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Penghentian pecahan kontinu pada titik pembagian manapun akan memberikan nilai pendekatan {{pi}}; dua pecahan 22/7 dan 355/113 secara historis digunakan sebagai pendekatan terhadap {{pi}}. Walauapun pecahan kontinu yang sederhana (seperti pada contoh di atas) untuk {{pi}} tidak memiliki pola-pola tertentu,<ref name="ReferenceA">
{{SloanesRef|sequencenumber=A001203|name=Continued fraction for Pi}} Retrieved 12 April 2012.</ref> matematikawan telah menemukan beberapa pecahan kontinu generalisasi yang memiliki pola tertentu, misalnya:<ref>{{cite journal|title=An Elegant Continued Fraction for {{pi}}|url=https://archive.org/details/sim_american-mathematical-monthly_1999-05_106_5/page/456|first=L. J.|last=Lange|journal=[[The American Mathematical Monthly]]|volume=106|issue=5|month=May|year=1999|pages=456–458|jstor=2589152|doi=10.2307/2589152|ref=harv}}</ref>
:<math>\pi=\textstyle \cfrac{4}{1+\textstyle \frac{1^2}{2+\textstyle \frac{3^2}{2+\textstyle \frac{5^2}{2+\textstyle \frac{7^2}{2+\textstyle \frac{9^2}{2+\ddots}}}}}}
=3+\textstyle \frac{1^2}{6+\textstyle \frac{3^2}{6+\textstyle \frac{5^2}{6+\textstyle \frac{7^2}{6+\textstyle \frac{9^2}{6+\ddots}}}}}
Baris 101:
Beberapa deret tak hingga untuk {{pi}} berkonvergen lebih cepat daripada yang lainnya. Matematikawan biasanya akan menggunakan deret yang lebih cepat berkonvergen untuk menghemat waktu sampai dengan tingkat akurasi tertentu.<ref name="Aconverge">{{cite journal|last=Borwein|first=J. M.|last2=Borwein|first2=P. B.|title=Ramanujan and Pi|year=1988|journal=Scientific American|volume=256|issue=2|pages=112–117|ref=harv|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> Deret tak terhingga untuk {{pi}} yang sederhana misalnya [[rumus Leibniz untuk π|deret Gregory-Leibniz]]:<ref>{{harvnb|Arndt|Haenel|2006|pp=69–72}}</ref>
:<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>
akan perlahan-lahan mendekati {{pi}}. Nilainya berkonvergen sangat lambat. Sampai dengan suku ke 500.000, deret ini hanya menghasilkan lima digit desimal yang benar untuk {{pi}}.<ref>{{cite journal|last=Borwein|first=J. M.|last2=Borwein|first2=P. B.|last3=Dilcher|first3=K.|year=1989|title=Pi, Euler Numbers, and Asymptotic Expansions|url=https://archive.org/details/sim_american-mathematical-monthly_1989-10_96_8/page/681|journal=American Mathematical Monthly|volume=96|issue=8|pages=681–687|doi=10.2307/2324715|ref=harv }}</ref>
Deret yang lebih cepat berkonvergen adalah (digunakan oleh Nilakantha pada abad ke-15):<ref>{{harvnb|Arndt|Haenel|2006|p=223}}</ref><ref group="n">(formula 16.10). Perhatikan bahwa (''n'' − 1)''n''(''n'' + 1) = ''n''<sup>3</sup> − ''n''.</ref><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=978-0-140-26149-3|ref=harv}}</ref>
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|alt2=Ratusan noktah secara acak menutupi suatu persegi dan suatu lingkaran yang disisipkan dalam persegi.
