Pi

A unit circle is rolled across a surface; after a full revolution, the center of the circle has moved π units forwards.

The number $π$ (/paɪ/ ^ⓘ; spelled out as pi) is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter. It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining $π$ , to avoid relying on the definition of the length of a curve.

The number $π$ is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as $⁠ 22 / 7 ⁠$ are commonly used to approximate it. Consequently, its decimal representation never ends, nor does it enter a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an algebraic equation involving only finite sums, products, powers, and integers. The transcendence of $π$ implies that it is impossible to solve the ancient problem of squaring the circle with a compass and straightedge. The decimal digits of $π$ appear to be evenly distributed, but no proof of this conjecture has been found.

Mathematicians have attempted to extend their understanding of $π$ , sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of $π$ for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate $π$ with arbitrary accuracy. In the 5th century AD, Chinese mathematicians approximated $π$ to seven digits, while Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula for $π$ , based on infinite series, was discovered a millennium later.

The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician William Jones in 1706. The invention of calculus soon led to the calculation of hundreds of digits of $π$ , enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and computer scientists have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of $π$ to hundreds of trillions of digits. These computations are motivated by the development of efficient algorithms to calculate numeric series, as well as the human quest to break records. The extensive computations involved have also been used to test the correctness of new computer processors.

Because it relates to a circle, $π$ is found in formulae in trigonometry and geometry, especially those concerning circles, ellipses and spheres. It is found as well in formulae from cosmology, fractals, thermodynamics, mechanics, and electromagnetism. It also appears in areas such as number theory, statistics, and in modern mathematical analysis: $π$ is ubiquitous.

Fundamentals

\pi

is normal

Transcendence

In addition to being irrational, $π$ is also a transcendental number, which means that it is not the solution of any non-constant polynomial equation with rational coefficients, such as ${\textstyle {\frac {x^{5}}{120}}-{\frac {x^{3}}{6}}+x=0}$ .^[17]^[b] This follows from the so-called Lindemann–Weierstrass theorem, which also establishes the transcendence of the constant $e$ .^[17]

The transcendence of $π$ has two important consequences: First, $π$ cannot be expressed using any finite combination of rational numbers and square roots or nth roots (such as ${\sqrt[{3}]{31}}$ or ${\sqrt {10}}$ ). Second, since no transcendental number can be constructed with compass and straightedge, it is not possible to "square the circle". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is exactly equal to the area of a given circle.^[18] Squaring a circle was one of the important geometry problems of the classical antiquity.^[19] Amateur mathematicians in modern times have sometimes attempted to square the circle and claim success – despite the fact that it is mathematically impossible.^[20]

An unsolved problem thus far is the question of whether or not the numbers $π$ and $e$ are algebraically independent ("relatively transcendental"). This would be resolved by Schanuel's conjecture^[21] – a currently unproven generalization of the Lindemann–Weierstrass theorem.^[22]

Continued fractions

As an irrational number, $π$ cannot be represented as a common fraction. However, like every number, $π$ can be represented by an infinite series of nested fractions, called a simple continued fraction:

$\pi =3+\textstyle {\cfrac {1}{7+\textstyle {\cfrac {1}{15+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{292+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{1+\ddots }}}}}}}}}}}}}}$

Truncating the continued fraction at any point yields a rational approximation for $π$ ; the first four of these are $3$ , $⁠ 22 / 7 ⁠$ , $⁠ 333 / 106 ⁠$ , and $⁠ 355 / 113 ⁠$ . These numbers are among the best-known and most widely used historical approximations of the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to $π$ than any other fraction with the same or a smaller denominator.^[23] Because $π$ is transcendental, it is by definition not algebraic and so cannot be a quadratic irrational. Therefore, $π$ cannot have a periodic continued fraction. Although the simple continued fraction for $π$ (with numerators all 1, shown above) also does not exhibit any other obvious pattern,^[24]^[25] several non-simple continued fractions do, such as:^[26]^[c]

${\begin{aligned}\pi &=3+{\cfrac {1^{2}}{6+{\cfrac {3^{2}}{6+{\cfrac {5^{2}}{6+{\cfrac {7^{2}}{6+\ddots }}}}}}}}={\cfrac {4}{1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+\ddots }}}}}}}}={\cfrac {4}{1+{\cfrac {1^{2}}{3+{\cfrac {2^{2}}{5+{\cfrac {3^{2}}{7+\ddots }}}}}}}}\end{aligned}}$

