Overview

During the nineteenth century several attempts were made to generalize the algebra of complex numbers, which provided an adequate description of displacements and rotations in the plane, to describe objects in three dimensions. A number of generalized or hypercomplex systems were found, but none that preserved the properties of multiplication that were common to the real and complex numbers. Over time the form of vector algebra developed and popularized by the American J. Willard Gibbs came to be accepted as standard in scientific use, although some of the issues raised by the earlier work have left a lasting impact on pure and applied mathematics.

Background

At the start of the eighteenth century, mathematicians were generally familiar with the appearance of the square root of negative integers in the solution of algebraic equations, but were distrustful of such results, considering...