Pi

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This article is about the number. For the Greek letter, see Pi (letter). For other uses of pi, π and Π, see Pi (disambiguation).

The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called

π

the mathematical constant $π$
Part of a series of articles on

Uses
Area of disk Circumference Use in other formulae
Properties
Irrationality Transcendence Less than 22/7
Value
Approximations Memorization
People
Archimedes Liu Hui Zu Chongzhi Madhava of Sangamagrama William Jones John Machin John Wrench Ludolph van Ceulen Aryabhata
History
Chronology Book
In culture
Legislation Holiday
Related topics
Squaring the circle Basel problem Tau ( $τ$ ) Other topics related to $π$
v d e

$π$ (sometimes written pi) is a mathematical constant that is the ratio of any Euclidean^[1] circle's circumference to its diameter. $π$ is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve $π$ , which makes it one of the most important mathematical constants.^[2] For instance, the area of a circle is equal to $π$ times the square of the radius of the circle.

$π$ is an irrational number, which means that its value cannot be expressed exactly as a fraction having integers in both the numerator and denominator (unlike 22/7). Consequently, its decimal representation never ends and never repeats. $π$ is also a transcendental number, which implies, among other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can render its value; proving this fact was a significant mathematical achievement of the 19th century.

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. Perhaps because of the simplicity of its definition, $π$ has become more entrenched in popular culture than almost any other mathematical concept,^[3] and is firm common ground between mathematicians and non-mathematicians.^[4] Reports on the latest, most-precise calculation of $π$ are common news items;^[5]^[6]^[7] the record as of September 2011, if verified, stands at 5 trillion decimal digits.^[8]

The Greek letter $π$ was first adopted for the number as an abbreviation of the Greek word for perimeter (περίμετρος), or as an abbreviation for "periphery/diameter", by William Jones in 1706. The constant is also known as Archimedes' Constant, after Archimedes of Syracuse who provided an approximation of the number during the 3rd century BC, although this name is uncommon today. Even rarer is the name Ludolphine number or Ludolph's Constant, after Ludolph van Ceulen, who computed a 35-digit approximation around the year 1600.

Fundamentals

Greek letter

Main article: Pi (letter)

Lower-case

π

denotes the constant. This mosaic is outside the mathematics building at the Technische Universität Berlin.

The Latin name of the Greek letter $π$ is pi.^[9] When referring to the constant, the symbol $π$ is pronounced like the English word "pie", the conventional English pronunciation of the Greek letter.^[10] The constant is named " $π$ " because " $π$ " is the first letter of the Greek word περιφέρεια "periphery"^[11] (or perhaps περίμετρος "perimeter", referring to the ratio of the perimeter to the diameter, which is constant for all circles^[1]). William Jones was the first to use the Greek letter in this way, in 1706,^[12] and it was later popularized by Leonhard Euler in 1737.^[13]^[14] William Jones 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 ... 3.14159, etc. = $π$ ...^[15]

When used as a symbol for the mathematical constant, the Greek letter ( $π$ ) is not capitalized at the beginning of a sentence. The capital letter $Π$ (Pi) has a completely different mathematical meaning; it is used for expressing the product of a sequence.

Geometric definition

When a circle's diameter is 1 unit, its circumference is

π

units.

In Euclidean plane geometry, $π$ is defined as the ratio of a circle's circumference $C$ to its diameter $d$ :^[1]

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

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 $C / d$ .

This definition depends on results of Euclidean geometry, such as the fact that all circles are similar, which can be 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 the trigonometric function cos( $x$ ) equals zero.^[16]

Because

π

is a transcendental number, squaring the circle is not possible in a finite number of steps using the classical tools of compass and straightedge

Irrationality and transcendence

Main article: Proof that π is irrational

$π$ is an irrational number, meaning that it cannot be written as the ratio of two integers. $π$ is also a transcendental number, meaning that there is no polynomial with rational coefficients for which $π$ is a root.^[17] 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 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.^[19]

Decimal representation

Estimating the value

Main article: Approximations of π

Numeral system	Approximation of $π$

Decimal	3.14159265358979323846264338327950288...
Hexadecimal	3.243F6A8885A308D31319...^[27]
Sexagesimal (used by ancients, including Ptolemy's Almagest)	3 ; 8′ 30″^[28] = 377/120
Rational approximations	3, 22/7, 333/106, 355/113, 52163/16604, 103993/33102, ...^[29] (listed in order of increasing accuracy)
Continued fraction	[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1...]^[30] (This fraction is not periodic. Shown in linear notation)
Generalized continued fraction expression	$\cfrac{4}{1 + \cfrac{1^2}{3 + \cfrac{2^2}{5 + \cfrac{3^2}{7 + \ddots}}}}$

3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 3809525720 1065485863 2788659361 5338182796 8230301952 0353018529 6899577362 2599413891 2497217752 8347913151 5574857242 4541506959
A calculation of $π$ accurate to 1120 decimal digits was obtained using a gear-driven calculator in 1948, by John Wrench and Levi Smith. This was the most accurate calculation of $π$ before electronic computers came into use.^[31]

The earliest numerical approximation of $π$ is almost certainly the value 3.^[32]^{[not in citation given]} 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.

$π$ 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,^[33] is to calculate the perimeter, $P$ _$n$, of a regular polygon with $n$ sides circumscribed around a circle with diameter $d$ . Then compute the limit of a sequence as $n$ increases to infinity:

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

This sequence converges because the more sides the polygon has, the smaller its maximum distance from the circle. 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:^[34]

$3\tfrac{10}{71} < \pi < 3\tfrac{10}{70}.$

$π$ can also be calculated using purely mathematical methods. 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 $π$ .^[35] The more terms included in a calculation, the closer to $π$ the result will get.

