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
- Ancient civilizations like Egyptians [14], Babylonians, China
([28], [51]), India [35],... were interested in
evaluating, for example, area or perimeter of circular fields. Of course in
this early history, p was not yet a constant and was
only implicit in all available documents.
Perhaps the most famous is the Rhind Papyrus which states the rule
used to compute the area of a circle: take away 1/9 of the diameter
and take the square of the remainder therefore implicitly p = (16/9)2.
- Archimedes of Syracuse (287-212 B.C.). He developed a method based on
inscribed and circumscribed polygons which will be of practical use
until the mid seventeenth. It's the first known algorithm to compute p to,
in principle, any required accuracy. By mean of regular polygons with 96 sides
[21], in his treatise Measurement of a Circle, he showed
that: the ratio of the circumference of any circle to its diameter is
less than 3+1/7 but greater than 3+10/71.
- Zu Chongzhi from China (430-501). He established that 3.1415926 < p < 3.1415927 using polygons [28].
- Al-Kashi from Samarkand (1380-1429). By mean of Archimedes' polygons,
he computed, in 1424, 2p to nine sexagesimal places and his estimation
(about 14 correct decimals) will remain unsurpassed for nearly 200 years
[1].
- Ludolph van Ceulen (1540-1610). Still using polygons, he made various
computations and finally found 35 decimal places before his death in 1610,
[9]. On his tombstone (today lost [22]), the following
approximation was engraved: when the diameter is 1, then the
circumference of the circle is greater than
|
314159265358979323846264338327950288
100000000000000000000000000000000000
|
|
|
but smaller than
|
314159265358979323846264338327950289
100000000000000000000000000000000000
|
. |
|
- Isaac Newton (1643-1727) and James Gregory (1638-1675) introduced
respectively the series expansion of the function arcsin (1669) and the
function arctan (1671) and opened the era of analytical methods to
compute p. After a 15 digits computation Newton wrote: I am
ashamed to tell you to how many figures I carried these computations, having
no other business at the time [5].
- John Machin (1680-1751). In 1706 [24], the Truly
Ingenious Mr. John Machin reached 100 decimal places with a fast converging
arctan formula which now bares his name. The same year, William Jones
(1675-1749) uses the symbol p to represent the ratio of the perimeter of a
circle to its diameter.
- Johann Heinrich Lambert (1728-1777). p is irrational
(1761, [29]).
- Adrien Marie Legendre (1752-1833). p2 is irrational
(1794, [30]).
- William Shanks (1812-1882). He spent a considerable part of his life to
compute various approximations of p including a final 707 digits
estimation ([41], [42]); this performance remains probably
the most impressive of this nature. It was not until 1946 that an unfortunate
mistake was discovered at the 528th place [15].
- Carl Louis Ferdinand von Lindemann (1852-1939). p is
transcendental (1882, [32]).
- François Genuys, Daniel Shanks with John Wrench and Jean Guilloud
with Martine Bouyer respectively reached 10,000 (1958, [18]), 100,000
(1961, [43]) and 1,000,000 (1973, [20]) digits by using
arctan formulae and classical series expansion computation.
- Richard Brent and Eugene Salamin. They published in 1976, two important
articles ([8], [39]) describing a new iterative
and quadratic algorithm to determine p. This opened the era of
fast algorithms, that is algorithms with complexity nearly
proportional to the number of computed decimal places.
Other methods of this nature and with higher order of convergence were later
developed by Peter and Jonathan Borwein [6].
- David Chudnovsky and Gregory Chudnovsky. Introduction of new very fast
series (consecutive to Ramanujan's work, [36], [11])
to establish various record on a home made supercomputer m-zero! The
first billion digits was achieved by them in 1989 (see [34]).
- Yasumada Kanada. Since 1980 he is one of the major actor in the race to
compute p to huge number of digits. Most of his calculations are made on
supercomputers and are based on modern high order iterative algorithms (see
[47], [25], [26], [46],...)
