History of π

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The mathematical constant π = 3.14159... has been subject to extensive study since ancient times.

This article gives a historical account on the increasing human knowledge about its mathematical properties. Here the aim is not to focus on the precision of known numerical approximations. There are more specialized articles about

Contents

[edit] History of π

[edit] Theory

  • That the ratio of the circumference to the diameter of a circle is the same for all circles, and that it is slightly more than 3, was known to ancient Egyptian, Babylonian, Indian and Greek geometers. The Indians and Greeks also knew that the area of a circle is πr2, where r is the radius.
  • Archimedes showed that the volume of a sphere is (4/3)πr3, where r is the radius, and that the surface area of a sphere is 4πr2, i.e., 4 times the area of the circle with the same radius. (Also it is notable that the derivative of volume of a sphere is the formula for the surface area of a sphere.)
\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots \pm \frac{1}{2n -1} \cdots
This can also be writen as follows:
\pi=4\sum^\infty_{k=0} \frac{ \left(-1\right)^k }{2k+1}
which is a realization of the power series expansion of the arctangent function. Madhava also used the first 21 terms of the related series:
\pi = \sqrt{12}\left(1-{1\over 3\cdot3}+{1\over5\cdot 3^2}-{1\over7\cdot 3^3}+\cdots\right)
to compute a finite-series approximation of π correct to 11 decimal places as 3.14159265359.
C \exp\left({-(x-900)^2 \over 900}\right)
where C is a constant that de Moivre could compute by numerical means. (This normal distribution was introduced in the 1738 edition of de Moivre's book The Doctrine of Chances.) As the number of tosses grows, the approximation can be made as close as desired (but "900" would be replaced by a larger number). De Moivre's friend James Stirling later showed that this constant is
{1 \over \sqrt{2\pi}}.

[edit] Computation

Main article: History of numerical approximations of π.
  • The Egyptian scribe Ahmes wrote the oldest known text to give an approximate value for π, citing a Middle Kingdom papyrus, corresponding to a value of 256 divided by 81 or 3.160.
  • As early as the 19th century BC, Babylonian mathematicians were using π = 25/8, which is within 0.53% of the exact value.
  • An early approximation of π is in the Bible, 1 King 7:23. The circumference is related to the diameter with a ratio of 30 to 10, which gives an estimate of 3 for π. Scholars estimate 1 King was written between 971-852 B.C.
  • By finding perimeters of circumscribed and inscribed regular polygons, Archimedes found that π is between 3 + 10/71 and 3 + 1/7.
  • The Chinese mathematician Liu Hui computed π to 3.141014 (good to three decimal places) in AD 263 and suggested that 3.14 was a good approximation.
  • According to I Ching (635–713), the Arc of 1/4 circle is 10, the chord is 9 so the pi is √2/0.9×2 = 3.1426968052735445…, while why the ratio of chord/Arc=0.9 is unknown.
  • The Indian mathematician and astronomer Aryabhata in the 5th century gave the approximation π = 62832/20000 = 3.1416, correct when rounded off to four decimal places.
  • The Chinese mathematician and astronomer Zu Chongzhi computed π to be between 3.1415926 and 3.1415927 and gave two approximations of π, 355/113 and 22/7, in the 5th century.
  • In the 14th century, the Indian mathematician and astronomer Madhava of Sangamagrama used
\frac{n^2 + 1}{4n^3 + 5n}
as an approximation of the remainder term of the infinite series expansion of \frac{\pi}{4}, after summing the series through n = 75, to find a rational approximation of π that was correct to 13 decimal places of accuracy.
\ 2\pi = 6.2831853071795865
which corresponds to
\ \pi = 3.14159265358979325
He achieved this level of accuracy by calculating the perimeter of a regular polygon with 3 × 218 sides.

[edit] History of the notation

The symbol "π" for Archimedes' constant was first introduced in 1706 by William Jones when he published A New Introduction to Mathematics, although the same symbol had been used earlier to indicate the circumference of a circle. The notation became standard after it was adopted by Leonhard Euler in 1737. In either case, 'π' is the first letter of περιφέρεια (periphereia, the Greek word for periphery) or περίμετρον (perimetron), meaning 'measure around' in Greek.

[edit] See also

[edit] Notes

  1. ^ Al-Kashi, author: Adolf P. Youschkevitch, chief editor: Boris A. Rosenfeld, p. 256

[edit] References

[edit] External links

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