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



Pi-Day in
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Home Version history Guestbook Who I am Some pictures (fr) Acknowledgements Last modifications Contact

Cette page en français This page in English


Pi room in the "palais de la découverte" in Paris

For a fistful of digits ...
History of records and methods


Since Archimede, mathematicians tried hard to compute as many digits of Pi they could, even if it appears a useless quest to some people... But without this obsession, one must realize that we would have missed the recent discovery of several algorithms such as the product of long integers by fast fourier transform. And think about the consequences of the BBP formulas ! Thus, discover, optimize and accelerate the computation of the digits of Pi is a lively mathematical field.

Index table

Algorithms useful for computation of the digits of Pi
Computation records of Pi over the ages...
Digits computed by hand then using a computer
Place computed (digits in base 2 not knowing the previous ones)
Continued fraction of Pi
Digits record in a classroom

Pi memory records
World records announcements
About the world record of Takahashi-Kanada from april 1999 (68 719 470 000 digits)
About the world record of Kanada from september 1999 (206,158,430,000 digits)
About the world record of Kanada from december 2002 (1,241,100,000,000 digits)
A few statistics on Pi
1 000 000 digits of Pi

Algorithms useful for computation of the digits of Pi

Computation records of Pi over the ages...
(in modern notations of course...)

note : 4*2+4*3 means Pi=4 arctan(1/2)+4arctan(1/3), it's the formula used to compute the digits (see Machin)

NAME

DATE

Approx. or method used

Good digits

Babylonians

-2000

3+1/8=3,125

1

Egyptians (scribe Ahmès)

-1650

(16/9)2=3,16045

1

Chinese

-1200

3

0

Bible

-550

3

0

Archimedes

-250

3,14185

3

Hon Han Shu

130

1

Ptolemy

150

377/120=3,14166

3

Chung Hing

250

1

Wang Fau

250

142/45=3,155

1

Liu Hui

264

3,14159

5

Siddhanta

380

3+177/1250=3,1416

3

Tsu Chung Chih

480?

355/113=3,141592

6

Aryabhata

499

3,14156

4

Brahmagupta

640

101/2=3,1622

1

Al-Khowarizmi

800

3,1416

3

Fibonacci

1220

3,141818

3

Al-Kashi

1429

6016I59II28III1IV34V51VI46VII14VIII50IX

14

Otho

1573

3,1415929

6

Viete

1593

3,1415926536

9

Romanus

1593

 

15

Van Ceulen

1596

Archimedes' method

20

Van Ceulen

1609

"

34

Grienberger

1630

"

39

Newton

1665

"

16

Sharp

1699

"

71

Seki

1700

"

10

Machin

1706

16*5-5*239 (Machin)

100

De Lagny

1719

4*2+4*3 (Euler)

112 (out of 127 computed)

Takebe Katahiro

1723

polygon with 1024 sides

41

Matsunaga

1739

 

50

Vega

1794

20*7+8arctan(3/79) (Euler 1755)

140

Rutherford

1824

16*5-4*70+4*99
(Euler 1764)

152 (out of 208)

Strassnitsky, Dahse

1844

4*2+4*5+4*8
(Strassnitsky 1844)

200

Clausen

1847

8*3+4*7 (Hutton 1776)

248

Lehmann

1853

8*3+4*7

261

Rutherford

1853

Machin's formula

440

Shanks

1874

Machin's formula

527 (sur 707)

Ferguson

1945

12*4+4*20+4*1985
(Loney 1893)

539

Ferguson

1947

 

620

Ferguson

1948

 

710

Ferguson and wrench

1948

 

808

Smith and Wrench

1949

 

1 120

Reitwiesner on the ENIAC

1949

Machin's formula

2 037

Nicholson and Jeenel

1954

arctan formulae

3 092

Felton

1957

32*10-4*239-16*515
(Klingenstierna 1730)

7 480

Genuys

01-1958

 

10 000

Felton

05-1958

48*18+32*57-20*239
(Gauss 1863)

10 021

Guilloud

1959

 

16 157

Shanks and Wrench

1961

24*8+8*57+4*239
(Störmer 1896) + Gauss formula

100 265

Guilloud and Filliatre

1966

 

250 000

Guilloud and Dichampt

1967

 

