Celebrate Pi Approximation Day 22/7

[QE]

Everyone should have this on the wall: http://unihedron.com/projects/pi/index.php
Or if you want to DIY:
http://unihedron.com/projects/pi/pi.pdf
http://unihedron.com/projects/pi/downloads/pi_purple.pdf

 

[Dr.Brain]

Aryabhata gave an accurate approximation for p. He wrote in the Aryabhatiya the following:-

Add four to one hundred, multiply by eight and then add sixty-two thousand. the result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given.

This gives p = 62832/20000 = 3.1416 which is a surprisingly accurate value. In fact p = 3.14159265 correct to 8 places. If obtaining a value this accurate is surprising, it is perhaps even more surprising that Aryabhata does not use his accurate value for p but prefers to use 10 = 3.1622 in practice. Aryabhata does not explain how he found this accurate value but, for example, Ahmad [5] considers this value as an approximation to half the perimeter of a regular polygon of 256 sides inscribed in the unit circle. However, in [9] Bruins shows that this result cannot be obtained from the doubling of the number of sides. Another interesting paper discussing this accurate value of p by Aryabhata is [22] where Jha writes:-

Aryabhata I's value of p is a very close approximation to the modern value and the most accurate among those of the ancients. There are reasons to believe that Aryabhata devised a particular method for finding this value. It is shown with sufficient grounds that Aryabhata himself used it, and several later Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata's value of p is of Greek origin is critically examined and is found to be without foundation. Aryabhata discovered this value independently and also realized that p is an irrational number. He had the Indian background, no doubt, but excelled all his predecessors in evaluating p. Thus the credit of discovering this exact value of p may be ascribed to the celebrated mathematician, Aryabhata I.




Source: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Aryabhata_I.html