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Boris Gourévitch
The world of Pi - V2.57
modif. 13/04/2013



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pict

Pi-Story or the history of the number Pi



Associated pages

Very short intro :

p  is a constant that finds it's origin from far antiquity. What a journey travelled up till this day !

The history of the constant spreads over four very distinct periods during which the spirits and the methods associated to p  were very differents. We remember basically :

1. p  in the antiquity

* Antiquity - 17th century : the geometric method of Archimedes  triumphs

2. Ah, finally some analysis !

* 18th century - start of 20th century: it's a time where it rains discoveries and formulae constantly !

3. The computers take over work...

* 20th century : with the equations by Ramanujan  and the computers, we can push at the calculus limits.

4. Let's take a walk in the decimals

* Nearing the millinium, Plouffe  starts a revolution !

 

A more serious intro !

The number p  occupies a particular place in the mathematical world, and always constantly reaffirmed. It can boast to have occupied the mathematicians mind since antiquity, and as Bergreen and the  Borwein brother [6], the calculation of it's decimals is probably the only problem to have appeared since the dawn of mathematics, and that is still current news in today's world. However, the associated motivations have sensibly evolve throughout the centuries.

First linked to the pratical and daily need of the ancients, the evaluations always more precise of the number p  then followed, nearly like a game, some succesive discovery and fruitful analysis formulae always more efficients from the renaisance until the 19th century.

The revolution of computers in the 20th century will then completely change the game: the attraction of the calculation did not come from the result itself, but from the method of obtaining it, from a technical mathematical and algorithmitic point of view. A new change of direction came since 1966, which tried to put in relation the analytical expression of p  and the profound structure of its decimals. Of this exploration field at the frontier of complexity theory was born a few still open conjecture on the "random" nature of numerous mathematical constant (p  ,ln(2)  , etc...). Other consequence that till here was never guessed at, like the possibility to obtain in certain bases the n-th digit without knowing the previous one with little time or memory, we have renewed the attraction of the calculation of digits or decimals of the number p  .

This page offers to come back on those four periods (Antiquity, 18th/19th century, 20th century, and since 1966) by explaining the trails of thoughts and methods of calculations that lead to knowing more than       9
1241.10  decimals of p  at the start of this third millenium ! We will achieve this essay by a little look the actual motivation for the calculation of the decimals of p  .

1 p  in the antiquity

With Pi, the time traveling machine is a reality.... But of course, for more conformability, the modern and usual language of mathematics are used here... Do not believe that the Babylonians wrote the numbers in decimals. They didn't even know about trigeonometry and used base 60  instead of base 10, in fraction notation (a = b+ c-+ -d2-
       60   60   ). Firthermore, the signs +, =,2  , V~  were only invented during the renaissance, the writtings of the antiquity resembled more to a writting language than anything else. The egyptians were a bit more advanced and used the decimal system, manipulating fractions with a numerator of 1, and had invented a sign + (two legs pointing left) and the - (two legs pointing right). This being said, let us go back to the origins of the legend...

1.1 From the height of the pyramids, p  looks down unto us !

I am sometimes asked in what year we discovered p  ! I do not know the answer to this question because it's the result of a long process. The ancients essentially needed it for the geometry and measuring the surfaces of the earth, or in architecture, for example to evaluate the height and the proportion of building. According to the greek historian Herodote, we could find numerous geometrical relations in the pyramids of Gizeh, to which he visited towards 450 BC (that's more than 2000 years after their construction still!). For example, a construction principle wanted that the aread of each lateral surface is equal to the squared area of the side equal to the height of the pyramid (which is accurate to roughly 99.3% for Kheops' pyramid!). Numerous constants appear in this nearly magical way, since the ratio of the height on the base of the pyramids with an error of 1,7% . This remarquable coincidence are sometimes contested but are still fascinating.

The most ancient object where p  intervenes more or less implicitely is a babylonian tablette in cuniform writting, discovered in 1936 and which goes back to the period 1900-1600 B.C. It evaluated the perimeter of an hexagon to 2245  times the one of a circonscibed circle (either [0,57,36]  in their base 60  from which we have seconds/minutes left), which comes down to estimating pBabylone = 3+ 18 = 3.125  . It's officially the oldest approximation known of p  !

Similarly, to come back to egyptians, the famous papyrus of Rhind written in hieratic writting and discovered in 1855 by A. H. Rhind at Louxor relating a serie of ancients problems recopied by the scribed Ahmes. The numbero50  affirm that the area of a disc of diameter 9  is equal to 64 , or the square of the diameter to which we removed 19  of it's length (     )
9 - 19.9 . This comes down to estimating p      = (16)2 = 3.1604...
 Egypte    9  . They probably came to this idea by aproximating the area of this disk by the one of an octagon  partition in squares and triangles of side 1, and which then gives an area of 63 (probleme number o 48). Note that the Egyptians only work with fractions with the form 1
n  and that they had opted for the best approximation of this kind this the diminution of 1
9  is better than the diminition of 1
8  . Notice anyway (let us be fair!) that those rudimentary tools furnished some value with errors of less than 1%. This same kind of method will be applied in India (600 B.C) or in China (150 A.D.)

In the antiquity, we visibly quickly notices that the ratio between the parameter of a circle and it's diameter was a constant. Furthermore we know that the Egyptians had understood that the ratio linked to the perimeter of a circle and the one link to it's area was the same.

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Rhind's papyrus conserved at the British Museum (1800-1650 B.C.)


We don't know if this was the case for the Babylonians, or even if those civilisation were conscious of having obtained a rought value of p  . We need to note that the diffusion of knoledge at that era was rather reduced, and numerous other approximation, sometime rather bad one did spring up after the discovery of those first value. Hence, after the Egyptians and the Babylonians, it's a bit empty... The Chinesse towards -1200 gives 3 as a value, this shows a certain lack of research on the subjects! The Bible, lacking a bit of divine inspiration for the occasion, also gives 3 as a value for p  towards -550 B.C... The passage on the founder Hiram and his chauldron stayed famous, here is the quote "He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it". 30/10=3, no mistakes. We will say a lot later that the quote concerned the inside contour of the chouldron to calm the debate, but the absence of precision since founders don't need it seems the oly justification... Luckly, here are the Greeks that will put a bit of order in all of this...

