2. pi Title Page Is pi useful ? pi in the antiquity With Archimedes To infinity Supremacy of arctan pi in India With Infnitesimal Ramanujan AGM and more SPIGOT Algorithm The Chudnovskys Individual digits Digit distribution High precession arithmetic Some examples 2000 digits of pi pi: binary, decimal & hex The Book : How to order END LINKS to Following pages : 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... 1 History and Algorithm of pi by Karl Helmut Schmidt Noli turbare circulos meos Archimedes The book contains overall description of the historical development of arithmetical methods for the calculation of pi. The CD accompanying this book gives an overview of many mathematical algorithms and some examples of how to get specific numbers or even individual digits of pi 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 practical and theoretical attention, to infinite series of products and sums, compounded fractions, and finally to the 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. Some mathematicians and amateurs alike did spent the most part of their lives for the exploitation and understanding of the phenomena pi. Pi is present in many areas, and offers substantial initiations for the study as well as general use, even to the specific point and analyses of modern mathematical theories. 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 the “Fast Fourier Transform”, deep penetration into pi could be achieved.

3. pi Title Page Is pi useful ? pi in the antiquity With Archimedes To infinity Supremacy of arctan pi in India With Infnitesimal Ramanujan AGM and more SPIGOT Algorithm The Chudnovskys Individual digits Digit distribution High precession arithmetic Some examples 2000 digits of pi pi: binary, decimal & hex The Book : How to order END LINKS to Following pages : 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... 2 Is pi useful ? If laws of mathematics or physics are valid in a specific area, then this laws are also valid in areas, which move relatively to the reference area.Albert Einstein The ratio of the circumference of a circle to is ratio is constant. Pi represents this ratio, and relates also to the ratio of the area of a circle to the square of its radius. In addition, pi results from the ratio of the sphere area to the square of the sphere diameter. Pi has puzzled and accompanied humanity for some thousand years. Practically in all cultures one may find some approximations for it: The Bibel shows a value = 3, 0 At Babylon and the Mesopotamia commonly use was 25 / 8 = 3, 125 The Egyptian Rhind Papyrus Rolls identify 256 / 81 = 3, 16 And even today many practicians use for pi 22 / 7 = 3, 14 At the beginning a specific value of pi was needed to construct circles and associated curves in architecture. Yet, scientists and mathematicians entered very early the quest of an answer to the direct translation of the area of a circle to a square – the famous search of the quadrature of the circle. 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, then the probability that a prime factor will be repeated is equal to the ratio of 6 / square of pi. Pi is not an irrational, but a transcendental number. In 1862, Lindeman gave a prove that pi is a transcendental number, which implies that the for so long search quadrature of a circle is impossible. 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 the most accurate machines, 10 places give the circumference of the earth to some Millimeter accuracy, and when 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? Why are we not satisfied with 50 or 100 decimal places of pi ? The direction and the way to get there are the target – The mountain will be climbed, because it just is there.

4. pi Title Page Is pi useful ? pi in the antiquity With Archimedes To infinity Supremacy of arctan pi in India With Infnitesimal Ramanujan AGM and more SPIGOT Algorithm The Chudnovskys Individual digits Digit distribution High precession arithmetic Some examples 2000 digits of pi pi: binary, decimal & hex The Book : How to order END LINKS to Following pages : 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... 3 pi in the antiquity Everything within the uiverse carries ist own specific number secret Chao-Hsiu Chen Over 2 million years mankind developed. Ninetynine percent of this time man was a collector of food and hunter. Besides weapons and tools to hunt, and later to work his fields numbers greater than 2 or even greater than 10 were unnecessary. A herd consisted of 2 animals or it had just many. 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 the first steps towards a simple arithmetic were taken, with the first forms of writing and documentation. Besides 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. Such rules of arithmetic brought along the discovery of relationship between specific subjects and its values. To double the volume resulted in doubling the weight. Certain relations of lengths of the sides of a triangle established a right angle. The ratios of the circumference to the diameter of circles were constant. 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 today extensively in Asia, uses for nothing or a way to use zero an empty row. The symbol for zero originated in India, and came together with the Hindu-Arabic presentation of numbers 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. About 1850 B.C. the Egyptian scribe Ahmes gave the earliest known record in the so-called Rhind Papyrus on how to calculate pi. His recipe results in a approximation using a equal-sided area with eight corners> the square of (64 / 81) = pi / 4 This shows for pi = 3.16

