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# History and Algorithm of pi

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### History and Algorithm of pi

1. 1. LINKS to History and Algorithm of pi Following pages : bypi Title PageIs pi useful ? Karl Helmut Schmidtpi in the antiquity Noli turbare circulos meos ArchimedesWith ArchimedesTo infinity The book contains overall description of the historical development of arithmetical methods for theSupremacy of arctan calculation of pi.pi in India The CD accompanying this book gives an overview of many mathematical algorithms and some examples ofWith Infnitesimal how to get specific numbers or even individual digits of piRamanujanAGM and more The number pi resides for quite a lot of mathematicians at the center of their interests within an important and large area of the total field of mathematics. Starting with geometry, which received substantial practicalSPIGOT Algorithm and theoretical attention, to infinite series of products and sums, compounded fractions, and finally to theThe Chudnovskys theory of mathematical complexity, series of coincidence, as well as the use of computers for the calculation and analysis of long listings of pi-digits.Individual digitsDigit distribution Some mathematicians and amateurs alike did spent the most part of their lives for the exploitation andHigh precession arithmetic understanding of the phenomena pi. Pi is present in many areas, and offers substantial initiations for theSome examples study as well as general use, even to the specific point and analyses of modern mathematical theories.2000 digits of pipi: binary, decimal & hex Especially, the last 50 years brought enormous progress in many mathematical fields by the use of fast calculation machines, the computers. Together with extremely fast mathematical algorithms, such as theThe Book : How to order “Fast Fourier Transform”, deep penetration into pi could be achieved.END3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 1
2. 2. If laws of mathematics or physics are valid in a specific area, LINKS to Is pi useful ? then this laws are also valid in areas, which move relatively to Following pages : the reference area. Albert Einsteinpi Title Page The ratio of the circumference of a circle to is ratio is constant. Pi represents this ratio, and relates also to theIs pi useful ? ratio of the area of a circle to the square of its radius. In addition, pi results from the ratio of the sphere area topi in the antiquity the square of the sphere diameter.With Archimedes Pi has puzzled and accompanied humanity for some thousand years. Practically in all cultures one may find some approximations for it:To infinity The Bibel shows a value = 3,0Supremacy of arctan At Babylon and the Mesopotamia commonly use was 25 / 8 = 3 , 125pi in India The Egyptian Rhind Papyrus Rolls identify 256 / 81 = 3 , 16With Infnitesimal And even today many practicians use for pi 22 / 7 = 3 , 14Ramanujan At the beginning a specific value of pi was needed to construct circles and associated curves in architecture.AGM and more Yet, scientists and mathematicians entered very early the quest of an answer to the direct translation of theSPIGOT Algorithm area of a circle to a square – the famous search of the quadrature of the circle.The Chudnovskys The fascination of pi is not limited to circles or curves, and its related calculation of sizes. Pi often appears in at unexpected places. For example, if one takes all primes, which result from the factorization of any number,Individual digits then the probability that a prime factor will be repeated is equal to the ratio of 6 / square of pi.Digit distribution Pi is not an irrational, but a transcendental number. In 1862 , Lindeman gave a prove that pi is aHigh precession arithmetic transcendental number, which implies that the for so long search quadrature of a circle is impossible.Some examples Nowadays millions of decimal, binary or hexadecimal digits of pi can be calculated. Now, why exists this great desire to search for records of billions and more digits, when 5 decimal place are sufficient to built the2000 digits of pi most accurate machines, 10 places give the circumference of the earth to some Millimeter accuracy, and whenpi: binary, decimal & hex 39 digits of pi are enough to calculate the circumference of the circle around the kwon universe to the accuracy of the diameter of a hydrogen atom?The Book : How to order Why are we not satisfied with 50 or 100 decimal places of pi ?END The direction and the way to get there are the target – The mountain will be climbed, because it just is there.3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 2