1.2 Eureka !! : The exhaustion principle with the greeks

The greek mathematicians were often considered as the first to really worry about proofs. The problems that preoccupied them stayed however mainly geometrical. Among them, the problem to build a square of the same area as a circle (the famous squaring a circle) was initiated, at least historicaly, by Anaxagore of Clazomène (500-428 B.C.) during a stay in prison for heriticity (he dared state in particular that the moon only reflected the light of the sun! It was no joking matter back then !). If numerous solution were then proposed, the problem became unsolvable for 23 century when Euclide added the conditions that it could solved uniquely :

  1. of an unmarque straight edge and a compas,
  2. in a finite number of stages (implicitely).

The link between those condition and the algebraic properties of numbers that flow from it was properly stated only in 1837 by Pierre Laurent Wantzel :

Theorem 1.1 Numbers finitely defined with the help of the ruler and compas are of size that can be defined by simple algebraic operations (additions, multiplications, extractions of roots....) in other words roots.

Even if numbers defined by radicals are algebraic (roots of polynomials with coefficients in Z  ), Abel proved in 1824 that not all algebraic numbers can always be expressed by radicals as soon as the degree of the polynomial reached 5, and Liouville showed in 1851 the existence of non algebraic numbers, i.e. transcendental. The problem of squaring the circle came quickly down in the middle of the 19th century to the problem of the transcendence of p  .

The distance which seperated the greek geometric conception and the formal expression of squaring the circle problem explained the slow growing of the mathematicians toward the final and rigourous proof of the transcendence of p  in 1882 by Lindemann , which would nail this old debated old of more than 23 centuries !

With the tools that the greek had, the solution proposed for the squaring the circle called most of the time for an infinite number of steps like the quadratic of Dinostrate, constructed initially by Hippias d’Elis towards 430 B.C. or even the exhaustion method.

This last one was generally attributed to Antiphon ( 430 avant J.C.) or Eudoxe of Cnide (408-355 B.C.), and consist of constructing a polygon whose number of sides will increase until it became very like a circle.This brilliant idea hit the problem of a lack of notion of limits in that era. We now know that it's not because a property is true for all integer n  that it is true at the limit (n -->  oo  ). In other words, even if a polygon of side n  is squarable, this does not mean that it's limit -  the circle - is also, contrary to Antiphon's idea. The brilliant Euclide (330-275 B.C.) write nevertheless that by considering a polygon with a great numbers of sides, we can "make the difference between the area to calculate and the area of the polygon that have constructed smaller than any positive quantity pre-assigned, no matter how small" (according to [12]), which is a conception suprisingly close to the formalisation of limits in mathematics of the 19th century. He deduce from it that the area of the circle was proportional to the square of its diameter (Elements 12.2)

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Two pages from Euclide's Elements


pict

Euclide (365-300 B.C.)


We need to wait however for Archimedes  (287-212 B.C.) and his paper “On the measure of the circle” for this idea to be efficiently applied to the evaluation of p  . The first of these 3 theorems states that the ratio of the area of a disk to the square of its radius is equal to the ratio of its perimeter to its diameter. The third problem explicitely states that

pict

pict

Archimedes (287-212 B.C.)


Let us take the exhaustion principle, it exhibites a formal relation between a polygon of n  sides with one of 2n  sides (see Archimedes's page). Starting from two hexagons respectively inscribed and circonscribed, he obtained the approximation 1 with two polygons of 96 sided ! This amazing calculation was made with no algebraic notation, coherant numeration (the greeks used an additive numerations like the romains), nor the knowledge of trigonometrie, and with only Euclide's geometry. He formalised it as follow: Let an  and bn  be the semi perimeters of polygons of 3.2n  sided respectively circonscribed and inscribed in a circle of radius 1. The hexagon gives a1 = 2 V~ 3  and b1 = 3  . We then have the recurrance formulae

        2a b                 V~ ------
an+1 = ---nn--   ,    bn+1 =  an+1bn
       an + bn
(2)

which gives p = limn-->o o  an = limn --> oo  bn  . Archimedes pushed his calculations up to n = 5  . The case n = 1  and n = 2  are illustrated in the picture below.

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n = 1  and n = 2  of Archimedes' iteration


The proof of the convergence and the adjacence of the two sequence is straight forward if we note that             (   )
an = 3.2n tan  3p.2n and            (   )
bn = 3.2nsin 3p.2n . We hence obtain

               (    )(       (    ))     3     (   )
an- bn = 3.2ntan -p--  1- cos --p-   = -p---+ o  1-
                 3.2n          3.2n     18.4n      4n
(3)

which assures a number of correct decimals equal to 3n5-  iterations roughly (3 decimals in 5 stages). We will qualify this convergence to be linear. The brilliance of Archimedes  was already well know in that era, that Tropfke [25] said that the value pGrecs = 3+ 17  quickly replaced in Alexandria the old value p  =(16)2
 9  , of more easy use but a lot less precise, and that it was spread till India and even in China in the 5th century. Ptolémée improved a bit the result using trigonometric tables

1.3 Traveling...

After Archimedes, the mathematician night fall on the East for 1500 years.... It is true that the numerical system of the romains for example is of not much use for calculation (try to multiply LVIII by XCVI !) and they did not leave much behind them in tems of mathematical works. But things happens else where. Even if the asian, arabian or indian mathematicians just applied the same method of Archimedes, or a slight variant for 20 century, they offer better and better approximation. Hence, in India, we also work since Aryabhata offers, towards 500 B.C., 3 exact decimals. But, it was in China where the decimal system was always used, that the progress was faster. Tsu Chung Chih seems to be the first to give the famous fraction 355/113 = 3,14159292... towards 480 A.D. i.e. 6 decimals. The Arabians and Persians are not left behind since in their Hexadecimal, Al Kashi calculated with virtuosity 10 digits i.e. 14 decimals in 1949.... Absolute record !

