

eulerarchive.maa.org

Unpublished work by Newton (1684), later independently discovered by others, and popularized by Euler (1755).
Roy, Ranjan (2021) [1st ed. 2011]. Series and Products in the Development of Mathematics. Vol. 1 (2 ed.). Cambridge University Press. pp. 215–216, 219–220.

Sandifer, Ed (2009). "Estimating π" (PDF). How Euler Did It. Reprinted in How Euler Did Even More. Mathematical Association of America. 2014. pp. 109–118.

Newton, Isaac (1971). Whiteside, Derek Thomas (ed.). The Mathematical Papers of Isaac Newton. Vol. 4, 1674–1684. Cambridge University Press. pp. 526–653.

Euler, Leonhard (1755). "§2.30". Institutiones Calculi Differentialis (in Latin). Academiae Imperialis Scientiarium Petropolitanae. p. 318. E 212.

Euler, Leonhard (1798) [written 1779]. "Investigatio quarundam serierum, quae ad rationem peripheriae circuli ad diametrum vero proxime definiendam maxime sunt accommodatae". Nova Acta Academiae Scientiarum Petropolitinae. 11: 133–149, 167–168. E 705.

Hwang Chien-Lih (2005), "An elementary derivation of Euler's series for the arctangent function", The Mathematical Gazette, 89 (516): 469–470, doi:10.1017/S0025557200178404, S2CID 123395287

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"Pi – 5 Trillion Digits".
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Yee, Alexander J.; Kondo, Shigeru (28 December 2013). "12.1 Trillion Digits of Pi".
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oeis.org (Global: 742^nd place; English: 538^th place)

Sloane, N. J. A. (ed.). "Sequence A002485 (Numerators of convergents to Pi)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
Sloane, N. J. A. (ed.). "Sequence A002486 (Denominators of convergents to Pi)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
For related results see The circle problem: number of points (x,y) in square lattice with x^2 + y^2 <= n.

pacific.edu (Global: low place; English: low place)

scholarlycommons.pacific.edu

Unpublished work by Newton (1684), later independently discovered by others, and popularized by Euler (1755).
Roy, Ranjan (2021) [1st ed. 2011]. Series and Products in the Development of Mathematics. Vol. 1 (2 ed.). Cambridge University Press. pp. 215–216, 219–220.

Sandifer, Ed (2009). "Estimating π" (PDF). How Euler Did It. Reprinted in How Euler Did Even More. Mathematical Association of America. 2014. pp. 109–118.

Newton, Isaac (1971). Whiteside, Derek Thomas (ed.). The Mathematical Papers of Isaac Newton. Vol. 4, 1674–1684. Cambridge University Press. pp. 526–653.

Euler, Leonhard (1755). "§2.30". Institutiones Calculi Differentialis (in Latin). Academiae Imperialis Scientiarium Petropolitanae. p. 318. E 212.

Euler, Leonhard (1798) [written 1779]. "Investigatio quarundam serierum, quae ad rationem peripheriae circuli ad diametrum vero proxime definiendam maxime sunt accommodatae". Nova Acta Academiae Scientiarum Petropolitinae. 11: 133–149, 167–168. E 705.

Hwang Chien-Lih (2005), "An elementary derivation of Euler's series for the arctangent function", The Mathematical Gazette, 89 (516): 469–470, doi:10.1017/S0025557200178404, S2CID 123395287

pi314.net (Global: low place; English: low place)

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Shanks, William (1873). "V. On the extension of the numerical value of π". Proceedings of the Royal Society of London. 21 (139–147). Royal Society Publishing: 318–319. doi:10.1098/rspl.1872.0066. S2CID 120851313.

