Fundamentals Of Hypercomplex Numbers

A survey of those systems of numbers that can be constructed by adding “imaginary units” to the real numbers. The simplest and most familiar example is the two-dimensional system of complex numbers. Much less familiar, but equally fascinating, are the systems of quaternions and Cayley numbers, of dimensions four and eight, respectively. These “algebras” still enable meaningful notions of addition, multiplication, and division, but only at a price: the loss of commutativity and (in the case of Cayley numbers) associativity. Things get even more bizarre when sedenions (dimension 16) and trigintaduonions (dimension 32) are brought into play. The latter part of the course is devoted to the theorems of Hurwitz and Frobenius on the existence of suitably behaved division algebras over the real numbers.