Approximate value and digits

Some approximations of pi include:

Digits in other number systems

Complex numbers and Euler's identity

Any complex number, say $z$ , can be expressed using a pair of real numbers. In the polar coordinate system, one number (radius or $r$ ) is used to represent $z$ 's distance from the originof the complex plane, and the other (angle or $φ$ ) the counter-clockwise rotation from the positive real line:^[30]

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

where $i$ is the imaginary unit satisfying $i^{2}=-1$ . The frequent appearance of $π$ in complex analysis can be related to the behaviour of the exponential function of a complex variable, described by Euler's formula:^[31]

$e^{i\varphi }=\cos \varphi +i\sin \varphi ,$

where the constant $e$ is the base of the natural logarithm. This formula establishes a correspondence between imaginary powers of $e$ and points on the unit circle centred at the origin of the complex plane. Setting $\varphi =\pi$ in Euler's formula results in Euler's identity, celebrated in mathematics due to it containing five important mathematical constants:^[31]^[32]

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

There are $n$ different complex numbers $z$ satisfying ⁠ $z^{n}=1$ ⁠, and these are called the " $n$ th roots of unity"^[33] and are given by the formula:

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

History

Surviving approximations of $π$ prior to the 2nd century CE are accurate to one or two decimal places at best. The earliest written approximations are found in Babylon and Egypt, both within one percent of the true value. In Babylon, a clay tablet dated 1900–1600 BCE has a geometrical statement that, by implication, treats $π$ as $⁠ 25 / 8 ⁠ = 3.125$ .^[34] In Egypt, the Rhind Papyrus, dated around 1650 BCE but copied from a document dated to 1850 BCE, has a formula for the area of a circle that treats $π$ as ${\textstyle {\bigl (}{\frac {16}{9}}{\bigr )}^{2}\approx 3.16}$ .^[25]^[34] Although some pyramidologists have theorized that the Great Pyramid of Giza was built with proportions related to $π$ , this theory is not widely accepted by scholars.^[35] In the Shulba Sutras of Indian mathematics, dating to an oral tradition from the 1st or 2nd millennium BCE, approximations are given which have been variously interpreted as approximately 3.08831, 3.08833, 3.004, 3, or 3.125.^[36]

Polygon approximation era

The first recorded algorithm for rigorously calculating the value of $π$ was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes, implementing the method of exhaustion.^[37] This polygonal algorithm dominated for over 1,000 years, and as a result $π$ is sometimes referred to as Archimedes's constant.^[38]Archimedes computed upper and lower bounds of $π$ by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that $⁠ 223 / 71 ⁠ < π < ⁠ 22 / 7 ⁠$ (that is, $3.1408 < π < 3.1429$ ).^[39] Archimedes' upper bound of $⁠ 22 / 7 ⁠$ may have led to a widespread popular belief that $π$ is equal to $⁠ 22 / 7 ⁠$ .^[40] Around 150 AD, Greco-Roman scientist Ptolemy, in his Almagest, gave a value for $π$ of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga.^[41]^[42]

In ancient China, values for $π$ included 3.1547 (around 1 AD), ${\sqrt {10}}$ (100 AD, approximately 3.1623), and $⁠ 142 / 45 ⁠$ (3rd century, approximately 3.1556).^[43] Around 265 AD, the Cao Weimathematician Liu Hui created a polygon-based iterative algorithm, with which he constructed a 3,072-sided polygon to approximate $π$ as 3.1416.^[44]^[45] Liu later invented a faster method of calculating $π$ and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4.^[44] Around 480 AD, Zu Chongzhi calculated that $3.1415926<\pi <3.1415927$ and suggested the approximations ${\textstyle \pi \approx {\frac {355}{113}}=3.14159292035\ldots }$ and ${\textstyle \pi \approx {\frac {22}{7}}=3.142857142857\ldots }$ , which he termed the milü ('close ratio') and yuelü('approximate ratio') respectively, iterating with Liu Hui's algorithm up to a 12,288-sided polygon. With a correct value for its seven first decimal digits, Zu's result remained the most accurate approximation of $π$ for the next 800 years.^[46]

