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 $\circledast$-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 $\circledast$-product, the commutative algebra $\mathbb{R}_\circledast^3$ is isomorphic to $\mathbb{R} \oplus \mathbb{C}$. Some geometric and algebraic properties of the $J_3$-numbers are discussed.