Quaternions for fun and profit

Derek Smith, Lafayette College

Abstract:
Here's an SAT ``analogy'' question from 1988.
Problem 42
Real numbers : complex numbers:
complex numbers : ?

The answer, (D) quaternions, was missed by more than
75% percent of the examinees, which I find troubling.  One
of the goals of my presentation will be to ensure that you
won't miss this question, should it arise on your GRE exam.
  Just as the complex numbers double the single dimension
of the real number line, the quaternions double again to give
dimension 4.  This might suggest that they skipped the
chance for application in our 3-dimensional world, but in fact
quaternions are often the algebraic tools of choice to represent
3-dimensional rotations, which leads to applications from
computer graphics to wireless communication.  In this
introductory presentation, I will introduce you to the mathematics
of the quaternion and related algebras and point out several
applications along the way.