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Pi

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File:Pi-unrolled-720.gif
When a circle's diameter is 1 unit, its circumference is pi units
List of numbersIrrational and suspected irrational numbers
γζ(3)√2√3√5φρδSαeπδ
  
Number systemEvaluation of \pi
Binary11.00100100001111110110…[1]
Decimal3.14159265358979323846264338327950288…
Hexadecimal3.243F6A8885A308D31319…[2]
Rational approximations3, 227, 333106, 355113, 103993/33102, ...[3]

(listed in order of increasing accuracy)

Continued fraction[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, … ][4]

(This continued fraction is not periodic. Shown in linear notation)

Trigonometry\pi radians = 180 degrees

π (sometimes written pi) is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. It is approximately equal to 3.141593 in the usual decimal notation (see the table for its representation in some other bases). Many formulae from mathematics, science, and engineering involve π, which is one of the most important mathematical and physical constants.[5]

π is an irrational number, which means that its value cannot be expressed exactly as a fraction m/n, where m and n are integers. Consequently, its decimal representation never ends or repeats. It is also a transcendental number, which implies, among other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can be equal to its value; proving this was a late achievement in mathematical history and a significant result of 19th century German mathematics. Throughout the history of mathematics, there has been much effort to determine π more accurately and to understand its nature; fascination with the number has even carried over into non-mathematical culture.

The Greek letter π, often spelled out pi in text, was adopted for the number from the Greek word for perimeter "περίμετρος", first by William Jones in 1707, and popularized by Leonhard Euler in 1737.[6]

Contents

Fundamentals

Lower-case π is used to symbolize the constant

The letter π

Circumference = π × diameter

The name of the Greek letter π is pi, and this spelling is commonly used in typographical contexts when the Greek letter is not available or its usage could be problematic. It is not capitalised (Π) even at the beginning of a sentence. When referring to this constant, the symbol π is always pronounced /ˈpaɪ/, "pie" in English, which is the conventional English pronunciation of the Greek letter. In Greek, the name of this letter is pronounced [pi].

The constant is named "π" because "π" is the first letter of the Greek words περιφέρεια (periphery) and περίμετρος (perimeter), probably referring to its use in the formula to find the circumference, or perimeter, of a circle.[7] π is Unicode character U+03C0 ("Greek small letter pi").[8]

Definition

Area of the circle = π × area of the shaded square

In Euclidean plane geometry, π is defined as the ratio of a circle's circumference to its diameter:[7]

 \pi = \frac{C}{d}.

The ratio C/d is constant, regardless of a circle's size. 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.

Alternatively π can be also defined as the ratio of a circle's area (A) to the area of a square whose side is equal to the radius:[7][9]

 \pi = \frac{A}{r^2}.

These definitions depend on results of Euclidean geometry, such as the fact that all circles are similar. This can be considered a problem when π occurs in areas of mathematics that otherwise do not involve geometry. For this reason, mathematicians often prefer to define π without reference to geometry, instead selecting one of its analytic properties as a definition. A common choice is to define π as twice the smallest positive x for which cos(x) = 0.[10] The formulas below illustrate other (equivalent) definitions.

Irrationality and transcendence

Squaring the circle: This was a problem proposed by the ancient geometers. In 1882, it was proven that \pi is transcendental, and consequently this figure cannot be constructed in a finite number of steps with an idealized compass and straightedge.

π is an irrational number, meaning that it cannot be written as the ratio of two integers. The belief in the irrationality of π is mentioned by Muhammad ibn Mūsā al-Khwārizmī[11] in the 9th century. Maimonides also mentions with certainty the irrationality of π in the 12th century.[12] This was proved in 1768 by Johann Heinrich Lambert.[13] In the 20th century, proofs were found that require no prerequisite knowledge beyond integral calculus. One of those, due to Ivan Niven, is widely known.[14][15] A somewhat earlier similar proof is by Mary Cartwright.[16]

π is also a transcendental number, meaning that there is no polynomial with rational coefficients for which π is a root.[17] This was proved by Ferdinand von Lindemann in 1882. 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.[18] This is historically significant, for squaring a circle is one of the easily understood elementary geometry problems left to us from antiquity; many amateurs in modern times have attempted to solve each of these problems, and their efforts are sometimes ingenious, but in this case, doomed to failure: a fact not always understood by the amateur involved.