pict pict pict



Aryabhata (476-550) Tsu Chung-Chi (430-501) Al-Kashi (1390-1450)




But the decimal notation starts to slowly impose itself in Europe in the Middle ages and it all natural that the West wakes up: it's Fibonacci, one of the few great mathematician of the era and knowledgable in decimal notation, which illustarate first of all and get p  =3,1418... meah...

pict

Fibonacci (1175-1250)


Note that we did not yet understand precisely the transcendant nature of Pi (which is quite normal...) since Nicolas de Cues  said in the 15th century

      3 ( V~ -   V~ -)
p  =  4    3+  6 ~ -  3.136                           (4)
Regiomontanus will prove (!) that this value is wrong... Nicolas de Cues  used a varient of the method by Archimède.

pict

Nicolas de Cues (1401-1464)


From time to time, a few try to be original. Hence, Descartes  (1596-1650) took the problem the other way and offered his method of isoperimeters which does not involve the perimeter of polygons but the diameter of those polygons, whose perimeter is fixed. Viete also derived a first infinite product in 1593 (without proving it, but well, it wasn't usual to in that time!) by looking at the area and not the perimeter: 

      V~ 2- V~ -2---  V~ ---2-------
p = 2  2  2 +  V~ 2      V~ --- V~ -
                  2 +   2+  2
(5)

But the convergence is so slow that he still prefered to use Archimède's method to calculate himself 9 decimals in 1593.

pict pict


René Descartes (1596-1650) Viete (1540-1603)



With simple 1000 years behind on the Chinese and Tsu Chung Chih, but it is his name who was remembered, Adrian Anthonisz does the average of the boxing values calculated by Archimède  and refind 355/113=3.14159292... In any case it's what his son claims by publishing the result in 1625. Those values where hold to be exact for p  . Since the greeks, we knew that there exist some numbers more complicated than the rationals (irrationals) like  V~ -
  2  . We felt that p  was of more complex nature, but we still tried to express it as a function of simpler numbers. During that time, the decimal numeratation quickly progress the decimal race that became a common sport and the records did not leave the west...

Pushing the calculation further than Archimedes, the record in this area belong to Ludolph Van Ceulen (1539-1610), who found 20 decimales in 1596 with the help of a polygon with    33
60.2  = 480  million sides, then 32 decimals with the help of a polygon of  62
2  sides in a postum plublication in 1615, so to finally be attributed and marked on his tombstone 35 decimals in 1621. What an obsession ! Might as well say he spend his life to this excersice, and for reward that p  is called von Ceulen's number in Germany !

pict

Ludolph Van-Ceulen (1540-1610)


In fact there was not much real progress in this period essentially due to the exclusie use of geometric approach, which started to find its limit in the calculation of the decimals of p  .

The squaring of the circle continued, but started to give true problems... Leaving the geometric era which did not give any solution, analysis will start looking at it, and we will see, with success. Note that the Académie des Sciences in France, had promise a reward for the solution, received at this period several wrong errors. As "Le petit Archimède", the best price in this domain is to someone named Liger which start to prove that  V~ --   V~ --
  24 =  25  , the rest follows...

2 Ah finally some analysis! (18th century - end 19th century)

The great turn happens with the manipulation more and more current of infinite sum and product in the mathematic world at the end of the midle ages.

2.1 The time of arguments

During this perid, some violent arguments appeared: one of the most famous of them concern Leibniz  (1646-1716) and Newton  (1642-1727) who fighted over the parentship of of the discovery of diferential calculus and loose a lot of energy.... There are still some who are skeptical (Rolle, for example, who did not believe in this revolution and will get angry at Varignon, but still give us one of the most famous thoerem of this theory), differential calculus will change mathematics forever... The reults apear rapidly and the research on p  will largely benifit from it. It's the first time that formulae does not translate directly the link between the geometry from which we define p  and p  itself. It's in fact one of the things that in my view is most fascinating in the study of p  . We are dealling with big formulae in which it is very difficult to recognise a geometric property, and yet we find p  . And the price on the research of a solution concerning the famous squaring the circle problem offered by the académie des Sciences always gave a big interest on the geometry. But all of this was not build in one day :

2.2 The premices of analysis

If there was a mathematician who symbolise a bit this passage of geometry to infinitesimal or the concious of infinity, its Wallis  (1616-1703).

pict pict


Wallis (1616-1703) Lord Brouncker (1620-1684)



His manipulation of infinite series and his research on the area of a quarter of a circle (starting from the integral (1 - x2)n  to reach n = 1/2  ) pushed him towards horizons yet unknow. After a proof that stayed famous due to its twist and turns, he find a very nice formula, the first infinite product of rationals converging to Pi :

      oo 
p = 2  prod --4n2--
     n=14n2- 1
(6)

The convergence is very slow, but it is maybe the first true serie converging to p  coming from analysis. Succeding it is Lord Brouncker  (1620-1684), a friend of Wallis, who, upon being asked, continue the research on continued fractions and change Wallis result into a famous developpment of 4p  which gives:

          1
p = 4------12----
     1+ 2+--3252-
          2+2+...
(7)

All of this is centered on infinitesimal but its only with differential calculus that techniques will start to flower.

2.3 The first series

Well before the european, we attributes to the Indians the first expression of p  under serie forms since we find in the writtings of the disciples of Nilakantha Somayaji (1444 - 1545 !) the formula

         oo 
p =  V~ 12  sum --(-1)n--
        n=0 (2n+ 1)3n
(8)

among eight other. Its simplicity and the speed of convergence (roughly n
2  correct decimals for n  terms) could have lead mathematicians of that era to know some tens of decimals of p  ! However, during this time, as we have seen above, its Al Kashi  of Samarkande who did an amazing feat by calculating 14 decimals of p  in 1429 with the help of a variant on Archimede's method and in sexagesimal system.