The Indian astronomer Aryabhata used a value of 3.1416 in his Āryabhaṭīya (499 AD).^[47]Around 1220, Fibonacci computed 3.1418 using a polygonal method devised independently of Archimedes.^[48] Italian author Danteapparently employed the value ${\textstyle 3+{\frac {\sqrt {2}}{10}}\approx 3.14142}$ .^[48]

The Persian astronomer Jamshīd al-Kāshī produced nine sexagesimal digits, roughly the equivalent of 16 decimal digits, in 1424, using a polygon with ${\textstyle 3\times 2^{28}}$ sides,^[49] which stood as the world record for about 180 years.^[50] French mathematician François Viètein 1579 achieved nine digits with a polygon of ${\textstyle 3\times 2^{17}}$ sides.^[50] Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593.^[50] In 1596, Dutch mathematician Ludolph van Ceulen reached 20 digits, a record he later increased to 35 digits (as a result, $π$ was called the "Ludolphian number" in Germany until the early 20th century).^[51] Dutch scientist Willebrord Snellius reached 34 digits in 1621,^[52] and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630 using 10⁴⁰ sides.^[53] Christiaan Huygens was able to arrive at 10 decimal places in 1654 using a slightly different method equivalent to Richardson extrapolation.^[54]

Infinite series

The calculation of $π$ was revolutionized by the development of infinite seriestechniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence. Infinite series allowed mathematicians to compute $π$ with much greater precision than Archimedes and others who used geometrical techniques.^[55]Although infinite series were exploited for $π$ most notably by European mathematicians such as James Gregory and Gottfried Wilhelm Leibniz, the approach also appeared in the Kerala school sometime in the 14th or 15th century.^[56]^[57] Around 1500, an infinite series that could be used to compute $π$ , written in the form of Sanskrit verse, was presented in Tantrasamgraha by Nilakantha Somayaji.^[56] The series are presented without proof, but proofs are presented in the later work Yuktibhāṣā, published around 1530. Several infinite series are described, including series for sine (which Nilakantha attributes to Madhava of Sangamagrama), cosine, and arctangent which are now sometimes referred to as Madhava series. The series for arctangent is sometimes called Gregory's series or the Gregory–Leibniz series.^[56] Madhava used infinite series to estimate $π$ to 11 digits around 1400.^[58]^[59]^[60]

In 1593, François Viète published what is now known as Viète's formula, an infinite product (rather than an infinite sum, which is more typically used in $π$ calculations):^[61]

${\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots$

In 1655, John Wallis published what is now known as the Wallis product, also an infinite product:^[62]

${\frac {\pi }{2}}={\Big (}{\frac {2}{1}}\cdot {\frac {2}{3}}{\Big )}\cdot {\Big (}{\frac {4}{3}}\cdot {\frac {4}{5}}{\Big )}\cdot {\Big (}{\frac {6}{5}}\cdot {\frac {6}{7}}{\Big )}\cdot {\Big (}{\frac {8}{7}}\cdot {\frac {8}{9}}{\Big )}\cdots$

A formal portrait of a man, with long hair — Isaac Newton used infinite series to compute $π$ to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".^[63]

In the 1660s, the English scientist Isaac Newton and German mathematician Gottfried Wilhelm Leibniz discovered calculus, which led to the development of many infinite series for approximating $π$ . Newton himself used an arcsine series to compute a 15-digit approximation of $π$ in 1665 or 1666, writing, "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."^[63]

In 1671, James Gregory, and independently, Leibniz in 1673, discovered the Taylor series expansion for arctangent:^[56]^[64]^[65]

$\arctan z=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots$

This series, sometimes called the Gregory–Leibniz series, equals ${\textstyle {\frac {\pi }{4}}}$ when evaluated with ⁠ $z=1$ ⁠.^[65]But for ⁠ ${1}$ ⁠, it converges impractically slowly (that is, approaches the answer very gradually), taking about ten times as many terms to calculate each additional digit.^[66]

In 1699, English mathematician Abraham Sharp used the Gregory–Leibniz series for ${\textstyle z={\frac {1}{\sqrt {3}}}}$ to compute $π$ to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.^[67]