Decimal representation

The decimal representation of π truncated to 50 decimal places is:[19]

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
See the links below and those at sequence A000796 in OEIS for more digits.

While the decimal representation of π has been computed to more than a trillion (1012) digits,[20] elementary applications, such as estimating the circumference of a circle, will rarely require more than a dozen decimal places. For example, the decimal representation of π truncated to 11 decimal places is good enough to estimate the circumference of any circle that fits inside the earth with an error of less than one millimetre, and the decimal representation of π truncated to 39 decimal places is sufficient to estimate the circumference of any circle that fits in the observable universe with precision comparable to the radius of a hydrogen atom.[21][22]

Because π is an irrational number, its decimal representation does not repeat, and therefore does not terminate. This sequence of non-repeating digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing ever more of these digits and investigating π's properties.[23] Despite much analytical work, and supercomputer calculations that have determined over 1 trillion digits of the decimal representation of π, no simple base-10 pattern in the digits has ever been found.[24] Digits of the decimal representation of π are available on many web pages, and there is software for calculating the decimal representation of π to billions of digits on any personal computer.

Estimating π

π can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter. Another geometry-based approach, attributed to Archimedes,[25] is to calculate the perimeter, Pn , of a regular polygon with n sides circumscribed around a circle with diameter d. Then

\pi = \lim_{n \to \infty}\frac{P_{n}}{d}.

That is, the more sides the polygon has, the closer the approximation approaches π. Archimedes determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon with the perimeter of a regular polygon with the same number of sides inscribed inside the circle. Using a polygon with 96 sides, he computed the fractional range: 3+1071 < π < 3+17.[26]

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:[27]

\pi = 4\sum^\infty_{k=0} \frac{(-1)^k}{2k+1} = \frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}\cdots.\!

While that series is easy to write and calculate, it is not immediately obvious why it yields π. In addition, this series converges so slowly that nearly 300 terms are needed to calculate π correctly to 2 decimal places.[28] However, by computing this series in a somewhat more clever way by taking the midpoints of partial sums, it can be made to converge much faster. Let

\pi_{0,1} = \frac{4}{1},\ \pi_{0,2} =\frac{4}{1}-\frac{4}{3},\ \pi_{0,3} =\frac{4}{1}-\frac{4}{3}+\frac{4}{5},\ \pi_{0,4} =\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}, \cdots\!

and then define

\pi_{i,j} = \frac{\pi_{i-1,j}+\pi_{i-1,j+1}}{2}\text{ for all }i,j\ge 1

then computing \pi_{10,10} will take similar computation time to computing 150 terms of the original series in a brute-force manner, and \pi_{10,10}=3.141592653\ldots, correct to 9 decimal places. This computation is an example of the van Wijngaarden transformation.[29]

History

The Great Pyramid of Giza is estimated to have originally been 280 cubits in height by 440 cubits in length at each side. The ratio of 440/280 is approximately equal to π/2.
Archimedes used the method of exhaustion to approximate the value of π.

The earliest evidenced conscious use of an accurate approximation for the length of a circumference with respect to its radius is of 3+1/7 in the designs of the Old Kingdom pyramids in Egypt. The Great Pyramid at Giza, constructed c.2550-2500 BC, was built with a perimeter of 1760 cubits and a height of 280 cubits; the ratio 1760/280 ≈ 2π. Egyptologists such as Professors Flinders Petrie [30] and I.E.S Edwards[31] have shown that these circular proportions were deliberately chosen for symbolic reasons by the Old Kingdom scribes and architects.[32][33] The same apotropaic proportions were used earlier at the Pyramid of Meidum c.2600 BC. This application is archaeologically evidenced, whereas textual evidence does not survive from this early period.

