History and Algorithm of pi

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

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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 Einstein
pi Title Page
The ratio of the circumference of a circle to is ratio is constant. Pi represents this ratio, and relates also to the
Is 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 to
pi 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 = 3,0
Supremacy of arctan
At Babylon and the Mesopotamia commonly use was 25 / 8 = 3 , 125
pi in India
The Egyptian Rhind Papyrus Rolls identify 256 / 81 = 3 , 16
With Infnitesimal
And even today many practicians use for pi 22 / 7 = 3 , 14
Ramanujan
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 the
SPIGOT 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 a
High 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 the
2000 digits of pi most accurate machines, 10 places give the circumference of the earth to some Millimeter accuracy, and when
pi: 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.
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Everything within the uiverse carries ist own specific
LINKS to number secret Chao-Hsiu Chen
Following pages :
pi in the antiquity
pi Title Page
Over 2 million years mankind developed. Ninetynine percent of this time man was a collector of food and
Is pi useful ? hunter. Besides weapons and tools to hunt, and later to work his fields numbers greater than 2 or even greater
pi 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 the
To infinity first steps towards a simple arithmetic were taken, with the first forms of writing and documentation. Besides
Supremacy 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 triangle
Ramanujan 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 today
SPIGOT 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 numbers
Individual 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 distribution
High precession arithmetic
About 1850 B.C. the Egyptian scribe Ahmes gave the earliest known record in the so-called Rhind Papyrus on
Some 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 / 4
pi: binary, decimal & hex This shows for pi = 3.16
END

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LINKS to Geometry is the best method to devote once free time

Following pages :
With Archimedes Plato

pi 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 theories
Is 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 equal
With 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 smaller
Supremacy 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 then
Ramanujan continuously doubled the number of sides up to the value of 96 sides. For the first time in history Archimedes
AGM 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 the
SPIGOT 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 halving
Individual digits angles, the practicality of number handling limited his approach to a 96-sided polygon. He thereby found the
value of pi to lie between
Digit distribution
3 10/71 < pi < 3 1/7 = 3,140845 < pi < 3,142857
The use of calculating the arithmetical mean pim = (a+b) / 2 would have given Archimedes the following
Some examples
approximation: pim = (3,140845 + 3,142857) / 2 = 3,141851
2000 digits of pi
1593 A.C. Viete, using the Archimedes method with a regular 393 216 sided polygon found the approximate
pi: binary, decimal & hex value for pi to be between
The Book : How to order 3,1415926535 < pi < 3,1415926537
END During many centuries after Archimedes no considerable improvements or new calculating methods were
established.
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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 Aquin
pi Title Page
Is pi useful ?
pi in the antiquity After Archimedes the first appreciable and mentionable activity in the filed of calculating pi within
With 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 the
Supremacy 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 Chudnovskys
Individual 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)
A practical evaluation of pi using this series is not feasible. This series converges to slowly.
Some examples
2000 digits of pi
In 1996 Newton calculated 15 correct decimal digits by the use of the formula
pi: binary, decimal & hex
π = (√3)/4 – 24  √(x–x2) dx This series corresponds in principal the arcsin x power series..
END

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LINKS to The supremacy of arcus tangens
Following pages : Our almighty teacher did invite the human beings to study and to imitate
pi Title Page the scientific structure of the infinite universe
Is pi useful ? Thomas Paine

pi in the antiquity
1706 John Machin developed his famous and very fast converging formula. By the use of this and the
With 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 one
pi 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. He
AGM and more also published the following
SPIGOT Algorithm π / 4 = arctan 1/2 + arctan 1/3
The 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 pi
pi: 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.
END

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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 Page
Is pi useful ? Since the antique very progressive mathematical investigations, establishment of arithmetical rules, and even
pi 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 document
To 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 a
Supremacy of arctan
square was defined as follows:
pi in India
Take the 8. part of a circle diameter and divide this in 29 parts
With Infnitesimal
Take then the 28.part and the 6.part of the remaining 29.part
Ramanujan
Then subtract the 8.part
AGM and more
As formula this results in Sq = d 9785/11136 From this pi = 4 Sq / d2 = 3,088
SPIGOT Algorithm
499 B.C. Arya-Bhata writes in the documents Siddhanta for the value of pi
The Chudnovskys
3 + 177/1250 = 3,141...
Individual digits
Yet, quite more interesting notes from India about pi are documents from the 15.century employing infinite
Digit distribution
power series. The Sanskrit-Documents Yukti-Dipika and Yukti-Bhasa give 8 power series for pi, including
High 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 :
π / 2 = √3 ∑ (–1)n / ((2n + 1) 3n )
Additional examples may be found in the book pi Geschichte und Algorithmen einer Zahl
END

