The First 10,000 digits of Pi

Of course, who needs to use the formula when you have about the first 10,000 digits right here!

3.1415926535897932384626433832795028841971693993751058209749445923078164062862
089986280348253421170679821480865132823066470938446095505822317253594081284811
174502841027019385211055596446229489549303819644288109756659334461284756482337
867831652712019091456485669234603486104543266482133936072602491412737245870066
063155881748815209209628292540917153643678925903600113305305488204665213841469
519415116094330572703657595919530921861173819326117931051185480744623799627495
673518857527248912279381830119491298336733624406566430860213949463952247371907
021798609437027705392171762931767523846748184676694051320005681271452635608277
857713427577896091736371787214684409012249534301465495853710507922796892589235
420199561121290219608640344181598136297747713099605187072113499999983729780499
510597317328160963185950244594553469083026425223082533446850352619311881710100
031378387528865875332083814206171776691473035982534904287554687311595628638823
537875937519577818577805321712268066130019278766111959092164201989380952572010
654858632788659361533818279682303019520353018529689957736225994138912497217752
834791315155748572424541506959508295331168617278558890750983817546374649393192
550604009277016711390098488240128583616035637076601047101819429555961989467678
374494482553797747268471040475346462080466842590694912933136770289891521047521
620569660240580381501935112533824300355876402474964732639141992726042699227967
823547816360093417216412199245863150302861829745557067498385054945885869269956
909272107975093029553211653449872027559602364806654991198818347977535663698074
265425278625518184175746728909777727938000816470600161452491921732172147723501
414419735685481613611573525521334757418494684385233239073941433345477624168625
189835694855620992192221842725502542568876717904946016534668049886272327917860
857843838279679766814541009538837863609506800642251252051173929848960841284886
269456042419652850222106611863067442786220391949450471237137869609563643719172
874677646575739624138908658326459958133904780275900994657640789512694683983525
957098258226205224894077267194782684826014769909026401363944374553050682034962
524517493996514314298091906592509372216964615157098583874105978859597729754989
301617539284681382686838689427741559918559252459539594310499725246808459872736
446958486538367362226260991246080512438843904512441365497627807977156914359977
001296160894416948685558484063534220722258284886481584560285060168427394522674
676788952521385225499546667278239864565961163548862305774564980355936345681743
241125150760694794510965960940252288797108931456691368672287489405601015033086
179286809208747609178249385890097149096759852613655497818931297848216829989487
226588048575640142704775551323796414515237462343645428584447952658678210511413
547357395231134271661021359695362314429524849371871101457654035902799344037420
073105785390621983874478084784896833214457138687519435064302184531910484810053
706146806749192781911979399520614196634287544406437451237181921799983910159195
618146751426912397489409071864942319615679452080951465502252316038819301420937
621378559566389377870830390697920773467221825625996615014215030680384477345492
026054146659252014974428507325186660021324340881907104863317346496514539057962
685610055081066587969981635747363840525714591028970641401109712062804390397595
156771577004203378699360072305587631763594218731251471205329281918261861258673
215791984148488291644706095752706957220917567116722910981690915280173506712748
583222871835209353965725121083579151369882091444210067510334671103141267111369
908658516398315019701651511685171437657618351556508849099898599823873455283316
355076479185358932261854896321329330898570642046752590709154814165498594616371
802709819943099244889575712828905923233260972997120844335732654893823911932597
463667305836041428138830320382490375898524374417029132765618093773444030707469
211201913020330380197621101100449293215160842444859637669838952286847831235526
582131449576857262433441893039686426243410773226978028073189154411010446823252
716201052652272111660396665573092547110557853763466820653109896526918620564769
312570586356620185581007293606598764861179104533488503461136576867532494416680
396265797877185560845529654126654085306143444318586769751456614068007002378776
591344017127494704205622305389945613140711270004078547332699390814546646458807
972708266830634328587856983052358089330657574067954571637752542021149557615814
002501262285941302164715509792592309907965473761255176567513575178296664547791
745011299614890304639947132962107340437518957359614589019389713111790429782856
475032031986915140287080859904801094121472213179476477726224142548545403321571
853061422881375850430633217518297986622371721591607716692547487389866549494501
146540628433663937900397692656721463853067360965712091807638327166416274888800
786925602902284721040317211860820419000422966171196377921337575114959501566049
631862947265473642523081770367515906735023507283540567040386743513622224771589
150495309844489333096340878076932599397805419341447377441842631298608099888687
