Pi

The circuit number (Pi) is a mathematical constant that is defined as the ratio of the circumference of a circle to its diameter. This ratio is independent of the size of the circle. is an irrational and transcendental number and is used in many areas of mathematics, also outside of geometry before. The decimal expansion of the circle number begins with

The county number and some of their properties were already known in antiquity, the name with the Greek letter pi ( ) ( by the first letter of the Greek word περιφέρεια - Peripheria, " border area " or περίμετρος - perimetros, " scale" ) in the 18th century by Leonhard Euler popular after it ( 1706 Synopsis palmariorum matheseos ) had been used previously, among others, by William Oughtred ( Theorematum in libris ArchiMedis de Sphaera et Cylindro Declaration, 1647) and William Jones.

  • 2.1 The practice everyday urges to initial estimates
  • 2.2 Archimedes of Syracuse 2.2.1 Umbeschreibung and Einbeschreibung up to 96 corners
  • 3.1 BBP series
  • 3.2 Calculation using area formula 3.2.1 program
  • 3.3.1 Buffonsches needle problem
  • 5.1 formulas that contain π 5.1.1 Formulas of Geometry
  • 5.1.2 Analysis of the formulas
  • 5.1.3 formulas of the theory of functions
  • 5.1.4 formulas of number theory
  • 5.1.5 Formulas of Physics
  • 6.1 Records and Curiosities
  • 6.2 film, music, culture and literature
  • 6.3 Pi - Sports
  • 6.4 Development of decimal places of π
  • 6.5 Alternative circle number τ

Basic mathematical data

Definition

There are several equivalent definitions for the district number. Commonly used is about the definition as

  • The ratio of the circumference of a circle to its diameter, or
  • The area of ​​a circle with the radius r = 1

In analysis it is more convenient to first define the cosine of its Taylor series and then the circle number than twice the smallest positive zero of the cosine (after Edmund Landau ).

Irrationality and transcendence

The number is an irrational number, which is a real, but not a rational number. This means that it can not be so represented as the ratio of two whole numbers as a fraction. The 1761 (or 1767) by Johann Heinrich Lambert proved.

In fact, the number is even transcendental, which means that there is no finite degree polynomial with rational coefficients, which has as a zero. This was first proven by Ferdinand von Lindemann in 1882. As a consequence, results from the fact that it is impossible to express only with whole numbers or fractions and roots. One consequence of this was that the exact quadrature of the circle with ruler and compass is not possible.

The first 100 decimal places

As an irrational number is, can their representation in any place value system to fully specify: The presentation is always infinitely long and not periodically. In the first 100 decimal places in the decimal expansion

Is no apparent regularity. Even more decimal satisfy statistical tests for randomness. See also the section on the question of normality.

Continued fraction expansion

An alternative way to represent real numbers, the continued fraction expansion. Since is irrational, this representation is also infinitely long.

In contrast to Euler's number e but continued fraction have so far been found of any regularities in the ( regular ).

The accuracy of 200 decimal digits is obtained with 194 part denominators ( sequence A001203 in OEIS ):

= [ 3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, 6, 6, 99, 1, 2, 2, 6, 3, 5, 1, 1, 6, 8, 1, 7, 1, 2, 3, 7, 1, 2, 1, 1, 12, 1, 1, 1, 3, 1, 1, 8, 1, 1, 2, 1, 6, 1, 1, 5, 2, 2, 3, 1, 2, 4, 4, 16, 1, 161, 45, 1, 22, 1, 2, 2, 1, 4, 1, 2, 24, 1, 2, 1, 3, 1, 2, 1, 1, 10, 2, 5, 4, 1, 2, 2, 8, 1, 5, 2, 2, 26, 1, 4, 1, 1, 8, 2, 42, 2, 1, 7, 3, 3, 1, 1, 7, 2, 4, 9, 7, 2, 3, 1, 57, 1, 18, 1, 9, 19, 1, 2, 18, 1, 3, 7, 30, 1, 1, 1, 3, 3, 3, 1, 2, 8, 1, 1, 2, 1, 15, 1, 2, 13, 1, 2, 1, 4, 1, 12, 1, 1, 3, 3, 28, 1, 10, 3, 2, 20, 1, 1, 1, 1, 4, 1, 1, 1, 5, 3, 2, 1, 6, 1, 4, ...]

