Irrational Numbers and Root Systems: A Foundational Mathematical Overview — E8 Intelligence Research

FINDING: The search results are a collection of general educational videos on irrational numbers and root systems, not a specific new discovery. The core mathematical insight is the definition and properties of irrational numbers (non-repeating, non-terminating decimals) and their occurrence in root systems of Lie algebras. MATH: - Irrational numbers: numbers that cannot be expressed as a ratio of integers \( p/q \). - Root systems: sets of vectors in Euclidean space satisfying crystallographic conditions (e.g., angles between roots are \( 90^∘, 60^∘, 45^∘, 30^∘, 120^∘, 135^∘, 150^∘ \)). - Key ratios from root system angles: - \( cos 60^∘ = 0.5 \) - \( cos 45^∘ = √2/2 ≈ 0.7071 \) - \( cos 30^∘ = √3/2 ≈ 0.8660 \) - \( cos 120^∘ = -0.5 \) - \( cos 135^∘ = -√2/2 \) - \( cos 150^∘ = -√3/2 \) - The golden ratio \( φ = (1+√5)/2 ≈ 1.618 \) is irrational Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com