hyprat: rational hypercomplex numbers

hyprat provides an immutable Hy class for exact, rational-valued hypercomplex numbers built via the Cayley-Dickson construction:

rank 0  ->  a plain fractions.Fraction        (a "real")
rank 1  ->  Hy(real, imag)                    (a "complex" number)
rank 2  ->  Hy(h1, h2), h1/h2 rank 1           (a "quaternion")
rank 3  ->  Hy(h3, h4), h3/h4 rank 2           (an "octonion")
rank n  ->  Hy(x, y),   x/y  rank (n-1)

All arithmetic (+ - * /), conjugation, norms, and inverses are defined recursively by the standard Cayley-Dickson formulas, so the same class correctly represents rational complex numbers, quaternions, octonions, and beyond – all with exact fractions.Fraction arithmetic, no floating-point rounding.

Installation

hyprat requires Python 3.9 or later and has no runtime dependencies.

pip install git+https://github.com/alreich/hyper_rationals.git

Optional extras: interop (SymPy, numpy-quaternion, quaternionic and NumPy, needed by the to_*/from_* interoperability methods and by to_matrix/from_matrix), test, docs and dev (everything). For example:

pip install "hyprat[interop] @ git+https://github.com/alreich/hyper_rationals.git"

Quickstart

>>> from hyprat import Hy
>>> z = Hy('5/2', '-16/5')
>>> str(z)
'(5/2-16/5j)'

>>> q = Hy(Hy(1, 2), Hy(3, 4))   # a rational quaternion
>>> str(q)
'(1+2i+3j+4k)'

>>> i = Hy(Hy(0, 1), Hy(0, 0))
>>> j = Hy(Hy(0, 0), Hy(1, 0))
>>> str(i * j)
'(k)'

>>> Hy.from_array([1, 2, 3, 4])   # same quaternion, from a flat list
Hy(Hy('1', '2'), Hy('3', '4'))

Contents

Indices and tables