Another continued fraction representation of

Convergents of the circle number

From their regular continued fraction representation arise as a best approximation breaks the circuit number of the following:

Spherical geometry

In the spherical geometry, the term circle number is not in use, because the ratio of circumference to diameter in this case is no longer dependent for all channels the same, but their size. For channels having a much smaller diameter than that of the ball on the surface of he "drawn" (approximately a circle with diameter of 1 m on the spherical surface ), the distance of the center point to the ( Euclidean ) circular plane is very small, and thus the Unlike the normal Euclidean geometry negligible, otherwise the variation, however, must be considered.

Designation

The county number is also known as Archimedes' constant and Ludolphsche after Ludolph van Ceulen number. The English mathematician William Jones used in his Synopsis Palmariorum Matheseos (1706 ) as the first Greek lowercase letters, to express the ratio of circumference to diameter.

Already some time previously used by the English mathematician William Oughtred in his first time in 1647 published writing Theorematum in libris ArchiMedis de SPHAERA & Cylyndro Declaration the term to express the ratio of half- circumference ( semiperipheria ) to radius ( Semidiameter ), ie, The same used names in 1664 also the English mathematician Isaac Barrow. David Gregory used (1697 ) for the ratio of circumference to radius.

Leonhard Euler first used in 1737 the Greek lowercase for the district number, having previously used p. Since then, due to the importance of Euler's this term is common.

History of the number π - of estimates to record hunt

Even before the Greeks sought people after this mysterious number, and although the estimates were always accurate, it succeeded for the first time the Greek mathematician Archimedes in 250 BC, to limit the number mathematically. In the subsequent history of the attempts to maximum approach to phased into a veritable record hunt, sometimes bizarre, and even self-sacrificing traits were assumed.

The everyday practice urges to initial estimates

For practical considerations, the people tried at a very early time, the phenomenon closer to a circle. Thus, the ratio of the circumference to the diameter of a circle for example important for the calculation of the length of the fitting of a wheel, the bonding of a ring or the volume of a keg.

Here, the oldest traditions always refer to concrete objects; whether the mathematical regularity has been detected, is unclear. So according to the Bible King Solomon made produce a round pool for the Israelite temple by the coppersmith Hiram of Tyre:

" Then he made ​​the sea. It was cast in bronze and measured 10 cubits from the one brim to the other; it was completely round and five cubits high. A string of 30 Ellen could encompass it round. "

Thus it can be concluded for the object described a ratio of circumference to diameter with the value 3. One can assume that an inaccurate measurement or tradition of circumference and diameter is present.

The value 3 was also used in ancient China. Specifically, the information provided in Egypt. The oldest known computer book in the world, the arithmetic book of Ahmes ( Rhind papyrus also, 17th century BC), called the value as an approximation for the Mesopotamians 3, or even in India they used in the Sulbasutras, the cord rules for the construction of altars, the value for the Indian mathematician Aryabhata determined the value of the mathematical constant for that time very accurately to 3.1416 in the 6th century.

Archimedes of Syracuse

For Archimedes of Syracuse (c. 287 BC to 212 BC) and yet for many mathematicians after him was unclear whether the calculation of not but eventually came to a conclusion, if so be a rational number what centuries of hunting can be understood on the number. Although the Greek philosopher was acquainted with the irrationality of the existence of such numbers, but Archimedes had no reason to exclude a rational representability calculating the area of a circle from the outset. Because there are quite all sides curvilinear surfaces that are even enclosed by circle parts that can be represented as a rational number as the lune of Hippocrates.

It was not until 1761/1767 was Johann Heinrich Lambert proved the irrationality of, even if the mathematicians had suspected it all along.