In 1706, John Machin used the Gregory–Leibniz series to produce an algorithm that converged much faster:^[68]^[69]^[70]

${\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}.$

Machin reached 100 digits of $π$ with this formula.^[71] Other mathematicians created variants, now known as Machin-like formulae, that were used to set several successive records for calculating digits of $π$ .^[72]^[71]

Isaac Newton accelerated the convergence of the Gregory–Leibniz series in 1684 (in an unpublished work; others independently discovered the result):^[73]

$\arctan x={\frac {x}{1+x^{2}}}+{\frac {2}{3}}{\frac {x^{3}}{(1+x^{2})^{2}}}+{\frac {2\cdot 4}{3\cdot 5}}{\frac {x^{5}}{(1+x^{2})^{3}}}+\cdots$

Leonhard Euler popularized this series in his 1755 differential calculus textbook, and later used it with Machin-like formulae, including ${\textstyle {\tfrac {\pi }{4}}=5\arctan {\tfrac {1}{7}}+2\arctan {\tfrac {3}{79}},}$ with which he computed 20 digits of $π$ in one hour.^[74]

Machin-like formulae remained the best-known method for calculating $π$ well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson – the best approximation achieved without the aid of a calculating device.^[75]

In 1844, a record was set by Zacharias Dase, who employed a Machin-like formula to calculate 200 decimals of $π$ in his head at the behest of German mathematician Carl Friedrich Gauss.^[76]

In 1853, British mathematician William Shanks calculated $π$ to 607 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect. Though he calculated an additional 100 digits in 1873, bringing the total up to 707, his previous mistake rendered all the new digits incorrect as well.^[77]

Rate of convergence

Some infinite series for $π$ converge faster than others. Given the choice of two infinite series for $π$ , mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate $π$ to any given accuracy.^[78] A simple infinite series for $π$ is the Gregory–Leibniz series:^[79]

$\pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}-\cdots$

As individual terms of this infinite series are added to the sum, the total gradually gets closer to $π$ , and – with a sufficient number of terms – can get as close to $π$ as desired. It converges quite slowly, though – after 500,000 terms, it produces only five correct decimal digits of $π$ .^[80]

An infinite series for $π$ (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory–Leibniz series is:^[81]^[82]

$\pi =3+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-{\frac {4}{8\times 9\times 10}}+\cdots$

The following table compares the convergence rates of these two series:

Infinite series for $π$	After 1st term	After 2nd term	After 3rd term	After 4th term	After 5th term	Converges to:
$\pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}+\cdots$	4.0000	2.6666 ...	3.4666 ...	2.8952 ...	3.3396 ...	$π$ = 3.1415 ...
$\pi ={3}+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-\cdots$	3.0000	3.1666 ...	3.1333 ...	3.1452 ...	3.1396 ...	$π$ = 3.1415 ...

After five terms, the sum of the Gregory–Leibniz series is within 0.2 of the correct value of $π$ , whereas the sum of Nilakantha's series is within 0.002 of the correct value. Nilakantha's series converges faster and is more useful for computing digits of $π$ . Series that converge even faster include Machin's series and Chudnovsky's series, the latter producing 14 correct decimal digits per term.^[78]

Irrationality and transcendence

Not all mathematical advances relating to $π$ were aimed at increasing the accuracy of approximations. When Euler solved the Basel problem in 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between $π$ and the prime numbers that later contributed to the development and study of the Riemann zeta function:^[83]

${\frac {\pi ^{2}}{6}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots$

Swiss scientist Johann Heinrich Lambert in 1768 proved that $π$ is irrational, meaning it is not equal to the quotient of any two integers.^[12] Lambert's proof exploited a continued-fraction representation of the tangent function.^[84] French mathematician Adrien-Marie Legendre proved in 1794 that $π$ ² is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that $π$ is transcendental,^[85] confirming a conjecture made by both Legendre and Euler.^[86]^[87]^[d]

Adoption of the symbol $π$

The earliest known use of the Greek letter

π

to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician William Jones in 1706.

Leonhard Eulerpopularized the use of the Greek letter

π

in works he published in 1736 and 1748.

