Unified View on Complex Numbers and Quaternions

Abstract :

In this paper, a novel view on complex numbers and quaternions is presented by introducing a five-dimensional complex space which is defined as the ‘union’ of the complex plane C and the quaternion space H. It is demonstrated how the complex 5-space can be visualized by R3 and by R2 for rotations with a fixed rotation axis. In these visualizations, the algebraic representations of a rotation, using a complex number, quaternions, and a rotation matrix, appear in an elementary-geometric setup which is generalizing the unit circle. The definition of the complex 5-space is based on an explicit distinction of four different imaginary units. The embedding of a rotation matrix into the three-dimensional view is achieved by the choice of an appropriate basis for the representing matrix of the rotation.

Keywords :

Rotational Displacements, Rotation Matrices, Complex Numbers, Quaternions