Umbeschreibung and Einbeschreibung up to 96 corners

Archimedes tried as well as other researchers, to approximate the circle with polygons, and to win in this way approximations. He proved that the circumference of a circle to its diameter behaves the same as the area of ​​the circle to the square of the radius, the respective ratio is thus in both cases the same size ( circle number ). With the circumscribed and inscribed polygons up to the 96 -gon he calculated upper and lower bounds for the circumference. He came to the for that time extremely important assessment determines that the relevant ratio would be somewhat smaller than, but greater than:

After Heron Archimedes possessed a more accurate estimate, but is over- delivers wrong:

Wilbur Knorr corrected to:

More and more accurately - by Zu Chongzhi about Ludolph van Ceulen to John Machin

As in many other social and cultural areas, there were also in mathematics in Western cultures for a very long time of stagnation after the end of antiquity and during the Middle Ages. Progress in approaching achieved in this period mainly Chinese and Persian scholars. In the third century Liu Hui determined from the 192 -gon the barriers 3.141024 3.142704 and, later, from the 3072 -gon with an approximate value 3.14159. To 480 calculated the Chinese mathematician and astronomer Zu Chong Zhi ( 430-501 ) for the circle constant 3.1415926 3.1415927 <<, ie the first 7 digits. He also knew the almost as good approximation break ( this is the third proximity fraction of the continued fraction expansion of ), found in Europe until the 16th century. In his 1424 self-contained works treatise on the circle of Persian scientists Jamshid Mas ʿ ud al - Kashi calculated with a 3,228 -Eck 2 already accurate to 16 digits.

In the 16th century, then the math awoke in Europe back from its long sleep. 1596 succeeded Ludolph van Ceulen to calculate the first 35 decimal places of. Supposedly he sacrificed 30 years of his life for this calculation. Van Ceulen but still contributed with no new thoughts on the calculation. He simply calculated by the Archimedes method further, but while Archimedes stopped at the 96 -gon, Ludolph put the bills continued to enrolled corner. The name ludolphsche number reminiscent of his performance.

The French mathematician François Viète varied in 1593 the Archimedean method of exhaustion by approximated the area of ​​a circle by a sequence enrolled corner. It deduced first a closed formula for in the form of an infinite product from:

The English mathematician John Wallis developed in 1655 was named after him wallissche product:

Wallis showed in 1655, this series Lord Brouncker, the first president of the " Royal Society ", which represented the equation as a continued fraction as follows:

Gradually, the calculations were more complicated, Leibniz contributed 1682 following series representation at:

See also circle number calculation according to Leibniz.

This was Indian mathematicians in the 15th century known. Leibniz discovered new for the European mathematics and proved the convergence of the infinite sum. The above series is due to a special case ( θ = 1) of the series expansion of the arctangent, which was the Scottish mathematician James Gregory in the 1670s:

It formed the basis of many of approximations in the following period. John Machin calculated with its formula of 1706, the first 100 points of. His equation

Can be used together with the taylor series expansion between the arc tangent function for quick calculations. This formula can be obtained in the real domain using the addition theorem of the inverse tangent, it's probably by considering the argument of the complex number

Leonhard Euler led in his published in 1748 Introductio in Analysin Infinitorum in the first volume already on 148 points exactly. From Euler discovered formulas (see also Riemann ζ - function):

Johann Heinrich Lambert in 1770 published a continued fraction, which today mostly in the form

Is written. Each step yields roughly 0.76555 decimal places, which is relatively high compared with other continued fractions, so that this continued fraction is particularly well suited for the calculation of.

Craftsmen used in times before the slide rule and calculator approximation and calculated so much in my head. The opposite error is about 0.04%. For everyday practical situations that was sufficient.

Another often used approximation was the break, after seven-digit precision. All such rational approximations for common is that they are partial evaluations of the continued fraction expansion of match, for example:

None of the previously developed formulas could for efficient computation of approximate values ​​for serve, even the amazing discovery of the Indian S. Ramanujan from 1914, based on studies of elliptic functions and modular functions, was this not yet suitable for

Such efficient method, but its implementation requires high-precision arithmetic, are iterative methods with a quadratic or even higher convergence.

