Table of Contents:
The Bridges of Konigsberg
· Euler's Solution
· Solution, problem 1
· Solution, problem 2
· Solution, problem 4
· Solution, problem 5
The Value of Pi
· A Chronological Table of Values
· Squaring the Circle
Prime Numbers
· Finding Prime Numbers
Famous Paradoxes
· Zeno's Paradox
· Cantor's Infinities
· Cantor's Infinities, Page 2
The Problem of Points
· Pascal's Generalization
· Summary and Problems
· Solution, Problem 1
· Solution, Problem 2
Proof of the Pythagorean Theorem
| Person/People | Year | Value |
| Babylonians | ~2000 B.C. | 3 1/8 |
| Egyptians | ~2000 B.C. | (16/9)^2= 3.1605 |
| Chinese | ~1200 B.C. | 3 |
| Old Testament | ~550 B.C. | 3 |
| Archimedes | ~300 B.C. | proves 3 10/71<Pi<3 1/7 uses 211875/67441=3.14163 |
| Ptolemy | ~200 A.D. | 377/120=3.14166... |
| Chung Huing | ~300 A.D. | sqrt(10)=3.16... |
| Wang Fau | 263 A.D. | 157/50=3.14 |
| Tsu Chung-Chi | ~500 A.D. | proves 3.1415926<Pi<3.1415929 |
| Aryabhatta | ~500 | 3.1416 |
| Brahmagupta | ~600 | sqrt(10) |
| Fibonacci | 1220 | 3.141818 |
| Ludolph van Ceulen | 1596 | Calculates Pi to 35 decimal places |
| Machin | 1706 | 100 decimal places |
| Lambert | 1766 | Proves Pi is irrational |
| Richter | 1855 | 500 decimal places |
| Lindeman | 1882 | Proves Pi is transcendental |
| Ferguson | 1947 | 808 decimal places |
| Pegasus Computer | 1957 | 7,840 decimal places |
| IBM 7090 | 1961 | 100,000 decimal places |
| CDC 6600 | 1967 | 500,000 decimal places |
So what happened between 1220 and 1596? Well, in the late 15th century European mathematicians (benefiting from the greater intellectual freedom that came with the end of the Middle Ages) figured out how to express Pi exactly as an infinite product. This facilitated the computation of much better approximations. As time passed, mathematicians made the expressions for Pi as an infinite product or sum more concise, and computational methods improved as well.
For example: François Viète (1540-1603) determined that:
John Wallis (1616-1703) showed that:
While Euler (1707-1783) derived his famous formula:
Today Pi is known to more than 10 billion decimal places.
to Finding the Value of Pi
to Java Applet on Squaring the Circle