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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
Lancelot
Is pi useful ? Hoyben
pi in the antiquity
With 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 calculus
To infinity
independent from each other at the same time. Newton’s fluxion and fluent rules are difficult to apply, which
Supremacy 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 India
With Infnitesimal
With the development of calculus the problem and the associated solution of the calculation of areas similar to
Ramanujan
the problem of Archimedes reappeared. The task is to evaluate and calculate the area limited by the curve
AGM 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 Algorithm
The Chudnovskys
Newton used a segment of a circle with the radius = 0.5 . His resulting power series converged relative rapid
Individual digits
to a solution. The first 24 partial sums already give 24 correct decimal digits of pi. Only within an hour
Digit distribution Newton calculated 20 correct decimal digits.
Some 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 the
2000 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.
END

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LINKS to ...climb to the paradise on the ladder of surprises

Following pages :
Ramanujan Ralph Waldo Emerson

pi 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 others
pi 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 signs
To 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 with
With 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 his
AGM and more time, brought him to the Trinity College Cambridge.
SPIGOT Algorithm Ramanujan formulated the “Riemann Series”, elliptical integrals, hypergeometrical series and functional
The 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 health
Individual 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 scientists
High 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 is
2000 digits of pi
1/π = (√8) / 9801 ∑ (4n)! (1103+26390 n) / ((n!)4 3964n )
representing a special solution for a related function of modular quantities.
Gosper calculated 17 million digits of π using this formula.
END

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LINKS to AGM : Algorithm using arithmetic-geometric-means
Following pages :
Mathematic is a fabulous science, yet, mathematicians don’t suit the
pi Title Page
henchman most of the time
Is pi useful ? Lichtenberg
pi in the antiquity The algorithm for the arithmetic-geometric-mean (AGM) was originally already used 1811 by Legendre in
With 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 an
To infinity iterative process fast convergence is achieved. Basically AGM is defined as
Supremacy of arctan AGM (x0 , y0 ) ≡ M [ (x0 + y0 ) / 2 ; √ (x0 * y0 ) ]
pi in India
With 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 on
Digit distribution the numerical evaluation for pi, which foresaw 33 million digits as a possible result.
Some 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 by
2000 digits of pi
Legendre.
From their research in the field of number theory the brothers Borwein offered general methods for the
The Book : How to order calculation of certain elementary mathematical constants.
END

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LINKS to
Following pages :
The Chudnovsky Brothers
pi Title Page
Shouting for worldly fame is only like a breeze
Is pi useful ? blowing from different directions and changing
pi in the antiquity thereby its direction Dante Alighieri

With Archimedes
To infinity
David und Gregory Chudnovsky, two brilliant brothers, even often quite excentric, both immigrated form the
Supremacy 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 NASA
pi in India Cray-Computer.
With Infnitesimal They constructed and built their own computer right at their apartment in Manhattan from generally available
Ramanujan 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
AGM and more parts, building blocks, interconnecting lines, cables and so on. Since power consumption had not been
SPIGOT 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 many
Individual digits millions of digits of pi. During May of 1989 they achieved 480 million of it, and 5 years later even 4 044 000
Digit 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 of
High precession arithmetic iteration produced 18 correct digits.
Some examples
2000 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=∞
END

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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 the
pi Title Page
logarithm e and pi was presented by Stanley Rabinowitz and Stan Wagon. This computing instruction
Is pi useful ? functions like a spigot from which individual digits appear without use of previous numbers
pi in the antiquity The “appearing” digits do not need great accurate arithmetic
The algorithm employs integer arithmetic with only 8 bit accuracy
With Archimedes
. By the law for the presentation of polyadic number systems z = ∑ ai bi for i = -m to n the formula for the
To infinity
development of this sum using a uniform number base b is then equal to
Supremacy 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 is
With Infnitesimal 139,812510 = 1* 102 + 3* 101 + 9* 100 + 8 10-1 + 1* 10-2 + 2* 10-3 + 5* 10-4
Ramanujan
Number systems with mixed number base, such as
AGM 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 Algorithm
The 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 to
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 Euler
2000 digits of pi and c = 1/1; 1/3; 2/5; 3/7; 4/9; … one finds
pi: 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; …)c
The Book : How to order The solution of this mixed base presentation follows a way similar to the Horner-Scheme. The
END corresponding algorithm equal the above mentioned Spigot program for the calculation of pi.