3. 3. Everything within the uiverse carries ist own specific LINKS to number secret Chao-Hsiu Chen Following pages : pi in the antiquitypi Title Page Over 2 million years mankind developed. Ninetynine percent of this time man was a collector of food andIs pi useful ? hunter. Besides weapons and tools to hunt, and later to work his fields numbers greater than 2 or even greaterpi in the antiquity than 10 were unnecessary. A herd consisted of 2 animals or it had just many.With Archimedes Only past the last glacial period , about 10 000 years B.C., brought through the union of groups and small settlements the necessity to have scales to measure quantities, distances and time periods. Through this theTo infinity first steps towards a simple arithmetic were taken, with the first forms of writing and documentation. BesidesSupremacy of arctan the Egyptian hieroglyphs, writings of the time 3000 B.C. of the area Elam and Mesopotamia have been found. Early tablets of clay report about arithmetic rules for the establishment and administration of property.pi in India Such rules of arithmetic brought along the discovery of relationship between specific subjects and its values.With Infnitesimal To double the volume resulted in doubling the weight. Certain relations of lengths of the sides of a triangleRamanujan established a right angle. The ratios of the circumference to the diameter of circles were constant.AGM and more The number systems of the Stone Age had no “Zero”, what made arithmetic quite difficult. The digit zero came relatively late. The Romans had none. The acabus, an old calculation device, which is still in use todaySPIGOT Algorithm extensively in Asia, uses for nothing or a way to use zero an empty row.The Chudnovskys The symbol for zero originated in India, and came together with the Hindu-Arabic presentation of numbersIndividual digits via Northafrica about 1200 A.C. to Europe. Only then the way to develop the real arithmetic with its rules and algorithms was established. Yet, the calculation of pi was still a long way off.Digit distributionHigh precession arithmetic About 1850 B.C. the Egyptian scribe Ahmes gave the earliest known record in the so-called Rhind Papyrus onSome examples how to calculate pi. His recipe results in a approximation using a equal-sided area with eight corners>2000 digits of pi the square of (64 / 81) = pi / 4pi: binary, decimal & hex This shows for pi = 3.16The Book : How to orderEND3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 3
4. 4. LINKS to Geometry is the best method to devote once free time Following pages : With Archimedes Platopi Title Page Archimedes (287-212 B.C.) developed the first mathematical analysis and its related algorithm to approximate pi. Archimedes based his thinking on the 12. Book of Euklid , which covered important theoriesIs pi useful ? about the ability to measure circles.pi in the antiquity Demonstration Nr. 7 The ratio of the perimeters of two regular polygons with equalWith Archimedes number of sides is equal to the ratio of their in- respectively circumscribed circles.To infinity Euklid established the theory, Archimedes developed the algorithm for the calculation of pi to any wanted accuracy. His algorithm is base on the fact that the circumference of regular polygon with n sides is smallerSupremacy of arctan than its perspective circumscribed circle, yet it is larger than its inscribed circle. If one takes n to be very large,pi in India the in- and circumscribed circle converge to a single value. With n equal to infinity the value for exact pi may be found.With Infnitesimal To calculate the circumference of a circle Archimedes started with a regular 6-sided polygon. He thenRamanujan continuously doubled the number of sides up to the value of 96 sides. For the first time in history ArchimedesAGM and more used the concept to approach calculation results with the method of “limits”. He found an approximate value of pi by calculating perimeters of in- and circumscribed regular polygons. Thereby he set an algorithm for theSPIGOT Algorithm calculation of pi to any desired accuracy. This calculation method of pi survived for many centuries.The Chudnovskys Since Archimedes was limited to the use of the formula of Pythagoras and the kwon method of halvingIndividual digits angles, the practicality of number handling limited his approach to a 96-sided polygon. He thereby found the value of pi to lie betweenDigit distribution 3 10/71 < pi < 3 1/7 = 3,140845 < pi < 3,142857High precession arithmetic The use of calculating the arithmetical mean pim = (a+b) / 2 would have given Archimedes the followingSome examples approximation: pim = (3,140845 + 3,142857) / 2 = 3,1418512000 digits of pi 1593 A.C. Viete, using the Archimedes method with a regular 393 216 sided polygon found the approximatepi: binary, decimal & hex value for pi to be betweenThe Book : How to order 3,1415926535 < pi < 3,1415926537END During many centuries after Archimedes no considerable improvements or new calculating methods were established.3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 4
