On a novel 3D hypercomplex number system

Abstract

This manuscript introduces $J_3$-numbers, a seemingly missing three-dimensional intermediate between complex numbers related to points in the Cartesian coordinate plane and Hamilton's quaternions in the 4D space. The current development is based on a rotoreflection operator $j$ in ${\mathbb{R}}^3$ that induces a novel $⊛$-multiplication of triples which turns out to be associative, distributive and commutative. This allows one to regard a point in ${\mathbb{R}}^3$ as the three-component $J_3$-number rather than a triple of real numbers. Being equipped with the $⊛$-product, the commutative algebra ${\mathbb{R}}_{⊛}^3$ is isomorphic to $\mathbb{R}\oplus \mathbb{C}$. Some geometric and algebraic properties of the $J_3$-numbers are discussed.

46 pages, 4 figures

References in corpus (2)

• Pairwise alignment of the DNA sequence using hypercomplex number representation
• Reversible Digital Filters Total Parametric Sensitivity Optimization using Non-canonical Hypercomplex Number Systems