5. pi Title Page Is pi useful ? pi in the antiquity With Archimedes To infinity Supremacy of arctan pi in India With Infnitesimal Ramanujan AGM and more SPIGOT Algorithm The Chudnovskys Individual digits Digit distribution High precession arithmetic Some examples 2000 digits of pi pi: binary, decimal & hex The Book : How to order END LINKS to Following pages : 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... 4 With Archimedes Geometry is the best method to devote once free time Plato 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 theories about the ability to measure circles. Demonstration Nr. 7 The ratio of the perimeters of two regular polygons with equal number of sides is equal to the ratio of their in- respectively circumscribed circles. 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 smaller than its perspective circumscribed circle, yet it is larger than its inscribed circle. If one takes n to be very large, the in- and circumscribed circle converge to a single value. With n equal to infinity the value for exact pi may be found. To calculate the circumference of a circle Archimedes started with a regular 6-sided polygon. He then continuously doubled the number of sides up to the value of 96 sides. For the first time in history Archimedes 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 the calculation of pi to any desired accuracy. This calculation method of pi survived for many centuries. Since Archimedes was limited to the use of the formula of Pythagoras and the kwon method of halving angles, the practicality of number handling limited his approach to a 96-sided polygon. He thereby found the value of pi to lie between 3 10 / 71 < pi < 3 1 / 7 = 3,140845 < pi < 3,142857 The use of calculating the arithmetical mean pi m = (a+b) / 2 would have given Archimedes the following approximation: pi m = (3,140845 + 3,142857) / 2 = 3,141851 1593 A.C. Viete, using the Archimedes method with a regular 393 216 sided polygon found the approximate value for pi to be between 3,1415926535 < pi < 3,1415926537 During many centuries after Archimedes no considerable improvements or new calculating methods were established.

6. pi Title Page Is pi useful ? pi in the antiquity With Archimedes To infinity Supremacy of arctan pi in India With Infnitesimal Ramanujan AGM and more SPIGOT Algorithm The Chudnovskys Individual digits Digit distribution High precession arithmetic Some examples 2000 digits of pi pi: binary, decimal & hex The Book : How to order END LINKS to Following pages : 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... 5 To infinity God is absolute Infinity, human beings are by nature finite and may not participate at infinity, and in no way understand it. Thomas von Aquin After Archimedes the first appreciable and mentionable activity in the filed of calculating pi within 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. Real progress came for the development of algorithm with the findings for the binomial series and the development of power series. Blaise Pascal, a brilliant mathematician, set with his well known PASCAL- triangle the basis for the infinitesimal arithmetic and thereby new ways to calculate pi. 1655 John Wallis published his famous formula for pi, which is the result of an infinite power series.  /2 = 2  (1 – 1/(2n +1) 2 ) Newton discovered in 1665 the binomial number series. 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. Leibniz inserted the value 1 for x and got the so-called Leibniz-Gregory-Series.  /4 =  (-1) n 1 / (2n+1) A practical evaluation of pi using this series is not feasible. This series converges to slowly. In 1996 Newton calculated 15 correct decimal digits by the use of the formula  = (  3)/4 – 24   (x–x 2 ) dx This series corresponds in principal the arcsin x power series..

