A cute proof that makes $e$ natural

TL;DR

The paper addresses the gap in intuition around Euler's number by presenting a compact, visual argument that unifies key properties of $e$, notably the equivalence between the limit $\bigl(1+\frac{1}{n}\bigr)^n$ and the fact that $e^x$ is its own derivative. It defines $e$ via a geometric, natural criterion—the unique base for which the exponential curve has slope $1$ at the origin—and then derives $(1+x/n)^n \to e^x$ and $\frac{d}{dx} e^x = e^x$ through a sequence of intuitive steps, including a concrete bridge using $\log_e$ as the inverse of exponentiation. The work also provides a Taylor-series-free route to the same conclusions, along with a calculus-level justification of uniqueness for the differential equation $y' = y$, and it situates the method within a broad survey of curricula and historical context. The contribution is a self-contained, open-access teaching reference that can be incorporated into secondary curricula to promote conceptual understanding of why $e$ is natural and how its defining properties interlock, with broader implications for math education practice.

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 $e$.

Figures (6)

• Figure 1: Commonly taught implications with short and intuitive proofs. The black implications are typically taught to Pre-Calculus students, the blue implications are typically taught to Calculus students, and the red implications are not commonly taught anymore, but were historically justified informally hundreds of years ago (e.g., by Jacob Bernoulli Bernoulli1690 when studying compound interest).
• Figure 2: Any horizontal stretch of an exponential curve produces another positive real base's exponential curve, so there's a unique positive real base whose exponential curve has a tangent line slope of 1 at its $y$-intercept.
• Figure 3: Geometric intuition for why $y = \log_e x$ has tangent slope 1 at $(1, 0)$.
• Figure 4: Derivatives of inverse functions. The right triangles with legs $a$ and 1 are mirror images of each other, and so the slopes of the red tangent line to $y = \log_e x$ and the blue tangent line to $y = e^x$ are reciprocals of each other (the "rise" and "run" switch places). This argument is a particular instance of the general proof that $(f^{-1}(x))' = \frac{1}{f'(f^{-1}(x))}$.
• Figure 5: Napier's construction. Two points move simultaneously, starting from $A$ and $C$ at the same initial speeds. When the point on the top line segment is at position $t$, the point on the bottom ray is also at position $t$.

Theorems & Definitions (16)

• proof
• proof
• proof
• proof : Intuitive "proof."
• proof : Rigorous completion of proof.
• proof : Proof that $(1 + \frac{x}{n})^n \to e^x$.
• Proposition A.1
• proof
• Proposition A.2
• proof