On the defects of quaternion theory and its alternative solutions
论四元数理论存在的缺陷及其替代方案

摘要

现在数学界一致公认四元数是复数集的推广，并在此基础上发展了八元数等超复数，四元数矩阵在数学及其它学科领域的理论研究中有着广泛的应用，是四元数体上重要定理，它在矩阵的学习过程中占有很重要的地位。然而本研究分析表明四元数作为复数集的拓广不满足数集扩展的必要性，把四元数作为复数集的拓广不满足数集的扩展原则，根据数集的扩展——原数集作为新数集的特例，原数集里的原有运算法则依然成立，即对应原理成立；在三维矢量ai+bj+ck中i、j、k是为了区分矢量在x轴、y轴、z轴上的分量而作的标记，可以规定i²=j²=k²=1或其他数值，但在复数a+bi中的i有着特殊的含义：i²=－1。在四元数ai+bj+ck+d中，当d=0时表示三维矢量，a、b、c分别代表在x轴、y轴、z轴上的分量，因此当c=0时，二维矢量应为ai+bj，a、b分别代表在x轴、y轴上的分量，但单位不一致，前者为i、j，后者为1、i，而i²=j²=－1≠1，因此这本身就具有一种不协调性，阐明了哈密尔顿四元数不能作为复数集的拓广，从而将矢量乘法与普通乘法区别开来，用几何积代替四元数，才满足数集的扩展原则和科学美的理念。新数集应是自然规律的产物，而不应是人为的约定；新数集的结论或保持与原数集一致，或者为原数集的推广，数学发展的方向就是运算类的扩充与数域的拓广。

Abstract

It is now universally acknowledged in the mathematical community that quaternions are an extension of the set of complex numbers, and based on this, hypercomplex numbers such as octonions have been developed. Quaternion matrices are widely used in theoretical research in mathematics and other disciplines, being important theorems over the quaternion division algebra, and they hold a crucial position in the learning process of matrices. However, the analysis of this study shows that treating quaternions as an extension of the complex number set does not meet the necessity of number set expansion, nor does it comply with the principles of number set extension. According to number set expansion rules—the original number set serves as a special case of the new number set, and the original operational rules in the original number set still hold, meaning the correspondence principle is satisfied. In the three-dimensional vector ai+bj+ck, i, j, and k are markers to distinguish the components of the vector on the x-axis, y-axis, and z-axis respectively, it is permissible to stipulate that i² = j² = k² = 1or assign other values to them. But in the complex number a+bi, i has a special meaning: i² =－1. In the quaternion ai+bj+ck+d, when d=0, it represents a three-dimensional vector, with a, b, and c representing the components on the x-axis, y-axis, and z-axis respectively. Therefore, when c = 0, the two-dimensional vector should be ai + bj, with a and b representing the components on the x-axis and y-axis respectively. However, there is an inconsistency in the units— the former uses i and j, while the latter uses 1 and i, and i² = j² =－1 ≠ 1. This itself presents an incoherence, illustrating that Hamilton's quaternions cannot be regarded as an extension of the complex number set. Thus, only by distinguishing vector multiplication from ordinary multiplication and replacing quaternions with geometric products can the principles of number set extension and the concept of scientific beauty be satisfied. A new number set should be a product of natural laws rather than artificial conventions; its conclusions should either be consistent with lower-level number sets or serve as their extensions. The direction of mathematical development lies in the expansion of operation types and the extension of number fields.