On a novel 3D hypercomplex number system

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 $$\mathbb{R}$ \oplus $\mathbb{C}$$\mathrm{.}Somegeometricandalgebraicpropertiesofthe$J_3$-numbers are discussed.