In Europe, it's war! Newton  and his battles against Leibniz  and his dx-
dy  ... One like the other have at least the merit of habing the vision of great changes in mathematics. The application of differential calculus finally allows to obtain or justify the limits of infinite serie as usual functions (through for example the famous Taylor's formula). During that time, James Gregory (1638-1675) make Europe benifit from his discovery of the development in integer series of the arctan function

            oo      n 2n+1
arctan(x) =  sum  (--1)-x----.
          n=0  (2n+ 1)
(9)

Applying x = 1  immediatly give a seire for Pi, but, funnily enough, Gregory missed it (or more probably, did not see the point in it) and it only finds itself in the work by Leibniz. In 1671, it's Abraham Sharp (1651-1742) which use the particular case          (   )
p = arctan  1 V~ --
6           3 , in other words the equation 8, so to obtain 71 correct decimals of p  in 1699, nearly 200 years after Nilakantha ! The development of arcsin was obtained by Newton towards 1665-1666, who took advantage to calculate 15 decimals of p  , having "nothing better to do" as he said it himself.

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Newton (1642-1727)


The function arctan is a good candidate for the calculation of the decimals of p  by hand because if we take it from the right side, we use only rational terms or at worst with a root like in 8. Hence John Machin  (1680-1752) easily reaches 100 decimals in 1706 by proposing the now famous formulae

           (  )        (    )
p-= 4.arctan  1  - arctan  -1-  .
4            5           239
(10)

pict

John Machin (1680-1752)


The proof or the research of this kind of formulae (from which four other examples are given further by the equation 12, 13, 23 and 24) are obtained by the following rule

Theorem 2.1 Let a1,a2, ... , ap  and b1,b2, ... , bp  be integers.            (  )
 sum p  arctan  bj  = kp
  j=1       aj  , k  (-  Z  , if and only if    ( prod p          )
 I m   j=1(aj + i.bj) = 0  .

Proof. If cj = aj + i.bj  (-  C  ,   ( prod p          )    sum p
ln   j=1 (aj + i.bj) =   j=1 [ln(| cj|)+ i.(arg(cj)+ 2kjp)]     sum p  [          (      (bj)       )]
 =  j=1 ln(|cj|)+ i. arctan aj  + 2kjp , kj  (-  Z  . Consequently     prod p
x =  j=1(aj + i.bj)  (-  R  iff  E kx  (-  Z  such that ln(x) = ln (| x| ) +i.kx.p  , iff  E k  (-  Z  such that            (  )
 sum p  arctan  bj  = kp
  j=1       aj  .  __

The speed of convergence of these formulae is constrained by the greatest term bj
aj  and according to equation 9, we obtain a linear convergence, in other word a number a× n  of decimals by using n  terms of Taylor's expansion of the funtion arctan  . From now on, the combination of arctan will be used as a basis for the race to decimals until 1985 !

2.4 The lord of the series

After that the mathematics predominance was installed in England under the impulsion of Newton, they come back to the old continent, in Swiss notably with the Bernoulli  and Euler. Its the great analysis period. Euler  searches everywhere and never tires of publishing to gives use numerous formulae. The most beautiful of them is with no doubt :

 oo  sum        2
   -1 = p--
n=1n2    6

which brings not much to the calculation of decimals but whose simplicity is remarquable... It's proof (see Euler page) is a one of a kind, completly not rigourous, but completly amazing!

pict

Euler (1707-1783)


A century late, a mathematician which according to me will indirectly bring the most to the research on p  is Joseph Fourier  (1768-1830). His theory on the decomposition of periodic functions into series, still to be finalised but who will be the object of rigourousity thorughtout the 19th century, and which is truly revolutionary (so much that certain mathematicians of the period like Poisson will try to fight back with loads of energy!). The result allows in fact to simply reprove about nearly all those formulae of our good old friend Euler. And of course find new quite interesting ones.

2.5 p  , probably...

p  , it's not only geometry and analysis ! Buffon  (1707-1788) prove to us with his famous needle problem that p  also intervened in the domain of probability. Of course, this is still due to the definition of p  as the component of the area and the perimeter of a circle, but the theorem by Cesàro (1859 - 1906) will confirm the inclusion of p  in the probabilities...

pict pict


Buffon (1707-1788) Cesàro (1859-1906)



2.6 The challenge

Let us go back to our chronicle race. Starting from Newton, they are - without insulting them - more second hand mathematicians who fight for the record of decimals. Anyway, the number of decimal already calculated in those time was greater than the needs of mathematicians or physicians. In fact we estimate that to calculate the circumference of the universe with the precision of a hydrogen atom, only 39 decimals of p  are required ! The motivations are hence more turned towards the search of periodicity or patterns in the decimals of p  . The periodicity of a number makes it rational (numbers written as a fraction ab  , a  , b  integers). However, at the turn of the 18th and 19th century, the problem of the irrationality of p  , in other words the impossibility to write it under rational form, still bother mathematicians. They suspect  its true for a long time but never managed to prove it. With the progress of analysis, Euler  show the irrationality of e  and Johann Lambert (1728-1777) brings an answer to the problem of p  in 1761. His rather heavy proof uses a expansion in continued fraction of the function tan  . Pi is truly irrational !

pict

Lambert (1728-1777)


In truth the maybe paradoxly more important result that we have found on the spread of the decimals of Pi such that this last remains a mystery to us today. The irrationality indicates that they are not periodic...

There's still the isue of transcendental, in other words the impossibility to represent p  as a combination of roots and powers of integers, or in equivalent terms, the fact that p  is not the root of any polynomials with integer coefficients, or even the impossibility of squaring a circle. Even with all the efforts of many mathematicians (and amateurs that are still trying to solve the squaring of the circle!), this fort stayed unbreachable until the end of the 19th century. And the crazy entries by amateur mathematicians obliged the académie des Sciences to refuse starting from 1775 the tentavives of proofs on the squaring the circle , immediate consequence of the transcendance.

2.7 p  falls in the shadow

The 19th century comes by... Bizarely, the most brilliant representant of the scientific genius of the century, Master Gauss  (1777-1855), even with his uncomparable talent, was not one interested at all by the research on p  , at least not direcly. We do need to credit him with a few arctan formulae, but nothing to worship him by !

