Quadratic Forms and their Number Systems and Geometry

TL;DR

This paper explores how quadratic forms can be used to derive various number systems like quaternions and split quaternions, and how these systems relate to geometric transformations in Euclidean and Lorentz spaces.

Contribution

It provides a method to find the associated number system for any quadratic form and derives related geometric tools such as bilinear forms and rotation matrices.

Findings

Derived bilinear forms and rotation matrices for general quadratic forms.

Established connections between quadratic forms and number systems like quaternions.

Provided a framework to generate geometric transformations from quadratic forms.

Abstract

Quaternions, split quaternions, and hybrid numbers are very well-known number systems. These number systems are used to make geometry in Euclidean and Lorentz spaces. These number systems can be obtained with the help of a quadratic form. This article examines how to find the corresponding number system for any quadratic form. As a result of this examination, bilinear form, vector product, skew-symmetric matrix, and rotation matrices corresponding to any number system were obtained.