Finding the Digits

How Do You Find the Digits of π?

You might be tempted to think that finding the digits of π has an obvious solution: just measure it! Well, let's try it. Get out a string or measuring tape of somekind, and using an actual circle, find the circumference and divide it by the diameter.

Experimental Results

You will definitely be able to tell that the ratio is a little more than 3. But how much more? If you make really good measurements, you might be able to find that it's about an extra tenth, but you would be hard pressed to get a better measurement than that. So, how is it that we have such an accurate representation of π?

Methods of Approximation

1)Archimedes method involving polygons:

Archimedes (287-212 BCE) came up with the approximation for π of 22/7 back in a time before computers or calculators or calculus. These were simple times, mathematically speaking, so how did he manage it?

He found this fractional approximation by inscribing a circle inside a 96 sided polygon and another 96 sided polygon within the circle. A polygon with this many sides looks very much like a circle, but it is easier to deal with. The perimeter and area of these figures are very easy to find, since they are basically made up of a bunch of identical triangles.

By finding the perimeter of each polygon and comparing it to their respective areas, Archimedes could put very accurate bounds on the ratio of π. Although he hadn't found it exactly, he pinned it down to a very small window.

2) Probabilistic model:

Randomly drop points in a unit square with a quarter of a circle of radius 1 inscribed in it. Then take 4*number of points in the circle section/total points dropped to get π. This method can either be done physically or simulated by a computer.

3)Buffon’s needle problem:

Drop needles on a flat piece of paper with parallel lines on it and count how many of the needles cross any parallel line on the paper. Then take:

(2 * #drops * needle length) / (#hits * distance between the parallel lines)

to get π.

4) Bounding Power Series:

Use two power series approximations of pi (a type of function) that are very close together, but one is an upperbound and one is a lowerbound. So as long as they agree, we can be certain that the digit of pi is correct.