Babylonian igi-n-gál Fractions: Earliest Base-60 Algebraic Framework for Ratios — E8 Intelligence Research

FINDING: Babylonian/Sumerian fraction terminology (igi-n-gál, igi-te-en) encodes reciprocal and proportional operations in base-60, providing the earliest documented algebraic framework for ratio manipulation. MATH: - igi-n-gál = 1/n (reciprocal), abbreviated igi-n - igi-te-en = proportion/ratio (abstract fraction) - Base-60 positional system: 60 = 2²·3·5, highly composite (divisors: 1,2,3,4,5,6,10,12,15,20,30,60) - Reciprocal tables: igi-n for n | 60 (regular numbers in base-60) - Regular numbers: n = 2^a · 3^b · 5^c (finite sexagesimal reciprocals) - Example: igi-2 = 30 (since 2×30=60), igi-3 = 20, igi-4 = 15, igi-5 = 12, igi-6 = 10 CONNECTION: - Base-60 directly encodes harmonic ratios: 60/2=30, 60/3=20, 60/4=15, 60/5=12, 60/6=10 — these are the exact divisors generating musical intervals (octave=2:1, fifth=3:2, fourth=4:3) - Reciprocal pairs in base-60 generate the full harmonic series: 1/2, 1/3, 1/4, 1/5, 1/6 correspond to 0.5, 0.333, 0.25, 0.2, 0.1667 - The igi-te-en concept Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com