7. pi Title Page Is pi useful ? pi in the antiquity With Archimedes To infinity Supremacy of arctan pi in India With Infnitesimal Ramanujan AGM and more SPIGOT Algorithm The Chudnovskys Individual digits Digit distribution High precession arithmetic Some examples 2000 digits of pi pi: binary, decimal & hex The Book : How to order END LINKS to Following pages : 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... 6 The supremacy of arcus tangens Our almighty teacher did invite the human beings to study and to imitate the scientific structure of the infinite universe Thomas Paine 1706 John Machin developed his famous and very fast converging formula. By the use of this and the previously stated power series of Gregory for arctan, Machin calculated 100 correct decimal digits of pi.  / 4 = 4 arctan (1/5) – arctan (1/239) John Machin found this via the formula for doubling tan 2 . Using his general presentation to dissect one value for arctan into two amounts. This brought so many additional formula of based upon the arctan power series. arctan u + arctan v = arctan (u+v) / (1–uv) 1738 Euler found a new method for calculating arctan value, which converged much faster then Gregory’s. He also published the following  / 4 = arctan 1 / 2 + arctan 1 / 3 Additional formula, such as shown below, were developed:  / 4 = 2 arctan (1/5) + arctan (1/7) + arctan (1/8)  / 4 = arctan (1/2) + arctan (1/4) + arctan (1/13)  / 4 = 3 arctan (1/4) + arctan (1/20) + arctan (1/1985)  / 4 = 22 arctan (1/28) + 2 arctan (1/443) – 5 arctan (1/1393) – 10 arctan (1/12943) The methods for calculating pi established by John Machin using arctan power series were extremely effective, 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.

8. pi Title Page Is pi useful ? pi in the antiquity With Archimedes To infinity Supremacy of arctan pi in India With Infnitesimal Ramanujan AGM and more SPIGOT Algorithm The Chudnovskys Individual digits Digit distribution High precession arithmetic Some examples 2000 digits of pi pi: binary, decimal & hex The Book : How to order END LINKS to Following pages : 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... 7 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. Henri Poincore’ Since the antique very progressive mathematical investigations, establishment of arithmetical rules, and even 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. A number of arithmetical rules, so-called cord-rules, were written down around 600 A.C. in a document 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 a square was defined as follows: Take the 8. part of a circle diameter and divide this in 29 parts Take then the 28.part and the 6.part of the remaining 29.part Then subtract the 8.part As formula this results in S q = d 9785 / 11136 From this pi = 4 S q / d 2 = 3,088 499 B.C. Arya-Bhata writes in the documents Siddhanta for the value of pi 3 + 177 / 1250 = 3,141... Yet, quite more interesting notes from India about pi are documents from the 15.century employing infinite power series. The Sanskrit-Documents Yukti-Dipika and Yukti-Bhasa give 8 power series for pi, including the so-called Leibniz-Series. 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. One example is :  / 2 =  3  (–1) n / ((2n + 1) 3 n ) Additional examples may be found in the book pi Geschichte und Algorithmen einer Zahl

9. pi Title Page Is pi useful ? pi in the antiquity With Archimedes To infinity Supremacy of arctan pi in India With Infnitesimal Ramanujan AGM and more SPIGOT Algorithm The Chudnovskys Individual digits Digit distribution High precession arithmetic Some examples 2000 digits of pi pi: binary, decimal & hex The Book : How to order END LINKS to Following pages : 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... 8 with Infinitesimal With the use of exact methods it is often extreme difficult if not impossible to solve certain equations, only by the use of iterations solutions may be found. Lancelot Hoyben 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 calculus independent from each other at the same time. Newton’s fluxion and fluent rules are difficult to apply, which 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. With the development of calculus the problem and the associated solution of the calculation of areas similar to the problem of Archimedes reappeared. The task is to evaluate and calculate the area limited by the curve 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. Newton used a segment of a circle with the radius = 0.5. His resulting power series converged relative rapid to a solution. The first 24 partial sums already give 24 correct decimal digits of pi. Only within an hour Newton calculated 20 correct decimal digits. 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 the value x=1. Leibniz’ merits within the filed of mathematics are versatile. For example, he published an article, in which he presented for the first time methods for basic binary arithmetic (+, -, *, /). This publication is considered the birth of radix-2 arithmetic.

