Hurwitz's Theorem: Optimal Rational Approximation Bound via Golden Ratio — E8 Intelligence Research

Abstract

FINDING: Hurwitz's theorem establishes that every irrational number has infinitely many rational approximations p/q satisfying |x − p/q| < 1/(√5 q²), with √5 being the optimal constant, attained only by the golden ratio φ = (1+√5)/2 and its equivalents. | MATH: Hurwitz constant = 1/√5 ≈ 0.4472; optimality condition: for any c > √5, there exist irrationals (e.g., φ) with only finitely many such approximations. φ = [1;1,1,1,…] = (1+√5)/2 ≈ 1.618; its conjugate = (1−√5)/2 ≈ −0.618; φ − 1 = 1/φ ≈ 0.618; φ² = φ + 1 ≈ 2.618. | CONNECTION: The golden ratio φ is the *worst* approximable irrational — its continued fraction partial quotients are all 1s, the slowest possible growth. This directly links to 0.618 (1/φ), 1.618 (φ), 2.618 (φ²), and the Markov constant √5 ≈ 2.236. The Hurwitz bound 1/√5 is the geometric mean of the two Markov constants for φ's Lagrange spectrum (√5, 2√2, √221/5, …). The continued fraction [1;1,1,…] is the simplest periodic orbit in the modular group PSL(2,ℤ), whose fu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com