Today I learned that the Indian astronomer and mathematician Aryabhata found an incredibly close approximation to pi as the ratio of 62832/20000 which he described as:
“Add four to 100, multiply by eight, and then add 62,000. By this rule, the circumference of a circle with a diameter of 20,000 can be approached.”
This gives an approximation of pi as 3.1416... really darn amazing.
As a modern student of physics (raised with a base-10 numbering system), this ratio and calculation doesn't make much sense. I'm sure it must have something to do with the numbering system of the time and several other factors of which I'm unaware, but I don't know what they might be.
It's a fantastic approximation... but can someone PLEASE explain that approach to me???
Noting that 62832/20000 is the same as 3927/1250 found by 3rd century Chinese mathematician Liu Hui (劉徽), it's possible that Aryabhata's method was the same.
Generally, early approximations of pi were found by finding the circumferences or areas of inscribed or circumscribed regular polygons - the largest regular polygon that would fit inside a circle, or the smallest one that would fit outside. For example, the two can be done at once to obtain an upper and lower bound for pi.
Liu Hui's method was to find the area for an inscribed n-gon (i.e., a polygon with n sides), and a (2n)-gon. The 2n-gon area provides a better approximation of pi. The key step towards getting a more accurate result was to use the difference between those two areas D_2n = A_2n - An. The "magic" trick is that A_2n + D_2n/3 is a good approximation for pi.
For n = 96, we have:
A_96 = 196209/62500
A_192 = 196314/62500
D_192 = 105/62500
pi = approx A_192 + D_192/3 = 196349/62500
Noting that 62500 is divisible by 50, and 196349 is almost divisible by 50, we can add 1 to the numerator, and divide both the numerator and denominator by 50, and have a simpler rational approximation of pi that is still very accurate:
We don't know if Liu Hui made this last approximation, so as to be able to divide by 50, but it does logically link his algorithm and the final ratio. The error in 3927/1250 is almost the same magnitude as the error in 196349/62500 (an overestimate rather than an underestimate, by almost the same amount), so this is a convenient simplification.
Some of the calculations can be simplified by choosing an appropriate radius for the circle. The above results are for a unit circle (r=1), but if r=10 was used, A_96 = 196209/625, etc. It is possible that Aryabhata numbers result from a different choice of radius, but it's impossible to be sure, if there was a final simplification of the ratio. It is more likely that a numerator of 20,000 was chose because it is easy to divide by - the equivalent modern choice would be 10,000, for pi = approx 31416/10000.
Fifth century Chinese mathematician Zu Chongzhi (祖沖之) used essentially the same method, going to larger n-gons, and obtained the most excellent approximation 355/113 - the error in this simpler ratio is about 27 times smaller than the error in 3927/1250.