Even if Pi continues to appears in many results, the 19th century is more directed towards algebra and arithmetics with Galois, Abel, Sophie Germain and the new theoricians of the non euclidian geometry such as Gauss, Beltrami, Lobatchevski and Bolyai. The calculation of decimals seems to run out of breath in the second half of the 19th century. All of this even with the extraordinary talent of Zacharias Dase, and the enthousiams of a few like Shanks. Employed by Strassnitsky to calculate 200 decimals, which he did in two month, Dase was the announcer of modern computers with his fantastic capacity to calculate (he could multiply in his head two numbers of 100 digits in 8hrs, with breaks, or even sleeping on it for a whole night, and starting again later!). It needs to be mentioned that the calculation of decimals by hand seems to have reach humain limits in 1874 with 707 decimals calculated by Shanks, which are in fact false starting from the 528`eme  . Ferguson will notice it nearly 70 years later when comparing them to the 808  that he obtained in 1948, with the first calculating machines (addtioners in fact, herited from the principle of those constructed by Leibniz since 1694). An error that will have hence lasted 92 years ! The room of the palais de la découverte in Paris which proudly showed off the result by Shanks decided it was a good idea to be refurnished !

pict

The room consacred to maths in the Palais de la Découverte in Paris.


And since the arctan formulae, we have still have not find in that era a way to go faster and further in the thoery like in practise...

Furthermore, the oldest mathematical problems was found to be solved in Lindemann in 1882 when he proved the transcendance of p  . The squaring of the circle is hence impossible... How are we going to exist this impass and remotivate ourself ?

It's from the depth of India at the end of the 19th century that someone grows and will remodel and reshape the century to come.... The era of algorithm and computing will start with him....

3 Computers take the relay

3.1 The Indian breath

During the first half of the 20th century, the mathematicians preoccupation were elsewhere. The theories elaborate by Cantor, Gödel, Kolmogorov, the topologie and the list of 23 problems by Hilbert opens sundunly the horizon which made classical analysis look ancient. It seemed to have found its natural evolution in the study of elliptical integral lead by Legendre, then Gauss, Abel and Jacobi during the first half of the 19th century.

Luckuly a genius born in the depth of India in 1887, Srinivasa Ramanujan , will bring it on himself to bring new life to research on p  .

pict

Srinivasa Ramanujan (1887 - 1920).


Himself passionated by this constant, he's a self taught man who spend his first 25 years of his life rebuilding the mathematics from a unique work containing 6165 theorems without proofs ("Synopsis of elementary results in pure and applied mathematics” by G.S.Carr). This state of mind lead him to often state results without proving them (which did not mean he did not understand where they came from!). Gifted with exceptional intuition allowed to progress with giant leapsteps in the tehory numbers and modular equation, he discovered some formulae from an other angles like this one published in 1914

         (                     ) -1
    9801   sum  oo  (4n)!(1103-+-26390n)
p =   V~ 8         (n!)4(396)4n        .
          n=0
(11)

There exists a funny story about this formula: his proof was nearly completed at the start of the 80s by the Borwein's brothers [7].

pict pict


Borwein Brothers



They just had to justified the 1103 coefficient, it should be an integer, but the length of the calculation and equations required were horible. Gosper  then effected in 1895 a "blind" calculation of 17 million decimals of p  with the help of this formula. As true as two integer close of less than a unit are equal, the concordance of the result of the calculation by Gosper  with the 10 millions of decimals already knew at that time constituted of a final proof for the formula Ramanujan  ! We suppose that Ramanujan  did the same procedure on a few decimals.

The Borwein  will get from the work of Ramanujan  a flow of remarquable algorithm still largely used today in the calculations of the decimals of p  (see paragraph 3.5). The complexity of the proof of such a formula is such that even a glance at the results, with no interdemiate proofs, take up a minimum of several pages in the remarquable work[13]. The reader interested to take a dive in the passionating world of modular equation, quartic bodies, and other elliptical integrals can refer themself to the "Bible" [7], of great pedagogie.

Ramanujan  was noticed by the English mathematician Hardy to which he wrote a letter to in 1913, went to England in 1914 and their fruitful collaboration lasted until 1919.

pict

Hardy (1877-1947)


He then went back to India were he died the following year, probably due to a lack of victamin. Considered as one of the greatest genius of the 20th cebtury, Ramanujan  left notebooks full with loads of formulae in non-standards notations, whose decoding is still hapening today (!), first started by Bruce Berndt, then the Borwein.

3.2 The race to the decimals start again

After Ramanujan  who died prematuratly in 1920, its a theoretical desert... until 1976. Hence, during this time, we calculate... After the war, the dawn of calculating machine make the race of decimals progress with giants leaps! Ferguson opens the way in 1946 by obtaining 620 decimals with the help of a desk calculator. The first calculation on computers was given to the famous ENIAC in 1949  which gives 2037 decimals in 70 hours with the help by the formula by Machin 10.

pict

ENIAC


In 1973, Guilloud and Bouyer reached the first million decimals on a CDC 7600 with the help of two famous arctan  formulae, the one by Gauss  12 and of Störmer 13

p           ( 1 )          ( 1 )          (  1 )
4-= 12.arctan  18  + 8.arctan  57  - 5.arctan  239
(12)

           (  )          (   )        (    )
p-= 6.arctan  1  + 2.arctan  1-  + arctan  -1-
4            8             57           239
(13)

The calculation in binary took respectivaly 22hr11 and 13hr40, and the conversion in decimal base 1h07. A book of 145 pages of decimals pulled from this calculation was qualified at the time as the "most boring book in the world"!

Note that process attach to the race of decimals is still the same as today: the calculations are fone with two distinc formulae then compared for the validation of the record. The figures 1 and 2 shows the evolution of the records of the calculation of the decimals of p  throughout the centuries.


PIC


Figure 1: Numbers of decimals calculated by hand in history. The scales are logarithmtic.



PIC


Figure 2: Number of decimals obtained with the help of calculators or computers in the 20th century. The scale of the decimals is logarithmtic.


3.3 Improvements in algorithms

During this time, the lack of efficient multiplication algorithm forced to break down each numbers into slices, for example B = B21020 + B11010 + B0  . The multiplication of numbers of size n  hence needed a time (or number of operations) proportional to n2  without taking into account the use of memory proportional to n  . With no theoretical and algorithmical improvement, the progress of records of decimals would have been very slow. It was at this time that things sped up.

