Old Babylonian Division via Regular Numbers and Sexagesimal Reciprocals — E8 Intelligence Research

Abstract

FINDING: Old Babylonian division used regular numbers (2^a·3^b·5^c) to convert division into multiplication via reciprocal tables, enabling exact sexagesimal arithmetic. | MATH: Regular numbers: \(2^a \cdot 3^b \cdot 5^c\) for integers \(a,b,c\). Sexagesimal reciprocals: e.g., \(2^{-1}=0;30\), \(3^{-1}=0;20\), \(5^{-1}=0;12\). Division algorithm: \(N \div D = N \times D^{-1}\) where \(D\) is regular. | CONNECTION: Base-60 arithmetic directly encodes the prime factors 2,3,5 — the same primes underlying the 5-fold and 6-fold crystallographic symmetries and the 60° angle of hexagonal lattices. The regular number set is the multiplicative semigroup generating the 60-cycle. | DEPTH: 8 — This is a foundational algorithmic insight linking number theory, computational efficiency, and geometric symmetry. The choice of base-60 is not arbitrary; it maximizes exact division for the smallest primes, mirroring the symmetry groups of icosahedral and hexagonal lattices. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com