Owen MareshApr 16, 2025

(this is a very likely sigbovik.org paper)

https://arxiv.org/abs/2504.10664
/A cute proof that makes e natural/
Po-Shen Loh

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 (1+1n)n to the fact that ex 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 cute proof that makes $e$ natural

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 $e$.

arXiv.org
Show threadtheHigherGeometerApr 16, 2025

@graveolensa "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." he he he.

EDIT: actually, this is serious. And I think the paper is really interesting from a teaching point of view.

Show threadOwen Maresh
@highergeometer it doesn't look like there are #sigbovik proceedings yet (http://sigbovik.org)
The Association for Computational Heresy

Apr 16, 2025 at 5:39amWeb
Show threadtheHigherGeometerApr 16, 2025
@graveolensa Actually, I think the paper is more serious than that. It doesn't mean 'natural' as in 'natural number', but more in the sense of 'inevitable'. The main section with the pedagogical discussion about introducing the facts related to e seems quite reasonable for high-school students.
Show threadOwen MareshApr 16, 2025
@highergeometer it reads like a sigbovik paper, it's april, and Po Shen-Lo is at CMU. Look at the references, there is an absurdity to going to the effort to get non-latin scripts correct? We'll see when the proceedings come out.