Quaternions were introduced in 1843 by the Irish mathematician Hamilton. They are numbers of the form a+bi+cj+dk where a,b,c,d are real numbers. They generalize complex numbers, with a more interesting rule of multiplication. Quaternionic multiplication can be seen as a precursor to the vector cross product in R³.

In this talk, I will describe the algebra of quaternions and an application to rotations in three-dimensional space. We will also see the connection between quaternions and more familiar algebraic objects such as matrices, as well as their role in studying spins of elementary particles such as the electron. En route, we will encounter an extraordinary historical saga of ambition, obscurity, and resurgence which will exemplify the very human face of mathematical discovery.