Modern numerical methods

BBP series

Discovered in 1995, Simon Plouffe, along with Peter Borwein and David Harold Bailey, a novel series representation ( BBP series ) for:

This series (also Bailey - Borwein - Plouffe formula called ) allows a simple way to compute the -th digit of a binary, hexadecimal, or any representation to a power of 2 based without first previous digits must be calculated.

Later BBP series were found for more:

Calculation by area formula

This calculation makes use of the connection, is that included in the formula of the circle area, however not in the area of the circumscribing formula square.

The formula for the area of ​​a circle with radius is

The area of ​​the square with side length is calculated as

For the ratio of the areas of a circle and its circumscribing square thus results

This can be written as four times this ratio.

Program

An example of an algorithm is indicated in the formula sheet is demonstrated, can be calculated using the approximation.

One puts on to the square of a grid, and calculates for each grid point, if it is also within the circle. The ratio of the lattice points inside the circle to the grid points within the square is multiplied by 4. The accuracy of the approximation of thus obtained depends on the mesh size and is controlled by means. With one obtains, for example, 3.16 and 3.1428 already. For the result is 3.14159, however, to put on, which is reflected in a square shape by the two-dimensional approach to the number of necessary arithmetic operations.

R = 10000 circular strike = 0 square strike = r ^ 2 for i = 0 to r -1    x = i 0.5    for j = 0 to r -1      y = j 0.5      if x ^ 2 y ^ 2 < = r ^ 2 then        circular strike = strike 1 circular output 4 * circular strike / square strike { 3.14159388 } Note: The above program has not been optimized for the fastest possible execution on a real computer system, but formulated as clearly as possible, for reasons of clarity. Furthermore, the circular surface is thus determined imprecise, as not the coordinates of the center for the respective area units are used, but the surface edge. By considering a full circle whose area is for the first and last lines to zero, the deviation for large is marginal.

The constant pi is for everyday use in computer programs typically already precomputed present, usually the associated value is specified with more points than it can accommodate the most efficient data types this computer language.

Statistical determination

A very interesting method for the determination of the statistical method. For the calculation can be random points in a square " rain " and calculates whether they are inside or outside an inscribed circle. The percentage of points inside is equal to

This method is a Monte Carlo algorithm; the accuracy of the achieved after a fixed number of steps of approximation can be specified, therefore, only with a certain probability of error. By the law of large numbers, but on average increases the accuracy with which the number of steps.

The following algorithm is written in the Java programming language:

Public static double approximiere_pi (int number of drops ) {    double pi = 0;    int in = 0;    int total = number of drops;      while ( number of drops > 0) { / / drop and add -generating function of their      double dotx = Math.random ();      double doty = Math.random ();        if ( dotx dotx * * doty doty < = 1) {        / / Point is inside the circle        within ;      Else { }        / / Point lies outside the circle      }        number of drops -;    }      pi = 4 * (double) within / total;    return pi; } Buffonsches needle problem

Another probability based and unusual method is the Buffon needle problem by Georges -Louis Leclerc de Buffon (1733 argued in 1777 published). Buffon threw sticks over his shoulder on a tiled floor. Then he counted how many times they met the joints. A more practical variant described Yakov Perelman in the book Entertaining geometry. Take a short, about 2 cm long needle - or another metal pin with similar length and diameter, preferably without tip - and draw on a sheet of paper a series of thin parallel bars that are around twice the length of the needle away from each other. Then you can very often (several hundreds or thousands ) from a certain height and fall noted that the needle intersects a line or not the needle on the sheet. It does not matter how you count the touch of a line through a needle finish. The total number of needle litters dividing by the number of cases in which the needle has cut a line results,

Wherein the length of the needles and the spacing of the lines marked on the paper. It follows easily an approximation of the needle can be curved or bent multiple times, in which case more than one intersection per litter is possible and has to be counted more than once. In the mid-19th century, the Swiss astronomer Rudolf Wolf came by 5000 needle throws to a value of

Geometric proximity construction

For the geometric construction of the number there is the approximate construction of Kochański, with which one can determine an approximate value of the mathematical constant with an error of less than 0.002 percent. It is therefore an approximation design for the ( not exactly possible ) quadrature of the circle.

