On a novel 3D hypercomplex number system

TL;DR

This paper introduces $J_3$-numbers, a new 3D hypercomplex system with a novel associative, commutative multiplication, bridging complex numbers and quaternions, and providing a new algebraic framework for 3D points.

Contribution

It develops a new 3D hypercomplex number system based on a rotoreflection operator, establishing a commutative algebra isomorphic to $ ext{Re} imes ext{C}$, with geometric and algebraic properties.

Findings

$J_3$-numbers form a commutative algebra with a novel multiplication.

The algebra is isomorphic to $ ext{Re} imes ext{C}$.

Geometric and algebraic properties of $J_3$-numbers are discussed.

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.