This article is about the mathematical constant.
For the Greek letter, see Pi
(letter). For other uses, see Pi (disambiguation).
Part of a series of articles on the
mathematical constant π
3.1415926535897932384626433...
Uses
• Area of a circle Circumference Use in other
formulae
Properties
• Irrationality Transcendence
Value
• Less than 22/7 Approximations
Memorization
People
• Archimedes Liu Hui Zu Chongzhi
Aryabhata Madhava Ludolph van Ceulen
Seki Takakazu Takebe Kenko William
Jones John Machin William Shanks
Srinivasa Ramanujan John Wrench
Chudnovsky brothers Yasumasa Kanada
History
• Chronology Book
In culture
• Legislation Pi Day
Related topics
• Squaring the circle Basel problem Six
nines in π Other topics related to π
• v te
It is known that π is a transcendental number:[2] it is not the root of any
polynomial with rational coefficients. The transcendence of π implies that it is
impossible to solve the ancient challenge of squaring the circle with a
compass and straightedge.
Ancient civilizations, including the Egyptians and Babylonians, required fairly
accurate approximations of π for practical computations. Around 250 BC, the
Greek mathematician Archimedes created an algorithm to approximate π with
arbitrary accuracy. In the 5th century AD, Chinese mathematics approximated
π to seven digits, while Indian mathematics made a five-digit approximation,
both using geometrical techniques. The first exact formula for π, based on
infinite series, was discovered a millennium later, when in the 14th century
the Madhava–Leibniz series was discovered in Indian mathematics.[4][5]
The invention of calculus soon led to the calculation of hundreds of digits of π,
enough for all practical scientific computations. Nevertheless, in the 20th and
21st centuries, mathematicians and computer scientists have pursued new
approaches that, when combined with increasing computational power,
extended the decimal representation of π to many trillions of digits.[6][7] The
primary motivation for these computations is as a test case to develop
efficient algorithms to calculate numeric series, as well as the quest to break
records.[8][9] The extensive calculations involved have also been used to test
supercomputers and high-precision multiplication algorithms.
Because its most elementary definition relates to the circle, π is found in
many formulae in trigonometry and geometry, especially those concerning
circles, ellipses, and spheres. In more modern mathematical analysis, the
number is instead defined using the spectral properties of the real number
system, as an eigenvalue or a period, without any reference to geometry. It
appears therefore in areas of mathematics and sciences having little to do
with geometry of circles, such as number theory and statistics, as well as in
almost all areas of physics. The ubiquity of π makes it one of the most widely
known mathematical constants—both inside and outside the scientific
community. Several books devoted to π have been published, and record-
setting calculations of the digits of π often result in news headlines. Adepts
have succeeded in memorizing the value of π to over 70,000 digits.