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api.semanticscholar.org

Jadhav, Dipak (1 January 2018). "On The Value Implied in the Data Referred To in the Mahābhārata for π". Vidyottama Sanatana: International Journal of Hindu Science and Religious Studies. 2 (1): 18. doi:10.25078/ijhsrs.v2i1.511. ISSN 2550-0651. S2CID 146074061.
Tweddle, Ian (1991). "John Machin and Robert Simson on Inverse-tangent Series for $π$ ". Archive for History of Exact Sciences. 42 (1): 1–14. doi:10.1007/BF00384331. JSTOR 41133896. S2CID 121087222.
Ferguson, D. F. (16 March 1946). "Value of π". Nature. 157 (3985): 342. Bibcode:1946Natur.157..342F. doi:10.1038/157342c0. ISSN 1476-4687. S2CID 4085398.
Shanks, William (1873). "V. On the extension of the numerical value of π". Proceedings of the Royal Society of London. 21 (139–147). Royal Society Publishing: 318–319. doi:10.1098/rspl.1872.0066. S2CID 120851313.
Unpublished work by Newton (1684), later independently discovered by others, and popularized by Euler (1755).
Roy, Ranjan (2021) [1st ed. 2011]. Series and Products in the Development of Mathematics. Vol. 1 (2 ed.). Cambridge University Press. pp. 215–216, 219–220.

Sandifer, Ed (2009). "Estimating π" (PDF). How Euler Did It. Reprinted in How Euler Did Even More. Mathematical Association of America. 2014. pp. 109–118.

Newton, Isaac (1971). Whiteside, Derek Thomas (ed.). The Mathematical Papers of Isaac Newton. Vol. 4, 1674–1684. Cambridge University Press. pp. 526–653.

Euler, Leonhard (1755). "§2.30". Institutiones Calculi Differentialis (in Latin). Academiae Imperialis Scientiarium Petropolitanae. p. 318. E 212.

Euler, Leonhard (1798) [written 1779]. "Investigatio quarundam serierum, quae ad rationem peripheriae circuli ad diametrum vero proxime definiendam maxime sunt accommodatae". Nova Acta Academiae Scientiarum Petropolitinae. 11: 133–149, 167–168. E 705.

Hwang Chien-Lih (2005), "An elementary derivation of Euler's series for the arctangent function", The Mathematical Gazette, 89 (516): 469–470, doi:10.1017/S0025557200178404, S2CID 123395287
Trueb, Peter (2020). The Borwein brothers, Pi and the AGM. Springer Proceedings in Mathematics & Statistics. Vol. 313. arXiv:1802.07558. doi:10.1007/978-3-030-36568-4. ISBN 978-3-030-36567-7. S2CID 214742997.
Dutka, J. (1982). "Wallis's product, Brouncker's continued fraction, and Leibniz's series". Archive for History of Exact Sciences. 26 (2): 115–126. doi:10.1007/BF00348349. S2CID 121628039.
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Vega, Géorge (1795) [1789]. "Detérmination de la demi-circonférence d'un cercle dont le diameter est $= 1$ , exprimée en $140$ figures decimals". Supplement. Nova Acta Academiae Scientiarum Petropolitanae. 11: 41–44.
Sandifer, Edward (2006). "Why 140 Digits of Pi Matter" (PDF). Jurij baron Vega in njegov čas: Zbornik ob 250-letnici rojstva [Baron Jurij Vega and His Times: Celebrating 250 Years]. Ljubljana: DMFA. ISBN 978-961-6137-98-0. LCCN 2008467244. OCLC 448882242. Archived from the original (PDF) on 28 August 2006. We should note that Vega's value contains an error in the 127th digit. Vega gives a 4 where there should be an [6], and all digits after that are incorrect.

web.archive.org (Global: 1^st place; English: 1^st place)

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Vega, Géorge (1795) [1789]. "Detérmination de la demi-circonférence d'un cercle dont le diameter est $= 1$ , exprimée en $140$ figures decimals". Supplement. Nova Acta Academiae Scientiarum Petropolitanae. 11: 41–44.
Sandifer, Edward (2006). "Why 140 Digits of Pi Matter" (PDF). Jurij baron Vega in njegov čas: Zbornik ob 250-letnici rojstva [Baron Jurij Vega and His Times: Celebrating 250 Years]. Ljubljana: DMFA. ISBN 978-961-6137-98-0. LCCN 2008467244. OCLC 448882242. Archived from the original (PDF) on 28 August 2006. We should note that Vega's value contains an error in the 127th digit. Vega gives a 4 where there should be an [6], and all digits after that are incorrect.
"Announcement at the Kanada lab web site". Super-computing.org. Archived from the original on 12 March 2011. Retrieved 11 December 2017.
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wired.com (Global: 193^rd place; English: 152^nd place)