The early history of π from textual sources roughly parallels the development of mathematics as a whole.[34] Some authors divide progress into three periods: the ancient period during which π was studied geometrically, the classical era following the development of calculus in Europe around the 17th century, and the age of digital computers.[35]

Antiquity

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 earliest known textually evidenced approximations date from around 1900 BC; they are 25/8 (Babylonia) and 256/81 (Egypt), both within 1% of the true value.[7] The Indian text Shatapatha Brahmana gives π as 339/108 ≈ 3.139.

Archimedes' Pi approximation
Liu Hui's π algorithm

Archimedes (287–212 BC) was the first to estimate π rigorously. He realized that its magnitude can be bounded from below and above by inscribing circles in regular polygons and calculating the outer and inner polygons' respective perimeters:[36]

By using the equivalent of 96-sided polygons, he proved that 3 + 10/71 < π < 3 + 1/7.[36] The average of these values is about 3.14185.

Ptolemy, in his Almagest, gives a value of 3.1416, which he may have obtained from Apollonius of Perga.[37]

Around AD 265, the Wei Kingdom mathematician Liu Hui provided a simple and rigorous iterative algorithm to calculate π to any degree of accuracy. He himself carried through the calculation to a 3072-gon and obtained an approximate value for π of 3.1416.[38] Later, Liu Hui invented a quick method of calculating π and obtained an approximate value of 3.14 with only a 96-gon,[38] by taking advantage of the fact that the difference in area of successive polygons forms a geometric series with a factor of 4.

Around 480, the Chinese mathematician Zu Chongzhi demonstrated that π ≈ 355/113, and showed that 3.1415926 < π < 3.1415927[38] using Liu Hui's algorithm applied to a 12288-gon. This value would remain the most accurate approximation of π available for the next 900 years.

Second Millennium AD

Until the second millennium AD, estimations of π were accurate to fewer than 10 decimal digits. The next major advances in the study of π came with the development of infinite series and subsequently with the discovery of calculus, which permit the estimation of π to any desired accuracy by considering sufficiently many terms of a relevant series. Around 1400, Madhava of Sangamagrama found the first known such series:

{\pi} = 4\sum^\infty_{k=0} \frac{(-1)^k}{2k+1} = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \cdots\!

This is now known as the Madhava–Leibniz series[39][40] or Gregory-Leibniz series since it was rediscovered by James Gregory and Gottfried Leibniz in the 17th century. Unfortunately, the rate of convergence is too slow to calculate many digits in practice; about 4,000 terms must be summed to improve upon Archimedes' estimate. However, by transforming the series into

\pi = \sqrt{12}\sum^\infty_{k=0} \frac{(-3)^{-k}}{2k+1} = \sqrt{12}\sum^\infty_{k=0} \frac{(-\frac{1}{3})^k}{2k+1} = \sqrt{12}\left(1-{1\over 3\cdot3}+{1\over5\cdot 3^2}-{1\over7\cdot 3^3}+\cdots\right),

Madhava was able to estimate π as 3.14159265359, which is correct to 11 decimal places. The record was beaten in 1424 by the Persian mathematician, Jamshīd al-Kāshī, who gave an estimate π that is correct to 16 decimal digits.

The first major European contribution since Archimedes was made by the German mathematician Ludolph van Ceulen (1540–1610), who used a geometric method to give an estimate of π that is correct to 35 decimal digits. He was so proud of the calculation, which required the greater part of his life, that he had the digits engraved into his tombstone.[41]

Around the same time, the methods of calculus and determination of infinite series and products for geometrical quantities began to emerge in Europe. The first such representation was the Viète's formula,

\frac2\pi = \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdot \cdots\!

found by François Viète in 1593. Another famous result is Wallis' product,

\frac{\pi}{2} = \prod^\infty_{k=1} \frac{(2k)^2}{(2k)^2-1} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots\ = \frac{4}{3} \cdot \frac{16}{15} \cdot \frac{36}{35} \cdot \frac{64}{63} \cdots\!

by John Wallis in 1655. Isaac Newton himself derived a series for π and calculated 15 digits, although he later confessed: "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."[42]

In 1706 John Machin was the first to compute 100 decimals of π, using the formula

\frac{\pi}{4} = 4 \, \arctan \frac{1}{5} - \arctan \frac{1}{239}\!

with

\arctan \, x = \sum^\infty_{k=0} \frac{(-1)^k x^{2k+1}}{2k+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots\!