500 000

Guilloud and Bouyer

1973

formulae Störmer+ Gauss

1 001 250

Miyoshi and Kanada

1981

 

2 000 036

Guilloud

1982

 

2 000 050

Tamura

1982

 

8 388 576

Kanada, Yoshino and Tamura

1982

 

16 777 206

Gosper

1985

Ramanujan's sum

17 526 200

Bailey

01-1986

Borwein's algorithms (order 2 and 4)

29 360 111

Kanada and Tamura

10-1986

Borwein's algorithms (order 2 and 4)

67 108 839

Kanada, Tamura, Kobo

01-1987

"

134 217 700

Kanada and Tamura

01-1988

"

201 326 551

Chudnovsky and Chudnovsky

05-1989

Ramanujan's type sums

480 000 000

Chudnovsky and Chudnovsky

06-1989

Ramanujan's type sums

525 229 270

Kanada and Tamura

07-1989

Borwein's algorithms (order 2 and 4)

536 870 898

Chudnovsky and Chudnovsky

08-1989

Ramanujan's type sums

1 011 196 691

Kanada and Tamura

11-1989

Borwein's algorithms (order 2 and 4)

1 073 741 799

Chudnovsky and Chudnovsky

08-1991

Ramanujan's type sums

2 260 000 000

Chudnovsky and Chudnovsky

05-1994

Ramanujan's type sums

4 044 000 000

Kanada

06-1995

Borwein's algorithms (order 2 and 4)

4 294 967 286

Kanada

10-1995

Borwein's algorithms (order 2 and 4)

6 442 450 938

Takahashi-Kanada

08-1997

Borwein's algorithms (order 2 and 4)

51,539,600,000

Takahashi-Kanada 04-1999

algorithms Brent/Salamin
and order 4 from Borwein

68,719,470,000

Takahashi-Kanada 20-09-1999
algorithms Brent/Salamin
and order 4 from Borwein
206,158,430,000
soit environ 3.236
Kanada 06-12-2002
Machin's type formulae
48*49+128*57-20*239+48*110443
176*57+28*239-48*682+96*12943
1,241,100,000,000

from D. Bailey, J. et P. Borwein, S. Plouffe and myself




Digits computed by hand then using a computer

Other plots from myself (in french but understandable I guess)




Place computed (digits in base 2 not knowing the previous ones)

Bailey-Borwein-Plouffe

1996

40 000 000 000

Bellard

7-6-1996

50 000 000 000

Bellard

10-7-196

100 000 000 000

Bellard

9-22-97

1 000 000 000 000

Colin Percival - Project Pihex

8-21-98

5 000 000 000 000

Colin Percival - Project Pihex
2-9-99
40,000,000,000,000
Colin Percival - Project Pihex

11-9-2000

1,000,000,000,000,000





Continued fraction for Pi

Main records :

Gosper

1977

17,001,303

H. Havermann

Juin 1999

20,000,000

H. Havermann
Mars 2002
180,000,000

Note that Havermann has a very interesting page about the fraction expansion of Pi



Digits record in a classroom

Ok, I invented this class of records ;-)... however, this was to promote the work of the 7th grade (US-equivalent) classsroom located in Bouchain, north of France. They meet every monday in this 2012-2013 year to add digits and there are 85 of them so far ! Congratulations to them !! (Click on images to get high density pictures).


 




Pi memory records

A Japanese holds the record, just try to imagine what it means as memory, that crazy !
Main Pi memory records:


Simon Plouffe

1975

4096

Hideaki Tomoyori

1979

15,151

Hiroyuki Goto (in 9h)

1995

42,000

Akira Haraguchi (details below)
2006
100,000

Akira Haraguchi, a 60-year old Japanese man, managed to recite 100 000 digits of Pi in 16h30 on October 3rd, 2006, breaking his own (unofficial) record of 83 431 digits established in 2005 ! First comment from Akira: "I don't think it's anything exceptional, I just emptied my mind of everything else and recited the numbers" ;-) Here is the BBC coverage, there the world-ranking of Pi memory (not up to date) and here the report about his previous record in 2005.

