A well-motivated proof that pi is irrational (2403.20140v1)

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.

• The paper presents an innovative approach that leverages orthogonal polynomials to motivate Niven's integral proof of π's irrationality.
• It reinterprets classic methods by connecting Taylor series and Legendre polynomials, enhancing intuitive understanding of the proof.
• The work fosters accessible pedagogical techniques and suggests analogous strategies for tackling other problems in number theory and analysis.

Insights into a Proof of the Irrationality of Pi

Timothy Y. Chow's expository work on the irrationality of π provides an insightful perspective aimed at demystifying Ivan Niven's concise yet seemingly 'magical' proof. Despite the well-known result of π's irrationality, Niven's proof, while short and precise, begins with an integral that appears to lack intuitive grounding. Chow's paper addresses this by leveraging orthogonal polynomials, thereby providing a clearer pathway for understanding and reconstructing the proof, even by mathematicians who do not consider themselves to be particularly brilliant.

Overview of the Approach

The primary innovation in Chow's exposition is the use of orthogonal polynomials to motivate the integral form used in Niven's proof. Orthogonal polynomials, which are a well-established tool in mathematical analysis and approximation theory, offer a structured approach to deriving rapidly converging approximations of transcendental numbers. Chow articulates a thought process that potentially leads a mathematician to discover the proof of π's irrationality without the need for ad hoc insight or genius-level deduction.

Structure of the Proof

Chow begins by illustrating the general philosophy behind irrationality proofs using the irrationality of $e$ demonstrated through its Taylor series expansion. The proof hinges on the rapid convergence of the series, which when manipulated correctly, leads to a contradiction under the assumption that $e$ is rational. Extending this philosophy to orthogonal polynomials, Chow introduces Legendre and other families of polynomials to provide alternative bases for expansions, which facilitate the integral expressions necessary for the irrationality arguments.

Chow’s exposition proceeds through the following stages:

1. Motivation and Examples: The paper begins by discussing the necessity of motivated proofs, contrasting concise proofs that are hard to internalize with verbose proofs that may be more intuitive.
2. Use of Legendre Polynomials: By expanding the exponential function $e^x$ in terms of Legendre polynomials, Chow shows how to construct an integral with smaller bounds, thus reinforcing the irrationality argument.
3. Transposing the Idea to Sine Functions: To address π specifically, the paper modifies the earlier exponential arguments to apply to trigonometric functions, in particular, using the sine series expansion.

Implications and Future Directions

The approach does not only illuminate the nuances of proving π's irrationality, but also exemplifies a methodological pathway that could be applicable to other similar problems in number theory and analysis. The stronger understanding facilitated by the orthogonal polynomial perspective might inspire analogous strategies in other domains where irrationality or transcendence must be established.

While the connection between orthogonal polynomials and irrationality is not unexpected given the historical context of tools like continued fractions and Padé approximants in transcendental number theory, Chow's exposition makes a compelling case for their utility in educational and expository settings. This could potentially pave the way for more accessible pedagogical approaches in advanced mathematical topics.

Conclusion

Timothy Y. Chow's paper stands as a significant contribution to mathematical exposition by demystifying a classic proof using standard tools in analysis. By connecting the integral approach in Niven's proof to the broader framework of orthogonal polynomials, it offers a motivated understanding that can be taught and communicated more effectively. For mathematicians and educators, this paper not only provides clarity on an important proof but also exemplifies the power of orthogonal polynomials in constructing rigorous mathematical arguments.