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[[Metode Monte Carlo]], yang mengevaluasi hasil dari banyak percobaan acak, dapat digunakan untuk membuat aproksimasi {{pi}}.<ref>{{harvnb|Arndt|Haenel|2006|p=39}}</ref> [[Jarum Buffon]] adalah salah satu tekniknya: Jika sebuah jarum dengan panjang {{math|''ℓ''}} dijatuhkan {{math|''n''}} kali di atas permukaan yang di atasnya digambar garis paralel yang dipisahkan sebesar {{math|''t''}} satuan, dan jika dari {{math|''x''}} kali ia jatuh melintasi garis ({{math|''x''}} > 0), maka aproksimasi {{pi}} dapat ditentukan berdasarkan perhitungan:<ref name="bn">{{cite journal|last=Ramaley|first=J. F.|title=Buffon's Noodle Problem|url=https://archive.org/details/sim_american-mathematical-monthly_1969-10_76_8/page/916|jstor=2317945|journal=The American Mathematical Monthly|volume=76|issue=8|date=October 1969|pages=916–918|doi=10.2307/2317945|ref=harv}}</ref>
: <math>\pi \approx \frac{2n\ell}{xt}</math>
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Under ideal conditions (uniform gentle slope on an homogeneously erodible substrate), the [[sinuosity]] of a [[meander]]ing river approaches {{pi}}. The sinuosity is the ratio between the actual length and the straight-line distance from source to mouth. Faster currents along the outside edges of a river's bends cause more erosion than along the inside edges, thus pushing the bends even farther out, and increasing the overall loopiness of the river. However, that loopiness eventually causes the river to double back on itself in places and "short-circuit", creating an [[ox-bow lake]] in the process. The balance between these two opposing factors leads to an average ratio of {{pi}} between the actual length and the direct distance between source and mouth.<ref>{{cite journal|journal=[[Science (journal)|Science]]|volume=271|issue=5256|date=22 March 1996|pages=1710–1713|doi=10.1126/science.271.5256.1710|title=River Meandering as a Self-Organization Process|url=https://archive.org/details/sim_science_1996-03-22_271_5256/page/1710|author=Hans-Henrik Stølum|bibcode=1996Sci...271.1710S|ref=harv}}</ref><ref>{{harvnb|Posamentier|Lehmann|2004|pp=140–141}}</ref>
The Wallis formula for pi can be obtained directly from the [[Calculus of variations|variational approach]] to the [[Bohr model|spectrum of the hydrogen atom]] in spaces of arbitrary dimensions greater than one, including the physical three dimensions.<ref>{{cite journal|doi=10.1063/1.4930800|author=T. Friedmann ; C.R. Hagen|title=Quantum mechanical derivation of the Wallis formula for pi|journal=Journal of Mathmatical Physics|volume=56|issue=11|year=2015}}</ref>
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* {{cite book|last=Posamentier|first=Alfred S.|last2=Lehmann|first2=Ingmar|title=Pi: A Biography of the World's Most Mysterious Number|publisher=Prometheus Books|year=2004|isbn=978-1-59102-200-8|ref=harv}}
* {{cite journal|last=Reitwiesner|first=George|title=An ENIAC Determination of pi and e to 2000 Decimal Places|journal=Mathematical Tables and Other Aids to Computation|year=1950|volume=4|issue= 29|pages=11–15|doi=10.2307/2002695|ref=harv }}
* {{cite journal|last=Roy|first=Ranjan|title=The Discovery of the Series Formula for pi by Leibniz, Gregory, and Nilakantha|url=https://archive.org/details/sim_mathematics-magazine_1990-12_63_5/page/291|journal=Mathematics Magazine|volume=63|issue= 5|year=1990|pages=291–306|doi=10.2307/2690896|ref=harv }}
* {{cite journal|last=Schepler|first=H. C.|title=The Chronology of Pi|journal=Mathematics Magazine|publisher=Mathematical Association of America|year=1950|volume=23|issue=3|pages= 165–170 (Jan/Feb), 216–228 (Mar/Apr), and 279–283 (May/Jun)|doi=10.2307/3029284|ref=harv }}. [<!-- http://www.jstor.org/stable/3029284 -->http://www.jstor.org/discover/10.2307/3029284 issue 3 Jan/Feb], [http://www.jstor.org/stable/3029832 issue 4 Mar/Apr], [http://www.jstor.org/stable/3029000 issue 5 May/Jun]
{{refend}}
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