- Alternative meanings: Pi (letter), π (movie), Pi
meson
The mathematical constant π
(written as "pi" when the Greek letter is not
available) is ubiquitous in many areas of mathematics and physics. In Euclidean plane
geometry, π may be defined as either the ratio of a circle's circumference to its diameter, or as the area of a circle of radius 1. Most modern textbooks define π analytically using trigonometric functions, e.g. as the smallest positive x for which sin(x) = 0, or as twice the smallest
positive x for which cos(x) = 0. All of these definitions are equivalent.
π is also known as Archimedes' constant (not to be
confused with Archimedes' number), Ludolph's constant or Ludolph's number.
Contrary to a common misconception, π is not a physical constant of nature, but rather a mathematical constant defined independently of any physical
measurements.
The first sixty-four decimal digits of π (sequence A000796 in OEIS) are:
- 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 592...
More digits of π may be found at the following Wikisource
links:Wikisource - Pi to 1,000 Places | 10,000 Places | 100,000 Places | 1,000,000 Places
Properties
π is an irrational number: that is, it cannot be written
as the ratio of two integers. This was proven in 1761 by Johann Heinrich Lambert. In fact,
the number is transcendental, as was proven by Ferdinand von Lindemann in 1882. This means that there is no polynomial with integer (or rational)
coefficients of which π is a root. As a consequence, it is impossible to express π using only a finite number of
integers, fractions and their roots.
This result establishes the impossibility of squaring the
circle: it is impossible to construct, using ruler and compass alone, a square whose area is equal to the area of a given circle. The
reason is that the coordinates of all points that can be constructed with ruler and compass are special algebraic numbers.
Formulae involving π
Geometry:
- Circumference of circle or
sphere of radius r: C = 2 π r
- Area of circle of radius r: A = π r2
- Area of ellipse with semiaxes a and b: A = π
ab
- Volume of sphere of radius r:
V = (4/3) π r3
- Surface area of sphere of radius r: A = 4 π
r2
- Volume of cylinder of height
h and radius r: V = (π r2 ) h
- Surface area of cylinder
of height h and radius r: A = ([π r2] 2 ) + ([2 π r] h
)
- Angles: 180 degrees is equivalent to π radians
Analysis:
-
- Leibniz' formula.
-
- Wallis' product.
-
- The Basel problem, first solved by Euler. See also Riemann zeta
function.
-
-
-
-
- Stirling's approximation.
-
-
- Euler's identity, called by Richard Feynman "the most remarkable formula in mathematics."
π has beautiful continued fractional
representations:
-
(You can see 12 other representations at [1] )
Number theory:
- The probability that two randomly chosen integers are relatively
prime is 6/π2.
- The probability that a randomly chosen integer is square-free is
6/π2.
- The average number of ways to write a positive integer as the sum of two perfect squares (order matters) is π/4.
Dynamical Systems / Ergodic theory:
-
- for almost every x0 in [0, 1] where the
xi are iterates of the Logistic map for
r=4.
Physics:
-
- Heisenberg's uncertainty principle.
-
- Einstein's field equation of general relativity.
-
- Coulomb's law.
Probability and Statistics:
-
- The probability density function for
the normal distribution.
-
- Buffon's needle
History
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 περιμετρος (perimetros),
meaning 'measure around' in Greek.
Here is a brief chronology of π, with World Records in bold:
- 20th Century BCE: Babylonians used 25/8 (=3.125) for
π
- 20th Century BCE: Egyptians (Rhind Papyrus) used π = (16/9)2
(=3.16045..)
- 12th Century BCE: Chinese used 3 for π
- 434 BC: Anaxagoras tried to square the circle with ruler and compass
- 3rd Century BCE: Archimedes found that 223/71 < π < 22/7
(3.1408.. < π < 3.1428..), and also found the approximation π = 211875/67441
(=3.14163..)
- 20 BCE: Vitruvius used 25/8
- 130 CE: Chang Hong used π = √10,
- 150 CE: Ptolemy used π = 377/120 (=3.14167)
- 250: Wang Fau uses π = 142/45 (=3.14555..)