Formulas of this type, now known as Machin-like formulas, were used to set several successive records and remained the best known method for calculating π well into the age of computers. A remarkable record was set by the calculating prodigy Zacharias Dase, who in 1844 employed a Machin-like formula to calculate 200 decimals of π in his head at the behest of Gauss. The best value at the end of the 19th century was due to William Shanks, who took 15 years to calculate π with 707 digits, although due to a mistake only the first 527 were correct. (To avoid such errors, modern record calculations of any kind are often performed twice, with two different formulas. If the results are the same, they are likely to be correct.)

Theoretical advances in the 18th century led to insights about π's nature that could not be achieved through numerical calculation alone. Johann Heinrich Lambert proved the irrationality of π in 1761, and Adrien-Marie Legendre also proved in 1794 π2 to be irrational. When Leonhard Euler in 1735 solved the famous Basel problem – finding the exact value of

 \sum^\infty_{k=1} \frac{1}{k^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots\!

which is π2/6, he established a deep connection between π and the prime numbers. Both Legendre and Euler speculated that π might be transcendental, which was finally proved in 1882 by Ferdinand von Lindemann.

William Jones' book A New Introduction to Mathematics from 1706 is said to be the first use of the Greek letter π for this constant, but the notation became particularly popular after Leonhard Euler adopted it in 1737.[43] He wrote:

There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to (16/5 − 4/239) − 1/3(16/53 − 4/2393) + ... = 3.14159... = π[7]

Computation in the computer age

Although practically a physicist needs only 39 digits of Pi to make a circle the size of the observable universe accurate to one atom of hydrogen, the number itself as a mathematical curiosity has created many challenges in different fields.

The advent of digital computers in the 20th century led to an increased rate of new π calculation records. John von Neumann et al. used ENIAC to compute 2037 digits of π in 1949, a calculation that took 70 hours.[44][45] Additional thousands of decimal places were obtained in the following decades, with the million-digit milestone passed in 1973. Progress was not only due to faster hardware, but also new algorithms. One of the most significant developments was the discovery of the fast Fourier transform (FFT) in the 1960s, which allows computers to perform arithmetic on extremely large numbers quickly.

In the beginning of the 20th century, the Indian mathematician Srinivasa Ramanujan found many new formulas for π, some remarkable for their elegance and mathematical depth.[46] One of his formulas is the series,

\frac{1}{\pi} = \frac{2 \sqrt 2}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!

and the related one found by the Chudnovsky brothers in 1987,

\frac{426880 \sqrt{10005}}{\pi} = \sum_{k=0}^\infty \frac{(6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 (-640320)^{3k}}\!

which deliver 14 digits per term.[46] The Chudnovskys used this formula to set several π computing records in the end of the 1980s, including the first calculation of over one billion (1,011,196,691) decimals in 1989. It remains the formula of choice for π calculating software that runs on personal computers, as opposed to the supercomputers used to set modern records.

Whereas series typically increase the accuracy with a fixed amount for each added term, there exist iterative algorithms that multiply the number of correct digits at each step, with the downside that each step generally requires an expensive calculation. A breakthrough was made in 1975, when Richard Brent and Eugene Salamin independently discovered the Brent–Salamin algorithm, which uses only arithmetic to double the number of correct digits at each step.[47] The algorithm consists of setting

a_0 = 1 \quad \quad \quad b_0 = \frac{1}{\sqrt 2} \quad \quad \quad t_0 = \frac{1}{4} \quad \quad \quad p_0 = 1\!

and iterating

a_{n+1} = \frac{a_n+b_n}{2} \quad \quad \quad b_{n+1} = \sqrt{a_n b_n}\!
t_{n+1} = t_n - p_n (a_n-a_{n+1})^2 \quad \quad \quad p_{n+1} = 2 p_n\!

until an and bn are close enough. Then the estimate for π is given by

\pi \approx \frac{(a_n + b_n)^2}{4 t_n}.\!