 


World records announcement (html)

1000 billion-th (10^12) binary digit of Pi by Fabrice Bellard- 22/09/97

5 000 billion-th digit of Pi is '0' by Colin Percival (project PiHex) - 21/08/98

40 000 billion-th digit of Pi is '0' by Colin Percival (project PiHex) - 9/02/99

The 1 000 000 billion-th binary digit of Pi (1015) is '0' by Colin Percival (project PiHex) - 11/09/00

51,539,600,000 digits of Pi by Kanada
08/06/97

206,158,430,208 (=3*2^36) digits of Pi by Kanada
04/10/99


About the world record of Takahashi-Kanada from april 1999 (68 719 470 000 digits)

Two computations on a HITACHI SR8000 based on two independant algorithms ( Brent/Salamin and fourth-order algorithm from the Borwein) generated 68,719,476,736 (=236) digits of Pi. Comparing the two results, they found 68,719,476,693 common digits. The new record has thus been established to 68,719,470,000 digits of Pi.


Main program:
Starting : April 2nd, 1999 8pm:14:38
End : April 4th, 1999 5am:08:41
Total time: 32:54:02
Memory used: 296 GB
Algorithm : Gauss-Legendre (Brent-Salamin)

Checking program :
Starting : 4th April 1999 5am:08:48
End : 5th April 1999 8pm:29:25
Total time: 39:20:37
Memory used : 280 GB
Algorithm : fourth order from Borwein




About the world record of Kanada from september 1999 (206,158,430,000 digits)

Two computations on a HITACHI SR8000 based on two independant algorithms ( Brent/Salamin and fourth-order algorithm from the Borwein) generated 206,158,430,208 (=3.2^36) digits of Pi. Comparing the two results, they found 206,158,430,163 common digits. The new record has thus been established to 206,158,430,000 digits of Pi.


Main program:
Starting : September 18th, 1999 7pm:00:52 (Japan time)
End : September 20th, 1999 8am:21:56
Total time: 37:21:04
Memory used: 865 GB (=6.758*128)
Algorithm : Gauss-Legendre (Brent-Salamin)

Checking program :
Starting : June 26th, 1999 1am:22:50
End : June 27th, 1999 11pm:30:40
Total time: 46:07:10
Memory used : 817 GB (=6,383*128)
Algorithm : fourth order from Borwein

 


About the world record of Kanada from december 2002 (1,241,100,000,000 digits)

Two computations lasting 600 hours on a HITACHI SR8000/MP with 1TB storage (1024Go), and based on two independent Machin-like formulas, generated 1,241,100,000 digits of Pi after that the result obtained in hexagesimal base was converted to base 10. Formulae used were :

This come back of amazingly simple formulae after using Brent-Salamin, Borwein algorithms or Ramanujan-like formulae for 15 years was highly unexpected. Actually, despite the algorithmic improvements, the complexity of these latter formulae had reached computer limits. Indeed, the widely use of root extractions and multiplications required the common use of very large scale Fast Fourier Transform (FFT). This algorithm needs huge memory. Kanada thus decided to be wiser and to use Machin-like formulas which need more arithmetical operations with much less very large scale FFT and so less memory. It seems that similar issues start to appear in the fastest computers in the world whose network and memory operations saturate faster than what was expected theoretically. Kanada estimates than his computation is twice faster than the previous one using Brent/Salamin and Borwein algorithms.

Some papers about this record :

Seattle Post Intelligencier

Kanada's lab with several articles in "Press Release"

Daily Times

Page of J. Borwein about this record

MathTrek (Ivan Peterson)

 




A few statistics on Pi

Frequency of digits on the first 50 000 000 000 digits:

'0' : 5000012647
'1' : 4999986263
'2' : 5000020237
'3' : 4999914405
'4' : 5000023598
'5' : 4999991499
'6' : 4999928368
'7' : 5000014860
'8' : 5000117637
'9' : 4999990486
Chi square = 5.60

Frequency of digits of 1/Pi on the first 50,000,000,000 digits:

'0' : 4999969955
'1' : 5000113699
'2' : 4999987893
'3' : 5000040906
'4' : 4999985863
'5' : 4999977583
'6' : 4999990916
'7' : 4999985552
'8' : 4999881183
'9' : 5000066450
Chi square = 7.04



On this page, you can also download 1 million digits !

1 000 000 digits of Pi

If you want more....
- Download 4 200 000 000 digits

Site ftp://pi.super-computing.org/


The first 1000 digits just for fun....
3,1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679
8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196
4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273
7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094
3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912
9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132
0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235
4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859
5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303
5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989...


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