## Four-Dimensional Numbers: The Quaternions

When complex numbers were first invented, many people were suspicious. What real significance could such "imaginary" numbers have? But it was only a short time before mathematicians and scientists discovered a tremendous number of applications for these numbers. The complex numbers turned out to be precisely the system that best described patters of flows in hydrodynamics and in electricity. These two-dimensional numbers have more than justified their existence.

One hundred years ago, Sir William Hamilton invented a number system of even higher dimension, the quaternions, a collection of four-dimensional numbers. The addition of quaternions is defined the same as addition of points in four-space, but the multiplication law is a complicated mixture of different formulas that come from vector calculus. Specifically we define (a, b, c, d)(x, y, u, v) = (ax - by - cu - dv, ay + bx + cv - du, au - bv + cx + dy, av + bu - cy + dx). As in the case of the complex numbers, there is a unit element (1, 0, 0, 0), and for every non-zero quaternion (a, b, c, d), there is a quaternion (x, y, u, v) such that (a, b, c, d)(x, y, u, v) = (1, 0, 0, 0).

But would such numbers ever have practical applications? As it happens, these are exactly the numbers used to describe the orbits of the motion of pairs of pendulums. And quaternions have been recognized in recent years as one of the most effective means of communicating information about rotations to a graphics computer. Thus some of the most abstract algebraic constructions in four-space can be visualized as geometric phenomena.