ABSTRACT
In the beginning numbers were for counting (possibly sheep at bedtime). But over the centuries, the concept gradually developed from the simple starting point of the counting numbers 1, 2, 3, 4, …, first to include zero, and then to embrace the idea of negative integers -1, -2, -3, -4, …. With the set of integers properly expanded to include both positive and negative numbers (and zero) we can always add and subtract, but we cannot always divide. There are, an infinite number of fractions, just as there are an infinite number of integers. Hamilton was convinced that in quaternions he had found the natural algebra of three dimensions. Its four-dimensional form seemed to him to be much more fundamental than any three-dimensional vector coordinate system because operations with quaternions were completely independent of any coordinate representation. Matrix algebra proved to be just the mathematical tool needed for the development of many aspects of quantum mechanics.
