Exercise 2175Number theory › IrrationalityM1Added 3 Nov 2026

Euler's number and transcendence

#e#transcendence#Euler

Statement

State whether $e$ is algebraic or transcendental, and outline Hermite's proof.

Hint

Hermite proved $e$ is transcendental by contradiction: assume $e$ is a root of a polynomial with integer coefficients, then derive a contradiction from the series expansion.

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Solution

Statement. $e$ is transcendental.

Sketch of Hermite's proof (1873). Suppose $\sum_{k=0}^n a_k e^k=0$ with $a_k\in\mathbb{Z}$ , $a_0a_n\neq0$ . Multiply by a carefully chosen integer to isolate a small positive quantity that is also a positive integer, contradiction.

Conclusion. $e$ is transcendental (not just irrational).

Significance. This was the first proof that a naturally occurring constant is transcendental, and inspired Lindemann's proof for $\pi$ .

Euler's number and transcendence

Statement

Solution

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