A cute proof that makes $e$ natural (2504.10664v3)

Abstract: The number $e$ has rich connections throughout mathematics, and has the honor of being the base of the natural logarithm. However, most students finish secondary school (and even university) without suitably memorable intuition for why $e$'s various mathematical properties are related. This article presents a solution. Various proofs for all of the mathematical facts in this article have been well-known for years. This exposition contributes a short, conceptual, intuitive, and visual proof (comprehensible to Pre-Calculus students) of the equivalence of two of the most commonly-known properties of $e$, connecting the continuously-compounded-interest limit $\big(1 + \frac{1}{n}\big)^n$ to the fact that $e^x$ is its own derivative. The exposition further deduces a host of commonly-taught properties of $e$, while minimizing pre-requisite knowledge, so that this article can be practically used for developing secondary school curricula. Since $e$ is such a well-trodden concept, it is hard to imagine that our visual proof is new, but it certainly is not widely known. The author checked 100 books across 7 countries, as well as YouTube videos totaling over 25 million views, and still has not found this method taught anywhere. This article seeks to popularize the 3-page explanation of $e$, while providing a unified, practical, and open-access reference for teaching about $e$.

A Conceptual and Intuitive Approach to Understanding Euler's Number $e$

The paper by Po-Shen Loh addresses a common issue in mathematical education — the lack of intuitive understanding among students for why the mathematical properties of Euler's number $e$ are related. Despite its fundamental importance across various sectors of mathematics and science, $e$ often remains an enigmatic concept to learners. The author proposes a novel, visually intuitive proof aimed at making the properties of $e$ accessible and interconnected to secondary school students and educators. This approach emphasizes the naturalness of $e$ through both its geometric and algebraic properties.

Overview of the Core Contributions

The central contribution of this paper is the exposition of a concise, three-page approach that explains the equivalence of some of the most well-known properties of $e$. This proof is presented in a manner that is digestible to students with a pre-calculus level of mathematical maturity, avoiding the sometimes overwhelming prerequisites of calculus. The key focus is on two primary phenomena related to $e$:

1. Continuously Compounded Interest: $e$ emerges as the limit of the expression $\big(1 + \frac{1}{n}\big)^n$ as $n$ approaches infinity.
2. Derivative Property: The function $e^x$ is uniquely significant in that it is its own derivative, meaning it describes the simplest form of growth where the rate of increase is directly proportional to current value.

Loh's approach is rooted in connecting these seemingly disparate properties using geometric intuition and minimal algebraic prerequisites. By starting with the tangent line slope of exponential functions, the exposition effectively bridges to continuously compounded interest, thus providing a cohesive understanding of $e$.

Detailed Examination of Key Insights

The author begins by redefining $e$ as the unique base of the exponential function whose tangent line at $x = 0$ has a slope of 1. This geometric perspective is then utilized to infer the limit definition by showing that the slope property implies the limit of the sequence $\big(1 + \frac{1}{n}\big)^n$ converges to $e$. This provides a solid groundwork for connecting the derivative property to its algebraic definition via calculus, all without stepping far into rigorous limits or differential equations which are often employed to justify properties of $e$.

A crucial aspect of this argument involves reflecting on the role of the natural logarithm as the inverse of the exponential function, particularly how both concepts can be seen as inherently natural from geometric considerations. The paper provides a visual argument, which, together with elementary calculus reasoning, ties these critical mathematical facts together seamlessly.

Practical and Educational Implications

This paper advocates for inserting such a visual and intuitive exposition into secondary school curricula, allowing it to serve as a foundational resource for educators. It also points to the gap present in current textbooks worldwide regarding intuitive explanations of $e$, urging educational materials to adopt and integrate these insights to enhance understanding and awareness.

By stripping away sophisticated calculus prerequisites, the author suggests that this approach not only demystifies $e$, making it accessible to a broader audience, but also aligns with teaching goals focused on promoting logical reasoning and curiosity-driven exploration in mathematics.

Speculation on Future Directions

While the paper provides a practical methodological contribution to mathematical pedagogy, it also hints at broader philosophical questions about the teaching of mathematics, particularly abstract concepts like $e$. The goal might be to inspire educators to use geometric and intuitive proofs to clarify other essential mathematical ideas traditionally viewed as complex or abstract.

Going forward, further research might explore how such proofs can be extended or adapted to other areas of mathematics or how educational strategies could evolve to prioritize an intuitive understanding of abstract concepts beyond $e$. This could lead to a shift in educational paradigms from rote memorization to one centered around conceptual clarity and logical interconnections.

In conclusion, Po-Shen Loh's work offers a refreshingly clear and conceptually rich pathway to understanding $e$, emphasizing a tangible and interconnected approach that could revitalize mathematical teaching and comprehension significantly.