The Pi story
Pi (π) has been known for almost 4000 years, but even if we calculated number of seconds in those 4000 years and calculated π to that number of places, we would still only be approximating its actual value.
Ancient Babylonians calculated area of a circle by taking 3 times square of its radius, which gave a value of pi = 3. One Babylonian tablet (1900–1680 BC) indicates a value of 3.125 for Ï€, which is a closer approximation.
Rhind Papyrus (1650 BC) gives us insight into mathematics of ancient Egypt. Egyptians calculated area of a circle by a formula that gave the approximate value of 3.1605 for π.
First calculation of Ï€ was done by Archimedes of Syracuse (287–212 BC), one of greatest mathematicians of the ancient world. Archimedes approximated area of a circle by using Pythagorean Theorem to find areas of two regular polygons: polygon inscribed within circle and polygon within which circle was circumscribed. Since actual area of circle lies between the areas of inscribed and circumscribed polygons, areas of polygons gave upper and lower bounds for area of circle. Archimedes knew that he had not found value of Ï€ but only an approximation within those limits. In this way, Archimedes showed that Ï€ is between 3 1/7 and 3 10/71.
A similar approach was used by Zu Chongzhi (429–501 CE), a brilliant Chinese mathematician and astronomer. Zu Chongzhi would not have been familiar with Archimedes’ method, but because his book has been lost, little is known of his work. He calculated value of ratio of circumference of a circle to its diameter to be 355/113. To compute this accuracy for Ï€, he must have started with an inscribed regular 24,576-gon and performed lengthy calculations involving hundreds of square roots carried out to 9 decimal places.
