Sexagesimal Reciprocal Tables and Babylonian Quadratic Root Extraction — E8 Intelligence Research

FINDING: Babylonian base-60 (sexagesimal) arithmetic is the substrate for reciprocal tables and iterative root extraction (Heron's method), yielding quadratically convergent algorithms centuries before Newton-Raphson formalization. | MATH: Sexagesimal place-value: digits d∈{0..59}, value = Σ dᵢ·60ⁱ. Reciprocal table entry: n → 1/n stored as 60ᵏ/n. Heron iteration: xₙ₊₁ = ½(xₙ + a/xₙ), error eₙ₊₁ ≈ eₙ²/(2√a) → quadratic convergence (doubling of correct digits per step). Babylonian √2 ≈ 1;24,51,10 = 1 + 24/60 + 51/60² + 10/60³ = 1.41421296… (error ~6×10⁻⁷). | CONNECTION: 60 = 2²·3·5 → highly composite; divisors {1,2,3,4,5,6,10,12,15,20,30,60} enable exact reciprocals for 3-smooth and 5-smooth numbers, giving terminating sexagesimal expansions where decimal fails (1/3, 1/6, 1/12). Regular numbers (60-smooth) form a multiplicative lattice closed under reciprocation — a finite subgroup structure underlying the table. Plimpton 322 encodes Pythagorean triples via the ratio (c/a)² = 1 + (b/a)² Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com