5. 5. LINKS to God is absolute Infinity, human beings are by nature finite Following pages : To infinity and may not participate at infinity, and in no way understand it. Thomas von Aquinpi Title PageIs pi useful ?pi in the antiquity After Archimedes the first appreciable and mentionable activity in the filed of calculating pi withinWith Archimedes medieval is the one of Francois Viete’ (1540 – 1603). His method is based on the relation of areas of n- sided to 2n-sided polygons.To infinity Real progress came for the development of algorithm with the findings for the binomial series and theSupremacy of arctan development of power series. Blaise Pascal, a brilliant mathematician, set with his well known PASCAL-pi in India triangle the basis for the infinitesimal arithmetic and thereby new ways to calculate pi.With Infnitesimal 1655 John Wallis published his famous formula for pi, which is the result of an infinite power series.Ramanujan π/2 = 2 Π (1 – 1/(2n +1)2)AGM and more Newton discovered in 1665 the binomial number series.SPIGOT Algorithm A little later, Gregory found the power series for tan á and his so famous solution for arctan using the infinite power series for the reversion of tangens.The ChudnovskysIndividual digits Leibniz inserted the value 1 for x and got the so-called Leibniz-Gregory-Series.Digit distribution π/4 = Π (-1)n 1 / (2n+1)High precession arithmetic A practical evaluation of pi using this series is not feasible. This series converges to slowly.Some examples2000 digits of pi In 1996 Newton calculated 15 correct decimal digits by the use of the formulapi: binary, decimal & hexThe Book : How to order π = (√3)/4 – 24  √(x–x2) dx This series corresponds in principal the arcsin x power series..END3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 5
6. 6. LINKS to The supremacy of arcus tangens Following pages : Our almighty teacher did invite the human beings to study and to imitatepi Title Page the scientific structure of the infinite universeIs pi useful ? Thomas Painepi in the antiquity 1706 John Machin developed his famous and very fast converging formula. By the use of this and theWith Archimedes previously stated power series of Gregory for arctan, Machin calculated 100 correct decimal digits of pi.To infinity π / 4 = 4 arctan (1/5) – arctan (1/239)Supremacy of arctan John Machin found this via the formula for doubling tan 2α. Using his general presentation to dissect onepi in India value for arctan into two amounts. This brought so many additional formula of based upon the arctan power series.With Infnitesimal arctan u + arctan v = arctan (u+v) / (1–uv)Ramanujan 1738 Euler found a new method for calculating arctan value, which converged much faster then Gregory’s. HeAGM and more also published the followingSPIGOT Algorithm π / 4 = arctan 1/2 + arctan 1/3The Chudnovskys Additional formula, such as shown below, were developed:Individual digits π / 4 = 2 arctan (1/5) + arctan (1/7) + arctan (1/8)Digit distribution π / 4 = arctan (1/2) + arctan (1/4) + arctan (1/13)High precession arithmetic π / 4 = 3 arctan (1/4) + arctan (1/20) + arctan (1/1985)Some examples π / 4 = 22 arctan (1/28) + 2 arctan (1/443) – 5 arctan (1/1393) – 10 arctan (1/12943)2000 digits of pipi: binary, decimal & hex The methods for calculating pi established by John Machin using arctan power series were extremely effective,The Book : How to order so that most calculations of many digits until the 20. century were based on this method. In other words, for centuries no real progress for the calculation of pi was made.END3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 6
