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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Approximations of pi | 2/10 | https://en.wikipedia.org/wiki/Approximations_of_pi | reference | science, encyclopedia | 2026-05-05T16:19:48.727542+00:00 | kb-cron |
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{\displaystyle \pi ={\sqrt {12}}\sum _{k=0}^{\infty }{\frac {(-3)^{-k}}{2k+1}}={\sqrt {12}}\sum _{k=0}^{\infty }{\frac {(-{\frac {1}{3}})^{k}}{2k+1}}={\sqrt {12}}\left(1-{1 \over 3\cdot 3}+{1 \over 5\cdot 3^{2}}-{1 \over 7\cdot 3^{3}}+\cdots \right)}
He used the first 21 terms to compute an approximation of π correct to 11 decimal places as 3.14159265359. He also improved the formula based on arctan(1) by including a correction:
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{\displaystyle \pi /4\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots -{\frac {(-1)^{n}}{2n-1}}\pm {\frac {n^{2}+1}{4n^{3}+5n}}}
It is not known how he came up with this correction. Using this he found an approximation of π to 13 decimal places of accuracy when n = 75. Indian mathematician Bhaskara II (12th century) used regular polygons with up to 384 sides to obtain another approximation of π, calculating it as 3.141666. Jamshīd al-Kāshī (Kāshānī) (15th century), a Persian astronomer and mathematician, correctly computed the fractional part of 2π to 9 sexagesimal digits in 1424, and translated this into 16 decimal digits after the decimal point:
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6.2831853071795864
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{\displaystyle 2\pi \approx 6.2831853071795864,}
which gives 16 correct digits for π after the decimal point:
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3.1415926535897932
{\displaystyle \pi \approx 3.1415926535897932}
He achieved this level of accuracy by calculating the perimeter of a regular polygon with 3 × 228 sides.
== 16th to 19th centuries == In the second half of the 16th century, the French mathematician François Viète discovered an infinite product that converged on π known as Viète's formula. The German-Dutch mathematician Ludolph van Ceulen (circa 1600) computed the first 35 decimal places of π with a 262-gon. He was so proud of this accomplishment that he had them inscribed on his tombstone. 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 π from a 96-sided polygon. In 1656, John Wallis published the Wallis product:
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{\displaystyle {\frac {\pi }{2}}=\prod _{n=1}^{\infty }{\frac {4n^{2}}{4n^{2}-1}}=\prod _{n=1}^{\infty }\left({\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}\right)={\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 )}\cdot \;\cdots }
In 1706, John Machin used Gregory's series (the Taylor series for arctangent) and the identity
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{\textstyle {\tfrac {1}{4}}\pi =4\operatorname {arccot} 5-\operatorname {arccot} 239}
to calculate 100 digits of π (see § Machin-like formula below). In 1719, Thomas de Lagny used a similar identity to calculate 127 digits (of which 112 were correct). In 1789, the Slovene mathematician Jurij Vega improved John Machin's formula to calculate the first 140 digits, of which the first 126 were correct. In 1841, William Rutherford calculated 208 digits, of which the first 152 were correct. 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 billion light-years) to a precision of less than one Planck length (at 1.6162×10−35 meters, the shortest unit of length expected to be directly measurable) using π expressed to just 62 decimal places. The English amateur mathematician William Shanks calculated π to 530 decimal places in January 1853, of which the first 527 were correct (the last few likely being incorrect due to round-off errors). He subsequently expanded his calculation to 607 decimal places in April 1853, but an error introduced right at the 530th decimal place rendered the rest of his calculation erroneous; due to the nature of Machin's formula, the error propagated back to the 528th decimal place, leaving only the first 527 digits correct once again. Twenty years later, Shanks expanded his calculation to 707 decimal places in April 1873. Due to this being an expansion of his previous calculation, most of the new digits were incorrect as well. Shanks was said to have calculated new digits all morning and would then spend all afternoon checking his morning's work. This was the longest expansion of π until the advent of the electronic digital computer three-quarters of a century later.
== 20th and 21st centuries ==
In 1910, the Indian mathematician Srinivasa Ramanujan found several rapidly converging infinite series of π, including
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{\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}}
which computes a further eight decimal places of π with each term in the series. His series are now the basis for the fastest algorithms currently used to calculate π. Evaluating the first term alone yields a value correct to seven decimal places:
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{\displaystyle \pi \approx {\frac {9801}{2206{\sqrt {2}}}}\approx 3.14159273}