In 1965, Cooley et Tukey introduced under it's modern form a method to reduce the complexity of the calculation of Fourier series known today under the name of Fast Fourier Transformation [11]. Schönhage and Strassen get from it an multiplication algorithm for big integers in complexity O (n.log(n).log(log(n)))  which is much better than  (  )
O n2 [24] !

In 1976, Salamin and Brent  managed independently to the same algorithm ([238])

pict

which has the extraordinary property of a quadratic convergence, in other words the number of exact decimals doubles at each iteration. In fact, Salamin  shows in [23] that if U
 n  is the approximation of p  after n  iteration of the algorithm 14, we get

           p22n+4   -p2n+1
|p - Un|< ---(----)2e      .
         M   1, V~ 12
(16)

This upper limits shows that the number of correct decimals of p  obtained from is stricly greater than (     )                     (        )
  -p-- 2n+1- n log  2- 2 log   ---4p V~ -
  log10           10       10 M(1,1/ 2) . M  is the arithmetic geometric average (see the page on Salamin).

pict

Richard Brent


Let us finaly add that the algorithm by Newton, offered towards 1669 (!), brings back the division and the extraction of square roots to multiplications and also offer a quadratic convergence. Might as well say that all the ingrediants are brought together to make the records explodes !

3.4 World competition

In fact the competition restart in 1981 with Kanada's team which was probably the first to apply this cocktail of algorithms (FFT , Newton, Brent-Salamin). Myioshi and Kanada obtained 2 000 036 decimals in 1981. Strating from this moment, the rythm speeds up, since the poor Guilloud, which tried to keep his record in 1982 by calculating 14 decimals more than the Japanese couple, find himself out of the competition at the end of 1982, when we know 16 777 206 decimals (Kanada, Yochino and Tamura). The fight then will mainly concern Kanada and, between 1991 and 1994, the two brothers David and Gregory Chudnovsky  which use their own serie of kind Ramanujan  and a computer who they themself construct the architecture. According to the legend, their New-York flat contained piles of papers in disorder and was heated with microchips ! Those isolated mathematicians but with recognised talent (Gregory is considered as an exceptional genius but an illness stops him from working at a uni) shows at least that some first class scientist are interested in the calculation of the decimals of p  [12].

3.5 A rainfall of formulae

The 80s see the birth of several very interesting formulae concerning p  . There was some formulae of kind Ramanujan, some infinite series converging linearly but fast enough that they are used in the records. These are refound and explained thanks to the decyphering of the Ramanujan's notebook by Bruce Berndt and the work of the Borwein  brother and Chudnovsky.

In other part, starting from modular equations implied that the theta functions (whose algorithm by Brent-Salamin  was a particular case), the Borwein brothers showed that we can construct algorithms that converges to any speed (quadratic, quartic, quintic...) towards Pi even if the complexity of the calculations increases as a consequence. A good compromise seemed to be the following formula, with quartic convergence [7] :

pict

It was used for the first time  by Bailey then Kanada in 1986. Bailey  used 12 iterations on a CRAY-2 to calculate 29 360 000 decimals of p  (in fact 45 millions are possible at that stage). It is ammusing to notice that there only need 100 multiplication, division and extraction of roots to reach it !

From 1982 to 2002, it is still formulae by Ramanujan  (like 11), the algorithm by Brent-Salamin  (14) or a variant, as well the quartic algorithm of Borwein  (17) which were used in the race of the calculation of decimals of p  . The first billion were obtained by the Chudnovsky brothers in 1989 and, with this method, Kanada, always him, reached 206 billion decimals of p  in 1999. Of course, Pi having an infinite number of decimals, it's not even a drop of water, but the mathematicians hoped still find, like the Chudnovsky brothers, some irregularity or property in the decimals of our favourite constant...

pict

Chudnovsky Brothers


pict

From left to right, Salamin, Kanada, Bailey and Gosper in 87. Ah, the crazy night betweens friends on p  !


4 BBP approche, more new stuff on p

4.1 How to progress ?

The multiplication, the division and extraction of roots now can be done in a nealry linear time, and knowing that we can not have less than a complexity O(n)  , we can not expect much more significant advance from this angle. We have seen that some extremly efficient algorithms existed for the calculation of the decimals of p  . Are we then doomed to see the record fall in function to the progression of computer performances? Yes and no!

The mathematicians often surprises where we least expect it. This is how on the 19th september 1995 at 0hr29 (!), after months of research in the dark, Simon Plouffe, David Bailey  and Peter Borwein  of Vancouver discovered the formula apparently normal and simple

      oo     (                              )
p =  sum  -1-   --4---  --2---- --1---- --1---
    k=016k   8k + 1   8k+ 4   8k+ 5   8k+ 6
(18)

called since then the BBP formula. The proof is nearly immediate if we notice that  sum                V~   sum     integral   V~ -            V~ -  integral   V~ 
 o o k=0 16k(18k+p)-= 2p o o k=0 01/ 2xp-1+8kdx =   2p 10/ 2x1p--x18dx  then by calculating the equivalent integral to 18. Euler  himself could have discovered it. In fact, it is to this day one of the most famous example of experimental maths since this formula was discovered by the algorithm PSLQ of research on linear relation between numbers. This encouraged the number of amateurs in mathematicians to start the research of such formulae thanks to PSLQ or LLL, a similar algorithm implemented on the Pari-GP software.

pict

Simon Plouffe


But our three canadian mathematician noticed above all that this expression is very close to a decomposition in base 16 of p  . In the article [3] of 1996 that is now famous, they show that we can use 18 to reach n  -th digit of p  in base 2p  (all the more in base 16) without calculating any of the previous, all of this in a nearly linear time O(n.log3n  ) and memory O(logn)  !! The memory required being minimal, they applied immediatly this result to calculate the 40  bilionth digit of p  in base 2. The revolution had started.

Plouffe  extend this result in all the bases b  in octobre 1996 [22] to the price of a time O(n3 log2 n)  and the clever uses of the serie

        oo  sum  k.2k(k!)2
p+ 3 =      (2k)!  .
       k=1
(19)

Some improvement in     2
O(n )  then in    2           2
O(n loglog n/log n)  are successively offered by Bellard  in 1997 [5] then Gourdon in 2003 [14]. Numerous other formulae of type BBP ( with powers of 2  or 3  ) have since then seen the day for the logarithm of integers, z(3)  , z(5)  , the Catalan's constant G  , and in general polylogarithm constants [917]. Unfortunatly, there propably does not exist any formula for p  with a power 10  , in other words in base 10  . The formula   (9-)     sum o o  -1--
ln 10 = -   k=1 10k.k  shows however we can calculate isolated decimal digits of certain constants.

Those fundamental results shows in particular that p  belongs to the class of complexity of Steven SCb  , regrouping the constants that we can calculate the digits in base b  in polynomial time and memory log  -polynomial, which was before judged to be improbable from a memory point of view.

Since then, Fabrice Bellard, a Polytechnician French studient, managed to reach the 1000 billionth digit in septembre 1997. Then Colin Percival reached the   15
10  -th binart digit of p  (a 0  !) by a collaboratif calculation on the internet of 1.2 million hours of shared CPU between 1734 computers spread in 56 countries between the 5th septembre 1998 and the 11th september 2000!

The principle of this calculation of the n  -th digit of p  is given on the page N-th digit.

4.2 A formula full of resources

The nearly direct access of the digits of those constants seeded the idea in mathematicians that we can find some properties on the spread of those digits , and that we can be near a proof for the normality of p  (and hence the irrationality of the Catalan's constant G  or the odd z  ). The normality of a number requires that each n  -uplet possible appears with a probability b1n  in the digits in base b  of this number. For example each digits between 0  and 9  appear on average once on 10  , each number between 0  and 99  appears once on 100  , and so on. In octobre 2000, Bailey  and Crandall reduced this condition on normality to the satisfaction on the following hypothesis :

Conjecture 4.1 Let       p(n)
r(n) = q(n)  (-  Q(n)  , n > 0  , with deg p < degq  and r  without poles on N  (its the polynomial part of the BBP formulae). Let a base b > 2  and x0 = 0  . The the sequence (xn)n (- N   defined by

xn = (b.xn-1 +rn) (mod 1)
(20)

has a finite attractor or is equally spread on [0,1]  .

pict

Richard E. Crandall


This kind of sequence is equivalent to the BBP representation of a constant since if      sum o o 
a =   k=1b1kr(k)  , then {bna}=  (xn + tn) mod 1  where      sum o o 
tn =  k=1 1bkr(k + n)  tends to 0 if n -->  oo  since degp < degq  .

The condition of a finite attractor represent intuitively the fact that for a sequence of always approaching a set of finite value starting from a certain rang, and in any order, as we are tending it towards a genral rational (whose decimal sequence are periodic.

Definition 4.2 Precisly, a sequence (xn)n (- N  (-  [0,1]  has a finite attractor W = {w1,w2,...,wP} if there exist e > 0  and such that for all k > 0  ,

||           ||
||xK+k - wt(k)|| < e  avec   1 < t(k) < P.
(21)

We can furthermore show that in our case (BBP formulae), this property is equivalent to the rationality of the limit of xn  [4]. The notion of evenly spreaded is more intuitive. It is checked if the proportion of the appearation of a sequence of values in [0,1]  in a given interval [c,d]  is exactly the length of that interval, basicly the values taken are uniformly spread in [0,1]  . (xn)n (- N   is evenly spread if

lim  Card-{xj  (- -[c,d], j-<-n} = d- c.
n--> oo           n
(22)

The evenly spread for this kind of sequence implies the normality [19]. Taking into account the various implied constants in the BBP representation, we scratch here some important improvement, especially for the odd z  . It is amusing to see the simplicity of the BBP formula that leads to all those consequences. Consult [420] to expand on those open questions.

5 An endless quest ?

5.1 Record to date

The 6th december 2002. Kanada, untirable competitor who holds or reclaim the record in the calculation of the decimals of p  for more than 20 years, calculated 1 241 100 000 000 decimals of p  . The surprise came from the fact that this time he only used two formula of the type Machin  like equation 10.

This comeback to suprisingly simple formulae after the use for fifteen years algorithms of Brent-Salamin, or Borwein  or series of Ramanujan  and Chudnovsky  was really unexpected. It seems that the perpetual uses of roots, multiplications and divisions needed for the use of FFT was on a great scale. This last requires a lot of memory to function. Kanada hence went back to some wiser method like the formulae of type Machin, which sensibly needs more arithmetic operation but less FFT and hence a lot less memory. Kanada estimated that his implementation is roughly two time faster than the previous one he used, the algorithm 14 of Brent-Salamin  and the one of quartic order by the Borwein  (eq. 17). The calculations were done in hexadecimal base, i.e. 1 030 700 000 000 digits. The used formulae were :

pict

The first was found in 1982 by a math profesor and composor, Takano, and the second was a discovery of Störmer in 1896.

This calculation by Kanada holds a second suprise: since the result was obtained in base 16  , after checking the equality between the two calculation, he then did a second check by directly calculating the 20 hexadecimal digits at the position 1 000 000 000 001 with the help of BBP formulae presented previously. The result B4466E8D21 5388C4E014, which required 21 hours of work, perfectly corresponded to the calculation done with the help of arctan  formulae. The hexadecimals digits were then converted in base ten, a not so trivial operation, reconverted into hexadecimals for verification, then the record was official [2].

The calculations (everything included) lasted around 600 hours on a HITACHI SR8000/MP with 1TB of memory (1024GB), i.e. the same amount of memory as for its previous record of 1999, even though 6 times more decimals were calculated!

5.2 Perspective and motivation

How to explain that the race to the decimals is still current news? The motivation to beat the record of decimals of p  have in fact never run out. We will mention mixed and mash the test of computer (a bug was discovered by Bailey  during his record of the calculation in 1986 on a Cray-2), the hypothesis of irregular appearance in the spread of calculated decimals (still nothing on that side) or the addition of yet more efficient implementation of the decimal calculation (the computation is running like a fever on the internet between several mathematician programer for the title of the fastest program in the world).

