So does anyone want to see a proof of the irrationality of pi?

(The one proof that I've read requires some basic differential calculus in order to follow it.)

hit us up brah.

OK! I'll try to go step by step through a proof that pi is irrational.

I'm taking this proof from

*An Introduction to Classical Real Analysis* by Karl Stromberg (1981). He attributes the proof to Ivan Niven, and his bibliography includes a 1956 monograph by Niven called

*Irrational Numbers*, so I assume he got it from there.

The method of proof is "proof by contradiction". That is, we'll start by assuming that pi is rational, then without making any other questionable assumptions, we'll derive something impossible from that premise.

So. First step: Assume that

*π* is rational. Then

*π*^{ 2} is rational. And we can find natural numbers (i.e., positive integers)

*a* and

*b* such that

*π*^{ 2} =

*a*/

*b*.

Next, let

*N* be a natural number that is big enough so that

*a*^{ N}/

*N*! < 1/

*π*.

Why can we do this? Because as

*N* grows,

*N*! ultimately grows faster than

*a*^{ N}. So we can make

*a*^{ N}/

*N*! arbitrarily small by picking a big enough value of

*N*.

(In fact, you can make an infinite series, summing the terms

*a*^{ n}/

*n*! as

*n* goes from 0 to infinity. This series converges to

*e*^{ a}, which is a real number for any real

*a*. In order for that convergence to come about, the individual terms have to get arbitrarily small. If anyone is unhappy with this part, we can have a little(?) digression about infinite series.)