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:^[36]

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

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 two decimal places.^[37] 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 the sequence

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

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 $π 10,10$ will take similar computation time to computing 150 terms of the original series in a brute-force manner, and $π 10,10 = 3.141592653...$ , correct to 9 decimal places. This computation is an example of the van Wijngaarden transformation.^[38]

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.^{[citation needed]} 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 355/113 (3.1415929...) is the best one that may be expressed with a three-digit or four-digit numerator and denominator; the next good approximation 52163/16604 (3.141592387...), which is also accurate to 7 significant figures, requires much bigger numbers, due to the large number 292 in the continued fraction expansion of $π$ .^[29] For extremely accurate approximations, either Ramanujan's approximation of $\sqrt[4]{\tfrac{2143}{22}}$ ^[39] (3.14159265258...) or 103993/33102^[29] (3.14159265301...) are used, which are both accurate to 10 significant figures.

History

The Great Pyramid at Giza, constructed c.2589–2566 BC, was built with a perimeter of 1760 cubits and a height of 280 cubits giving the ratio 1760/280 ≈ 2 $π$ . The same apotropaic proportions were used earlier at the Pyramid of Meidum c.2613-2589 BC and later in the pyramids of Abusir c.2453-2422. Some Egyptologists consider this to have been the result of deliberate design proportion. Verner wrote, "We can conclude that although the ancient Egyptians could not precisely define the value of $π$ , in practice they used it".^[40] Petrie, author of Pyramids and Temples of Gizeh concluded: "but these relations of areas and of circular ratio are so systematic that we should grant that they were in the builders design".^[41] Others have argued that the Ancient Egyptians had no concept of $π$ and would not have thought to encode it in their monuments. They argued that creation of the pyramid may instead be based on simple ratios of the sides of right-angled triangles (the seked).^[42], however, the Rhind Papyrus in fact shows that the seked was derived from the base and height dimensions, and not the converse^[43], so that the use of the seked system does not negate the conclusions regarding the original dimension and proportion design choices. This means that the so called 'pi theory' remains legitimate, and it has been accepted by many authorities including Petrie, Edwards and Verner^[44].

The early history of $π$ from textual sources roughly parallels the development of mathematics as a whole.^[45]

Antiquity

The earliest known textually evidenced approximations of pi date from around 1900 BC. They are found in the Egyptian Rhind Papyrus 256/81 ≈ 3.160 and on Babylonian tablets 25/8 = 3.125, both within 1 percent of the true value.^[1]

The Indian text Shatapatha Brahmana (composed between the 8th to 6th centuries BCE, Iron Age India)^[46] gives $π$ as 339/108 ≈ 3.139. It has been suggested that passages in the 1 Kings 7:23 and 2 Chronicles 4:2 discussing a ceremonial pool in the temple of King Solomon with a diameter of ten cubits and a circumference of thirty cubits show that the writers considered $π$ to have had an approximate value of three, which various authors have tried to explain away through various suggestions such as a hexagonal pool or an outward curving rim.^[47]

Estimating

π

with inscribed polygons

Estimating

π

with circumscribed and inscribed polygons

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:^[32] By using the equivalent of 96-sided polygons, he proved that $3\tfrac{10}{71} < \pi < 3\tfrac{10}{70}.$ ^[32] 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.^[48]

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 (i.e. a 3072-sided polygon) and obtained an approximate value for $π$ of 3.1416.^[49] Later, Liu Hui invented a quick method of calculating $π$ and obtained an approximate value of 3.14 with only a 96-gon,^[49] 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 (≈ 3.1415929 ), and showed that 3.1415926 < $π$ < 3.1415927^[49] 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.

Maimonides mentions with certainty the irrationality of $π$ in the 12th century.^[50] This was proved in 1768 by Johann Heinrich Lambert.^[51] 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.^[52]^[53] A somewhat earlier similar proof is by Mary Cartwright.^[54]

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:

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

This is now known as the Madhava–Leibniz series^[55]^[56] 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

$\begin{align} \pi &= \sqrt{12}\sum^\infty_{k=0} \frac{(-3)^{-k}}{2k+1} = \sqrt{12}\sum^\infty_{k=0} \frac{(-1)^k}{3^k(2k+1)} \\ &= \sqrt{12}\left({1\over 3^0\cdot1}-{1\over 3^1\cdot3}+{1\over 3^2\cdot5}-{1\over 3^3\cdot7}+\cdots\right), \end{align}$

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.^[57]

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.^[58] $π$ is sometimes called "Ludolph's Constant", though not as often as it is called "Archimedes' Constant."^[59]

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 derived the arcsin series for $π$ in 1665–66 and calculated 15 digits:

$\begin{align} \pi &= 6 \arcsin \frac {1} {2} = 3 \sum_{n=0}^\infty \frac {\binom {2n} n} {16^n(2n+1)} \\ & = 6 \left( \frac {1} {2^1 \cdot 1} + \left( \frac {1} {2} \right) \frac {1} {2^3 \cdot 3} + \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {1} {2^5 \cdot 5} + \left( \frac{1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6 } \right) \frac{1} {2^7 \cdot 7} + \cdots \right) \\ & = 3 + \frac{1}{8} + \frac{9}{640} + \frac{15}{7168} + \frac{35}{98304} + \frac{189}{2883584} + \frac{693}{54525952} + \frac{429}{167772160} + \cdots \end{align}$

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."^[60]^[61]

In 1706 John Machin was the first to compute 100 decimals of $π$ , using the arctan series in thee 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 CC-BY-SA.

Pi

Pi

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