- 263: Liu Hui used π =3.14159..
- 480: Zu Chongzhi (430-501) found that 3.1415926 < π < 3.1415927
- 499: Aryabhatta used the value 62832/20000 = 3.1416
- 598: Brahmagupta, in India, used the value π = √10 (=3.162..)
- 800: Al Khwarizmi used 3.1416
- 12th Century: Bhaskara (b. 1114) used π = 3.14156
- 1220: Fibonacci used the value 3.141818
- 1400: Madhava used π = 3.1415926359
- 1424: Jamshid Masud Al Kashi (d. 1429) calculated π to 16 decimal places
- 1573: Valenthus Otho calculated π up to 6 decimal places
- 1593: François Viète calculated π up to 9 decimal
places
- 1593: Dutchman Adriaen van Roomen calculated π to 15 decimal places
- 1596: Ludolph van Ceulen calculated π to 20
decimal places
- 1615: Ludolph van Ceulen posthumously published π to 32 decimal places
- 1621: Willebrord Snel, a pupil of Van Ceulen, gave π to 35 decimal places
- 1665: Isaac Newton, 16 digits
- 1699: Abraham Sharp, 71 decimal places
- 1700: Seki Kowa, 10 digits
- 1706: Machin, 100 digits
- 1706: William Jones, a British mathematician, introduced the symbol π, which would later be taken up by Euler
- 1730: Kamata, 25 digits.
- 1719: De Lagny calculates 127 decimal places, of which 112 are correct
- 1723: Takebe, 41 decimal places
- 1734: With Leonhard Euler's usage of Jones' symbolism, the Greek
letter π becomes widely accepted
- 1739: Matsunaga, 50 decimal places
- 1761: Johann Heinrich Lambert proved that π
is irrational
- 1775: Euler points out the possibility that π might be transcendental
- 1789: Jurij Vega, 140 decimal places, of which 137 are
correct
- 1794: Adrien-Marie Legendre showed that π2 (and hence π)
is irrational, and mentioned the possibility that π might be transcendental.
- 1841: Rutherford calculated 208 decimal places, of which 152 are correct
- 1844: Zacharias Dase and Strassnitzky calculated 200 decimal places
- 1847: Thomas Clausen, 248 decimal places
- 1853: Lehmann, 261 digits
- 1853: Rutherford, 440 digits
- 1853: William Shanks, 527 decimal places
- 1855: Richter, 500 decimal places
- 1874: Shanks, 707 digits, of which 527 are correct
- 1882: Lindemann proved that π is transcendental
- 1953: Mahler showed that π is not a Liouville number
Computers reached:
- 1946: D F Ferguson, 620 decimal places
- 1947: D.F. Ferguson, 710 decimal places
- 1947: J. W. Wrench, Jr, and L. R. Smith, to 808 decimal places
- 1947: 2037 decimal places
- 1955: 3 089 decimal places
- 1961: 100 000 decimal places
- 1966: 250 000 decimal places
- 1967: 500 000 decimal places
- 1974: 1 000 000 decimal places
- 1992: 2 180 000 000 decimal places
- 1995: Kanada > 6 000 000 000 decimal places
- 1997: Kanada and Takahashi > 51 500 000 000 decimal places
- 1999: Kanada and Takahashi > 206 000 000 000 decimal places
Numerical approximations of π
Due to the transcendental nature of π, there are no nice closed expressions for π. Therefore numerical calculations
must use approximations to the number. For many purposes, 3.14 or 22/7 is
close enough, although engineers often use 3.1416 (5 significant
figures) or 3.14159 (6 significant figures) for more accuracy. The approximations 22/7 and 355/113, with 3 and 7 significant
figures respectively, are obtained from the simple continued
fraction expansion of π.
An Egyptian scribe called Ahmes is the source of the oldest known text to give an
approximate value for π. The Rhind Papyrus dates from the 17th century B.C.E. and describes the value in such a way that the
result obtained comes out to 256 divided by 81 or 3.160.
