Babylonian Base-60 Math: Exact Fractions and the World's Oldest Trigonometry — E8 Intelligence Research

FINDING: Babylonian base-60 (sexagesimal) system encodes exact fractional divisions (1/60, 1/3600) and the Plimpton 322 tablet reveals the world's oldest trigonometric table, likely based on exact ratios rather than approximations. MATH: - Base-60 positional notation: \( \frac{1}{60} = 0;01_{60} \), \( \frac{1}{3600} = 0;00\,01_{60} \) — exact finite sexagesimal expansions for all fractions whose denominator is \( 2^a \cdot 3^b \cdot 5^c \) (since 60 = \( 2^2 \cdot 3 \cdot 5 \)). - Plimpton 322: Pythagorean triples \( a^2 + b^2 = c^2 \) generated by \( a = 2pq \), \( b = p^2 - q^2 \), \( c = p^2 + q^2 \) (with \( p, q \) in sexagesimal), yielding exact ratios \( \frac{b}{a} \) and \( \frac{c}{a} \) — not sine/cosine approximations, but exact secant/tangent values. - Sexagesimal harmonic structure: \( 60 = 3 \times 20 = 4 \times 15 = 5 \times 12 \) — rich in divisors (12 divisors), enabling exact division into 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 parts without repeating fractions. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com