The Hyperbolic Number Plane

Bicomplex Quantum Mechanics: II. The Hilbert Space

“…However, recently some interest have been deployed to study quantum mechanics for associative and commutative algebras beyond the paradigm of algebras without zero divisors [7,8,9]. This leads to a wide spectrum of possibilities, among which we have the hyperbolic numbers D ∼ = Cl R (0, 1) (also called duplex numbers) [10], the bicomplex numbers T ∼ = Cl C (1, 0) ∼ = Cl C (0, 1) [11] and, more generally, the multicomplex numbers [12,13].…”

Some Conditions Which Force Euclidean Nearrings to Be Rings

“…These are the elements of the two dimensional Euclidean ring where the multiplication is given by vw = (viwi + V2W2,viW2 + V2W1). However, it is easy to check that the map <p defined by <p(v) = (i;i -v 2 ,v\ + V2) is an isomorphism from the ring of hyperbolic numbers onto the direct sum R © R. A nice exposition of the hyperbolic numbers together with a list of related papers is given in [9].…”

The Missing Spectral Basis in Algebra and Number Theory

“…The hyperbolic numbers H := R[x]/ < x 2 − 1 >, mentioned earlier, might also seem unnatural because we introduce new square roots of unity when they already exist in R, i.e., ±1. In [8], the hyperbolic numbers are developed alongside their sister complex numbers, and they are also shown to have utility in solving the general cubic equation.…”