A Proof of Irrationality of $\pi$ by Contraposition

TL;DR

This paper presents a novel proof of the irrationality of pi by contraposition, building on Niven's proof, and also offers similar proofs for the irrationality of sqrt(2) and sqrt(3).

Contribution

It introduces a new proof technique for irrationality using contraposition, extending classical proofs to a different logical framework.

Findings

Proof of irrationality of pi via contraposition.

Proofs of irrationality for sqrt(2) and sqrt(3).

Extension of classical proofs using a new logical approach.

Abstract

We point out that the proof of irrationality of $\pi$ by Niven can be modified to a proof by contraposition. As a warm-up, we also give a proof of irrationality of $\sqrt{2}$ and $\sqrt{3}$ in a similar way.