Each of these reason taken seperatly is not sufficient. It would be more truthful to say that its the conjection of all those reason, without taking into account the passion started by the "personality" of p  , which feeds and renew the competition. It is tru that the mordern implementation of arithmetic libraries on large numbers (based on the FFT  , and Newton  technics seen previously) helped a lot in the race of decimals of p  . It is also true that putting together efficient program is not in everyone's reach and that you need to carefully know the architecture of your calculator to have any hope of an interesting performance. It's a challange for numerous computer scientist gifted in maths (or the other way round!). We have also seen that the recent discovery of the BBP formula 18 opened up a large field of theoretical investigation.

I will add that the popalarity of p  plays an important role: it is this justification that is maybe rarely mentioned but which is taken in full with simple amators of matheamticians member of the lovers of p  sects, or even visible by sifting through maths forums over the worlds: the theory of numbers, classical analysis and the constants have an accesible base, at least in appearance, for the non-mathematicians and stay the greatest giver of little excersices of some funny properties of numbers. Its through this bias that is born many vocations of mathematics, and we just need to consult mathematicians web pages like Bailey, Plouffe , Kanada or the Borwein brothers to be convinced that they keep at over fifty years a refreshing passion for the constants, visible part of the math iceberg. The difficulty of abstraction of other mathematics domains like algebraic geometry confines them more to a more knowledgeable audience and they do not benifits from as many nearly universal symbols like p  or the prime numbers can be in number theory. We can compare hence the quest of decimals of p  with the one of the greatest Mersenne prime for example, excersice much in fashion at the moment (on the day I wrote this text, the record has fallen again with the 43rd known Mersenne prime !).

But I should propably give the last word to Genuys to whom we asked, after his record of 10000 decimales in 1958, why he chosed p  and who simply replied "It was for me a programing excersice, I could have chosen an other number but e  was too easy and c  (Euler's constant) too hard! Plus there was the record!” [21].

pict

François Genuys in 2004


References

[1]   J. Arndt, C. Haenel, ”p  Unleashed”, Springer-Verlag, Berlin Heidelberg New York, 2001.

[2]   D. H. Bailey, “Some Background on Kanada’s Recent Pi Calculation”, may 2003, preprint, http://www.nersc.gov/~dhbailey/dhbpapers/dhb-kanada.pdf.

[3]   D.H. Bailey, P.B. Borwein, S. Plouffe, “On the Rapid Computation of Various Polylogarithmic Constants”, Mathematics of Computation, 1997, vol. 66, pp. 903-913.

[4]   D. H. Bailey, R. E. Crandall, "On the Random Character of Fundamental Constant Expansions", Experimental Mathematics, 2001, vol. 10, no. 2, pp. 175-190.

[5]   F. Bellard, “Computation of the n’th digit of pi in any base in O(n2)  , unpublished, 1997, http://fabrice.bellard.free.fr/pi/pi_n2.ps.

[6]   L. Berggren, J. Borwein, P. Borwein, ”Pi : A Source Book”, Springer-Verlag, 2nd ed., 1997.

[7]   J. Borwein, P. Borwein, “PI and the AGM: A Study in Analytic Number Theory and Computational Complexity”, Wiley, 1987.

[8]   R. Brent, “Fast multiple-precision evaluation of elementary fonctions”, Journal of the Association of Computing Machinery, 1976, pp.242-251.

[9]   D. J. Broadhurst, “Polylogarithmic ladders, hypergeometric series and the ten millionth digits of z(3) and z(5)”, 1998, preprint.

[10]   S. A. Cook, S. O. Aanderaa, “On the minimum computation of functions”, Trans. AMS, 1969, vol. 142, pp291-314.

[11]   J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series”, Math. Comp. 1965, vol 19, pp. 297-301.

[12]   J.-P. Delahaye, ”Le Fascinant Nombre p  ”, Bibliothèque Pour La Science, Belin, 1997.

[13]   P. Eymard, J.-P. Lafon, ”Autour du Nombre p  ”, Hermann, Paris, 1999.

[14]   X. Gourdon, “Computation of the n-th decimal digit of p  with low memory”, preprint, 2003, http://numbers.computation.free.fr/Constants/Algorithms/nthdecimaldigit.pdf.

[15]   X. Gourdon, P. Sebah, “N-th digit computation”, preprint, 2003, http://numbers.computation.free.fr/Constants/Algorithms/nthdigit.pdf.

[16]   M. D. Householder, “The Numerical Treatment of a Single Nonlinear Equation”, 1970, MacGraw-Hill, New-York.

[17]   G. Huvent, “Formules BBP”, 2001, Preprint.http://ano.univ-lille1.fr/seminaries/expo_huvent01.pdf.

[18]   D. E. Knuth, “The Art of Computer Programming. Vol. 2: Seminumerical Algorithms”, Addison-Wesley, Reading, MA, 1981.

[19]   L. Kuipers and H.Niederreiter, “Uniform distribution of sequences”, Wiley-Interscience, New York, 1974.

[20]   J. C. Lagarias, “On the normalityof fundamental constants", Experiment. Math., 2001, vol. 10, no 2.

[21]   “Le nombre p  ”, Le Petit Archimède, Association pour le Dévelopement de la Culture Scientifique, Amiens, 1980.

[22]   S. Plouffe, “On the computation of the n  -th decimal digit of various transcendental numbers”, unpublished, 11/1996.

[23]   E. Salamin, “Computation of p  using arithmetic-geometric mean”, Mathematics of computation, 1976, vol. 30, pp. 565-570.

[24]   A. Schönhage, V. Strassen, “Schennelle Multiplikation grosser Zahlen”, Computing, 1971, vol. 7, pp. 281-292.

[25]   J. Tropfke, “Geschichte der Elementarmathematik”, Vierter Band, Ebene Geometrie, Dritte Auflage, Walter De Gruyter & Co, 1940.


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