Babylonian Tablets Reveal Ancient Trigonometry and Proto-Proof of √2 Irrationality — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Babylonian tablet Plimpton 322 is a 3700-year-old exact sexagesimal trigonometric table, not a school exercise; a separate Babylonian geometry tablet series encodes an implicit proto-proof of √2 irrationality via reciprocal-pair algorithms. | MATH: Sexagesimal (base-60) reciprocal pairs p, q with p/q generating Pythagorean triples (a,b,c) where a = p2 − q2, b = 2pq, c = p2 + q2; tablet columns list b2/a2 (or tan2θ) in exact sexagesimal fractions. For √2: geometry tablets use the standard approximation 1;24,51,10 = 1 + 24/60 + 51/3600 + 10/216000 = 305470/216000 ≈ 1.41421296 (error ~6×10⁻⁷), paired with its reciprocal 0;42,25,35. The arxiv paper (1603.06656) argues two tablets contain a *reductio* style argument: assuming √2 = p/q in lowest terms leads to an infinite descent in the reciprocal-pair algorithm — a proto-irrationality proof. | CONNECTION: The sexagesimal system is base-60, whose divisors include 2,3,5
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
- Babylonian Tablets Reveal Proto-Proofs of √2 Irrationality via Reciprocal Algorithms — E8 Intelligence Research
- Base-60 Cuneiform Reveals Advanced Babylonian Trigonometry and Modular Math — E8 Intelligence Research
- Babylonian Tablet Plimpton 322: Ancient Sexagesimal Trigonometry Predates Greeks — E8 Intelligence Research
- Babylonian Trigonometry Linked to Rational Circle Parametrization via Base-60 Divisions — E8 Intelligence Research
- Sumerian Base-60 Trigonometry: Plimpton 322's Exact Rational Triples Outperform Modern Sine Methods — E8 Intelligence Research
BibTeX
@misc{caldin2026babylonian,
title = {{Babylonian Tablets Reveal Ancient Trigonometry and Proto-Proof of √2 Irrationality — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23447080},
url = {https://zenodo.org/records/23447080},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Babylonian Tablets Reveal Ancient Trigonometry and Proto-Proof of √2 Irrationality — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23447080
UR - https://zenodo.org/records/23447080
AB - FINDING: Babylonian tablet Plimpton 322 is a 3700-year-old exact sexagesimal trigonometric table, not a school exercise; a separate Babylonian geometry tablet series encodes an implicit proto-proof of √2 irrationality via reciprocal-pair algorithms. | MATH: Sexagesimal (base-60) reciprocal pairs p, q with p/q generating Pythagorean triples (a,b,c) where a = p^2 − q^2, b = 2pq, c = p^2 + q^2; tablet columns list b^2/a^2 (or tan^2θ) in exact sexagesimal fractions. For √2: geometry tablets use the standard approximation 1;24,51,10 = 1 + 24/60 + 51/3600 + 10/216000 = 305470/216000 ≈ 1.41421296 (error ~6×10⁻⁷), paired with its reciprocal 0;42,25,35. The arxiv paper (1603.06656) argues two tablets contain a *reductio* style argument: assuming √2 = p/q in lowest terms leads to an infinite descent in the reciprocal-pair algorithm — a proto-irrationality proof. | CONNECTION: The sexagesimal system is base-60, whose divisors include 2,3,5 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -