Hamilton’s Quaternions

Abstract

This chapter deals with the search for fields that extend the field of complex numbers to an even more encompassing field. Since we can view the set of complex numbers as a two-dimensional real vector space, it makes sense to begin by looking for a field that arises from a three-dimensional real vector space. It turns out, however, that no such field exists. In contrast, we discover that a field arising from a four-dimensional real vector space exists, provided that we abandon commutativity of multiplication. In this way, we are led to the construction of the skew field of Hamilton’s quaternions. In the appendix to the last chapter, we ask whether there can exist a number system that even extends Hamilton’s quaternions. It turns out that also giving up associativity of multiplication, there is precisely one additional extension, Cayley’s octonions, which brings to a close our investigation of number systems.