7. 7. LINKS to pi in India The scientist does not study the nature because this is just possible, he studies it for his enjoyment and the wonderful beauty he sees. Following pages : Henri Poincore’pi Title PageIs pi useful ? Since the antique very progressive mathematical investigations, establishment of arithmetical rules, and evenpi in the antiquity analytical results came out of India. Indian mathematician were also quite successful in the search for an answer to the mysterious ratio. In many mathematical writings, some over 4000 years old, pi had shown up.With Archimedes A number of arithmetical rules, so-called cord-rules, were written down around 600 A.C. in a documentTo infinity named Salvasutra. Such rules were used to construct altars as well as buildings. In addition they dealt with the calculation of circle areas respectively the conversion of a circle to a square. The length of the side of aSupremacy of arctan square was defined as follows:pi in India Take the 8. part of a circle diameter and divide this in 29 partsWith Infnitesimal Take then the 28.part and the 6.part of the remaining 29.partRamanujan Then subtract the 8.partAGM and more As formula this results in Sq = d 9785/11136 From this pi = 4 Sq / d2 = 3,088SPIGOT Algorithm 499 B.C. Arya-Bhata writes in the documents Siddhanta for the value of piThe Chudnovskys 3 + 177/1250 = 3,141...Individual digits Yet, quite more interesting notes from India about pi are documents from the 15.century employing infiniteDigit distribution power series. The Sanskrit-Documents Yukti-Dipika and Yukti-Bhasa give 8 power series for pi, includingHigh precession arithmetic the so-called Leibniz-Series.Some examples NilaKantha (1444-1545) published these series in the document Tantra Sangrahan. Some of these series are after all some hundred years older than found by European mathematicians.2000 digits of pi One example is :pi: binary, decimal & hex π / 2 = √3 ∑ (–1)n / ((2n + 1) 3n )The Book : How to order Additional examples may be found in the book pi Geschichte und Algorithmen einer ZahlEND3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 7
8. 8. LINKS to With the use of exact methods it is often extreme difficult Following pages : with Infinitesimal if not impossible to solve certain equations, only by the use of iterations solutions may be found.pi Title Page LancelotIs pi useful ? Hoybenpi in the antiquityWith Archimedes One of the most important progress in the filed of mathematics was the development of infinitesimal calculus by Barrow, Newton and Leibniz. Isaac Newton and Gottfried Wilhelm Leibniz developed calculusTo infinity independent from each other at the same time. Newton’s fluxion and fluent rules are difficult to apply, whichSupremacy of arctan did not help to make practical use of them. Leibniz introduced the now-a-days used nomenclature for the differentiation quotient and y/dx and integral  f(x) dx .pi in IndiaWith Infnitesimal With the development of calculus the problem and the associated solution of the calculation of areas similar toRamanujan the problem of Archimedes reappeared. The task is to evaluate and calculate the area limited by the curveAGM and more defined by y=f(x) , and by the x-axis. Finding the solution for this area is especially well suited for the multi digit calculation of pi via an integral and the use of an infinite power series.SPIGOT AlgorithmThe Chudnovskys Newton used a segment of a circle with the radius = 0.5 . His resulting power series converged relative rapidIndividual digits to a solution. The first 24 partial sums already give 24 correct decimal digits of pi. Only within an hourDigit distribution Newton calculated 20 correct decimal digits.High precession arithmeticSome examples Leibniz offered a solution via the use of polar coordinates and an associated integral calculation. His result equaled the answer previously provided by Gregory, if one inserts in the Gregory infinite power series the2000 digits of pi value x=1. Leibniz’ merits within the filed of mathematics are versatile. For example, he published an article,pi: binary, decimal & hex in which he presented for the first time methods for basic binary arithmetic (+, -, *, /). This publication is considered the birth of radix-2 arithmetic.The Book : How to orderEND3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 8
9. 9. LINKS to ...climb to the paradise on the ladder of surprises Following pages : Ramanujan Ralph Waldo Emersonpi Title Page During the 18. and 19. centuries a number of quite famous mathematicians lived. Boole, Cantor, Cauchy,Is pi useful ? Chebychev, Fourier, Langrange, Laplace, Mersenne, Plank, Poisson, Riemann, Taylor, Turing and otherspi in the antiquity developed excellent new theories, and offered corresponding results within the field of general mathematics. Yet, practically nothing new in the field of calculations for pi was brought forward.With Archimedes Srinivasa Ramanujan born 1887 in Erode, a small town in Southern India, showed very early in his life signsTo infinity of a mathematical genius. At the age of 12 he had mastered the extensive publication “Plane Trigonometry”,Supremacy of arctan being 15 years old he studied from “Relations of elementary results of pure mathematics” . This was his total mathematical education.pi in India Despite of his limited training he succeeded in reformulating and expanding on the general number theory withWith Infnitesimal new theory and formulas. After publishing his astonishing and brilliant results on “Bernoulli Numbers”Ramanujan Ramanujan achieved international attention and scientific recognition. He researched Modular Equations and he is unsurpassed with his results for singularities. Godfrey H. Hardy, the most respected mathematician of hisAGM and more time, brought him to the Trinity College Cambridge.SPIGOT Algorithm Ramanujan formulated the “Riemann Series”, elliptical integrals, hypergeometrical series and functionalThe Chudnovskys equations for the “Zeta-Function”. Like so many great mathematician he worked on pi, he defined precise expressions for the calculation of pi and developed many approximation values. His fame grew, but his healthIndividual digits failed. He died 1920 in India.Digit distribution Ramanujan bestowed a range of unpublished notebooks. 70 years after his death, quite an number of scientistsHigh precession arithmetic and mathematicians search for an understanding of his fascinating formulas to apply them in to-days problem solutions and for use in developing better algorithm for computers.Some examples The most famous presentation for the calculation of pi using an infinite series of sums by Ramanujan is2000 digits of pi 1/π = (√8) / 9801 ∑ (4n)! (1103+26390 n) / ((n!)4 3964n )pi: binary, decimal & hex representing a special solution for a related function of modular quantities.The Book : How to order Gosper calculated 17 million digits of π using this formula.END3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 9
10. 10. LINKS to AGM : Algorithm using arithmetic-geometric-means Following pages : Mathematic is a fabulous science, yet, mathematicians don’t suit thepi Title Page henchman most of the timeIs pi useful ? Lichtenbergpi in the antiquity The algorithm for the arithmetic-geometric-mean (AGM) was originally already used 1811 by Legendre inWith Archimedes his works to simplify and to evaluate elliptical integrals. Independently Gauss discovered AGM being only 14 years old in 1799. Gauss described in precise details the calculation and application of AGM. By the use of anTo infinity iterative process fast convergence is achieved. Basically AGM is defined asSupremacy of arctan AGM (x0 , y0 ) ≡ M [ (x0 + y0 ) / 2 ; √ (x0 * y0 ) ]pi in IndiaWith Infnitesimal This fast convergence is best shown on the following example:Ramanujan With x0 = 1 and y0 = 0,8 x1 = 0,9 y1 = 0,894427190999915…AGM and more x2 = 0,897213595499957… y2 = 0,897209268732734…SPIGOT Algorithm x3 = 0,897211432116346… y3 = 0,897211432113738… x4 = 0,897211432115042… y4 = 0,897211432115042…The Chudnovskys 1976 E. Salamin and R.P. Brent did independently of each other rediscover AGM for the calculation of pi,Individual digits and developed a very fast converging algorithm for computer usage. Salamin gave at that time an estimate onDigit distribution the numerical evaluation for pi, which foresaw 33 million digits as a possible result.High precession arithmeticSome examples J.M. Borwein and P.B. Borwein made many additional theoretical studies and analyses, and developed a range of effective algorithms for the calculation of pi. All based on the original formulas developed by2000 digits of pi Legendre.pi: binary, decimal & hex From their research in the field of number theory the brothers Borwein offered general methods for theThe Book : How to order calculation of certain elementary mathematical constants.END3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 10