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 = Ö{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 | 1 001 250 | arctan(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 | 201 326 551 | GL2,
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],...).
|
| |
= 4arctan |
æ è
|
1
5
|
ö ø
|
-arctan |
æ è
|
1
239
|
ö ø
|
, Machin (M) |
| |
= 8arctan |
æ è
|
1
10
|
ö ø
|
-arctan |
æ è
|
1
239
|
ö ø
|
-4arctan |
æ è
|
1
515
|
ö ø
|
, Klingenstierna (K) |
| |
= arctan |
æ è
|
1
2
|
ö ø
|
+arctan |
æ è
|
1
5
|
ö ø
|
+arctan |
æ è
|
1
8
|
ö ø
|
, Strassnitzky (SD) |
| |
= 12arctan |
æ è
|
1
18
|
ö ø
|
+8arctan |
æ è
|
1
57
|
ö ø
|
-5arctan |
æ è
|
1
239
|
ö ø
|
, Gauss (G) |
| |
= 4arctan |
æ è
|
1
5
|
ö ø
|
-arctan |
æ è
|
1
70
|
ö ø
|
+arctan |
æ è
|
1
99
|
ö ø
|
, Euler (E1) |
| |
= 5arctan |
æ è
|
1
7
|
ö ø
|
+2arctan |
æ è
|
3
79
|
ö ø
|
, Euler (E2) |
| |
= arctan |
æ è
|
1
2
|
ö ø
|
+arctan |
æ è
|
1
3
|
ö ø
|
, Euler (E3) |
| |
= 2arctan |
æ è
|
1
3
|
ö ø
|
+arctan |
æ è
|
1
7
|
ö ø
|
, Hutton (H) |
| |
= 3arctan |
æ è
|
1
4
|
ö ø
|
+arctan |
æ è
|
1
20
|
ö ø
|
+arctan |
æ è
|
1
1985
|
ö ø
|
, Loney (L) |
| |
= 6arctan |
æ è
|
1
8
|
ö ø
|
+2arctan |
æ è
|
1
57
|
ö ø
|
+arctan |
æ è
|
1
239
|
ö ø
|
, Störmer (S1) |
| |
= 12arctan |
æ è
|
1
49
|
ö ø
|
+32arctan |
æ è
|
1
57
|
ö ø
|
-5arctan |
æ è
|
1
239
|
ö ø
|
+12arctan |
æ è
|
1
110443
|
ö ø
|
(S2) |
| |
= 44arctan |
æ è
|
1
57
|
ö ø
|
+7arctan |
æ è
|
1
239
|
ö ø
|
-12arctan |
æ è
|
1
682
|
ö ø
|
+24arctan |
æ è
|
1
12943
|
ö ø
|
(S3) |
|
|
4.2 Other series
|
= |
3Ö3
4
|
+24 |
ó õ
|
1/4
0
|
| Ö
|
x-x2
|
dx |
| |
= |
3Ö3
4
|
+24 |
æ è
|
1
12
|
- |
1
5.25
|
- |
1
28.27
|
- |
1
72.29
|
... |
ö ø
|
, 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],...
|
= |
2Ö2
9801
|
|
¥ å
k=0
|
|
(4k)!
(k!)444k
|
|
(1103+26390k)
994k
|
, Ramanujan [36]
(Ra) |
| |
=12 |
¥ å
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],...
Set x0=1,y0=1/Ö2,a0=1/2 and:
|
ì ï ï í
ï ï î
|
|
|
ak+1=ak-2k+1( xk+12-yk+12) |
|
|
|
|
then ([6], [8], [39]):
p = |
lim
k®¥
|
( 2xk2/ak) . (GL2) |
| |
Set x0=Ö2,y0=0,a0=2+Ö2 and:
|
ì ï ï ï ï í
ï ï ï ï î
|
xk+1= |
æ è
| Ö
|
xk
|
+1/ | Ö
|
xk
|
ö ø
|
/2 |
|
yk+1= | Ö
|
xk
|
|
æ è
|
1+yk
yk+xk
|
ö ø
|
|
|
ak+1=akyk+1 |
æ è
|
1+xk+1
1+yk+1
|
ö ø
|
|
|
|
|
|
then ([6]):
Set y0=Ö2-1,a0=6-4Ö2 and:
ì í
î
|
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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