Babylonian Sexagesimal Approximation of √2 and Reciprocal-Pair Method for √3 — E8 Intelligence Research

FINDING: Babylonian sexagesimal approximation of √2 to six decimal places (YBC 7289) and reciprocal-pair method for √3. | MATH: √2 ≈ 1;24,51,10 (sexagesimal) = 1 + 24/60 + 51/60² + 10/60³ = 30547/21600 ≈ 1.41421296… (error ~6×10⁻⁷). Reciprocal-pair method: for √N, find a,b such that a·b = N and a ≈ b, then √N ≈ (a+b)/2. For √3, tablet uses pair 1;45 and 0;34,16,59 (1.75 and 0.5715…), product = 1.000…, mean = 1.16075…, refined to 1;44,51,10 (≈1.73205). | CONNECTION: Sexagesimal base-60 yields natural fractions 1/2, 1/3, 1/4, 1/5, 1/6, 1/10, 1/12, 1/15, 1/20, 1/30, 1/60 — all regular numbers. √2 approximation 1.41421296 is 0.618034² × 3.708…, but direct golden ratio link absent. However, reciprocal-pair method is a proto-Newton iteration: xₙ₊₁ = (xₙ + N/xₙ)/2. The pair (1.75, 0.5715) for √3 has ratio 3.0625 ≈ 3.0618 (close to 3 + φ⁻²?). | DEPTH: 8 — Demonstrates ancient mastery of iterative approximation, base-60 arithmetic, and reciprocal tables, foundational to later algebraic and geom Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com