Formulas, applications, open questions

Formulas containing π

Formulas of geometry

In geometry, the properties of a circle number stand out immediately.

  • Circumference of a circle with radius:
  • Area of ​​a circle with radius:
  • Volume of a sphere with radius:
  • Surface of a sphere with radius:
  • Volume of a cylinder with radius and height:
  • A volume defined by the rotation of the graph to the axis of rotation with the body and limits:

Formulas of Analysis

In the field of analysis also plays a role in many contexts, for example in

  • The integral representation, Karl Weierstrass in 1841 used to define.
  • The infinite series: ( Euler, see also Riemann zeta function)
  • The Gaussian normal distribution: or other representation:
  • Stirling's formula as an approximation of the Faculty of endowment:
  • The Fourier transform:
  • Euler's identity:

This identity as a combination of circuit number, which is also transcendent Euler's number, the imaginary unit and the two basic numbers 0 and 1 is considered as one of the " most beautiful mathematical formulas ."

Formulas of the theory of functions

Also in the theory of functions and complex analysis emerges the circle number. Below in

  • The integral formula of Cauchy:

Formulas of number theory

  • The relative frequency that two randomly chosen integers which lie below a barrier, are relatively prime, sought with against.

Formulas of physics

In physics plays next

  • The circular motion ( angular velocity equal times rotational frequency )

Especially in a role waves because there is received over the sine and cosine function. Thus, for example, so

  • In quantum mechanics ( Heisenberg uncertainty principle ).

Also

  • In the calculation of the buckling load,
  • In the friction of particles in liquids ( Stokes' law )

Applications and the benefit of present calculations

The approximate values ​​and procedures to circle number were long time especially for the applied sciences, such as civil engineering, very valuable; the newer approximations, however, already have so many places that a practical benefit is hardly given.

For example, suffice to calculate the circumference of a millimeter accuracy

  • , at a radius of 30 meters of four decimal places,
  • The radius of the earth ten decimal places,
  • At a radius with the distance Earth-Sun 15 decimal places.

How many sites are well necessary to compute the greatest imaginable in our universe real circuit with the greatest imaginable accuracy? The light of the Big Bang in the form of the microwave background radiation reaches us from a distance, as the product of the age of (about 1.38 x 1010 a) the speed of light (about 299,792 km · s- 1 or 9.45 · 1015 m · a-1) results in, or about 1.30 · 1026 m. The circle with this radius thus has a circumference of about 8.20 · 1026 m. The smallest physically meaningful unit of length is the Planck length of about 1.616 · 10-35 m. The circle therefore consists of 1.3245 · 1062 Planck lengths. In order him out of the given radius (assuming this would be a Planck length exactly known) to calculate the accuracy of a Planck length, so already 63 decimal places would be enough of.

In August 2010, the record was numerical calculations at about 5 trillion decimal places. The practical use of these calculations lies in the possibility of computer hardware and software to test because even small miscalculations too many of the wrong places.

Open question of normality

A current, particularly current mathematical question as to whether it is a normal number, ie whether they, for example, in a binary (or any other n -ary ) number representation contains any finite binary or other group of digits alike - as statistics, would expect if one were to generate a number completely randomly.

Ultimately, this would mean, for example, that all previous and future books written must contain somewhere in encoded binary form. See also the Infinite Monkey Theorem.

Bailey and Crandal showed in 2000 with the above mentioned Bailey - Borwein - Plouffe formula that normality can be reduced from base-2 to a conjecture of chaos theory.

