There are many cool identities involving the number . (Proof: trivially follows from looking at wikipedia page on ). Now, with more serious tone, I think the following is quite remarkable:
This is known as Machin’s formula. Apparently, this identity (and its variations) provides a very fast algorithm to calculate many digits of . There are probably multiple way of arriving at Machin’s formula (using tangent addition formula and such). There is a really nice approach outlined as exercise in Fundamentals of Complex Analysis by Saff and Snider, which will be the content of this blogpost.
Proof of Machin’s formula: Let’s take a look at the argument of the complex number in two different ways. Since the argument of a product is the sum of the arguments, we get
Now, we will calculate argument differently, namely, using brute force to expand the parenthesis. Okay here we go
which we can multiply out to get
But now, argument of can be readily computed:
Now combining with the other expression for argument of , we deduce that
which can be rearranged to give Machin’s formula: