Module arithmetic 

Exact arithmetic for Atlas computations

This module provides exact rational arithmetic with NO floating point. All computations use exact integers and rationals to preserve mathematical structure.

Key Types

• Rational - Exact rational numbers (alias for Ratio<i64>)
• HalfInteger - Half-integers for E₈ coordinates (multiples of 1/2)
• Vector8 - 8-dimensional vectors with exact coordinates
• RationalMatrix - Matrices with exact rational entries (for Weyl groups)

Design Principles

1. No floating point - All arithmetic is exact
2. Fraction preservation - Rationals never lose precision
3. Overflow detection - All operations check for overflow
4. Type safety - Prevent mixing incompatible number types

Structs

HalfInteger

Half-integer: numbers of the form n/2 where n ∈ ℤ

RationalMatrix

Matrix with exact rational entries

RationalVector

N-dimensional vector with exact rational coordinates

Vector8

8-dimensional vector with exact coordinates

Type Aliases

Rational

Exact rational number (fraction)