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FUNDAMENTALS

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The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercase Greek letter π, sometimes spelled out as pi, and derived from the first letter of the Greek word perimetros, meaning circumference.In English, π is pronounced as "pie" . In mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation.

Definition

π is commonly defined as the ratio of a circle's circumference C to its diameter

The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle, it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curve (non-Euclidean) geometry, these new circles will no longer satisfy the formula π = C/d.

Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be formally defined independently of geometry using limits—a concept in calculus.[14] For example, one may directly compute the arc length of the top half of the unit circle, given in Cartesian coordinates by the equation x2 + y2 = 1, as the integral

An integral such as this was adopted as the definition of π by Karl Weierstrass, who defined it directly as an integral in 1841.

Integration is no longer commonly used in a first analytical definition because, as Remmert 2012 explains, differential calculus typically precedes integral calculus in the university curriculum, so it is desirable to have a definition of π that does not rely on the latter. One such definition, due to Richard Baltzer and popularized by Edmund Landau, is the following: π is twice the smallest positive number at which the cosine function equals 0. The cosine can be defined independently of geometry as a power series, or as the solution of a differential equation. In a similar spirit, π can be defined using properties of the complex exponential, exp z, of a complex variable z. Like the cosine, the complex exponential can be defined in one of several ways. The set of complex numbers at which exp z is equal to one is then an (imaginary) arithmetic progression of the form

and there is a unique positive real number π with this property.

A more abstract variation on the same idea, making use of sophisticated mathematical concepts of topology and algebra, is the following theorem: there is a unique (up to automorphism) continuous isomorphism from the group R/Z of real numbers under addition modulo integers (the circle group), onto the multiplicative group of complex numbers of absolute value one. The number π is then defined as half the magnitude of the derivative of this homomorphism.

Irrationality and normality

lπ is an irrational number, meaning that it cannot be written as the ratio of two integers. Fractions such as 22 / 7 and 355 / 113 are commonly used to approximate π, but no common fraction (ratio of whole numbers) can be its exact value.Because π is irrational, it has an infinite number of digits in its decimal representation, and does not settle into an infinitely repeating pattern of digits. There are several proofs that π is irrational; they generally require calculus and rely on the reductio ad absurdum technique. The degree to which π can be approximated by rational numbers (called the irrationality measure) is not precisely known; estimates have established that the irrationality measure is larger than the measure of e or ln 2 but smaller than the measure of Liouville numbers.

The digits of π have no apparent pattern and have passed tests for statistical randomness, including tests for normality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often. The conjecture that π is normal has not been proven or disproven.

Since the advent of computers, a large number of digits of π have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of π, and found them consistent with normality; for example, the frequencies of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found.[26] Any random sequence of digits contains arbitrarily long subsequences that appear non-random, by the infinite monkey theorem.

Thus, because the sequence of π's digits passes statistical tests for randomness, it contains some sequences of digits that may appear non-random, such as a sequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation of π. This is also called the "Feynman point" in mathematical folklore, after Richard Feynman, although no connection to Feynman is known.

Transcendence

In addition to being irrational, π is also a transcendental number,[3] which means that it is not the solution of any non-constant polynomial equation with rational coefficients.

The transcendence of π has two important consequences: First, π cannot be expressed using any finite combination of rational numbers and square roots or n-th roots. 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. Squaring a circle was one of the important geometry problems of the classical antiquity. Amateur mathematicians in modern times have sometimes attempted to square the circle and claim success—despite the fact that it is mathematically impossible.

Continued fractions

Like all irrational numbers, π cannot be represented as a common fraction (also known as a simple or vulgar fraction), by the very definition of irrational number . But every irrational number, including π, can be represented by an infinite series of nested fractions, called a continued fraction:

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.

Because π is known to be 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 (shown above) also does not exhibit any other obvious pattern, mathematicians have discovered several generalized continued fractions that do, such as:

Approximate value and digits

Some approximations of pi include: Integers: 3 Fractions: Approximate fractions include (in order of increasing accuracy) 22 / 7 , 333 / 106 , 355 / 113 , 52163 / 16604 , 103993 / 33102 , 104348 / 33215 , and 245850922 / 78256779 . Digits: The first 50 decimal digits are 3.14159265358979323846264338327950288419716939937510...

Digits in other number systems

The first 48 binary (base 2) digits (called bits) are 11.001001000011111101101010100010001000010110100011... The first 20 digits in hexadecimal (base 16) are 3.243F6A8885A308D31319... The first five sexagesimal (base 60) digits are 3;8,29,44,0,47

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 origin of the complex plane, and the other (angle or φ) the counter-clockwise rotation from the positive real line:

where i is the imaginary unit satisfying i2 = −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:

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 centered at the origin of the complex plane. Setting φ = π in Euler's formula results in Euler's identity, celebrated in mathematics due to it containing the five most important mathematical constants:

There are n different complex numbers z satisfying zn = 1, and these are called the "n-th roots of unity" and are given by the formula:

Modern Quest for more Digits

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. 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.[117][118] The record, always relying on an arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits were reached in 1973

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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 algorithms that could multiply large numbers very rapidly. Such algorithms are particularly important in modern π computations because most of the computer's time is devoted to multiplication. They include the Karatsuba algorithm, Toom–Cook multiplication, and Fourier transform-based methods.