11. 11. LINKS to Following pages : The Chudnovsky Brotherspi Title Page Shouting for worldly fame is only like a breezeIs pi useful ? blowing from different directions and changingpi in the antiquity thereby its direction Dante AlighieriWith ArchimedesTo infinity David und Gregory Chudnovsky, two brilliant brothers, even often quite excentric, both immigrated form theSupremacy of arctan previous USSR to the United States, did not follow the general trend to calculate many millions of digits of pi with very efficient and capable computer available at large universities or research centers such as the NASApi in India Cray-Computer.With Infnitesimal They constructed and built their own computer right at their apartment in Manhattan from generally availableRamanujan parts. These components and cables they ordered from mailing houses. Over time this computer occupied almost every available place in their apartment. Everything there disappeared below mountains of computerAGM and more parts, building blocks, interconnecting lines, cables and so on. Since power consumption had not beenSPIGOT Algorithm optimized, most likely it had been even impossible, extensive heat did develop, some even thought of hell like proportions.The Chudnovskys Despite of all the Chudnovsky brothers made very successful progress in the field of calculation of manyIndividual digits millions of digits of pi. During May of 1989 they achieved 480 million of it, and 5 years later even 4 044 000Digit distribution 000 correct decimal digits. They did use none of the very fast converging algorithm such as the Salamin-Brent one or one of the Borwein versions, but they employed a infinite power series of Ramanujan. Each step ofHigh precession arithmetic iteration produced 18 correct digits.Some examples2000 digits of pi 1/π = (6541681608 / 6403201/2 ) ∑[ (13591409 / 545140134) +k ] (–1)k (6k)! / [(3k)! (k!)3 6403203k]pi: binary, decimal & hex für k=0 bis k=∞The Book : How to orderEND3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 11
12. 12. LINKS to Patience is the power from which we achieve the best Spigot Algorithm Confucius Following pages : A very interesting Algorithm for the calculation of certain number values such as √2 , the basis of thepi Title Page logarithm e and pi was presented by Stanley Rabinowitz and Stan Wagon. This computing instructionIs pi useful ? functions like a spigot from which individual digits appear without use of previous numberspi in the antiquity The “appearing” digits do not need great accurate arithmetic The algorithm employs integer arithmetic with only 8 bit accuracyWith Archimedes . By the law for the presentation of polyadic number systems z = ∑ ai bi for i = -m to n the formula for theTo infinity development of this sum using a uniform number base b is then equal toSupremacy of arctan ...+ a3 b3 + a2 b2 + a1 b1 + a0 b0 + a-1 b-1 + a-2 b-2 + …pi in India an example for the decimal system isWith Infnitesimal 139,812510 = 1* 102 + 3* 101 + 9* 100 + 8 10-1 + 1* 10-2 + 2* 10-3 + 5* 10-4Ramanujan Number systems with mixed number base, such asAGM and more 3 weeks + 4 days + 1 hour + 49 minutes + 7 seconds + 99 hundreds of a sec. Pound + 18 Shilling + 11 Pence = 3010 + 1820 + 1112 (the old UK monetary system)SPIGOT AlgorithmThe Chudnovskys May be presented with a differing number base ci ...+ a3 b3 + a2 c22 + a1 c1 1 + a0 c00 + a-1 c-1-1 + a-2 c-2-2 + …Individual digits Now, an interesting answer appears, if one uses the mixed number base c = 1/1; 1/2,; 1/3; ¼; 1/5; … .Digit distribution The math constant e = 2,718281… diverts toHigh precession arithmetic e = 1 + 1/1(1 + 1/2 (1 + 1/3 (1 + 1/4 (1 + 1/5 (1 + … )))Some examples Naturally, it is also feasible to present pi on a mixed base. Using the Leibniz-Series as converted by Euler2000 digits of pi and c = 1/1; 1/3; 2/5; 3/7; 4/9; … one findspi: binary, decimal & hex π/2 = 1 + 1/3(1 + 2/5(1 + 3/7 (1 + 4/9 ( 1 + …))) or π = (2; 2; 2; 2; 2; 2; …)cThe Book : How to order The solution of this mixed base presentation follows a way similar to the Horner-Scheme. TheEND corresponding algorithm equal the above mentioned Spigot program for the calculation of pi.3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 12
13. 13. LINKS to Calculation of individual digits of pi Following pages :pi Title Page Much is not sufficient, the quality is the clue The AutorIs pi useful ?pi in the antiquity At the beginning of 1995 David Bailey and Simon Plouffe published and surprised with an absolutely newWith Archimedes development for the calculation of digits for pi. Without the need to determine any previous digits theyTo infinity calculated any individual hexadecimal digits. For this the usedSupremacy of arctan π = ∑ 1 / 16 n [ 4 / (8n+1) – 2 / (8n+4) – 1 / (8n+5) – 1 / (8n+6) ] for n=0 to n=∞pi in India This remarkable formula was found by intensive computer search and the use of the PSQL Integer Relation Algorithm. This so new formula was praised as very much astonishing, since after some thousand years someWith Infnitesimal new fundamentals were discovered.Ramanujan Bailey, Borwein and Plouffe