Using this scheme, 25 iterations suffice to reach 45 million correct decimals. A similar algorithm that quadruples the accuracy in each step has been found by Jonathan and Peter Borwein.[48] The methods have been used by Yasumasa Kanada and team to set most of the π calculation records since 1980, up to a calculation of 206,158,430,000 decimals of π in 1999. As of January 2010, the record is almost 2.7 trillion digits.[49] This beats the previous record of 2,576,980,370,000 decimals, set by Daisuke Takahashi on the T2K-Tsukuba System, a supercomputer at the University of Tsukuba northeast of Tokyo.[50]

An important recent development was the Bailey–Borwein–Plouffe formula (BBP formula), discovered by Simon Plouffe and named after the authors of the paper in which the formula was first published, David H. Bailey, Peter Borwein, and Simon Plouffe.[51] The formula,

\pi = \sum_{k=0}^\infty \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6}\right),

is remarkable because it allows extracting any individual hexadecimal or binary digit of π without calculating all the preceding ones.[51] Between 1998 and 2000, the distributed computing project PiHex used a modification of the BBP formula due to Fabrice Bellard to compute the quadrillionth (1,000,000,000,000,000:th) bit of π, which turned out to be 0.[52]

If a formula of the form

\pi = \sum_{k=0}^\infty \frac{1}{b^{ck}} \frac{p(k)}{q(k)},

were found where b and c are positive integers and p and q are polynomials with fixed degree and integer coefficients (as in the BPP formula above), this would be one the most efficient ways of computing any digit of π at any position in base bc without computing all the preceding digits in that base, in a time just depending on the size of the integer k and on the fixed degree of the polynomials. Plouffe also describes such formulas as the interesting ones for computing numbers of class SC*, in a logarithmically polynomial space and almost linear time, depending only on the size (order of magnitude) of the integer k, and requiring modest computing resources. The previous formula (found by Plouffe for π with b=2 and c=4, but also found for log(9/10) and for a few other irrational constants), implies that π is a SC* number.

In 2006, Simon Plouffe, using the integer relation algorithm PSLQ, found a series of formulas.[53] Let q = eπ (Gelfond's constant), then

\frac{\pi}{24} = \sum_{n=1}^\infty \frac{1}{n} \left(\frac{3}{q^n-1} - \frac{4}{q^{2n}-1} + \frac{1}{q^{4n}-1}\right)
\frac{\pi^3}{180} = \sum_{n=1}^\infty \frac{1}{n^3} \left(\frac{4}{q^n-1} - \frac{5}{q^{2n}-1} + \frac{1}{q^{4n}-1}\right)

and others of form,

\pi^k = \sum_{n=1}^\infty \frac{1}{n^k} \left(\frac{a}{q^n-1} + \frac{b}{q^{2n}-1} + \frac{c}{q^{4n}-1}\right)

where k is an odd number, and abc are rational numbers.

In the previous formula, if k is of the form 4m + 3, then the formula has the particularly simple form,

p\pi^k = \sum_{n=1}^\infty \frac{1}{n^k} \left(\frac{2^{k-1}}{q^n-1} - \frac{2^{k-1}+1}{q^{2n}-1} + \frac{1}{q^{4n}-1}\right)

for some rational number p where the denominator is a highly factorable number, though no rigorous proof has yet been given.

Pi and continued fraction

The sequence of partial denominators of the simple continued fraction of π does not show any obvious pattern:[4]

\pi=[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,1,84,\cdots]

or

\pi=3+\frac{1}{7+\textstyle \frac{1}{15+\textstyle \frac{1}{1+\textstyle \frac{1}{292+\textstyle \frac{1}{1+\textstyle \frac{1}{1+\textstyle \frac{1}{1+\textstyle \frac{1}{2+\textstyle \frac{1}{1+\textstyle \frac{1}{3+\textstyle \frac{1}{1+\textstyle \frac{1}{14+\textstyle \frac{1}{2+\textstyle \frac{1}{1+\textstyle \frac{1}{1+\ddots}}}}}}}}}}}}}}}

However, there are generalized continued fractions for π with a perfectly regular structure, such as:[54]

\pi=\frac{4}{1+\textstyle \frac{1^2}{2+\textstyle \frac{3^2}{2+\textstyle \frac{5^2}{2+\textstyle \frac{7^2}{2+\textstyle \frac{9^2}{2+\ddots}}}}}}=3+\frac{1^2}{6+\textstyle \frac{3^2}{6+\textstyle \frac{5^2}{6+\textstyle \frac{7^2}{6+\textstyle \frac{9^2}{6+\ddots}}}}}=\frac{4}{1+\textstyle \frac{1^2}{3+\textstyle \frac{2^2}{5+\textstyle \frac{3^2}{7+\textstyle \frac{4^2}{9+\ddots}}}}}

Memorizing digits

Recent decades have seen a surge in the record for number of digits memorized.

Even long before computers have calculated π, memorizing a record number of digits became an obsession for some people.In 2006, Akira Haraguchi, a retired Japanese engineer, claimed to have recited 100,000 decimal places.[55] 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.[56] It took him 24 hours and 4 minutes to recite to the 67,890th decimal place of π without an error.[57]

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, originally devised by Sir James Jeans: How I need (or: want) a drink, alcoholic in nature (or: of course), after the heavy lectures (or: chapters) involving quantum mechanics.[58][59] 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 3835 digits of π in this manner.[60] 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. In other languages there are similar methods of memorization. However, this method proves inefficient for large memorizations of π. Other methods include remembering patterns in the numbers and the method of loci.[61][62]

Advanced properties

Numerical approximations

Due to the transcendental nature of π, there are no closed form expressions for the number in terms of algebraic numbers and functions.[17] Formulas for calculating π using elementary arithmetic typically include series or summation notation (such as "..."), which indicates that the formula is really a formula for an infinite sequence of approximations to π.[63] The more terms included in a calculation, the closer to π the result will get.

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 next good approximation 103993/33102 (3.14159265301...) requires much bigger numbers, due to the large number 292 in the continued fraction expansion.[3]

The earliest numerical approximation of π is almost certainly the value 3.[36] 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.

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 integer base, not just base 10.[64] 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,[65] although it is clear that at least two such digits must occur infinitely often, since otherwise π would be rational, which it is not.

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.[66]

It is also unknown whether π and e are algebraically independent, although Yuri Nesterenko proved the algebraic independence of {π, eπ, Γ(1/4)} in 1996.[67]

Use in mathematics and science

π is ubiquitous in mathematics, appearing even in places that lack an obvious connection to the circles of Euclidean geometry.[68]

Geometry and trigonometry

For any circle with radius r and diameter d = 2r, the circumference is πd and the area is πr2. Further, π appears in formulas for areas and volumes of many other geometrical shapes based on circles, such as ellipses, spheres, cones, and tori.[69] Accordingly, π appears in definite integrals that describe circumference, area or volume of shapes generated by circles. In the basic case, half the area of the unit disk is given by:[70]\int_{-1}^1 \sqrt{1-x^2}\,dx = \frac{\pi}{2}and

\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\,dx = \pi

gives half the circumference of the unit circle.[69] More complicated shapes can be integrated as solids of revolution.[71]

From the unit-circle definition of the trigonometric functions also follows that the sine and cosine have period 2π. That is, for all x and integers n, sin(x) = sin(x + 2πn) and cos(x) = cos(x + 2πn). Because sin(0) = 0, sin(2πn) = 0 for all integers n. Also, the angle measure of 180° is equal to π radians. In other words, 1° = (π/180) radians.

In modern mathematics, π is often defined using trigonometric functions, for example as the smallest positive x for which sin x = 0, to avoid unnecessary dependence on the subtleties of Euclidean geometry and integration. Equivalently, π can be defined using the inverse trigonometric functions, for example as π = 2 arccos(0) or π = 4 arctan(1). Expanding inverse trigonometric functions as power series is the easiest way to derive infinite series for π.

Complex numbers and calculus

Euler's formula depicted on the complex plane. Increasing the angle φ to π radians (180°) yields Euler's identity.