413260472156951623965864573021631598193195167353812974167729478672422924654366
800980676928238280689964004824354037014163149658979409243237896907069779422362
508221688957383798623001593776471651228935786015881617557829735233446042815126
272037343146531977774160319906655418763979293344195215413418994854447345673831
624993419131814809277771038638773431772075456545322077709212019051660962804909
263601975988281613323166636528619326686336062735676303544776280350450777235547
105859548702790814356240145171806246436267945612753181340783303362542327839449
753824372058353114771199260638133467768796959703098339130771098704085913374641
442822772634659470474587847787201927715280731767907707157213444730605700733492
436931138350493163128404251219256517980694113528013147013047816437885185290928
545201165839341965621349143415956258658655705526904965209858033850722426482939
728584783163057777560688876446248246857926039535277348030480290058760758251047
470916439613626760449256274204208320856611906254543372131535958450687724602901
618766795240616342522577195429162991930645537799140373404328752628889639958794
757291746426357455254079091451357111369410911939325191076020825202618798531887
705842972591677813149699009019211697173727847684726860849003377024242916513005
005168323364350389517029893922334517220138128069650117844087451960121228599371
623130171144484640903890644954440061986907548516026327505298349187407866808818
338510228334508504860825039302133219715518430635455007668282949304137765527939
751754613953984683393638304746119966538581538420568533862186725233402830871123
282789212507712629463229563989898935821167456270102183564622013496715188190973
038119800497340723961036854066431939509790190699639552453005450580685501956730
229219139339185680344903982059551002263535361920419947455385938102343955449597
783779023742161727111723643435439478221818528624085140066604433258885698670543
154706965747458550332323342107301545940516553790686627333799585115625784322988
273723198987571415957811196358330059408730681216028764962867446047746491599505
497374256269010490377819868359381465741268049256487985561453723478673303904688
383436346553794986419270563872931748723320837601123029911367938627089438799362
016295154133714248928307220126901475466847653576164773794675200490757155527819
653621323926406160136358155907422020203187277605277219005561484255518792530343
513984425322341576233610642506390497500865627109535919465897514131034822769306
247435363256916078154781811528436679570611086153315044521274739245449454236828
860613408414863776700961207151249140430272538607648236341433462351897576645216
413767969031495019108575984423919862916421939949072362346468441173940326591840
443780513338945257423995082965912285085558215725031071257012668302402929525220
118726767562204154205161841634847565169998116141010029960783869092916030288400
269104140792886215078424516709087000699282120660418371806535567252532567532861
291042487761825829765157959847035622262934860034158722980534989650226291748788
202734209222245339856264766914905562842503912757710284027998066365825488926488
025456610172967026640765590429099456815065265305371829412703369313785178609040
708667114965583434347693385781711386455873678123014587687126603489139095620099
393610310291616152881384379099042317473363948045759314931405297634757481193567
091101377517210080315590248530906692037671922033229094334676851422144773793937
517034436619910403375111735471918550464490263655128162288244625759163330391072
253837421821408835086573917715096828874782656995995744906617583441375223970968
340800535598491754173818839994469748676265516582765848358845314277568790029095
170283529716344562129640435231176006651012412006597558512761785838292041974844
236080071930457618932349229279650198751872127267507981255470958904556357921221
033346697499235630254947802490114195212382815309114079073860251522742995818072
471625916685451333123948049470791191532673430282441860414263639548000448002670
496248201792896476697583183271314251702969234889627668440323260927524960357996
469256504936818360900323809293459588970695365349406034021665443755890045632882
250545255640564482465151875471196218443965825337543885690941130315095261793780
029741207665147939425902989695946995565761218656196733786236256125216320862869
222103274889218654364802296780705765615144632046927906821207388377814233562823
608963208068222468012248261177185896381409183903673672220888321513755600372798
394004152970028783076670944474560134556417254370906979396122571429894671543578
468788614445812314593571984922528471605049221242470141214780573455105008019086
996033027634787081081754501193071412233908663938339529425786905076431006383519
834389341596131854347546495569781038293097164651438407007073604112373599843452
251610507027056235266012764848308407611830130527932054274628654036036745328651
057065874882256981579367897669742205750596834408697350201410206723585020072452
256326513410559240190274216248439140359989535394590944070469120914093870012645
600162374288021092764579310657922955249887275846101264836999892256959688159205
60010165525637568

 

Table of computation of Pi from 2000 BC to now

  Everyone could use the table of computation of Pi!!!

