In this post I will present a beautiful proof owing to Ivan Niven on the irrationality of . An analytical definition of is that is twice the smallest positive for which equals . (This definition is better then the usual circumference/diameter one since it is not dependent on Euclidean axioms).
Theorem (Niven): is irrational.
Proof: Let, by way of contradiction with . Now let
where we will specify later. Now with and so . Not only this, for . For we have with and so is integral in this case as well. Also note that and so for any we have following which i.e. is also integral.
Now it follows by elementary calculus that and so . Now is integral because and are both integers for any . It is also easy to see that for we have and hence . But now and as grows faster then any exponential so this upper bound for the integral can be made arbitrarily small. Hence the value of the integral lies between and . This is the contradiction which proves the result.
One final word. Even though the proof is credited to Ivan Niven, it is very much possible that the idea was given by Hermite much earlier and all did Niven was to tweak Hermite’s idea to give this proof.