pi

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Pi or π is the ratio of a circle's circumference to its diameter in Euclidean geometry, approximately 3.14159. Pi is a mathematical constant and a transcendental (and therefore irrational) real number, with many uses in mathematics, physics, and engineering. It is also known as Archimedes' constant (not to be confused with an Archimedes number) and as Ludolph's number.
 List of numbers - Irrational numbers ζ(3) - √2 - √3 - √5 - φ - α - e - π - δ Binary 11.00100100001111110110... Decimal 3.14159265358979323846... Hexadecimal 3.243F6A8885A308D31319... Continued fraction Note that this continued fraction is not periodic.
When a circle's diameter is 1, its circumference is π.

The letter π

Main article: pi (letter)
The name of the Greek letter π is pi, and this spelling is used in typographical contexts where the Greek letter is not available or where its usage could be problematic. When referring to this constant, the symbol π is always pronounced like "pie" in English, the conventional English pronunciation of the letter. In Greek, the name of this letter is pronounced slightly differently.

The constant is named "π" because it is the first letter of the Greek words περιφέρεια 'periphery'[1] and περίμετρος 'perimeter', i.e. 'circumference'.

π is Unicode character U+03C0 ("Greek small letter pi").

Circumference = π × diameter

Definition

In Euclidean plane geometry, π is defined as the ratio of a circle's circumference to its diameter:

Note that the ratio c/d does not depend on the size of the circle. For example, if a circle has twice the diameter d of another circle it will also have twice the circumference c, preserving the ratio c/d. This fact is a consequence of the similarity of all circles.

Alternatively π can be also defined as the ratio of a circle's area to the area of a square whose side is the radius:

The constant π may be defined in other ways that avoid the concepts of arc length and area, for example, as twice the smallest positive x for which cos(x) = 0.[2] The formulæ below illustrate other (equivalent) definitions.

Numerical value

The numerical value of π truncated to 50 decimal places is:

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510

See the links below and those at sequence in OEIS for more digits.

While the value of pi has been computed to more than a trillion (1012) digits,[3] practical science and engineering will rarely require more than 10 decimal places. As an example, computing the circumference of the Earth's equator from its radius using only 10 decimal places of pi yields an error of less than 0.2 millimeters. A value truncated to 39 decimal places is sufficient to compute the circumference of the visible universe to a precision comparable to the size of a hydrogen atom.[4]

Most circular objects worthy of physical study, particularly on the scale of planetary radii, have imperfections and eccentricities which account for a greater error in calculation than would be yielded by calculations using approximations of pi. The exact value of π has an infinite decimal expansion: its decimal expansion never ends and does not repeat, since π is an irrational number (and indeed, a transcendental number). This infinite sequence of digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing more digits and investigating the number's properties. Despite much analytical work, and supercomputer calculations that have determined over 1 trillion digits of π, no simple pattern in the digits has ever been found. Digits of π are available on many web pages, and there is software for calculating π to billions of digits on any personal computer. See history of numerical approximations of π.

Calculating π

Main article: Computing π

π can be empirically measured by drawing a large circle, then measuring its diameter and circumference, since the circumference of a circle is always π times its diameter.

π can also be calculated using purely mathematical methods. Most formulae used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in trigonometry and calculus. However, some are quite simple, such as this form of the Gregory-Leibniz series:

While that series is easy to write and calculate, it is not immediately obvious why it yields π. In fact, this series converges so slowly that 300 terms are not sufficient enough to calculate π correctly to 2 decimal places. A more intuitive approach is to draw an imaginary circle of radius r centered at the origin. Then any point (x,y) whose distance d from the origin is less than r, as given by the pythagorean theorem, will be inside the circle:

Finding a collection of points inside the circle allows the circle's area A to be approximated. For example, by using integer coordinate points for a big r. Since the area A of a circle is π times the radius squared, π can be approximated by using:

Properties

The constant 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. See Proof that π is irrational for an elementary proof due to Ivan Niven.

Furthermore, π is also transcendental, as was proven by Ferdinand von Lindemann in 1882. This means that there is no polynomial with rational coefficients of which π is a root. An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.

History

Main article: History of π

Use of the symbol π

Often William Jones' book A New Introduction to Mathematics from 1706 is cited as the first text where the Greek letter π was used for this constant, but this notation became particularly popular after Leonhard Euler adopted it in 1737 (cf History of π).

Pi was chosen as the symbol for the number representing the relationship between circumference and diameter, because the letter π in Greek, pronounced like the letter "p", stands for perimeter.

Early approximations

Main article: History of numerical approximations of π

The value of π has been known in some form since antiquity. As early as the 19th century BC, Babylonian mathematicians were using π = 258, which is within 0.5% of the true value.

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.

It is sometimes claimed that the Bible states that π = 3, based on a passage in 1 Kings 7:23 giving measurements for a round basin as having a 10 cubit diameter and a 30 cubit circumference. The discrepancy has been explained in various ways by different exegetes. Rabbi Nehemiah explained it by the diameter being measured from outside rim to outside rim while the circumference was the inner brim; but it may suffice that the measurements are given in round numbers.

Bryson of Heraclea and Antiphon were the first to place an upper and lower bound on pi in a manner similar to that used by Archimedes, but considering area instead of perimeter.

Archimedes of Syracuse discovered, by considering the perimeters of 96-sided polygons inscribing a circle and inscribed by it, that π is between 22371 and 227. The average of these two values is roughly 3.1419.

The Chinese mathematician Liu Hui computed π to 3.141014 in AD 263 and suggested that 3.14 was a good approximation.

The Indian mathematician and astronomer Aryabhata in the 5th century gave the approximation π = 6283220000 = 3.1416, correct when rounded off to four decimal places. He also said that this was a value that "approached" the correct number, which was interpreted in the 15th c. as meaning that is irrational, a concept which would not be known in Europe till the 18th c.

The Chinese mathematician and astronomer Zu Chongzhi computed π to be between 3.1415926 and 3.1415927 and gave two approximations of π, 355113 and 227, in the 5th century.

The Indian mathematician and astronomer Madhava of Sangamagrama in the 14th century computed the value of π after transforming the power series of arctan(1)=π4 into the form

::

and using the first 21 terms of this series to compute a rational approximation of π correct to 11 decimal places as 3.14159265359. By adding a remainder term to the original power series of π4, he was able to compute π to an accuracy of 13 decimal places.

The Persian astronomer Ghyath ad-din Jamshid Kashani (1350–1439) correctly computed π to 9 digits in the base of 60, which is equivalent to 16 decimal digits as:

2π = 6.2831853071795865

By 1610, the German mathematician Ludolph van Ceulen had finished computing the first 35 decimal places of π. It is said that he was so proud of this accomplishment that he had them inscribed on his tombstone.

In 1789, the Slovene mathematician Jurij Vega improved John Machin's formula from 1706 and calculated the first 140 decimal places for π, of which the first 126 were correct [1], and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places of which the first 152 were correct.

The English amateur mathematician William Shanks, a man of independent means, spent over 20 years calculating π to 707 decimal places (accomplished in 1873). He published his value of pi in a book, which was promptly dubbed "the world's most boring book". In 1944, D. F. Ferguson found that Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were incorrect. By 1947, Ferguson had recalculated pi to 808 decimal places (with the aid of a mechanical desk calculator).

Numerical approximations

Main article: History of numerical approximations of π
Due to the transcendental nature of π, there are no closed form expressions for the number in terms of algebraic numbers and functions. Formulae for calculating π using elementary arithmetic invariably include notation such as "...", which indicates that the formula is really a formula for an infinite sequence of approximations to π. The more terms included in a calculation, the closer to π the result will get, but none of the results will be π exactly.

Consequently, numerical calculations must use approximations of π. 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 precision. The approximations 22/7 and 355/113, with 3 and 7 significant figures respectively, are obtained from the simple continued fraction expansion of π. The approximation 355113 (3.1415929…) is the best one that may be expressed with a three-digit or four-digit numerator and denominator.

The earliest numerical approximation of π is almost certainly the value 3. In cases where little precision is required, it may be an acceptable substitute. That 3 is an underestimate follows from the fact that it is the ratio of the perimeter of an inscribed regular hexagon to the diameter of the circle.

Formulae

Geometry

The constant π appears in many formulæ in geometry involving circles and spheres.

Geometrical shape Formula
Circumference of circle of radius r and diameter d
Area of circle of radius r
Area of ellipse with semiaxes a and b
Volume of sphere of radius r and diameter d
Surface area of sphere of radius r and diameter d
Volume of cylinder of height h and radius r
Surface area of cylinder of height h and radius r
Volume of cone of height h and radius r
Surface area of cone of height h and radius r

All of these formulae are a consequence of the formula for circumference. For example, the area of a circle of radius R can be accumulated by summing annuli of infinitesimal width using the integral . The others concern a surface or solid of revolution.

Also, the angle measure of 180° (degrees) is equal to π radians.

Analysis

Many formulas in analysis contain π, including infinite series (and infinite product) representations, integrals, and so-called special functions.
:
:
:
:
:
:
:
:
:
• Symmetric formula (see Sondow, 1997)
:
:
:
where the nth factor is the 2nth root of the product
:
:
:
:and generally, is a rational multiple of for positive integer n
:
:
:
:
:
:
where the path of integration is a closed curve around the origin, traversed in the standard anticlockwise direction.

Number theory

Some results from number theory:
In the above three statements, "probability", "average", and "random" are taken in a limiting sense, i.e. we consider the probability for the set of integers {1, 2, 3,…, N}, and then take the limit as N approaches infinity.
• The product of (1 − 1/p2) over the primes, p, is 6/π2.
The theory of elliptic curves and complex multiplication derives the approximation
which is valid to about 30 digits.

Dynamical systems and ergodic theory

Consider the recurrence relation
Then for almost every initial value x0 in the unit interval [0,1],
This recurrence relation is the logistic map with parameter r = 4, known from dynamical systems theory. See also: ergodic theory.

Physics

The number π appears routinely in equations describing fundamental principles of the Universe, due in no small part to its relationship to the nature of the circle and, correspondingly, spherical coordinate systems.
• Kepler's third law constant:

Probability and statistics

In probability and statistics, there are many distributions whose formulæ contain π, including:

Note that since , for any pdf f(x), the above formulæ can be used to produce other integral formulae for π.

A semi-interesting empirical approximation of π is based on Buffon's needle problem. Consider dropping a needle of length L repeatedly on a surface containing parallel lines drawn S units apart (with S > L). If the needle is dropped n times and x of those times it comes to rest crossing a line (x > 0), then one may approximate π using:
[As a practical matter, this approximation is poor and converges very slowly.]

Another approximation of π is to throw points randomly into a quarter of a circle with radius 1 that is inscribed in a square of length 1. π, the area of a unit circle, is then approximated as 4×(points in the quarter circle) ÷ (total points).

Efficient methods

In the early years of the computer, the first expansion of π to 100,000 decimal places was computed by Maryland mathematician Dr. Daniel Shanks and his team at the United States Naval Research Laboratory (N.R.L.) in 1961.

Daniel Shanks and his team used two different power series for calculating the digits of π. For one it was known that any error would produce a value slightly high, and for the other, it was known that any error would produce a value slightly low. And hence, as long as the two series produced the same digits, there was a very high confidence that they were correct. The first 100,000 digits of π were published by the Naval Research Laboratory.

None of the formulæ given above can serve as an efficient way of approximating π. For fast calculations, one may use a formula 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

Formulæ of this kind are known as Machin-like formulae.

