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علماء الرياضيات | A chronology of pi |
 615
 06:55 مساءاً
date: |
Author : D H Bailey, J M Borwein, P B Borwein, and S Plouffle 
Book or Source : The quest for Pi 
Page and Part : ...|
Read More
Date: 11-10-2015
909
Date: 8-10-2015
1312
Date: 14-10-2015
894
|
Pre computer calculations of π
|
Mathematician |
Date |
Places |
Comments |
|
|
|
1 |
Rhind papyrus |
2000 BC |
1 |
3.16045 (= 4(8/9)2) |
|
|
2 |
Archimedes |
250 BC |
3 |
3.1418 (average of the bounds) |
|
|
3 |
Vitruvius |
20 BC |
1 |
3.125 (= 25/8) |
|
|
4 |
Chang Hong |
130 |
1 |
3.1622 (= √10) |
|
|
5 |
Ptolemy |
150 |
3 |
3.14166 |
|
|
6 |
Wang Fan |
250 |
1 |
3.155555 (= 142/45) |
|
|
7 |
Liu Hui |
263 |
5 |
3.14159 |
|
|
8, |
Zu Chongzhi |
480 |
7 |
3.141592920 (= 355/113) |
|
|
9 |
Aryabhata |
499 |
4 |
3.1416 (= 62832/20000) |
|
|
10 |
Brahmagupta |
640 |
1 |
3.1622 (= √10) |
|
|
11 |
Al-Khwarizmi |
800 |
4 |
3.1416 |
|
|
12 |
Fibonacci |
1220 |
3 |
3.141818 |
|
|
13 |
Madhava |
1400 |
11 |
3.14159265359 |
|
|
14 |
Al-Kashi |
1430 |
14 |
3.14159265358979 |
|
|
15 |
Otho |
1573 |
6 |
3.1415929 |
|
|
16 |
Viète |
1593 |
9 |
3.1415926536 |
|
|
17 |
Romanus |
1593 |
15 |
3.141592653589793 |
|
|
18 |
Van Ceulen |
1596 |
20 |
3.14159265358979323846 |
|
|
19 |
Van Ceulen |
1596 |
35 |
3.1415926535897932384626433832795029 |
|
|
20 |
Newton |
1665 |
16 |
3.1415926535897932 |
|
|
21 |
Sharp |
1699 |
71 |
|
|
|
22 |
Seki Kowa |
1700 |
10 |
||
|
23 |
Kamata |
1730 |
25 |
||
|
24 |
Machin |
1706 |
100 |
|
|
|
25 |
De Lagny |
1719 |
127 |
Only 112 correct |
|
|
26 |
Takebe |
1723 |
41 |
|
|
|
27 |
Matsunaga |
1739 |
50 |
|
|
|
28 |
von Vega |
1794 |
140 |
Only 136 correct |
|
|
29 |
Rutherford |
1824 |
208 |
Only 152 correct |
|
|
30 |
Strassnitzky, Dase |
1844 |
200 |
|
|
|
31 |
Clausen |
1847 |
248 |
|
|
|
32 |
Lehmann |
1853 |
261 |
|
|
|
33 |
Rutherford |
1853 |
440 |
|
|
|
34 |
Shanks |
1874 |
707 |
Only 527 correct |
|
|
35 |
Ferguson |
1946 |
620 |
|
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
|
Mathematician |
Date |
Places |
Type of computer |
||
|
Ferguson |
Jan 1947 |
710 |
Desk calculator |
||
|
Ferguson, Wrench |
Sept 1947 |
808 |
Desk calculator |
||
|
Smith, Wrench |
1949 |
1120 |
Desk calculator |
||
|
Reitwiesner et al. |
1949 |
2037 |
ENIAC |
||
|
Nicholson, Jeenel |
1954 |
3092 |
NORAC |
||
|
Felton |
1957 |
7480 |
PEGASUS |
||
|
Genuys |
Jan 1958 |
10000 |
IBM 704 |
||
|
Felton |
May 1958 |
10021 |
PEGASUS |
||
|
Guilloud |
1959 |
16167 |
IBM 704 |
||
|
Shanks, Wrench |
1961 |
100265 |
IBM 7090 |
||
|
Guilloud, Filliatre |
1966 |
250000 |
IBM 7030 |
||
|
Guilloud, Dichampt |
1967 |
500000 |
CDC 6600 |
||
|
Guilloud, Bouyer |
1973 |
1001250 |
CDC 7600 |
||
|
Miyoshi, Kanada |
1981 |
2000036 |
FACOM M-200 |
||
|
Guilloud |
1982 |
2000050 |
|||
|
Tamura |
1982 |
2097144 |
MELCOM 900II |
||
|
Tamura, Kanada |
1982 |
4194288 |
HITACHI M-280H |
||
|
Tamura, Kanada |
1982 |
8388576 |
HITACHI M-280H |
||
|
Kanada, Yoshino, Tamura |
1982 |
16777206 |
HITACHI M-280H |
||
|
Ushiro, Kanada |
Oct 1983 |
10013395 |
HITACHI S-810/20 |
||
|
Gosper |
Oct 1985 |
17526200 |
SYMBOLICS 3670 |
||
|
Bailey |
Jan 1986 |
29360111 |
CRAY-2 |
||
|
Kanada, Tamura |
Sept 1986 |
33554414 |
HITACHI S-810/20 |
||
|
Kanada, Tamura |
Oct 1986 |
67108839 |
HITACHI S-810/20 |
||
|
Kanada, Tamura, Kubo |
Jan 1987 |
134217700 |
NEC SX-2 |
||
|
Kanada, Tamura |
Jan 1988 |
201326551 |
HITACHI S-820/80 |
||
|
Chudnovskys |
May 1989 |
480000000 |
|||
|
Chudnovskys |
June 1989 |
525229270 |
|||
|
Kanada, Tamura |
July 1989 |
536870898 |
|||
|
Chudnovskys |
Aug 1989 |
1011196691 |
|||
|
Kanada, Tamura |
Nov 1989 |
1073741799 |
|||
|
Chudnovskys |
Aug 1991 |
2260000000 |
|||
|
Chudnovskys |
May 1994 |
4044000000 |
|||
|
Kanada, Tamura |
June 1995 |
3221225466 |
|||
|
Kanada |
Aug 1995 |
4294967286 |
|||
|
Kanada |
Oct 1995 |
6442450938 |
|||
|
Kanada, Takahashi |
Aug 1997 |
51539600000 |
HITACHI SR2201 |
||
|
Kanada, Takahashi |
Sept 1999 |
206158430000 |
HITACHI 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 nth hexadecimal digit of π to be computed without the preceeding n- 1 digits.
C. Plouffe discovered a new algorithm to compute the nth digit of π in any base in 1997.
.D H Bailey, J M Borwein, P B Borwein, and S Plouffle, The quest for Pi, The Mathematical Intelligencer 19 (1997), 50-5
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