10. pi Title Page Is pi useful ? pi in the antiquity With Archimedes To infinity Supremacy of arctan pi in India With Infnitesimal Ramanujan AGM and more SPIGOT Algorithm The Chudnovskys Individual digits Digit distribution High precession arithmetic Some examples 2000 digits of pi pi: binary, decimal & hex The Book : How to order END LINKS to Following pages : 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... 9 Ramanujan Ramanujan...climb to the paradise on the ladder of surprises Ralph Waldo Emerson During the 18. and 19. centuries a number of quite famous mathematicians lived. Boole, Cantor, Cauchy, Chebychev, Fourier, Langrange, Laplace, Mersenne, Plank, Poisson, Riemann, Taylor, Turing and others 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. Srinivasa Ramanujan born 1887 in Erode, a small town in Southern India, showed very early in his life signs of a mathematical genius. At the age of 12 he had mastered the extensive publication “Plane Trigonometry”, being 15 years old he studied from “Relations of elementary results of pure mathematics”. This was his total mathematical education. Despite of his limited training he succeeded in reformulating and expanding on the general number theory with new theory and formulas. After publishing his astonishing and brilliant results on “Bernoulli Numbers” 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 his time, brought him to the Trinity College Cambridge. Ramanujan formulated the “Riemann Series”, elliptical integrals, hypergeometrical series and functional 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 health failed. He died 1920 in India. Ramanujan bestowed a range of unpublished notebooks. 70 years after his death, quite an number of scientists 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. The most famous presentation for the calculation of pi using an infinite series of sums by Ramanujan is 1/  = (  8) / 9801  (4n)! (1103+26390 n) / ((n!) 4 396 4n ) representing a special solution for a related function of modular quantities. Gosper calculated 17 million digits of  using this formula.

11. pi Title Page Is pi useful ? pi in the antiquity With Archimedes To infinity Supremacy of arctan pi in India With Infnitesimal Ramanujan AGM and more SPIGOT Algorithm The Chudnovskys Individual digits Digit distribution High precession arithmetic Some examples 2000 digits of pi pi: binary, decimal & hex The Book : How to order END LINKS to Following pages : 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... 10 AGM : Algorithm using arithmetic-geometric-means Mathematic is a fabulous science, yet, mathematicians don’t suit the henchman most of the time Lichtenberg The algorithm for the arithmetic-geometric-mean (AGM) was originally already used 1811 by Legendre in 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 an iterative process fast convergence is achieved. Basically AGM is defined as AGM (x 0, y 0 )  M [ (x 0 + y 0 ) / 2 ;  (x 0 * y 0 ) ] This fast convergence is best shown on the following example: With x 0 = 1 and y 0 = 0,8 x 1 = 0,9y 1 = 0,894427190999915… x 2 = 0,897213595499957…y 2 = 0,897209268732734… x 3 = 0,897211432116346…y 3 = 0,897211432113738… x 4 = 0,897211432115042…y 4 = 0,897211432115042… 1976 E. Salamin and R.P. Brent did independently of each other rediscover AGM for the calculation of pi, and developed a very fast converging algorithm for computer usage. Salamin gave at that time an estimate on the numerical evaluation for pi, which foresaw 33 million digits as a possible result. 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 by Legendre. From their research in the field of number theory the brothers Borwein offered general methods for the calculation of certain elementary mathematical constants.

12. pi Title Page Is pi useful ? pi in the antiquity With Archimedes To infinity Supremacy of arctan pi in India With Infnitesimal Ramanujan AGM and more SPIGOT Algorithm The Chudnovskys Individual digits Digit distribution High precession arithmetic Some examples 2000 digits of pi pi: binary, decimal & hex The Book : How to order END LINKS to Following pages : 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... 11 The Chudnovsky Brothers Shouting for worldly fame is only like a breeze blowing from different directions and changing thereby its direction Dante Alighieri David und Gregory Chudnovsky, two brilliant brothers, even often quite excentric, both immigrated form the 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 NASA Cray-Computer. They constructed and built their own computer right at their apartment in Manhattan from generally available 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 computer parts, building blocks, interconnecting lines, cables and so on. Since power consumption had not been optimized, most likely it had been even impossible, extensive heat did develop, some even thought of hell like proportions. Despite of all the Chudnovsky brothers made very successful progress in the field of calculation of many millions of digits of pi. During May of 1989 they achieved 480 million of it, and 5 years later even 4 044 000 000 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 of iteration produced 18 correct digits. 1/  = (6541681608 / 640320 1/2 )  [ (13591409 / 545140134) +k ] (–1) k (6k)! / [(3k)! (k!) 3 640320 3k ] für k=0 bis k= 

