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 pages about

the history of numerical approximations of π, which says more about the ongoing hunt for billions of decimal places of π
the chronology of computations of π with an overview of "world records" concerning these computations.

1 History of π
- 1.1 Theory
- 1.2 Computation
2 History of the notation
3 See also
4 References and links

[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.)

The Indian mathematician and astronomer Madhava of Sangamagrama in the 14th century found the following infinite series expansion of π:

$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots \pm \frac{1}{2n -1} \cdots$

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.

In the 18th century, Abraham de Moivre found that when a fair coin is tossed 1800 times, the probability that the number of heads is x is approximately

$C \exp\left({-(x-900)^2 \over 900}\right)$

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

${1 \over \sqrt{2\pi}}.$

In 1761, Johann Heinrich Lambert showed that π is an irrational number by showing that $tan a$ is irrational if $a$ is rational, and since $tanπ / 4 = 1$ , it follows that π is irrational.

In 1882, Ferdinand von Lindemann proved that π is a transcendental number. It had earlier been proved that if π is transcendental, then it is impossible to solve the ancient Greek geometers' problem of squaring the circle.

In 1953, Kurt Mahler proved that π is not a Liouville number.

[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.
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.

In 1424, the Persian Muslim mathematician and astronomer Ghyath ad-din Jamshid Kashani (1380–1429) correctly computed 2π to 9 sexagesimal (base 60) digits.^[1] This figure is equivalent to 16 decimal (base 10) digits as

$\ 2\pi = 6.2831853071795865$

which corresponds to

$\ \pi = 3.14159265358979325$

He achieved this level of accuracy by calculating the perimeter of a regular polygon with 3 × 2¹⁸ sides.

With the appearance of computers, a hunt on millions and billions of decimal places of π has started and is still ongoing. See history of numerical approximations of π for a detailed account.

[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 either case, 'π' is the first letter of περιφέρεια (periphereia, the Greek word for periphery) or περίμετρον (perimetron), meaning 'measure around' in Greek.