The iterative algorithms were independently published in 1975–1976 by physicist Eugene Salamin and scientist Richard Brent.[122] These avoid reliance on infinite series. An iterative algorithm repeats a specific calculation, each iteration using the outputs from prior steps as its inputs, and produces a result in each step that converges to the desired value. The approach was actually invented over 160 years earlier by Carl Friedrich Gauss, in what is now termed the arithmetic–geometric mean method (AGM method) or Gauss–Legendre algorithm. As modified by Salamin and Brent, it is also referred to as the Brent–Salamin algorithm.

The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally multiply the number of correct digits at each step. For example, the Brent-Salamin algorithm doubles the number of digits in each iteration. In 1984, brothers John and Peter Borwein produced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step.[123] Iterative methods were used by Japanese mathematician Yasumasa Kanada to set several records for computing π between 1995 and 2002. This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.

Outside mathemetich

Although not a physical constant, π appears routinely in equations describing fundamental principles of the universe, often because of π's relationship to the circle and to spherical coordinate systems. A simple formula from the field of classical mechanics gives the approximate period T of a simple pendulum of length L, swinging with a small amplitude (g is the earth's gravitational acceleration)

One of the key formulae of quantum mechanics is Heisenberg's uncertainty principle, which shows that the uncertainty in the measurement of a particle's position (Δx) and momentum (Δp) cannot both be arbitrarily small at the same time (where h is Planck's constant)

The fact that π is approximately equal to 3 plays a role in the relatively long lifetime of orthopositronium. The inverse lifetime to lowest order in the fine-structure constan α is {\displaystyle {\frac {1}{\tau }}=2{\frac {\pi ^{2}-9}{9\pi }}m\alpha ^{6},}{\frac {1}{\tau }}=2{\frac {\pi ^{2}-9}{9\pi }}m\alpha ^{6}, where m is the mass of the electron.π is present in some structural engineering formulae, such as the buckling formula derived by Euler, which gives the maximum axial load F that a long, slender column of length L, modulus of elasticity E, and area moment of inertia I can carry without buckling. {\displaystyle F={\frac {\pi ^{2}EI}{L^{2}}}.}F={\frac {\pi ^{2}EI}{L^{2}}}. The field of fluid dynamics contains π in Stokes' law, which approximates the frictional force F exerted on small, spherical objects of radius R, moving with velocity v in a fluid with dynamic viscosity η:[204] {\displaystyle F=6\pi \eta Rv.}{\displaystyle F=6\pi \eta Rv.} In electromagnetics, the vacuum permeability constant μ0 appears in Maxwell's equations, which describe the properties of electric and magnetic fields and electromagnetic radiation. Before 20 May 2019, it was defined as exactly

Memorizing digits Main article: Piphilology Piphilology is the practice of memorizing large numbers of digits of π,[207] and world-records are kept by the Guinness World Records. The record for memorizing digits of π, certified by Guinness World Records, is 70,000 digits, recited in India by Rajveer Meena in 9 hours and 27 minutes on 21 March 2015.[208] In 2006, Akira Haraguchi, a retired Japanese engineer, claimed to have recited 100,000 decimal places, but the claim was not verified by Guinness World Records.[209] One common technique is to memorize a story or poem in which the word lengths represent the digits of π: The first word has three letters, the second word has one, the third has four, the fourth has one, the fifth has five, and so on. Such memorization aids are called mnemonics. An early example of a mnemonic for pi, originally devised by English scientist James Jeans, is "How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics."[207] When a poem is used, it is sometimes referred to as a piem. Poems for memorizing π have been composed in several languages in addition to English.[207] Record-setting π memorizers typically do not rely on poems, but instead use methods such as remembering number patterns and the method of loci.[210]

In popular culture Pi Pie at Delft University A pi pie. The circular shape of pie makes it a frequent subject of pi puns. Perhaps because of the simplicity of its definition and its ubiquitous presence in formulae, π has been represented in popular culture more than other mathematical constructs.[213] In the 2008 Open University and BBC documentary co-production, The Story of Maths, aired in October 2008 on BBC Four, British mathematician Marcus du Sautoy shows a visualization of the – historically first exact – formula for calculating π when visiting India and exploring its contributions to trigonometry.[214] In the Palais de la Découverte (a science museum in Paris) there is a circular room known as the pi room. On its wall are inscribed 707 digits of π. The digits are large wooden characters attached to the dome-like ceiling. The digits were based on an 1853 calculation by English mathematician William Shanks, which included an error beginning at the 528th digit. The error was detected in 1946 and corrected in 1949.[215]