The Numbers of Geometry and the Geometry of Numbers: A Brief Exploration of Hypercomplex Numbers

几何的数与数的几何：超复数的浅探究 苏剑林 · 2014-01-11 · Mathematics algebra geometry complex number matrix

This is also one of my term papers... The full text runs 17 pages, covering the construction of quaternions, elementary applications, etc. The appendix includes the equivalence of determinants and volume, the description of three-dimensional rotations, and more. Written using LaTeX (LaTeX will make you fall in love with mathematical writing).

The Numbers of Geometry and the Geometry of Numbers

—A Brief Exploration of Hypercomplex Numbers

Abstract

Today, whether in mathematics or physics, high-dimensional problems are handled using vector analysis as the basic tool, and quaternions are hardly to be found in mathematical physics anymore. Historically, however, the development of quaternions was of considerable importance. Quaternion arithmetic was in fact the "progenitor" of vector analysis—the concepts of the dot product and cross product first appeared in quaternion operations—and the birth of quaternions also marked the beginning of non-commutative algebra. Even today, quaternions remain the simplest tool for describing three-dimensional rotations in computing. Moreover, as a generalization of complex numbers, quaternions offer a way of thinking about generalizing certain problems involving complex numbers.

This paper combines matrices and geometry in a suitable way, using the property of the matrix determinant $\det (AB) =(\det A)(\det B)$ to derive the generation rule for quaternions and higher-dimensional hypercomplex numbers, and discusses some of their properties along with their applications in describing rotations. Some proof details and less fully developed ideas have been placed in the appendix. more

Table of Contents

1 Background 2
2 Determinants 3
3 Quaternions 3
3.1 Definition of Complex Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 Complex Numbers and Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.3 Triples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.4 Quaternions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1
1 Background 2
3.4.1 The Geometric Approach. . . . . . . . . . . . . . . . . . . . . . . . . 6
3.4.2 The Algebraic Approach. . . . . . . . . . . . . . . . . . . . . . . . . 8
3.5 Some Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.5.1 Basic Operations. . . . . . . . . . . . . . . . . . . . . . . . . 9
3.5.2 Spatial Rotation. . . . . . . . . . . . . . . . . . . . . . . . . 10
3.5.3 Non-commutative Addition. . . . . . . . . . . . . . . . . . . . . . . . 11
4 Octonions 11
5 Concluding Remarks 12
A Appendix 12
A.1 The Equivalence of Determinants and Volume. . . . . . . . . . . . . . . . 12
A.2 Describing Rotation with Matrices. . . . . . . . . . . . . . . . . . . . . . . 13
A.3 Redefining the Modulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
A.4 Quaternion Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
References 17

"The Numbers of Geometry and the Geometry of Numbers: A Brief Exploration of Hypercomplex Numbers".pdf