# A three dimensional modification of the Gaussian number field

## TL;DR

This paper introduces a new three-dimensional algebraic structure on vectors in three-dimensional space, extending Gaussian numbers and revealing its algebraic and geometric properties with potential applications.

## Contribution

It defines a novel associative, commutative, and distributive multiplication on 3D vectors, linking it to hyperbolic and Gaussian complex numbers, and describes its algebraic structure.

## Key findings

- The algebra is associative, commutative, and distributive.
- Contains a subring isomorphic to hyperbolic complex numbers.
- Isomorphic to the direct product of complex numbers and real numbers.

## Abstract

For vectors in $\mathbb{E}_3$ we introduce an associative, commutative and distributive multiplication. We describe the related algebraic and geometrical properties, and hint some applications. Based on properties of hyperbolic (Clifford) complex numbers, we prove that the resulting algebra $\mathbb{T}$ is an associative algebra over a field and contains a subring isomorphic to hyperbolic complex numbers. Moreover, the algebra $\mathbb{T}$ is isomorphic to direct product $\mathbb{C} \times \mathbb{R}$, and so it contains a subalgebra isomorphic to the Gaussian complex plane.