Using Random Numbers to Estimate Pi

A friend recently asked that I post something about the mathematical constant, pi. It is the ratio of a circle's circumference to it diameter. To fifty digits, pi is 

3.14159265358979323846264338327950288419716939937510...

There have been many methods established for calculating pi. One of the more interesting methods is derived from Euler's solution to the zeta function:

Euler showed that this function, in its limit, is:

This result has been used in number theory to establish the probability (P) of two random numbers being relatively prime (having no common factors greater than 1) is approximately:

Now, if we generate a large number of pairs of random numbers, we can count how many are relatively prime and then use the proportion to estimate pi using the formula:

I wrote a Google Sheet to make this approximation using 1000 pairs of random numbers. A link to this sheet is here and you can try for yourself. To refresh or change the 1000 pairs of numbers, simply update a blank cell in the sheet. In my first estimate, there were 612 of the 1000 pairs that were relatively prime yielding a proportion, P, of 0.612. This gave an approximation of pi of 3.13.

Pi Approximation - Google Sheets