p  and its computation through the ages

 

(Click here for a for a Postscript version of this page and here for a pdf version)

 

The value of p has engaged the attention of many mathematicians and calculators from the time of Archimedes to the present day, and has been computed from so many different formulae, that a complete account of its calculation would almost amount to a history of mathematics.

- James Glaisher (1848-1928)

The history of pi is a quaint little mirror of the history of man.

- Petr Beckmann

1  Computing the constant p

Understanding the nature of the constant p, as well as trying to estimate its value to more and more decimal places has engaged a phenomenal energy from mathematicians from all periods of history and from most civilizations. In this small overview, we have tried to collect as many as possible major calculations of the most famous mathematical constant, including the methods used and references whenever there are available.

1.1  Milestones of p's computation

Other enumerations of p's computations can also be found in: [2], [3], [5], [13], [40], [53], ... in which we got many valuable informations.

The two following sections are enumerating the main computations respectively before and during computer era. The Method column usually refers to the last section; for example arctan(M) means that the arctangent relation (M) or Machin's formula was used.

2  History of p calculations before computer era

Author Year Exact digits Method Comment
Egyptians 2000 B.C. 1 unknown p = (16/9)2
Babylonians 2000 B.C. 1 p = 3+1/8
Bible 550 B.C.? 0 p = 3
Archimedes, [21] 250 B.C. 2 polygon p = 22/7, 96 sides
Ptolemy 150 3 p = 3+8/60+30/602
Liu Hui, [28] 263 5 polygon 3072  sides
Zu Chongzhi, [28] 480 7 polygon Also p = 355/113
Aryabhata 499 4 polygon p = 62832/20000
Brahmagupta 640 1 p = {10}
Al-Khwarizmi 830 4 p = 62832/20000
Fibonacci 1220 3 polygon p = 3.141818
Al-Kashi, [1] 1424 14 polygon 6.227 sides
Otho 1573 6 p = 355/113
Viète 1579 9 polygon 6.216 sides
van Roomen 1593 15 polygon 230 sides
van Ceulen 1596 20 polygon 60.233sides
van Ceulen, [9] 1610 35 polygon 262 sides
van Roijen Snell, [44] 1621 34 polygon 230 sides
Grienberger 1630 39 polygon
Newton 1671 15 series(N)
Sharp 1699 71 arctan(Sh)
Machin, [24] 1706 100 arctan(M)
De Lagny, [27] 1719 112 arctan(Sh) 127 computed
Takebe Kenko 1722 40 series
Matsunaga 1739 49 series
Euler 1755 20 arctan(E2) In one hour!
Vega 1789 126 arctan(H) 143 computed
Vega, [50] 1794 136 arctan(H) 140 computed
Rutherford, [38] 1841 152 arctan(E1) 208 computed
Dahse, [12] 1844 200 arctan(SD)
Clausen 1847 248 arctan(H & M)
Lehmann 1853 261 arctan(E3)
Shanks, [41] 1853 527 arctan(M) 607 computed
Rutherford 1853 440 arctan(M)
Richter 1854 500 unknown
Shanks, [42] 1873 527 arctan(M) 707 computed
Tseng Chi-hung 1877 100 arctan(E3)
Uhler 1900 282 arctan(M)
Duarte 1902 200 arctan(M)
Uhler, [49] 1940 333
Ferguson 1944-1945 530 arctan(L)
Ferguson, [15] 07-1946 620 arctan(L) Last hand calculation

3  History of p calculations during computer era

Author Year Exact digits Method Computer
Ferguson 01-1947 710 arctan(L)
Ferguson & Wrench Jr 09-1947 808 arctan(M)
Smith & Wrench Jr, [52] 06-1949 1 120 arctan(M)
Reitwiesner et al., [37] 09-1949 2 037 arctan(M) ENIAC
Nicholson & Jeenel, [33] 11-1954 3 092 arctan(M) NORC
Felton 03-1957 7 480 arctan(K & G) Pegasus
Genuys, [18] 01-1958 10 000 arctan(M) IBM 704
Felton 05-1958 10 020 arctan(K & G) Pegasus
Guilloud 07-1959 16 167 arctan(M) IBM 704
Shanks & Wrench Jr, [43] 07-1961 100 265 arctan(S1 & G) IBM 7090
Guilloud & Filliatre 02-1966 250 000 arctan(S1 & G) IBM 7030
Guilloud & Dichampt 02-1967 500 000 arctan(S1 & G) CDC 6600
Guilloud & Bouyer, [20] 05-1973 1 001 250arctan(S1 & G) CDC 7600
Kanada & Miyoshi 1981 2 000 036 arctan(K & M) FACOM M-200
Guilloud 1982 2 000 050 unknown unknown
Tamura 1982 2 097 144 GL2 MELCOM 900II
Tamura & Kanada, [47] 1982 4 194 288 GL2 Hitachi M-280H
Tamura & Kanada 1982 8 388 576 GL2 Hitachi M-280H
Kanada et al. 1983 16 777 206 GL2 Hitachi M-280H
Kanada et al., [25] 10-1983 10 013 395 arctan(G), GL2 Hitachi S-810/20
Gosper 10-1985 17 526 200 series(Ra), B4 Symbolics 3670
Bailey, [4] 01-1986 29 360 111 B2, B4 CRAY-2
Kanada & Tamura 09-1986 33 554 414 GL2, B4 Hitachi S-810/20
Kanada & Tamura 10-1986 67 108 839 GL2 Hitachi S-810/20
Kanada et al. 01-1987 134 214 700 GL2, B4 NEC SX-2
Kanada & Tamura, [26] 01-1988 201 326 551GL2, B4 Hitachi S-820/80
Chudnovskys 05-1989 480 000 000 series CRAY-2
Chudnovskys 06-1989 525 229 270 series IBM 3090
Kanada & Tamura 07-1989 536 870 898 GL2 Hitachi S-820/80
Chudnovskys 08-1989 1 011 196 691 series(CH) IBM 3090 & CRAY-2
Kanada & Tamura 11-1989 1 073 741 799 GL2, B4 Hitachi S-820/80
Chudnovskys, [34] 08-1991 2 260 000 000 series(CH?) m-zero
Chudnovskys 05-1994 4 044 000 000 series(CH) m-zero
Kanada & Takahashi 06-1995 3 221 220 000 GL2, B4 Hitachi S-3800/480
Kanada & Takahashi 08-1995 4 294 967 286 GL2, B4 Hitachi S-3800/480
Kanada & Takahashi 10-1995 6 442 450 000 GL2, B4 Hitachi S-3800/480
Chudnovskys 03-1996 8 000 000 000 series(CH?) m-zero ?
Kanada & Takahashi 04-1997 17 179 869 142 GL2, B4 Hitachi SR2201
Kanada & Takahashi, [46] 06-1997 51 539 600 000 GL2, B4 Hitachi SR2201
Kanada & Takahashi 04-1999 68 719 470 000 GL2, B4 Hitachi SR8000
Kanada & Takahashi 09-1999 206 158 430 000 GL2, B4 Hitachi SR8000
Kanada et al. 12-2002 1 241 100 000 000 arctan(S2 & S3) Hitachi SR8000/MP