13. pi Title Page Is pi useful ? pi in the antiquity With Archimedes To infinity Supremacy of arctan pi in India With Infnitesimal Ramanujan AGM and more SPIGOT Algorithm The Chudnovskys Individual digits Digit distribution High precession arithmetic Some examples 2000 digits of pi pi: binary, decimal & hex The Book : How to order END LINKS to Following pages : 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... 12 Spigot Algorithm Patience is the power from which we achieve the best Confucius A very interesting Algorithm for the calculation of certain number values such as  2, the basis of the logarithm e and pi was presented by Stanley Rabinowitz and Stan Wagon. This computing instruction functions like a spigot from which individual digits appear without use of previous numbers The “appearing” digits do not need great accurate arithmetic The algorithm employs integer arithmetic with only 8 bit accuracy. By the law for the presentation of polyadic number systems z =  a i b i for i = -m to n the formula for the development of this sum using a uniform number base b is then equal to...+ a 3 b 3 + a 2 b 2 + a 1 b 1 + a 0 b 0 + a -1 b -1 + a -2 b -2 + … an example for the decimal system is 139,8125 10 = 1* 10 2 + 3* 10 1 + 9* 10 0 + 8 10 -1 + 1* 10 -2 + 2* 10 -3 + 5* 10 -4 Number systems with mixed number base, such as 3 weeks + 4 days + 1 hour + 49 minutes + 7 seconds + 99 hundreds of a sec. Pound + 18 Shilling + 11 Pence = 30 10 + 18 20 + 11 12 (the old UK monetary system) May be presented with a differing number base c i...+ a 3 b 3 + a 2 c 2 2 + a 1 c 1 1 + a 0 c 0 0 + a -1 c -1 -1 + a -2 c -2 -2 + … Now, an interesting answer appears, if one uses the mixed number base c = 1/1; 1/2,; 1/3; ¼; 1/5; …. The math constant e = 2,718281… diverts to e = 1 + 1/1(1 + 1/2 (1 + 1/3 (1 + 1/4 (1 + 1/5 (1 + … ))) Naturally, it is also feasible to present pi on a mixed base. Using the Leibniz-Series as converted by Euler and c = 1/1; 1/3; 2/5; 3/7; 4/9; … one finds  /2 = 1 + 1/3(1 + 2/5(1 + 3/7 (1 + 4/9 ( 1 + …))) or  = (2; 2; 2; 2; 2; 2; …) c The solution of this mixed base presentation follows a way similar to the Horner-Scheme. The corresponding algorithm equal the above mentioned Spigot program for the calculation of pi.

14. pi Title Page Is pi useful ? pi in the antiquity With Archimedes To infinity Supremacy of arctan pi in India With Infnitesimal Ramanujan AGM and more SPIGOT Algorithm The Chudnovskys Individual digits Digit distribution High precession arithmetic Some examples 2000 digits of pi pi: binary, decimal & hex The Book : How to order END LINKS to Following pages : 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... 13 Calculation of individual digits of pi Much is not sufficient, the quality is the clue The Autor At the beginning of 1995 David Bailey and Simon Plouffe published and surprised with an absolutely new development for the calculation of digits for pi. Without the need to determine any previous digits they calculated any individual hexadecimal digits. For this the used  =  1 / 16 n [ 4 / (8n+1) – 2 / (8n+4) – 1 / (8n+5) – 1 / (8n+6) ] for n=0 to n=  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 some new fundamentals were discovered. Bailey, Borwein and Plouffe found during the month of November 1995 the 40* 10 9 digit in HEX : 921C73C6838FB2 1996 Simon Plouffe solved then the task to calculate the n-th decimal digit of some irrational as well as transcendental numbers such as ,  3, integer powers of the Riemann Zeta Function Zeta(3), log(2), and others. For a long time this was considered to be impossible or at least extremely difficult. The basis for this calculations was the following formula already developed by Euler  + 3 =  n 2 n /  2n über n  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. There exists some more infinite sums based upon C(mn,n), which are suitable for the calculation of individual digit of any number base.

