Stefan Rönn

Title: Two complex pair algebras

Abstract: We present the algebras of quaternions and bicomplex numbers
using pairs of complex numbers (complex pairs). Especially the
quaternions have applications in graphical data processing, while the
bicomplex numbers are of interest from a function theoretic
viewpoint. Both types of numbers can be seen as generalizations of the
ordinary complex numbers, but in the process of generalization
algebraic properties are lost. In quaternionic algebra the
multiplication operation is non-commutative and in bicomplex algebra
one has an infinite set of so-called singular bicomplex numbers, which
lack an inverse. We focus in particular on the following issues:
representations of quaternions and bicomplex numbers, basic algebraic
properties, conjugations, norms, inverses and division operations.

Reference: Rönn S 2001, "Bicomplex algebra and function theory",
arXiv:math.CV/0101200