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A well-motivated proof that pi is irrational

Timothy Y. Chow

Abstract

Ivan Niven's succinct proof that pi is irrational is easy to verify, but it begins with a magical formula that appears to come out of nowhere, and whose origin remains mysterious even after one goes through the proof. The goal of this expository paper is to describe a thought process by which a mathematician might come up with the proof from scratch, without having to be a genius. Compared to previous expositions of Niven's proof, perhaps the main novelty in the present account is an explicit appeal to the theory of orthogonal polynomials, which leads naturally to the consideration of certain integrals whose relevance is otherwise not immediately obvious.

A well-motivated proof that pi is irrational

Abstract

Ivan Niven's succinct proof that pi is irrational is easy to verify, but it begins with a magical formula that appears to come out of nowhere, and whose origin remains mysterious even after one goes through the proof. The goal of this expository paper is to describe a thought process by which a mathematician might come up with the proof from scratch, without having to be a genius. Compared to previous expositions of Niven's proof, perhaps the main novelty in the present account is an explicit appeal to the theory of orthogonal polynomials, which leads naturally to the consideration of certain integrals whose relevance is otherwise not immediately obvious.
Paper Structure (9 sections, 21 equations)

This paper contains 9 sections, 21 equations.

Table of Contents

  1. Introduction
  2. The fundamental theorem of transcendental number theory
  3. The irrationality of $e^r$
  4. Orthogonal polynomials to the rescue
  5. Legendre polynomials
  6. A shorter and more elementary proof
  7. On to the irrationality of $\pi$
  8. Cheat sheet
  9. Historical remarks and acknowledgments