Many other expressions for π were developed and published by Indian mathematician Srinivasa Ramanujan. He worked with mathematician Godfrey Harold Hardy in England for a number of years.

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.

The first one million digits of π and 1/π are available from Project Gutenberg (see external links below). The record as of December 2002 by Yasumasa Kanada of Tokyo University 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 formulæ 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. (Normality of π will always depend on the infinite string of digits on the end, not on any finite computation.)

In 1997, David H. Bailey, Peter Borwein and Simon Plouffe published a paper (Bailey, 1997) on a new formula for π as an infinite series:

This formula permits one to fairly readily 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).

Fabrice Bellard claims to have beaten the efficiency record set by Bailey, Borwein, and Plouffe with his formula to calculate binary digits of π [2]:

Other formulæ that have been used to compute estimates of π include:

Newton.

Srinivasa Ramanujan.

This converges extraordinarily rapidly. Ramanujan's work is the basis for the fastest algorithms used, as of the turn of the millennium, to calculate π.

David Chudnovsky and Gregory Chudnovsky.

Miscellaneous formulæ

The base 60 representation of π, correct to eight significant figures (in base 10) is:

In addition, the following expressions approximate π:
• accurate to 9 decimal places: [5]
:
• accurate to 9 places:
:
Ramanujan claimed he had a dream in which the goddess Namagiri appeared and told him the true value of π. [6]
• accurate to 3 decimal places: <ref name="mathworld:approximations" />
:
• accurate to 2 decimal places:
:
Karl Popper conjectured that Plato knew this expression; that he believed it to be exactly π; and that this is responsible for some of Plato's confidence in the omnicompetence of mathematical geometry — and Plato's repeated discussion of special right triangles that are either isosceles or halves of equilateral triangles.
• The continued fraction representation of π can be used to generate successively better rational approximations, which start off: 22/7, 333/106, 355/113…. These approximations are the best possible rational approximations of π relative to the size of their denominators.

Memorizing digits

Main article: Piphilology
Recent decades have seen a surge in the record number of digits memorized.

Even long before computers have calculated π, memorizing a record number of digits became an obsession for some people. A Japanese man named Akira Haraguchi claims to have memorized 100,000 decimal places. This, however, has yet to be verified by Guinness World Records. The Guinness-recognized record for remembered digits of π is 67,890 digits, held by Lu Chao, a 24-year-old graduate student from China.[7] It took him 24 hours and 4 minutes to recite to the 67,890th decimal place of π without an error.[8]

There are many ways to memorize π, including the use of piems, which are poems that represent π in a way such that the length of each word (in letters) represents a digit. Here is an example of a piem: How I need a drink, alcoholic in nature (or: of course), after the heavy lectures involving quantum mechanics. Notice how the first word has 3 letters, the second word has 1, the third has 4, the fourth has 1, the fifth has 5, and so on. The Cadaeic Cadenza contains the first 3834 digits of π in this manner. Piems are related to the entire field of humorous yet serious study that involves the use of mnemonic techniques to remember the digits of π, known as piphilology. See for examples. In other languages there are similar methods of memorization. However, this method proves inefficient for large memorizations of pi. Other methods include remembering patterns in the numbers (for instance, the year 1971 appears in the first fifty digits of pi).

Open questions

The most pressing open question about π is whether it is a normal number -- 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", and that this is true in every base, not just base 10. Current knowledge on this point is very weak; e.g., it is not 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 formulae 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.

It is also unknown whether π and e are algebraically independent. However it is known that at least one of πe and π + e is transcendental (see Lindemann–Weierstrass theorem).

Naturality

In non-Euclidean geometry the sum of the angles of a triangle may be more or less than π radians, and the ratio of a circle's circumference to its diameter may also differ from π. This does not change the definition of π, but it does affect many formulæ in which π appears. So, in particular, π is not affected by the shape of the universe; it is not a physical constant but a mathematical constant defined independently of any physical measurements. Nonetheless, it occurs often in physics.