15. pi Title Page Is pi useful ? pi in the antiquity With Archimedes To infinity Supremacy of arctan pi in India With Infnitesimal Ramanujan AGM and more SPIGOT Algorithm The Chudnovskys Individual digits Digit distribution High precession arithmetic Some examples 2000 digits of pi pi: binary, decimal & hex The Book : How to order END LINKS to Following pages : 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... 14 Digit Distribution Some mathematician consider the decimal expansion of pi a random series, but to modern numerologist it is rich with remarkable patterns. Dr.I.J.Matrix (Martin Gardner) Over centuries pi was intensively investigated for its characteristics and patterns. In general it seems that the digits arrange in a row randomly. Yet, the change of only one decimal digit results in a complete different number; it is then no longer pi.Many investigations deal with the search for patterns of repetition or specific number series. The digit ZERO (0) shows for the first time at the 32. decimal position. The sum of the first 20 decimal digits is 100. Adding the first 144 decimal digits one gets the sum 666. The 3 decimals ending at position 315, had the sequence 315. The first 0 shows at position32the first ONE1 at position 1 0030711 94 000601111 153 0000133901111 12700 000001753411111 Quite often one may see some interesting patterns : 11011at decimal position3844 1000114201 8777817234 2020207285 665556610143 The frequency (F) of every decimal digit (0 to 9) of the first 29 millions of the decimal digits of pi shows the following: Digit 0 1 2 3 4 5 Rel. Frequency 0,0999440 0,0999333 0,1000306 0,0999964 0,1001093 0,1000466 Digit 6 7 8 9 Rel. Frequency 0,0999337 0,1000207 0,0999914 0,1000040 Not proven, but it is generally assumed that all 10 digits are equally distributed, since the relative frequency approaches 0.1.

16. pi Title Page Is pi useful ? pi in the antiquity With Archimedes To infinity Supremacy of arctan pi in India With Infnitesimal Ramanujan AGM and more SPIGOT Algorithm The Chudnovskys Individual digits Digit distribution High precession arithmetic Some examples 2000 digits of pi pi: binary, decimal & hex The Book : How to order END LINKS to Following pages : 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... 15 Highly precise Computer Arithmetic For one person science is the high, heavenly goddess, where as another person sees it as an efficient cow providing butter Friedrich von Schiller 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 accurate 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 be questioned continuously. The Supercomputer Cray-2 at the NASA AMES Research Center used by David H. Bailey and others for the calculation of any millions of digits of pi is very fast. His main memory can handle 2 28 computer words with 64 information bits each. For floating point arithmetic the Cray-2 uses a FORTRAN compiler in vector mode, which is about 20 times faster than scalar mode. Integer arithmetic uses optimized FFT program routines (Fast Fourier Routines) for the multiplication of numbers with very many 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.Otto Forster of the Universität München was used.. This interpreter may be down-loaded from the INTERNET of the FTP-server of the Mathematical Institute LMU Munich. 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. Integer arithmetic permits the use of number up to 2 65535, equivalent to about 24065 digits base 10. Aribas uses elements of Modulo-2, Lisp, C, Fortran and other computer program languages.