4  List of the main used methods

In this section are expressed the main identities used to compute p just after the geometric period which was based on the computation of the perimeter (or area) of regular polygons with many sides.

4.1  Machin like formulae

There are numerous formulae to compute p by mean of arctan functions (see [3], [23], [31], [45], [48],...).
 p

6
= arctan
 1

3

(Sh)
 p

4
= 4arctan
 1

5

-arctan
 1

239

, Machin (M)
 p

4
= 8arctan
 1

10

-arctan
 1

239

-4arctan
 1

515

Klingenstierna (K)
 p

4
= arctan
 1

2

+arctan
 1

5

+arctan
 1

8

Strassnitzky (SD)
 p

4
= 12arctan
 1

18

+8arctan
 1

57

-5arctan
 1

239

, Gauss (G)
 p

4
= 4arctan
 1

5

-arctan
 1

70

+arctan
 1

99

, Euler (E1)
p

4
= 5arctan
 1

7

+2arctan
 3

79

, Euler (E2)
 p

4
= arctan
 1

2

+arctan
 1

3

, Euler (E3)
 p

4
= 2arctan
 1

3

+arctan
 1

7

, Hutton (H)
 p

4
= 3arctan
 1

4

+arctan
 1

20

+arctan
 1

1985

Loney (L)
 p

4
= 6arctan
 1

8

+2arctan
 1

57

+arctan
 1

239

, Störmer (S1)
 p

4
= 12arctan
 1

49

+32arctan
 1

57

-5arctan
 1

239

+12arctan
 1

110443

   (S2)
 p

4
= 44arctan
 1

57

+7arctan
 1

239

-12arctan
 1

682

+24arctan
 1

12943

     (S3)

4.2  Other series

4.2.1  Newton

 
p
=  33

4
+24
1/4

0 


 

x-x2
 
dx
=  33

4
+24
 1

12
-  1

5.25
-  1

28.27
-  1

72.29
...
, Newton (N)

4.2.2  Ramanujan like series

The important point is that evaluating such series to huge number of digits requires to develop specific algorithms. Such algorithms are now well known and are based on idea related to binary splitting. To learn more about those consult: [6], [7], [19],...

 
 1

p
=  22

9801


k=0 
 (4k)!

(k!)444k
 (1103+26390k)

994k
,  Ramanujan [36] (Ra)
 1

p
=12

k=0 
(-1)k  (6k)!

(3k)!(k!)3
 (13591409+545140134k)

6403203k+3/2
, Chudnovsky   (CH)

4.3  Iterative algorithms

The main difficulty with the following iterative procedures is to compute to a high accuracy inverses and square roots of a real number. By mean of FFT based methods to compute products of numbers with many decimal places this is now possible in a quite efficient way. To find how to compute those operations you can consult [3], [6], [8], [19],...

4.3.1  Gauss-Legendre (or Brent-Salamin) quadratic

Set x0=1,y0=1/2,a0=1/2 and:

 






xk+1=( xk+yk) /2
yk+1=

 

xkyk
 
ak+1=ak-2k+1( xk+12-yk+12)
then ([6], [8], [39]):
p =
lim
k 
( 2xk2/ak) . (GL2)

4.3.2  Borwein quadratic

Set x0=2,y0=0,a0=2+2 and:

 










xk+1=


 

xk
 
+1/

 

xk
 

/2
yk+1=

 

xk
 

 1+yk

yk+xk

ak+1=akyk+1
 1+xk+1

1+yk+1

then ([6]):
p =
lim
k 
ak. (B2)

4.3.3  Borwein quartic

Set y0=2-1,a0=6-42 and:


yk+1=( 1-(1-yk4)1/4) /(1+(1-yk4)1/4)
ak+1=(1+yk+1)4ak-22k+3yk+1(1+yk+1+yk+12)
then ([6]):
p =
lim
k 
(1/ak ). (B4)

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[2]
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[4]
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[17]
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[18]
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[26]
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[32]
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