The Chinese mathematician Liu Hui computed π to 3.141014 (incorrect in the
fourth decimal digit) in 263 C.E. and suggested that 3.14 was a good approximation.
The Chinese mathematician and astronomer Zu Chongzhi computed π to
3.1415926 to 3.1415927 and gave two approximations of π 355/113 and 22/7 in 5th century.
The Iranian mathematician and astronomer, Ghyath ad-din Jamshid Kashani, 1350-1439, computed π to 9 digits in the base
of 60, which is equivalent to 16 decimal digit as:
- 2 π = 6.2831853071795865
The German mathematician Ludolph van Ceulen (circa
1600) computed the first 35 decimals. He was so proud of this accomplishment that he had
them inscribed on his tombstone.
The Slovene mathematician Jurij Vega in 1789 calculated the first 140 decimal places for π of which the first 137 were correct and held the world record
for 52 years until 1841, when William Rutherford calculated
208 decimal places of which the first 152 were correct. Vega improved John
Machin's formula from 1706 and his method is still mentioned today.
None of the formulas given above can serve as an efficient way of approximating π. For fast calculations, one may use
formulas such as Machin's:
-
together with the Taylor series expansion of the function
arctan(x). This formula is most easily verified using polar
coordinates of complex numbers, starting with
-
Formulas of this kind are known as Machin-like formulas.
Extremely long decimal expansions of π are typically computed with the Gauss-Legendre algorithm and Borwein's algorithm; the Salamin-Brent algorithm which was invented in 1976 has also
been used in the past.
The first one million digits of π and 1/π are available from Project Gutenberg (see external links below). The current record (December 2002) stands at 1,241,100,000,000 digits, which were computed in September 2002
on a 64-node Hitachi supercomputer with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly
twice as many as the computer used for the previous record (206 billion digits). The following Machin-like formulas were used for
this:
-
- K. Takano (1982).
-
- F. C. W. Störmer (1896).
These approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers
and (obviously) for establishing new π calculation records.
In 1996 David H. Bailey,
together with Peter Borwein and
Simon Plouffe, discovered a new formula for π as an infinite series:
-
This formula permits one to easily compute the kth binary or hexadecimal digit of π, without
having to compute the preceding k − 1 digits. Bailey's website
contains the derivation as well as implementations in various programming languages. The PiHex project computed 64-bits
around the quadrillionth bit of π (which turns out to be 0).
Other formulas that have been used to compute estimates of π include:
-
- Newton.
-
- Ramanujan.
-
- David Chudnovsky and
Gregory
Chudnovsky.
-
- Euler.
Open questions
The most pressing open question about π is whether it is normal,
i.e. whether any digit block occurs in the expansion of π just as often as one would statistically expect if the digits had
been produced completely "randomly". This must be true in any base, not just in base 10. Current knowledge in this direction is
very weak; e.g., it isn't even known which of the digits 0,...,9 occur infinitely often in the decimal expansion of π.
Bailey and Crandall showed in 2000 that the existence of the above mentioned
Bailey-Borwein-Plouffe formula and similar formulas imply that the normality in base 2 of π and various other constants can
be reduced to a plausible conjecture of chaos theory. See Bailey's above mentioned web site for details.
The nature of π
In non-Euclidean geometry the sum of the angles of a
triangle may be more or less than π, and the ratio
of a circle's circumference to its diameter may also differ from π. This doesn't change the definition of π, but it
does affect many formulae in which π appears. So, in particular, π is not affected by the shape of the universe; it is a mathematical constant, not a
physical constant.
Pi culture
There is an entire field of humorous yet serious study that involves the use of mnemonic techniques to remember the digits of π, which is known as Piphilology. For
example, part of the school cheer
of MIT is: "Cosine, secant, tangent, sine! 3 point 1 4 1 5 9!" See Piphilology for more examples.
March 14 (3/14) marks Pi Day which is
celebrated by many lovers of π. On July 22, Pi Approximation Day is celebrated (22/7 is a popular approximation of π).
Related articles
External links
|