17. pi Title Page Is pi useful ? pi in the antiquity With Archimedes To infinity Supremacy of arctan pi in India With Infnitesimal Ramanujan AGM and more SPIGOT Algorithm The Chudnovskys Individual digits Digit distribution High precession arithmetic Some examples 2000 digits of pi pi: binary, decimal & hex The Book : How to order END LINKS to Following pages : 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... 16 1000 Dezimal Stellen von pi  Decimal Digits 1 bis 1000 3. 1415926535_8979323846_2643383279_5028841971_6939937510_ 5820974944_5923078164_0628620899_8628034825_3421170679_ 8214808651_3282306647_0938446095_5058223172_5359408128_ 4811174502_8410270193_8521105559_6446229489_5493038196_ 4428810975_6659334461_2847564823_3786783165_2712019091_ 4564856692_3460348610_4543266482_1339360726_0249141273_ 7245870066_0631558817_4881520920_9628292540_9171536436_ 7892590360_0113305305_4882046652_1384146951_9415116094_ 3305727036_5759591953_0921861173_8193261179_3105118548_ 0744623799_6274956735_1885752724_8912279381_8301194912_ 9833673362_4406566430_8602139494_6395224737_1907021798_ 6094370277_0539217176_2931767523_8467481846_7669405132_ 0005681271_4526356082_7785771342_7577896091_7363717872_ 1468440901_2249534301_4654958537_1050792279_6892589235_ 4201995611_2129021960_8640344181_5981362977_4771309960_ 5187072113_4999999837_2978049951_0597317328_1609631859_ 5024459455_3469083026_4252230825_3344685035_2619311881_ 7101000313_7838752886_5875332083_8142061717_7669147303_ 5982534904_2875546873_1159562863_8823537875_9375195778_ 1857780532_1712268066_1300192787_6611195909_2164201989_ Decimal Digits 1001 bis 2000 3809525720_1065485863_2788659361_5338182796_8230301952_ 0353018529_6899577362_2599413891_2497217752_8347913151_ 5574857242_4541506959_5082953311_6861727855_8890750983_ 8175463746_4939319255_0604009277_0167113900_9848824012_ 8583616035_6370766010_4710181942_9555961989_4676783744_ 9448255379_7747268471_0404753464_6208046684_2590694912_ 9331367702_8989152104_7521620569_6602405803_8150193511_ 2533824300_3558764024_7496473263_9141992726_0426992279_ 6782354781_6360093417_2164121992_4586315030_2861829745_ 5570674983_8505494588_5869269956_9092721079_7509302955_ 3211653449_8720275596_0236480665_4991198818_3479775356_ 6369807426_5425278625_5181841757_4672890977_7727938000_ 8164706001_6145249192_1732172147_7235014144_1973568548_ 1613611573_5255213347_5741849468_4385233239_0739414333_ 4547762416_8625189835_6948556209_9219222184_2725502542_ 5688767179_0494601653_4668049886_2723279178_6085784383_ 8279679766_8145410095_3883786360_9506800642_2512520511_ 7392984896_0841284886_2694560424_1965285022_2106611863_ 0674427862_2039194945_0471237137_8696095636_4371917287_ 4677646575_7396241389_0865832645_9958133904_7802759009_

18. pi Title Page Is pi useful ? pi in the antiquity With Archimedes To infinity Supremacy of arctan pi in India With Infnitesimal Ramanujan AGM and more SPIGOT Algorithm The Chudnovskys Individual digits Digit distribution High precession arithmetic Some examples 2000 digits of pi pi: binary, decimal & hex The Book : How to order END LINKS to Following pages : 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... 17 Pi digits in binary, decimal and hex Binary : 480 digits 11. 00100100 00111111 01101010 10001000 10000101 10100011 00001000 11010011 00010011 00011001 10001010 00101110 00000011 01110000 01110011 01000100 10100100 00001001 00111000 00100010 00101001 10011111 00110001 11010000 00001000 00101110 11111010 10011000 11101100 01001110 01101100 10001001 01000101 00101000 00100001 11100110 00111000 11010000 00010011 01110111 10111110 01010100 01100110 11001111 00110100 11101001 00001100 01101100 11000000 10101100 00101001 10110111 11001001 01111100 01010000 11011101 00111111 10000100 11010101 10110101 Decimal : 500 digits 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 Hexadecimal : 480 digits 3. 243F6A88 85A308D3 13198A2E 03707344 A4093822 299F31D0 082EFA98 EC4E6C89 452821E6 38D01377 BE5466CF 34E90C6C C0AC29B7 C97C50DD 3F84D5B5 B5470917 9216D5D9 8979FB1B D1310BA6 98DFB5AC 2FFD72DB D01ADFB7B8E1AFED 6A267E96 BA7C9045 F12C7F99 24A19947 B3916CF7 0801F2E2 858EFC16 636920D8 732FE90D BC3A9442 ECC19381 729F4C5F 6574E198 30FBBC58 3EF6975C 4CED66B9 361B921D 9B591887 138A3C7A 2FB68DB2 798A23C2 065092C0 BF910A90 8C77C3C8 61EEC346 18ACD015 ACA52B18 D6E9DDBB 787749ED 52FA928E 1D2E34A7 3497F6DA 3BAB12DE BE6B7A9B 1D0FBA91 2AE475E2 6236E10E

19. pi Title Page Is pi useful ? pi in the antiquity With Archimedes To infinity Supremacy of arctan pi in India With Infnitesimal Ramanujan AGM and more SPIGOT Algorithm The Chudnovskys Individual digits Digit distribution High precession arithmetic Some examples 2000 digits of pi pi: binary, decimal & hex The Book : How to order END LINKS to Following pages : 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... 18 Press Escape die Präsentation zu benden oder Click die gewünschte Schaltfläche