The first recorded use of the symbol $π$ in circle geometry is in Oughtred's Clavis Mathematicae (1648),^[88]^[89] where the Greek letters $π$ and δ were combined into the fraction ⁠ ${\tfrac {\pi }{\delta }}$ ⁠ for denoting the ratios semiperimeter to semidiameter and perimeter to diameter, that is, what is presently denoted as $π$ .^[1]^[90]^[91]^[92] (Before then, mathematicians sometimes used letters such as c or p instead.^[89]) Barrow likewise used the same notation,^[93] while Gregory instead used ${\textstyle {\frac {\pi }{\rho }}}$ to represent $6.28...$ .^[94]^[91]

The earliest known use of the Greek letter $π$ alone to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician William Jones in his 1706 work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics.^[68]^[95] The Greek letter appears on p. 243 in the phrase " ${\textstyle {\tfrac {1}{2}}}$ Periphery ( $π$ )", calculated for a circle with radius one. However, Jones writes that his equations for $π$ are from the "ready pen of the truly ingenious Mr. John Machin", leading to speculation that Machin may have employed the Greek letter before Jones.^[89] Jones' notation was not immediately adopted by other mathematicians, with the fraction notation still being used as late as 1767.^[90]^[96]

Euler started using the single-letter form beginning with his 1727 Essay Explaining the Properties of Air, though he used $π = 6.28...$ , the ratio of periphery to radius, in this and some later writing.^[97] Euler first used $π$ = 3.14... in his 1736 work Mechanica,^[98] and continued in his widely read 1748 work Introductio in analysin infinitorum (he wrote: "for the sake of brevity we will write this number as $π$ ; thus $π$ is equal to half the circumference of a circle of radius $1$ ").^[99] Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly, and the practice was universally adopted thereafter in the Western world,^[89] though the definition still varied between $3.14...$ and $6.28...$ as late as 1761.^[100]

Modern quest for more digits

Motives for computing $π$

For most numerical calculations involving $π$ , a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmological calculations, because that is the accuracy necessary to calculate the circumference of the observable universe with a precision of one atom. Accounting for additional digits needed to compensate for computational round-off errors, Arndt concludes that a few hundred digits would suffice for any scientific application. Despite this, people have worked strenuously to compute $π$ to thousands and millions of digits.^[101] This effort may be partly ascribed to the human compulsion to break records, and such achievements with $π$ often make headlines around the world.^[102] They also have practical benefits, such as testing supercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms) – and within pure mathematics itself, providing data for evaluating the randomness of the digits of $π$ .^[103]^[104]

Computer era and iterative algorithms

The development of computers in the mid-20th century again revolutionized the hunt for digits of $π$ . Mathematicians John Wrench and Levi Smith reached 1,120 digits in 1949 using a desk calculator.^[105] Using an inverse tangent (arctan) infinite series, a team led by George Reitwiesner and John von Neumann that same year achieved 2,037 digits with a calculation that took 70 hours of computer time on the ENIAC computer.^[106]^[107] The record, always relying on an arctan series, was broken repeatedly (3,089 digits in 1955,^[108] 7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits was reached in 1973.^[106]

Two additional developments around 1980 once again accelerated the ability to compute $π$ . First, the discovery of new iterative algorithms for computing $π$ , which were much faster than the infinite series; and second, the invention of fast multiplication algorithmsthat could multiply large numbers very rapidly.^[109] Such algorithms are particularly important in modern $π$ computations because most of the computer's time is devoted to multiplication.^[110] They include the Karatsuba algorithm, Toom–Cook multiplication, and Fourier transform-based methods.^[111]

The Gauss–Legendre iterative algorithm:
Initialize $\textstyle a_{0}=1,\quad b_{0}={\frac {1}{\sqrt {2}}},\quad t_{0}={\frac {1}{4}},\quad p_{0}=1.$ Iterate $\textstyle a_{n+1}={\frac {a_{n}+b_{n}}{2}},\quad \quad b_{n+1}={\sqrt {a_{n}b_{n}}},$ $\textstyle t_{n+1}=t_{n}-p_{n}(a_{n}-a_{n+1})^{2},\quad \quad p_{n+1}=2p_{n}.$ Then an estimate for $π$ is given by $\textstyle \pi \approx {\frac {(a_{n}+b_{n})^{2}}{4t_{n}}}.$