Babylonians                 2000? BCE           1   3.125  = 3 + 1/8
Egyptians                   2000? BCE           1   3.16045 
China                       1200? BCE           1   3 
Bible (1 Kings 7:23)         550? BCE           1   3  
Archimedes                   250? BCE           3   3.1418 (ave.)  
Hon Han Shu                  130 AD             1   3.1622  = sqrt(10) ?  
Ptolemy                      150                3   3.14166  
Chung Hing                   250?               1   3.16227 = sqrt(10)  
Wang Fau                     250?               1   3.15555  = 142/45 
Liu Hui                      263                5   3.14159  
Siddhanta                    380                3   3.1416  
Tsu Ch'ung Chi               480?               7   3.1415926  
Aryabhata                    499                4   3.14156  
Brahmagupta                  640?               1   3.162277 = sqrt(10)  
Al-Khowarizmi                800                4   3.1416  
Fibonacci                   1220                3   3.141818  
Al-Kashi                    1429               14    
Otho                        1573                6   3.1415929  
Viete                       1593                9   3.1415926536 (ave.)  
Romanus                     1593               15    
Van Ceulen                  1596               20    
Van Ceulen                  1615               35    
Newton                      1665               16    
Sharp                       1699               71    
Seki                        1700?              10    
Kamata                      1730?              25    
Machin                      1706              100    
De Lagny                    1719              127   (112 correct)  
Takebe                      1723               41    
Matsunaga                   1739               50    
Vega                        1794              140    
Rutherford                  1824              208   (152 correct)  
Strassnitzky and Dase       1844              200    
Clausen                     1847              248    
Lehmann                     1853              261    
Rutherford                  1853              440    
Shanks                      1874              707   (527 correct)  
The 20'th century
Ferguson                    1946                   620  
Ferguson                    Jan. 1947              710  
Ferguson and Wrench         Sep. 1947              808  
Smith and Wrench            1949                 1,120  
Reitwiesner et al. (ENIAC)  1949                 2,037  
Nicholson and Jeenel        1954                 3,092  
Felton                      1957                 7,480  
Genuys                      Jan. 1958           10,000  
Felton                      May  1958           10,021  
Guilloud                    1959                16,167  
Shanks and Wrench           1961               100,265  
Guilloud and Filliatre      1966               250,000  
Guilloud and Dichampt       1967               500,000  
Guilloud and Bouyer         1973             1,001,250  
Miyoshi and Kanada          1981             2,000,036  
Guilloud                    1982             2,000,050  
Tamura                      1982             2,097,144  
Tamura and Kanada           1982             4,194,288  
Tamura and Kanada           1982             8,388,576  
Kanada, Yoshino and Tamura  1982            16,777,206  
Ushiro and Kanada           Oct. 1983       10,013,395  
Gosper                      1985            17,526,200  
Bailey                      Jan. 1986       29,360,111  
Kanada and Tamura           Sep. 1986       33,554,414  
Kanada and Tamura           Oct. 1986       67,108,839  
Kanada, Tamura, Kubo et al  Jan. 1987      134,217,700  
Kanada and Tamura           Jan. 1988      201,326,551  
Chudnovskys                 May  1989      480,000,000  
Chudnovskys                 Jun. 1989      525,229,270  
Kanada and Tamura           Jul. 1989      536,870,898  
Kanada and Tamura           Nov. 1989    1,073,741,799  
Chudnovskys                 Aug. 1989    1,011,196,691  
Chudnovskys                 Aug. 1991    2,260,000,000  
Chudnovskys                 May  1994    4,044,000,000  
Takahashi and Kanada        Jun. 1995    3,221,225,466  
Takahashi and Kanada        Aug. 1995    4,294,967,286
Takahashi and Kanada        Oct. 1995    6,442,450,938

