The Irrationality of e and π

2026/07/19 by L. Lerner · 1 voice
#math.HO

paper · pdf

Abstract

Unlike an ordinary fraction with an infinite numerator and denominator, an infinite continued fraction must be irrational. Euler was the first to show that the base of the natural logarithm e is irrational, by numerically estimating its continued fraction, and showing the infinite sequence of convergents p_n/q_n so obtained converged to e using the Ricatti equation. Hermite showed that the recurrence relations for these convergents correspond to the recurrence relations between certain improper integrals, so proving the continued fraction tends to e in the limit of infinite n. Here we provide a motivation for the integrals involved and obtain closed form integral representations for p_n and q_n for all n. An interesting feature, is that the above results provide a short cut to the standard proof of the irrationality of π.

Discussions

• L. Lerner: The Irrationality of $e$ and $\pi$ https://arxiv.org/abs/2607.19418 https://arxiv.org/pdf/2607.19418 https://arxiv.org/html/2607.19418 [bsky, 0 points, 0 comments]

Related

• On the Connection Between Irrationality Measures and Polynomial Continued Fractions
• Continued fractions of cubic irrationalities
• Rational function approximations of the special function e^xE₁(x) and applications to irrationality of Euler-Gompertz constant δ
• Irrationality and Transcendence of Alternating Series Via Continued\n Fractions
• Irrationality exponents of semi-regular continued fractions