For example, consider Coulomb's law (SI units)
.
Here, 4πr2 is just the surface area of sphere of radius r. In this form, it is a convenient way of describing the inverse square relationship of the force at a distance r from a point source. It would of course be possible to describe this law in other, but less convenient ways, or in some cases more convenient. If Planck charge is used, it can be written as

and thus eliminate the need for π.

References

Footnotes

1. ^ OED: probably περιφέρεια or periphery
2. ^ Rudin p.183
3. ^ Current publisized world record of pi. Retrieved on 2007-10-14.
4. ^ Statistical estimation of pi using random vectors. Retrieved on 2007-08-12.
5. ^ Eric W. Weisstein, Pi Approximations at MathWorld.
6. ^ Robert Kanigel (1991), The Man Who Knew Infinity: a life of the genius Ramanujan ISBN 0-671-75061-5
7. ^ [3]
8. ^ [4]

Digits
General
Pi (uppercase Π, lower case π) is the sixteenth letter of the Greek alphabet. In the system of Greek numerals it has a value of 80.

In Modern Greek, the name of the letter is pronounced /pi/
Pi, or π, is the mathematical constant equal to a circle's circumference divided by its diameter. Pi may also refer to:

Letter Pi (π)

• Pi (letter), the Greek letter
• Lowercase π:

This article or section is in need of attention from an expert on the subject.
circle is the set of all points in a plane at a fixed distance, called the radius, from a given point, the centre.

Circles are simple closed curves which divide the plane into an interior and exterior.
The circumference is the distance around a closed curve. Circumference is a kind of perimeter.

Circle

The circumference of a circle can be calculated from its diameter using the formula:

diameter (Greek words diairo = divide and metro = measure) of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle.
Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's text Elements is the earliest known systematic discussion of geometry.
A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. Unlike physical constants, mathematical constants are defined independently of any physical measurement.
In mathematics, a transcendental number is a real or complex number which is not algebraic, that is, not a solution of a non-zero polynomial equation, with rational coefficients.

The most prominent examples of transcendental numbers are π and e.
In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero.
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
Physics is the science of matter[1] and its motion[2][3], as well as space and time[4][5] —the science that deals with concepts such as force, energy, mass, and charge.
Engineering is the applied science of acquiring and applying knowledge to design, analysis, and/or construction of works for practical purposes. The American Engineers' Council for Professional Development, also known as ECPD,[1] (later ABET [2]
Archimedes of Syracuse (Greek: Άρχιμήδης c. 287 BC – c. 212 BC) was an ancient Greek mathematician, physicist and engineer.
An Archimedes number (not to be confused with Archimedes' constant, π), named after the ancient Greek scientist Archimedes, to determine the motion of fluids due to density differences, is a dimensionless number in the form:

where:

Ludolph van Ceulen (28 January 1540 – 31 December 1610) was a German mathematician. Born in Hildesheim, Germany. Like many Germans during the Catholic Inquisitions, he emigrated to the Netherlands.

He moved to Delft to teach fencing and mathematics.

Rational numbers

Notable rational numbers

Natural numbers

0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero.
In mathematics, Apéry's constant is a curious number that occurs in a variety of situations. It is defined as the number ζ(3),

where ζ is the Riemann zeta function.
square root of 2, also known as Pythagoras' constant, often denoted by

is the positive real number that, when multiplied by itself, gives the number 2.
The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It denoted by

The first sixty significant digits of its decimal expansion are:
1.

The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. This number appears in the formula for the golden ratio. It can be denoted in surd form as:

It is an irrational algebraic number.
golden section is a line segment sectioned into two according to the golden ratio. The total length a+b is to the longer segment a as a
The Feigenbaum constants are two mathematical constants named after the mathematician Mitchell Feigenbaum. Both express ratios in a bifurcation diagram.

e is the unique real number such that the value of the derivative (slope of the tangent line) of f(x) = ex at the point x = 0 is exactly 1.