found during the month of November 1995 the 40* 109 digit in HEX :AGM and more 921C73C6838FB2SPIGOT Algorithm 1996 Simon Plouffe solved then the task to calculate the n-th decimal digit of some irrational as well as transcendental numbers such as π, π3, integer powers of the Riemann Zeta Function Zeta(3), log(2), andThe Chudnovskys others. For a long time this was considered to be impossible or at least extremely difficult.Individual digitsDigit distribution The basis for this calculations was the following formula already developed by EulerHigh precession arithmetic π + 3 = ∑ n 2 n / [ 2n über n ]Some examples The success of this Euler formula lies in the solution of the „Central“ Binomial-Coefficient 2n over n for all prime factors, which are smaller than 2n over n .2000 digits of pi There exists some more infinite sums based upon C(mn,n) , which are suitable for the calculation ofpi: binary, decimal & hex individual digit of any number base.The Book : How to orderEND3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 13
14. 14. LINKS to Some mathematician consider the decimal expansion of pi a random Digit Distribution series, but to modern numerologist it is rich with remarkable patterns. Following pages : Dr.I.J.Matrix (Martin Gardner)pi Title Page Over centuries pi was intensively investigated for its characteristics and patterns. In general it seems that theIs pi useful ? digits arrange in a row randomly. Yet, the change of only one decimal digit results in a complete differentpi in the antiquity number; it is then no longer pi.Many investigations deal with the search for patterns of repetition or specific number series.With Archimedes The digit ZERO (0) shows for the first time at the 32. decimal position.To infinity The sum of the first 20 decimal digits is 100.Supremacy of arctan Adding the first 144 decimal digits one gets the sum 666. The 3 decimals ending at position 315, had the sequence 315.pi in India The first 0 shows at position 32 the first ONE 1 at position 1With Infnitesimal 00 307 11 94Ramanujan 000 601 111 153 0000 13390 1111 12700AGM and more 00000 17534 11111SPIGOT Algorithm Quite often one may see some interesting patterns :The Chudnovskys 11011 at decimal position 3844 10001 14201Individual digits 87778 17234Digit distribution 202020 7285 6655566 10143High precession arithmetic The frequency (F) of every decimal digit (0 to 9) of the first 29 millions of the decimal digits of pi shows theSome examples following: Digit 0 1 2 3 4 52000 digits of pi Rel. Frequency 0,0999440 0,0999333 0,1000306 0,0999964 0,1001093 0,1000466pi: binary, decimal & hexThe Book : How to order Digit 6 7 8 9 Rel. Frequency 0,0999337 0,1000207 0,0999914 0,1000040END Not proven, but it is generally assumed that all 10 digits are equally distributed, since the relative frequency3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... approaches 0.1 . 14
15. 15. LINKS to Highly precise Computer Arithmetic Following pages : For one person science is the high, heavenly goddess, where aspi Title Page another person sees it as an efficient cow providing butterIs pi useful ? Friedrich von Schillerpi in the antiquityWith Archimedes Basically one may use floating point or integer arithmetic within a computer algorithm. One of the most important element for high precision computer calculation is the availability of specific, very fast and accurateTo infinity programs. Evidently one may attack this problem by using extreme long calculation times. Yet, there always remains the risk, that a hidden and not yet discovered hard-ware fault appears, and the results would have to beSupremacy of arctan questioned continuously.pi in India The Supercomputer Cray-2 at the NASA AMES Research Center used by David H. Bailey and others for theWith Infnitesimal calculation of any millions of digits of pi is very fast. His main memory can handle 228 computer words with 64 information bits each. For floating point arithmetic the Cray-2 uses a FORTRAN compiler in vector mode,Ramanujan which is about 20 times faster than scalar mode.AGM and moreSPIGOT Algorithm Integer arithmetic uses optimized FFT program routines (Fast Fourier Routines) for the multiplication ofThe Chudnovskys numbers with very many digits.Individual digits The author did program many of the algorithm and routines listed in the Book, and ran them on a normal PC with Pentium processor. For the integer arithmetic the ARIBAS Interpreter for Arithmetic of Professor Dr.OttoDigit distribution Forster of the Universität München was used.. This interpreter may be down-loaded from the INTERNET ofHigh precession arithmetic the FTP-server of the Mathematical Institute LMU Munich.Some examples2000 digits of pi Aribas is an interactive interpreter for integer arithmetic of large numbers. Of course, it may also be used for floating point arithmetic; the accuracy is then reduced to 192 bits, equivalent to about 56 decimal digits.pi: binary, decimal & hex Integer arithmetic permits the use of number up to 265535 , equivalent to about 24065 digits base 10.The Book : How to order Aribas uses elements of Modulo-2, Lisp, C, Fortran and other computer program languages.END3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 15