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LINKS to
Calculation of individual digits of pi
Following pages :
pi Title Page Much is not sufficient, the quality is the clue
The Autor
Is pi useful ?
pi in the antiquity
At the beginning of 1995 David Bailey and Simon Plouffe published and surprised with an absolutely new
With Archimedes development for the calculation of digits for pi. Without the need to determine any previous digits they
To infinity calculated any individual hexadecimal digits. For this the used

Supremacy 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 some
With 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 921C73C6838FB2

SPIGOT 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), and
The Chudnovskys others. For a long time this was considered to be impossible or at least extremely difficult.
Individual digits
Digit distribution The basis for this calculations was the following formula already developed by Euler
High 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 of
individual digit of any number base.
END

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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 the
Is pi useful ? digits arrange in a row randomly. Yet, the change of only one decimal digit results in a complete different
pi 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 1
With Infnitesimal 00 307 11 94
Ramanujan 000 601 111 153
0000 13390 1111 12700
AGM and more 00000 17534 11111
SPIGOT Algorithm Quite often one may see some interesting patterns :
The Chudnovskys 11011 at decimal position 3844
10001 14201
Individual digits 87778 17234
Digit distribution 202020 7285
6655566 10143
The frequency (F) of every decimal digit (0 to 9) of the first 29 millions of the decimal digits of pi shows the
Some examples following: Digit
0 1 2 3 4 5
2000 digits of pi
Rel. Frequency 0,0999440 0,0999333 0,1000306 0,0999964 0,1001093 0,1000466
The Book : How to order Digit 6 7 8 9
Rel. Frequency 0,0999337 0,1000207 0,0999914 0,1000040
END
Not proven, but it is generally assumed that all 10 digits are equally distributed, since the relative frequency
3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ...
approaches 0.1 . 14

LINKS to
Highly precise Computer Arithmetic
Following pages :
For one person science is the high, heavenly goddess, where as
pi Title Page
another person sees it as an efficient cow providing butter
Is pi useful ? Friedrich von Schiller
pi in the antiquity
With 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 accurate
To 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 be
Supremacy 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 the
With 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 more
SPIGOT Algorithm
Integer arithmetic uses optimized FFT program routines (Fast Fourier Routines) for the multiplication of
The 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.Otto
Digit distribution
Forster of the Universität München was used.. This interpreter may be down-loaded from the INTERNET of
High precession arithmetic the FTP-server of the Mathematical Institute LMU Munich.
Some examples
2000 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.
Integer arithmetic permits the use of number up to 265535 , equivalent to about 24065 digits base 10.
Aribas uses elements of Modulo-2, Lisp, C, Fortran and other computer program languages.
END

3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 15

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_
4201995611_2129021960_8640344181_5981362977_4771309960_
Some examples
5187072113_4999999837_2978049951_0597317328_1609631859_
2000 digits of pi
5024459455_3469083026_4252230825_3344685035_2619311881_
7101000313_7838752886_5875332083_8142061717_7669147303_
5982534904_2875546873_1159562863_8823537875_9375195778_
END
1857780532_1712268066_1300192787_6611195909_2164201989_
3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 16

LINKS to Pi digits in binary, decimal and hex
Following pages : Binary : 480 digits
pi Title Page 11.
Is pi useful ? 00100100 00111111 01101010 10001000 10000101 10100011 00001000 11010011 00010011
pi in the antiquity 00011001 10001010 00101110 00000011 01110000 01110011 01000100 10100100 00001001
00111000 00100010 00101001 10011111 00110001 11010000 00001000 00101110 11111010
With Archimedes 10011000 11101100 01001110 01101100 10001001 01000101 00101000 00100001 11100110
To infinity 00111000 11010000 00010011 01110111 10111110 01010100 01100110 11001111 00110100
11101001 00001100 01101100 11000000 10101100 00101001 10110111 11001001 01111100
Supremacy of arctan 01010000 11011101 00111111 10000100 11010101 10110101
pi in India
With Infnitesimal Decimal : 500 digits
3.
Ramanujan
1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164
AGM and more
0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172
SPIGOT Algorithm 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975
6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482
The Chudnovskys
1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436
Individual digits 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953
0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381
Digit distribution
8301194912
Some examples
Hexadecimal : 480 digits
2000 digits of pi 3.
pi: binary, decimal & hex 243F6A88 85A308D3 13198A2E 03707344 A4093822 299F31D0 082EFA98 EC4E6C89
452821E6 38D01377 BE5466CF 34E90C6C C0AC29B7 C97C50DD 3F84D5B5 B5470917
9216D5D9 8979FB1B D1310BA6 98DFB5AC 2FFD72DB D01ADFB7B8E1AFED 6A267E96
END BA7C9045 F12C7F99 24A19947 B3916CF7 0801F2E2 858EFC16 636920D8 732FE90D
BC3A9442 ECC19381 729F4C5F 6574E198 30FBBC58 3EF6975C 4CED66B9 361B921D
3.1415926535 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

LINKS to
Following pages :
pi Title Page
Is pi useful ?
Press Escape die Präsentation zu benden
pi in the antiquity
With Archimedes
To infinity
Supremacy of arctan
pi in India
oder Click die gewünschte Schaltfläche
With Infnitesimal
Ramanujan
AGM and more
SPIGOT Algorithm
The Chudnovskys
Individual digits
Digit distribution
Some examples
2000 digits of pi
END

3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... 18

History and Algorithm of pi

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