A complex number z can be expressed in polar coordinates as follows:

z = r\cdot(\cos\varphi + i\sin\varphi)

The frequent appearance of π in complex analysis can be related to the behavior of the exponential function of a complex variable, described by Euler's formula

e^{i\varphi} = \cos \varphi + i\sin \varphi \!

where i is the imaginary unit satisfying i2 = −1 and e ≈ 2.71828 is Euler's number. This formula implies that imaginary powers of e describe rotations on the unit circle in the complex plane; these rotations have a period of 360° = 2π. In particular, the 180° rotation φ = π results in the remarkable Euler's identity

e^{i \pi} = -1.\!

e^{i \pi} + 1 = 0.\!

Euler's identity is famous for linking several basic mathematical constants and operators.

There are n different n-th roots of unity

e^{2 \pi i k/n} \qquad (k = 0, 1, 2, \dots, n - 1).

The Gaussian integral

\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}.

A consequence is that the gamma function of a half-integer is a rational multiple of √π.

Physics

Although not a physical constant, 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. Using units such as Planck units can sometimes eliminate π from formulae.

\Lambda = {{8\pi G} \over {3c^2}} \rho
 \Delta x\, \Delta p \ge \frac{h}{4\pi}
 R_{ik} - {g_{ik} R \over 2} + \Lambda g_{ik} = {8 \pi G \over c^4} T_{ik}
 F = \frac{\left|q_1q_2\right|}{4 \pi \varepsilon_0 r^2}
 \mu_0 = 4 \pi \cdot 10^{-7}\,\mathrm{N/A^2}\,
\frac{P^2}{a^3}={(2\pi)^2 \over G (M+m)}

Probability and statistics

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

f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-(x-\mu )^2/(2\sigma^2)}
f(x) = \frac{1}{\pi (1 + x^2)}.

Note that since \int_{-\infty}^{\infty} f(x)\,dx = 1 for any probability density function f(x), the above formulas can be used to produce other integral formulas for π.[79]

Buffon's needle problem is sometimes quoted as a empirical approximation of π in "popular mathematics" works. 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 the Monte Carlo method:[80][81][82][83]\pi \approx \frac{2nL}{xS}.Though this result is mathematically impeccable, it cannot be used to determine more than very few digits of π by experiment. Reliably getting just three digits (including the initial "3") right requires millions of throws,[80] and the number of throws grows exponentially with the number of digits desired. Furthermore, any error in the measurement of the lengths L and S will transfer directly to an error in the approximated π. For example, a difference of a single atom in the length of a 10-centimeter needle would show up around the 9th digit of the result. In practice, uncertainties in determining whether the needle actually crosses a line when it appears to exactly touch it will limit the attainable accuracy to much less than 9 digits.

Geomorphology and chaos theory

Under ideal conditions (uniform gentle slope on an homogeneously erodable substrate), the ratio between the actual length of a river and its straight-line from source to mouth length tends to approach π.[84] Albert Einstein was the first to suggest that rivers have a tendency towards an ever more loopy path because the slightest curve will lead to faster currents on the outer side, which in turn will result in more erosion and a sharper bend. The sharper the bend, the faster the currents on the outer edge, the more the erosion, the more the river will twist and so on. However, increasing loopiness will result in rivers doubling back on themselves and effectively short-circuiting, creating an ox-bow lake. The balance between these two opposing factors leads to an average ratio of π between the actual length and the direct distance between source and mouth.[85]

In popular culture

A whimsical "Pi plate"

Probably because of the simplicity of its definition, the concept of pi and, especially its decimal expression, have become entrenched in popular culture to a degree far greater than almost any other mathematical construct.[86] It is, perhaps, the most common ground between mathematicians and non-mathematicians.[87] Reports on the latest, most-precise calculation of π (and related stunts) are common news items.[88][89][90]

Pi Day (March 14, from 3.14) is observed in many schools.[91] At least one cheer at the Massachusetts Institute of Technology includes "3.14159!"[92]

On November 7, 2005, alternative musician Kate Bush released the album, Aerial. The album contains the song "π" whose lyrics consist principally of Bush singing the digits of π to music, beginning with "3.14"[93]

In Carl Sagan's novel Contact, pi played a key role in the story and suggested that there was a message buried deep within the digits of pi placed there by whoever created the universe. This part of the story was left out of the film adaption of the novel.

See also

References

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