16. 16. LINKS to 1000 Dezimal Stellen von pi π Decimal Digits 1 bis 1000 Following pages :pi Title Page 3. 1415926535_8979323846_2643383279_5028841971_6939937510_Is pi useful ? 5820974944_5923078164_0628620899_8628034825_3421170679_pi in the antiquity 8214808651_3282306647_0938446095_5058223172_5359408128_With Archimedes 4811174502_8410270193_8521105559_6446229489_5493038196_To infinity 4428810975_6659334461_2847564823_3786783165_2712019091_Supremacy of arctan 4564856692_3460348610_4543266482_1339360726_0249141273_pi in India 7245870066_0631558817_4881520920_9628292540_9171536436_With Infnitesimal 7892590360_0113305305_4882046652_1384146951_9415116094_Ramanujan 3305727036_5759591953_0921861173_8193261179_3105118548_AGM and more 0744623799_6274956735_1885752724_8912279381_8301194912_SPIGOT Algorithm 9833673362_4406566430_8602139494_6395224737_1907021798_The Chudnovskys 6094370277_0539217176_2931767523_8467481846_7669405132_Individual digits 0005681271_4526356082_7785771342_7577896091_7363717872_Digit distribution 1468440901_2249534301_4654958537_1050792279_6892589235_High precession arithmetic 4201995611_2129021960_8640344181_5981362977_4771309960_Some examples 5187072113_4999999837_2978049951_0597317328_1609631859_2000 digits of pi 5024459455_3469083026_4252230825_3344685035_2619311881_pi: binary, decimal & hex 7101000313_7838752886_5875332083_8142061717_7669147303_The Book : How to order 5982534904_2875546873_1159562863_8823537875_9375195778_END 1857780532_1712268066_1300192787_6611195909_2164201989_3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 16
17. 17. LINKS to Pi digits in binary, decimal and hex Following pages : Binary : 480 digitspi Title Page 11.Is pi useful ? 00100100 00111111 01101010 10001000 10000101 10100011 00001000 11010011 00010011pi in the antiquity 00011001 10001010 00101110 00000011 01110000 01110011 01000100 10100100 00001001 00111000 00100010 00101001 10011111 00110001 11010000 00001000 00101110 11111010With Archimedes 10011000 11101100 01001110 01101100 10001001 01000101 00101000 00100001 11100110To infinity 00111000 11010000 00010011 01110111 10111110 01010100 01100110 11001111 00110100 11101001 00001100 01101100 11000000 10101100 00101001 10110111 11001001 01111100Supremacy of arctan 01010000 11011101 00111111 10000100 11010101 10110101pi in IndiaWith Infnitesimal Decimal : 500 digits 3.Ramanujan 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164AGM and more 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172SPIGOT Algorithm 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482The Chudnovskys 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436Individual digits 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381Digit distribution 8301194912High precession arithmeticSome examples Hexadecimal : 480 digits2000 digits of pi 3.pi: binary, decimal & hex 243F6A88 85A308D3 13198A2E 03707344 A4093822 299F31D0 082EFA98 EC4E6C89 452821E6 38D01377 BE5466CF 34E90C6C C0AC29B7 C97C50DD 3F84D5B5 B5470917The Book : How to order 9216D5D9 8979FB1B D1310BA6 98DFB5AC 2FFD72DB D01ADFB7B8E1AFED 6A267E96END BA7C9045 F12C7F99 24A19947 B3916CF7 0801F2E2 858EFC16 636920D8 732FE90D BC3A9442 ECC19381 729F4C5F 6574E198 30FBBC58 3EF6975C 4CED66B9 361B921D3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 342117067961EEC346 9B591887 138A3C7A 2FB68DB2 798A23C2 065092C0 BF910A90 8C77C3C8 ... 17 18ACD015 ACA52B18 D6E9DDBB 787749ED 52FA928E 1D2E34A7 3497F6DA 3BAB12DE
18. 18. LINKS to Following pages :pi Title PageIs pi useful ? Press Escape die Präsentation zu bendenpi in the antiquityWith ArchimedesTo infinitySupremacy of arctanpi in India oder Click die gewünschte SchaltflächeWith InfnitesimalRamanujanAGM and moreSPIGOT AlgorithmThe ChudnovskysIndividual digitsDigit distributionHigh precession arithmeticSome examples2000 digits of pipi: binary, decimal & hexThe Book : How to orderEND3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 18