A chronology of π

Pre computer calculations of π

1Rhind papyrus2000 BC13.16045 (= 4(89)24(\large\frac{8}{9}\normalsize )^{2}) Click for note 1
2Archimedes250 BC33.1418 (average of the bounds) Click for note 2
3Vitruvius20 BC13.125 (= 258\large\frac{25}{8}\normalsize) Click for note 3
4Chang Hong13013.1622 (= √10) Click for note 4
5Ptolemy15033.14166 Click for note 5
6Wang Fan25013.155555 (= 14245\large\frac{142}{45}\normalsize) Click for note 6
7Liu Hui26353.14159 Click for note 7
8,Zu Chongzhi48073.141592920 (= 355113\large\frac{355}{113}\normalsize) Click for note 8
9Aryabhata49943.1416 (= 6283220000\large\frac{62832}{20000}\normalsize) Click for note 9
10Brahmagupta64013.1622 (= √10) Click for note 10
11Al-Khwarizmi80043.1416 Click for note 11
12Fibonacci122033.141818 Click for note 12
13Madhava1400113.14159265359 Click for note 13
14Al-Kashi1430163.1415926535897932 Click for note 14
15Otho157363.1415929 Click for note 15
16Viète159393.1415926536 Click for note 16
17Romanus1593153.141592653589793 Click for note 17
18Van Ceulen1596203.14159265358979323846 Click for note 18
19Van Ceulen1596353.1415926535897932384626433832795029 Click for note 19
20Newton1665163.1415926535897932 Click for note 20
21Sharp169971 Click for note 21
22Seki Kowa170010
24Machin1706100 Click for note 24
25De Lagny1719127Only 112 correct Click for note 25
26Takebe172341 Click for note 26
27Matsunaga173950 Click for note 27
28von Vega1794140Only 136 correct Click for note 28
29Rutherford1824208Only 152 correct Click for note 29
30Strassnitzky, Dase1844200 Click for note 30
31Clausen1847248 Click for note 31
32Lehmann1853261 Click for note 32
33Rutherford1853440 Click for note 33
34Shanks1874707Only 527 correct Click for note 34
35Ferguson1946620 Click for note 35
General Remarks:
A. In early work it was not known that the ratio of the area of a circle to the square of its radius and the ratio of the circumference to the diameter are the same. Some early texts use different approximations for these two "different" constants. For example, in the Indian text the Sulba Sutras the ratio for the area is given as 3.088 while the ratio for the circumference is given as 3.2.

B. Euclid gives in the Elements XII Proposition 2:
Circles are to one another as the squares on their diameters.
He makes no attempt to calculate the ratio.

Computer calculations of π

MathematicianDatePlacesType of computer
FergusonJan 1947710Desk calculator
Ferguson, WrenchSept 1947808Desk calculator
Smith, Wrench19491120Desk calculator
Reitwiesner et al.19492037ENIAC
Nicholson, Jeenel19543092NORAC
GenuysJan 195810000IBM 704
FeltonMay 195810021PEGASUS
Guilloud195916167IBM 704
Shanks, Wrench1961100265IBM 7090
Guilloud, Filliatre1966250000IBM 7030
Guilloud, Dichampt1967500000CDC 6600
Guilloud, Bouyer19731001250CDC 7600
Miyoshi, Kanada19812000036FACOM M-200
Tamura19822097144MELCOM 900II
Tamura, Kanada19824194288HITACHI M-280H
Tamura, Kanada19828388576HITACHI M-280H
Kanada, Yoshino, Tamura198216777206HITACHI M-280H
Ushiro, KanadaOct 198310013395HITACHI S-810/20
GosperOct 198517526200SYMBOLICS 3670
BaileyJan 198629360111CRAY-2
Kanada, TamuraSept 198633554414HITACHI S-810/20
Kanada, TamuraOct 198667108839HITACHI S-810/20
Kanada, Tamura, KuboJan 1987134217700NEC SX-2
Kanada, TamuraJan 1988201326551HITACHI S-820/80
ChudnovskysMay 1989480000000
ChudnovskysJune 1989525229270
Kanada, TamuraJuly 1989536870898
ChudnovskysAug 19891011196691
Kanada, TamuraNov 19891073741799
ChudnovskysAug 19912260000000
ChudnovskysMay 19944044000000
Kanada, TamuraJune 19953221225466
KanadaAug 19954294967286
KanadaOct 19956442450938
Kanada, TakahashiAug 199751539600000HITACHI SR2201
Kanada, TakahashiSept 1999206158430000HITACHI SR8000

General Remarks:

A. Calculating π to many decimal places was used as a test for new computers in the early days.

B. There is an algorithm by Bailey, Borwein and Plouffe, published in 1996, which allows the nnth hexadecimal digit of π to be computed without the preceeding n1n - 1 digits.

C. Plouffe discovered a new algorithm to compute the nnth digit of π in any base in 1997.

References (show)

  1. D H Bailey, J M Borwein, P B Borwein, and S Plouffle, The quest for Pi, The Mathematical Intelligencer 19 (1997), 50-57.

Written by J J O'Connor and E F Robertson
Last Update September 2000