The Nombre pi , noted by the Greek letter of the same name π (always into tiny) is the constant relationship between the Circonférence of a Cercle and its Diamètre. It is also called constant of Archimedes .
A value approached in is
π is a irrational Nombre, i.e. it is not the ratio of two integers. In fact, this number is transcendent. This means that there does not exist Polynôme not no one with whole coefficients of which π is a root.
The transcendence of π establishes impossibility of solving the problem of the Quadrature of the circle: it is impossible to build, using the rule and of the compass only, one square whose surface is rigorously equal to the surface of a given disc.
The number very early was a source of inspiration for many mathematicians, and this as much in Algèbre that in analyzes. Thus, as of Antiquity, the Greek scientists, in particular scientists, are leaning on the properties of this number at the time of study on problems of Géométrie.
The oldest value of whose veracity is attested comes from a Babylonian shelf in wedge-shaped writing, discovered in 1936. This shelf goes back to 2000 before J. - C. the Babylonian would have arrived there while comparing the Périmètre circle with that of the Hexagone registered, equal to three times the diameter; they deduced one from them from the first known values of : (= 3,125).
Discovered in 1855, the papyrus of Rhind contains the text, recopied about year 1650 before our era by the Egyptian scribe Ahmès, of an older handbook of problems teaching still. One finds trace of a calculation which implies that is evaluated with (≈ 3,160…).
Formulas including π
The interesting formulas including are innumerable and appear in almost all the fields of mathematics and sciences.
Pi appears in many formulas of Géométrie implying the Cercle S and the Sphère S
The surface of a cylinder circumscribed with the sphere and of the same height is the same one (bases of the cylinder excluded).
is also found in the calculation of the areas and volumes of the hypersphères (with more than 3 dimensions). The measurement of Angle 180° (in degree S) is equal to Radian S.
- By using the trigonometrical identities, and , one can express and by successive fitments of square roots. One obtains the formulas which follow for .
*π can then be expressed in the form of a infinite iteration of square roots:
- , where K is the number of encased square roots
- * Another expression of , which can simply result from the first of these two equalities (to multiply by , led to the infinite Produit according to (formula of François Viète, 1593).
- , where K is the number of encased square roots
(formula of Leibniz inspired by James Gregory)
and more generally, Euler indicated that ζ (2n) is a rational multiple of for a positive entirety N
(function gamma of Euler)
π can be written in the form of generalized continuous fractions remarkable:
(there are other representations on)
where is the indicating function of Euler (cf also the series Farey).
(surface of a unit quadrant)
Theory of the numbers
The frequency of appearance of pairs of natural entireties first between them among the pairs of entireties lain between 0 and NR tends towards when NR tends towards the infinite one.
The median number of ways of writing two unspecified positive entireties ranging between 0 and NR as the sum of two square perfect, by holding suspense account, tends towards when NR tends towards the infinite one.
Dynamic systems/ergodic Theory
The probability so that two natural entireties are first between them is , with the direction where if one randomly draws two natural entireties ranging between 1 and NR (where NR is a fixed natural entirety not no one) according to the uniform law, this probability tends towards when NR tends towards the infinite one.
almost everywhere on 1 where the X I are reiterated logistic plan for R = 4.
Calculation of the value of pi
Because of its irrational nature , the π number does not have a decimal Développement finished or periodic. It results from it that one can calculate of it only one decimal writing approximate. For example, a value approached with 100 decimals is
- 3,1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 4587513258…
History of the calculation of piWith the Babylonians used approximation 25/8 and the Egyptians ((16/9) 2 = 3,16049…) who was a rather good approximation. One has the testimony of a better approximation only at third century BC towards 250 av. J. - C. with the treaty of Archimedes on the measurement of the circle . Thanks to a method consisting in framing a circle between two continuations Polygon S, of which the double number on sides to each iteration, Archimedes obtained: ≈ is, known as in a very anachronistic way, a precision of and 2 decimals exact.
In Persia in 1429, Al-Kashi calculated 14 decimals of . In 1596, always with geometrical methods, the Dutch Ludolph van Ceulen calculated 20 decimals, then 34 in 1609. It was so proud of its exploit (it devoted a good part of its life to it) which it asked so that the number is engraved on its tomb.
James Gregory (1638 - 1675) discovers the following formula
Isaac Newton calculated 16 decimals in 1665, John Machin 100 in 1706. About 1760, Euler calculated 20 decimals in one hour (to be compared with about thirty decimals obtained by Van Ceulen in addition to 10 years of calculation).
The Slovenien mathematician Jurij Vega calculated in 1789 the first 140 decimals π among which 137 were correct. This record will hold more than 50 years. It improved the formula that John Machin had found in 1706 and its method is always mentioned today.
The mathematician William Shanks spent 20 years of his life to calculate the decimals of pi. He calculated 707 of them, but only the 528 first were correct. At the time of the World Fair of Paris of 1937, those were unfortunately engraved in the room π of the Palais of Discovered the. The error was detected only in 1945.
The calculation of the decimals of pi packed at the 20th century with the appearance of data processing: : 2037 are calculated in 1949 by the American calculator ENIAC: 10000 decimals are obtained in 1958: 100000 in 1961: 1000000 in 1973: 10000000 in 1982: 100000000 in 1989, then: 1000000000 the same year. In 2002: 1241100000000 decimals were known.
Methods of calculating of pi
Formulas of ThingThe Formula of Thing used by John Thing, similar to formulas still used today, allows a fast calculation:
It obtained it with a development in Série of Taylor of the function arctan ('' X ''). This formula can be checked easily in polar coordinates in the plane complex, with
The very precise approximations of π are generally calculated with the Algorithme of Gauss-Legendre and the Algorithme of Borwein; the Algorithme of Salamin-Brent, invented in 1976 was also used for very great numbers of decimals.
