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5.3.4. Pi (π ≈ 3.1416)
Few people would doubt that pi (π) is one of the most important numbers, if not the most important number, in mathematics. Pi is defined as the ratio of the circumference to the diameter of a circle. Its numerical value to 10 decimal places is 3.1415926536. Pi is the most precisely-known irrational number. In 2002, a team led by Yasumasa Kanada computed the value of pi to 1,241,100,000,000 decimal places. That's enough to fill a library with over a million medium-sided books! [1]
So why is π such an important number? Part of the reason is that since antiquity, people have been fascinated with circles. A circle, after all, is the most symmetric possible two-dimensional geometric figure, having complete rotational symmetry. Circles appear all over the place, from atoms, to wheels, to balls, to planets, to stars, to planetary orbits. Pi not only occurs in the formula for the circumference of a circle, but also in the formula for its area. Furthermore, pi occurs in the formulas for the surface area and volume of a sphere.
Below are a few important formulas involving pi:
- (5.3.4.1) C = πd = 2πr (circumference of a circle in terms of its diameter or radius)
- (5.3.4.2) A = πr2 (area of a circle in terms of its radius)
- (5.3.4.3) S = 4πr2 (surface area of a sphere in terms of its radius)
- (5.3.4.4) V = 4πr3/3 (volume of a sphere in terms of its radius)
Since pi is defined in terms of a circle, it may seem that this number is only used in connection with circles and spheres. Surprisingly, this is not at all the case. Pi occurs in numerous formulas throughout math and science, most of them having nothing to do with circles. For instance, pi is used in statistics, in particular, in the formula for the normal distribution. Pi also occurs frequently in connection with trigonometric functions, which although they can be defined in terms of circles, are usually used in applications not directly involving them.
Although there is no practical reason for having to know pi to more than 40 digits or so, mathematicians have long been fascinated with the challenge of computing it to as many digits as possible. There are several reasons for this. Mainly, the challenge is akin to climbing Mount Everest or setting an air speed record - the challenge is simply to do it because it's there. There are other more practical reasons as well. For instance, the accuracy of computer software can be tested by computing pi to a large number of digits, since it's value is known and can be checked. Also, mathematicians are interested statistical properties of the decimal expansion of pi, such as the distribution of its digits.
The following table shows the history of some records in the computation of pi. All records from 1946 onward were performed with the aid of a computer. [1, 2]
Year |
Discoverer |
number of digits |
| 250? BC | Archimedes | 3 |
| 263 | Lui Hui | 5 |
| 480? | Tsu Ch'ung Chi | 7 |
| 1429 | Al-Kashi | 14 |
| 1593 | Romanus | 15 |
| 1596 | Van Ceulen | 20 |
| 1615 | Van Ceulen | 35 |
| 1699 | Sharp | 71 |
| 1706 | Machin | 100 |
| 1719 | De Lagni | 127 (112 correct) |
| 1794 | Vega | 140 |
| 1824 | Rutherford | 208 (152 correct) |
| 1844 | Strassnitzky and Dase | 200 |
| 1847 | Claussen | 248 |
| 1853 | Lehmann | 261 |
| 1853 | Rutherford | 440 |
| 1874 | Shanks | 707 (527 correct) |
| 1946 | Ferguson | 620 |
| 1949 | Smith and Wrench | 1120 |
| 1949 | Reitwiesner et. al. | 2037 |
| 1954 | Nicholson and Jeenel | 3092 |
| 1957 | Felton | 7480 |
| 1958 | Genuys | 10,000 |
| 1959 | Guilloud | 16,167 |
| 1961 | Shanks and Wrench | 100,265 |
| 1966 | Guilloud and Filliatre | 250,000 |
| 1967 | Guilloud and Dichampt | 500,000 |
| 1973 | Guilloud and Bouyer | 1,001,250 |
| 1981 | Miyoshi and Kanada | 2,000,036 |
| 1982 | Kanada, Yoshimo, and Tamura | 16,777,206 |
| 1985 | Gosper | 17,526,200 |
| 1986 | Bailey | 29,360,111 |
| 1994 | Chudnovskys | 4,044,000,000 |
| 1999 | Kanada and Takahashi | 206,158,430,000 |
| 2002 | Kanada, Ushiro, and Koroda | 1,241,100,000,000 |
Table 5.3.4.5: History of Some Computational Records of Pi
References:
- http://oldweb.cecm.sfu.ca/projects/ISC/Pihistory.html (history of the computation of pi)
- http://www.cecm.sfu.ca/~jborwein/pi-slides.pdf (The Life of Pi: History and Computation)
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