Allain, Rhett (18 March 2011). "What is the Best Fractional Representation of Pi". Wired. Retrieved 16 March 2020.

wolfram.com (Global: 513^th place; English: 537^th place)

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Other representations are available at The Wolfram Functions Site.

mathworld.wolfram.com

Weisstein, Eric W. "BBP Formula". MathWorld.

worldcat.org (Global: 5^th place; English: 5^th place)

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Jadhav, Dipak (1 January 2018). "On The Value Implied in the Data Referred To in the Mahābhārata for π". Vidyottama Sanatana: International Journal of Hindu Science and Religious Studies. 2 (1): 18. doi:10.25078/ijhsrs.v2i1.511. ISSN 2550-0651. S2CID 146074061.
Vega, Géorge (1795) [1789]. "Detérmination de la demi-circonférence d'un cercle dont le diameter est $= 1$ , exprimée en $140$ figures decimals". Supplement. Nova Acta Academiae Scientiarum Petropolitanae. 11: 41–44.
Sandifer, Edward (2006). "Why 140 Digits of Pi Matter" (PDF). Jurij baron Vega in njegov čas: Zbornik ob 250-letnici rojstva [Baron Jurij Vega and His Times: Celebrating 250 Years]. Ljubljana: DMFA. ISBN 978-961-6137-98-0. LCCN 2008467244. OCLC 448882242. Archived from the original (PDF) on 28 August 2006. We should note that Vega's value contains an error in the 127th digit. Vega gives a 4 where there should be an [6], and all digits after that are incorrect.
Ferguson, D. F. (16 March 1946). "Value of π". Nature. 157 (3985): 342. Bibcode:1946Natur.157..342F. doi:10.1038/157342c0. ISSN 1476-4687. S2CID 4085398.
Tsaban, Boaz; Garber, David (February 1998). "On the rabbinical approximation of $π$ " (PDF). Historia Mathematica. 25 (1): 75–84. doi:10.1006/hmat.1997.2185. ISSN 0315-0860. Retrieved 14 July 2009.
Rabinowitz, Stanley; Wagon, Stan (1995). "A Spigot Algorithm for the Digits of π". The American Mathematical Monthly. 102 (3): 195–203. doi:10.2307/2975006. ISSN 0002-9890. JSTOR 2975006.

Approximations of π (English Wikipedia)

academia.edu (Global: 121st place; English: 142nd place)

americanscientist.org (Global: 7,231st place; English: 5,635th place)

ams.org (Global: 451st place; English: 277th place)

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archive.org (Global: 6th place; English: 6th place)

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cloud.google.com (Global: 9,964th place; English: 6,992nd place)

demon.co.uk (Global: 3,292nd place; English: 2,192nd place)

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doi.org (Global: 2nd place; English: 2nd place)

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fhgr.ch (Global: low place; English: low place)

free.fr (Global: 321st place; English: 724th place)

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harvard.edu (Global: 18th place; English: 17th place)

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hathitrust.org (Global: 441st place; English: 311th place)

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illinois.edu (Global: 1,349th place; English: 866th place)

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jstor.org (Global: 26th place; English: 20th place)

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lbl.gov (Global: 2,509th place; English: 2,329th place)

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livemint.com (Global: 390th place; English: 227th place)

loc.gov (Global: 70th place; English: 63rd place)

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maa.org (Global: 3,479th place; English: 2,444th place)

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mathforum.org (Global: low place; English: low place)

mschneider.cc (Global: low place; English: low place)

newscientist.com (Global: 774th place; English: 716th place)

northwestern.edu (Global: 2,302nd place; English: 1,389th place)