One can see: 1000000 of decimals of and on the Project Gutenberg (see external bonds).
The current record is of: 1241100000000 of decimals, given after 600 hours of calculation in November 2002 on a Supercomputer parallel Hitachi with 64 nodes, with 1 Terabyte of main Memory, which could carry out: 2000 billion operations in floating decimal point a second, is close to twice as much as for the preceding record (206 billion decimals); the following formulas of Thing were used for that:
(K. Takano, 1982)
(F.C.W. Störmer, 1896)
These approximations are so large that they do not have any practical use, if is not to test the new supercomputers.
Other methods and algorithms are currently being studied and implemented like the use in parallel of computers connected on the network Internet.
The calculation isolated from the decimals of piIn 1995 David Bailey, in collaboration with Peter Borwein and Simon Plouffe, discovered a new formula of π, a series (often called Formule BBP):
This formula makes it possible to easily calculate N E binary or hexadecimal decimal of π, without having to calculate the preceding decimals. The site of Bailey contains of it derivation and the implementation in many languages of Programmation. Thanks to a formula derived from the Formula BBP, it: 4000000000000000e figure of π in bases 2 was obtained in 2001.
One year later, Simon Plouffe develops an algorithm allowing the calculation of N E decimal of π, but this time into decimal. It is described in a short article available since the page of Simon Plouffe (http://www.lacim.uqam.ca/~plouffe/Simon/articlepi.html). Unfortunately, this algorithm which currently makes it possible to determine in bases 10 a precise figure and isolated from π is slower than that which consists in calculating all the preceding decimal digits.
Other formulasOther formulas were used to calculate π of which:
(formula due to Ramanujan)
(formula due to David and Gregory Chudnovsky)
To retain pi
A means popular Mnémotechnique (but little practices) is the poem:
That I like to make learn a number useful for the wise ones!
- Immortal Archimedes, artist, engineer,
- Which of your judgment can snuff the value?
- For me your problem had similar advantages.
- Jadis, mysterious, a problem blocked
- All the admirable process, imposing work
- That Pythagore discovered with the former Greeks.
- O squaring! Old torment of the philosopher
- Insoluble roundness, too a long time you have
- Défié Pythagore and its imitateurs.
- How to integrate circular plane space?
- To form a triangle to which it will be equivalent?
- New invention: Archimedes will register
- Dedans a hexagon; will appreciate its surface
- Fonction of the ray. Not too will not be held to with it:
- Will duplicate each former element;
- Toujours of the sphere calculated will approach;
- Will define limit; finally, the arc, the limiting device
- Of this worrying circle, too rebellious enemy
- Professor, teach his problem with zeal
- Immortal Archimedes, artist, engineer,
The number of letters of each word corresponds to a decimal, except for the figure " 0" whose coding corresponds to a word of 10 letters.
In 2005, a 59 year old Japanese, Akira Haraguchi, succeeded in aligning by heart: 83431 decimals of pi in 13 hours. He reiterated his record later one year (2006) while memorizing and reciting publicly: 100000 decimals during 16 hours. This exploit was approved by the Livre Guinness of the records.
An important open-ended question is to know if π is a normal Nombre, i.e. if any succession of N figures appears in the decimal value of π with the same probability as another succession of N figures, as one would expect it for an infinite and completely random succession of figures. That would have in truth being checked in any bases and not only in 10 bases. Bailey and Crandall showed in 2000 that the existence of the formula Bailey-Borwein-Plouffe above and similar formulas involves normality in bases 2 π.
In the same spirit, one does not know if π is a Nombre universe, i.e. a number which one can find any succession of figures finite length it does not matter the probability of appearance of this one.
One does not even know which are the figures of the decimal development of which the number of appearances is infinite.
Nature of π
In nonEuclidean Geometry, the sum of the angles of a triangle can be higher or lower than π, and the report/ratio of the circumference of the circle to its diameter can also be different from π. That does not change the value of π, but that affects the formulas in which this number appears. In particular, the shape of the Universe does not affect the value of π: it is a constant mathematics, not a physical value.
- Day of π
- the library of Babel
- Jean-Paul Delahaye, fascinating It number π , Belin Editions, For Science - ISBN 2-9029-1825-9
- Pierre Eymard, Jean-Pierre Lafon, Around number pi , Hermann Editions, Paris, 1999 - ISBN 2705614435
- Jörg ARNDT & Christoph HAENEL: With the continuation of , Vuibert Editions, 2006 - ISBN 2-7117-7170-9
a humorous reading of the Bible, entitled " God Said pi = 3" resulted in re-examining the decimal value of π. It is about an interpretation Créationniste of a passage of the First book of the Kings, chapter 7 verses 23 to 26: http://gospelofreason.wordpress.com/2007/06/13/god-said-pi-3-stand-by-your-beliefs-dammit/
- pi314.net: a site dedicated to pi
- www.nombrepi.com: another site dedicated to number pi
- Peripheria: the French-speaking gate on number pi
- : Constant pi in the dictionary of the numbers
- Site allowing a research of figures in the first 200.000.000 decimals
- Exposed on pi " How to approximate the constant? " , demonstrations of the formula of Viète, the theorem of Buffon, use of the radioactivity
- the site Wolfram Mathematics compiles off many formulas for π
- Finding the been worth pi
- PlanetMath: Pi
- File of the decimals of π, the first 32 million decimals available (in English)
- program to extract the decimals from pi
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