The n'th binary digit
Bailey, Borwein, Plouffe    Nov. 1995    40,000,000,000 (hexa 921C73C6838FB2)
Bellard                     Jul. 1996   200,000,000,000 (hexa 1A10A49B3E2B82A4404F9193AD4EB6) 
Bellard                     Oct. 1996   400,000,000,000 (hexa 9C381872D27596F81D0E48B95A6C46)

 

Pi Facts

I bet you don't know this much about pi!

  • Pi is the number of times a circle's diameter will fit around its circumference
  • Pi occurs in hundreds of equations in many sciences including those describing the DNA double helix, a rainbow, ripples spreading from where a raindrop fell into water, superstrings, Einstein's Gravitational Field Equation, normal distribution, distribution of primes, geometry problems, waves, navigation....
  • It is easy to prove that if you have a circle that fits exactly inside a square, then

p = 4 x (Area of circle) / (Area of square)

  • Pi does not have to be written in decimal (base 10) notation (3.14159265....). Here it is in binary (base 2) notation:

11.0010010000111111011010101000100010000101101000110000100011010011

You can do lots more stuff with Pi when it is in binary format - like drawing weird pictures of it, or even listening to it. As Pi has an infinite number of places, it is quite possible that any message you liked could be heard somewhere in Pi. It has even been suggested it contains the VOICE OF GOD. In Carl Sagan's book 'Contact' the places of Pi (in base 11) are found to contain a message from the beings that built the universe

  • Satan does not appear in Pi too quickly: The first time '666' appears is at position 2440
  • Half the circumference of a circle with radius 1 is exactly Pi. The area inside that circle is also exactly Pi!
  • It is impossible to 'square the circle'. i.e.: You can't draw a square with the same area as a circle using standard / Euclidean straightedge and compass construction in a finite number of steps. The Greeks were obsessed with trying to do this
  • In ancient Greece the symbol p was used to denote the number 80
  • The digits of Pi appear 'random' and yet they describe something essential to the universe - No pattern emerges and yet it is predestined. The sequence of digits has so far passed all known tests for randomness. 'Approximate Entropy' can be used to establish just how random a number is - it turns out that Pi is more random than the square root of 2, which itself is more random than the square root of 3.
  • There is no zero in the first 31 digits of Pi
  • The fraction (22 / 7) is a well-used approximation of Pi. It is accurate to 0.04025%
  • Another fraction used as an approximation to Pi is (355 / 113) which is accurate to 0.00000849%
  • A more accurate but less easily remembered fraction is (104348 / 33215) as an approximation to Pi. This is accurate to 0.00000001056%
  • If you have a uniform grid of parallel lines, unit distance apart and you drop a needle length k < 1 onto the grid, then the probability the needle falls across a line is 2k / p
  • Taking the first 6,000,000,000 decimal places of Pi, this is the distribution:

0 occurs 599963005 times

1 occurs 600033260 times

2 occurs 599999169 times

3 occurs 600000243 times

4 occurs 599957439 times

5 occurs 600017176 times

6 occurs 600016588 times

7 occurs 600009044 times

8 occurs 599987038 times

9 occurs 600017038 times

This shows NO unusual deviation from expected 'random' behavior

  • The first 39 digits of Pi suffice for any application imaginable. i.e.: Calculations involving circles the size of the Universe will have an accuracy to the size of a PROTON if Pi is taken to 39 decimal places
  • Pi is irrational. An irrational number is a number that cannot be expressed in the form (a / b) where a and b are integers
  • Pi is a 'transcendental' number. This means that it is not the solution to any finite polynomial (e.g.: lots of numbers added in a series) with whole number coefficients. This is why it is impossible to square the circle
  • It is not known if Pi is 'normal'. That is if all numbers appear equally often forever. For all we know after a Billion Billion decimal places we get a random series of 1's and 0's (00101011101011000010111010....) No one has proved that Pi is not normal so people generally assume that it is
  • There are MANY irrational numbers very close to integers of the form ep n0.5where n is an integer. For example e p 1630.5 is seriously close to 262,537,412,640,768,744
  • The probability of 2 large random numbers having no common factor (coprime) is 6/(p 2)
  • Indiana State Legislature House Bill No. 246, 1897, set the value of Pi to a to 3. Duh! Actually the wording of the bill is really unclear and can interpreted in various ways. Other interpretations include 3.2, 4 and 9.2376. The bill goes on in a confused way to contradict other areas of elementary geometry, and to contradict itself. It got passed all the way up to Senate level, then a Mathematics Professor noticed it by chance, informed the Senate, and it was postponed indefinitely at its second reading.
  • Pi was calculated to 2,260,321,363 decimal places in 1991 by the Chudnovsky brothers in New York
  • The Babylonians found the first known value for Pi in around 2000BC -They used (25/8)
  • The Egyptians used Pi = 3 but improved this to (22 / 7). They also used (256/81). If you imagine a circle in the Great Pyramid at Giza in Egypt like this:

Then the circle's circumference is twice the base length of the pyramid, and the circle's area is equal to the pyramid's vertical sectional area through the peak. The ratio of the perimeter of the base of the Great Pyramid to its height is twice Pi. The same ratio for the Pyramid of the Sun in Mexico is four times Pi. Both are built to an accuracy of a few inches.

  • The D&M Pyramid (a geological feature on Mars) is situated at 40.868 degrees North, which is exactly equal to arcTan (e/p ). Many of its internal angles also share trigonometric configurations of e and Pi.
  • In around 200 BC Archimedes found that Pi was between (223 / 71) and (22 / 7). His error was no more than 0.008227 %. He did this by approximating a circle as a 96 sided polygon
  • The Bible uses a value of Pi of 3. Here is a verse from I Kings 7,23:

And he made a molten sea, ten cubits from one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about

  • The first person to use the Greek letter p was Welshman William Jones in 1706. He used it as an abbreviation for the 'periphery' of a circle with unit diameter. Euler adopted the symbol and it quickly became a standard notation
  • The Pi memory champion is Hiroyoki Gotu (21 years old) who memorized an amazing 42,000 digits. Woooooooooooooooooh!
  • There are many formulae for Pi. They are completely arithmetical despite the fact that Pi arises from Geometry:

Wallis: p = (8 / 3) . (4.4.6.6.8.8.10.10.12...) / (3.5.5.7.7.9.9.11.11...)

This is very slow to converge - After 7000 terms it is accurate to 7 decimal places (averaging terms 7000 and 7001)

Gregory (also attributed to Leibniz): p = 4 - (4/3) + (4/5) - (4/7) + (4/9) - (4/11) + ...

This is about the same - After 7000 terms it is also accurate to 7 decimal places

p = Sqrt ( 12 - (12/22) + (12/32) - (12/42) + ...

This is slightly faster - After just 1100 terms it is accurate to at least 9 decimal places

There are many other such formulas

         Ramanujan developed a formula for Pi which adds 8 decimal places each term!

This is the formula generally used by supercomputers to calculate Pi

         David Bailey, Peter Borwein and Simon Plouffe recently computed the ten billionth digit in the hexadecimal expansion of Pi. They used an astonishing formula:

which enables one to calculate the n-th digit of Pi without being forced to calculate all the preceding n -1 digits. No one had previously even conjectured that such a digit-extraction algorithm for Pi was possible

  • Choose 2 random numbers x and y between 1 and -1. Do this N times. If for M of those N times x2 + y2 < 1 then as N approaches infinity, p = 4(M/N)
  • The area of a circle is p r2
  • The volume of a sphere is 4/3p r3 and its surface area is 4p r2
  • The circle is the shape with the least perimeter length to area ratio (for a given shape area). Centuries ago mathematicians were also philosophers. They considered the circle to be the 'perfect' shape because of this. The sphere is the 3D shape with the least surface area to volume ratio (for a given volume)
  • Most people would say that a circle has no corners - but it is more accurate to say that it has an infinite number of corners
  • Pi is approximately 6/5 2 where is the GOLDEN MEAN - another interesting number that appears all over nature and in art
  • Pi is of course the ratio of a circle's circumference to its diameter. If we bring everything up one dimension to get a '3D value for Pi'... The ratio of a sphere's surface area to the area of the circle seen if you cut the sphere in half is EXACTLY 4
  • If you take 10 million random digits, statistically on average you would expect 200 cases where you get 5 digits in a row the same. If you take 10 million digits of Pi - guess what... you get exactly 200
  • Euler showed that eip + 1 = 0 Where i is the square-root of -1
  • Landau showed that p /2 is the value of x between 1 and 2 for which cos x vanishes
  • p = n sin(180/n). Try it with different values of n. Your calculator will need to be in degrees (not radians) mode. This equation gets more accurate with larger values of n
  • The following are all NEARLY Pi:

101/2

Cube root of 31

666/212

10/p

(97 + 9/22)1/4

9/5 + (9/5)1/2

(19 (7)1/2) / 16

(2)1/2 + (3)1/2

1.1 x 1.2 x 1.4 x 1.7

(296/167) 2

  • Kochansky found that Pi is NEARLY a root of the equation 9x4 - 240x2 + 1492
  • A year is about p x107 seconds
  • Ludolph Van Ceulen (1540 - 1610) spent most of his life working out Pi to 35 decimal places. Pi is sometimes known as Ludolph's Constant
  • It is not known if the following are 'irrational':

p + e

p /e

lnp

  • If you approximate the circle with a radius of 1 as a 100 sided polygon, then its area is only accurate to 1 decimal place or 0.0658%
  • At position 762 there are six nines in a row. This is known as the Feynman Point
  • Pi in base Pi is 10
  • p = 4(1/2)!2
  • All permutations of 3 arbitrary digits appear somewhere in Pi
  • In 1931 a Cleveland businessman published a book announcing that Pi is exactly 256/81
  • Starting with the conventional 5-by-5 magic square, and then substituting the nth digit of pi for each number n in the square, we obtain a new array of numbers. The sum of the numbers in every column is duplicated by a sum of numbers in every row.
  • Write the letters of the English alphabet, in capitals, clockwise around a circle, and cross out the letters that have right-left symmetry, A, H, I, M, etc. The letters that remain group themselves in sets of 3, 1, 4, 1, 6"
  • At one time it was thought there was an illness attached to trying to 'square a circle' called Morbus Cyclometricus.
  • After saying (correctly) that Pi / 2 is the value of x between 1 and 2 for which Cos x vanishes, Edmund Landau was dismissed from his position in 1934 for teaching in an 'un-German' style.
  • In the following series of natural numbers, constructed by taking successively larger strings of digits from the beginning of the decimal expansion of the number Pi: 3, 31, 314, 31415, 314159, 3141592, etc. The first thousand numbers of the series include only 4 primes.
  • The earliest known reference to Pi is on a Middle Kingdom papyrus scroll, written around 1650 BC by Ahmes the scribe.
  • Decimal places 7, 22, 113 and 355 of Pi are all the number 2. (22/7 and 355/113 are good approximations of Pi)
  • The sequence 314159 re-appears in the decimal expansion of Pi at place 176451. This sequence appears 7 times in the first 10 million places (not including right at the start)
  • A.C. Aitken from Edimbourg University is able, for the first 2000 decimals of Pi, to tell you what number appears in position x
  • (p + 20)i is almost equal to -1 (where i is the square-root of minus one)
  • If you approximate the circle as a square then the value you get for Pi is about 10% out. It just goes to show that you shouldn't approximate the circle as a square. Well you wouldn't make square wheels would you?
  • Here's a Pi limerick:

Three point one four one five nine two

Its been around forever - it's not new

It appears everywhere

In here and in there

Its irrational I know but its true!

 

 

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Pi

We have known about pi for four millenniums, calculated millions of its digits, and even created a holiday just for it. It is used everyday by students and highly skilled math theorists alike. Because it is one of the most important numbers in the world, we decided it should have at least its own webpage.

Calculating Pi

There is a fairly simple formula you can use if you want to calculate pi.

Arc Tangent(1) = Pi/4

so that would simplify out to

4[Arc Tangent(1)] = Pi

 

 

 
   
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