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numberworld.org (Global: low place; English: low place)

oeis.org (Global: 742nd place; English: 538th place)

pacific.edu (Global: low place; English: low place)

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pi314.net (Global: low place; English: low place)

projecteuclid.org (Global: 3,707th place; English: 2,409th place)

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spektrum.de (Global: 1,960th place; English: low place)

st-and.ac.uk (Global: 3,503rd place; English: 3,612th place)

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st-andrews.ac.uk (Global: 1,547th place; English: 1,410th place)

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stackexchange.com (Global: 1,983rd place; English: 1,330th place)

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storagereview.com (Global: low place; English: low place)

super-computing.org (Global: low place; English: low place)

theguardian.com (Global: 12th place; English: 11th place)

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blog.timothymullican.com

u-tokyo.ac.jp (Global: 1,856th place; English: 4,678th place)

pi2.cc.u-tokyo.ac.jp

uz.ac.zw (Global: low place; English: low place)

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wcsu.edu (Global: low place; English: low place)

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web.archive.org (Global: 1st place; English: 1st place)

wired.com (Global: 193rd place; English: 152nd place)

wolfram.com (Global: 513th place; English: 537th place)

academia.edu (Global: 121^st place; English: 142^nd place)

americanscientist.org (Global: 7,231^st place; English: 5,635^th place)

ams.org (Global: 451^st place; English: 277^th place)

archive.org (Global: 6^th place; English: 6^th place)

arxiv.org (Global: 69^th place; English: 59^th place)

biu.ac.il (Global: 9,276^th place; English: 9,572^nd place)

books.google.com (Global: 3^rd place; English: 3^rd place)

britannica.com (Global: 40^th place; English: 58^th place)

cloud.google.com (Global: 9,964^th place; English: 6,992^nd place)

demon.co.uk (Global: 3,292^nd place; English: 2,192^nd place)

doi.org (Global: 2^nd place; English: 2^nd place)

extremetech.com (Global: 4,667^th place; English: 3,322^nd place)

free.fr (Global: 321^st place; English: 724^th place)

guinnessworldrecords.com (Global: 550^th place; English: 453^rd place)

gutenberg.org (Global: 489^th place; English: 377^th place)

harvard.edu (Global: 18^th place; English: 17^th place)

hathitrust.org (Global: 441^st place; English: 311^th place)

illinois.edu (Global: 1,349^th place; English: 866^th place)

jstor.org (Global: 26^th place; English: 20^th place)

knaw.nl (Global: 2,678^th place; English: 2,423^rd place)

lbl.gov (Global: 2,509^th place; English: 2,329^th place)

livemint.com (Global: 390^th place; English: 227^th place)

loc.gov (Global: 70^th place; English: 63^rd place)

maa.org (Global: 3,479^th place; English: 2,444^th place)

newscientist.com (Global: 774^th place; English: 716^th place)

northwestern.edu (Global: 2,302^nd place; English: 1,389^th place)

oeis.org (Global: 742^nd place; English: 538^th place)

projecteuclid.org (Global: 3,707^th place; English: 2,409^th place)

royalsocietypublishing.org (Global: 1,993^rd place; English: 3,231^st place)

semanticscholar.org (Global: 11^th place; English: 8^th place)

spektrum.de (Global: 1,960^th place; English: low place)

st-and.ac.uk (Global: 3,503^rd place; English: 3,612^th place)

st-andrews.ac.uk (Global: 1,547^th place; English: 1,410^th place)

stackexchange.com (Global: 1,983^rd place; English: 1,330^th place)

theguardian.com (Global: 12^th place; English: 11^th place)

u-tokyo.ac.jp (Global: 1,856^th place; English: 4,678^th place)

web.archive.org (Global: 1^st place; English: 1^st place)

wired.com (Global: 193^rd place; English: 152^nd place)

wolfram.com (Global: 513^th place; English: 537^th place)

worldcat.org (Global: 5^th place; English: 5^th place)