Physicists at Purdue University have studied in 2005, the first 100 million digits of their randomness and compared with commercial random number generators. The researchers Ephraim Fischbach and his assistant Shu -Ju Tu were able to discover any hidden patterns in number. Thus Fischbachs opinion the number is actually a good source of randomness. However, some random number generators cut even better than decrease.

So far it is not even known if not for example, only the digits 5 and 6 occur at a certain point.

Others

Records and Curiosities

  • Friends of the number commemorate one on March 14, the circle with the number Pi Day because of the American date notation 3/14. Secondly, a - Näherungstag is celebrated on 22 July, by which the approximation is honored 22/7 by Archimedes.
  • In 1897, was proposed in the U.S. state of Indiana with the Indiana Pi Bill bill ( "for to act introducing a new mathematical truth" ) for the number by law 4 or 3.2. The amateur mathematician Edwin J. Goodwin was sure to have found the quadrature of the circle. He suggested the government to trade the right to waive all royalties from the application of his discovery in mathematical education and training, if his discovery would become law. Only after the disclosure by a " seasoned " mathematician who happened to read about the proposed legislation in the newspaper, adjourned the second chamber of Parliament, the House of Representatives (Parliament), already unanimously adopted draft indefinitely. The Guinness Book of World Records knows this story slightly differently: " The least accurate value. In 1897 the General Assembly adopted by Indiana law (Bill No. 246 ), after the value of de jure four. "
  • The version number of the typesetting program TeX by Donald E. Knuth, contrary to the usual conventions of software development incremented since the 1990s so that it approaches slowly. The current version of the 2014 has the number 3.14159265.
  • Scientists send radio telescopes the circle number into space. They are of the opinion that other civilizations must know this number if they can catch the signal.
  • The current record in the Pi - reading is 108,000 decimal places in 30 hours. The world record attempt began on 3 June 2005 at 18:00 clock and was completed on time on 5 June 2005 at 0:00 clock successfully. Over 360 volunteer readers read every 300 decimal places. Was organized the world record from Mathematikum in Giessen.

Film, music, culture and literature

Pi - Sports

From the learning of Pi has become a sport. The memorization of pi is considered the best way to make memorizing long numbers to the test. The Chinese Chao Lu is the official world record holder with confirmed 67,890 decimal places, which he flawlessly recited in a time of 24 hours and 4 minutes on 20 November 2005. It is guided both by the Guinness Book of Records as well as the Pi World Ranking List as the record holder.

The unofficial world record for memorization of pi was in October 2006 at 100,000 jobs, set up by Akira Haraguchi. The Japanese broke so its also still unofficial old record of 83 431 decimal places. The German record is held by Jan Harms with 9,140 points. For the memorization of pi specific mnemonic techniques are applied. The technique differs here to taste the memory of the artist, his talents and the amount of decimal places to memorierenden.

For memorizing the first digits of Pi there are simple shopping systems, to Pi - sport - shopping rules.

Development of decimal places of π

The record of the calculation of 2010 held a few months of living in Paris Fabrice Bellard software developers with 2.699.999.990.000 ( around 2.7 trillion ) positions. For the calculation Bellard used a standard Core i7 PC. The calculation took a total of 131 days: 103 days for binary, 13 days for a plausibility check, 12 days for the conversion to the decimal and three more days to verify, so you can expect a correct result with very high probability.

Alternative circle number τ

The American mathematician Robert Palais proposed in a 2001 issue of Mathematics Magazine " The Mathematical Intelligencer " before, for, rather than the conventional ratio of circumference and diameter of a circle, in the future the ratio of circumference and radius (respectively) as a fundamental constant to to use. His argument based on the fact that in many mathematical formulas, the factor appearing in front of the circle number. Another argument is the fact that is a solid angle, in radians, instead of a half angle, which has a less arbitrarily. A new normalized wave number for which notation Michael Hartl and Peter Harremoës the Greek letter ( tau ) suggested would reduce these formulas. With this convention, then applies. The proposal has so far not enforced.

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