Geometric Numbers and Number Geometry: A Brief Exploration of Hypercomplex Numbers

Abstract

Today, whether in high-dimensional problems of mathematics or physics, vector analysis is used as the basic tool; the shadow of quaternions is rarely found in mathematical physics. However, historically, the development of quaternions has significant meaning. Quaternion operations are actually the "ancestor" of vector analysis; the concepts of the vector dot product and cross product first appeared in quaternion operations. The birth of quaternions also marked the beginning of non-commutative algebra. Even now, quaternions remain the simplest tool for computers to describe three-dimensional spatial rotation problems. Furthermore, as an extension of complex numbers, quaternions provide ideas for the generalization of certain complex number problems.

This paper appropriately combines matrices with geometry, utilizing the property of the matrix determinant \det(AB) = (\det A)(\det B) to derive the generation laws of quaternions and higher-dimensional hypercomplex numbers, and discusses some of their properties as well as their applications in describing rotations. Some proof details and incomplete ideas are placed in the appendix.

Table of Contents

1 Background
2 Determinants
3 Quaternions
3.1 Definition of Complex Numbers
3.2 Complex Numbers and Matrices
3.3 Ternions
3.4 Quaternions
3.4.1 Geometric Method
3.4.2 Algebraic Method
3.5 Some Properties
3.5.1 Basic Operations
3.5.2 Spatial Rotation
3.5.3 Non-commutative Addition
4 Octonions
5 Conclusion
A Appendix
A.1 Equivalence of Determinants and Volume
A.2 Matrix Description of Rotation
A.3 Changing the Definition of Norm
A.4 Quaternion Calculus
References

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