The Bicomplex Numbers

Abstract

We start directly by defining the set ${\Bbb B}{\Bbb C}$ of bicomplex numbers by

$$ {\Bbb B}{\Bbb C}: = \left\{ {z_1 + jz_2 \left| {z_1 ,\,z_2 \in {\Bbb C}} \right.} \right\}, $$

where C is the set of complex numbers with the imaginary unit i, and where i and j ≠ i are commuting imaginary units, i.e., $\textbf{ij} = \textbf{ji},\,\textbf{i}^2 = \textbf{j}^2 = - 1$. Thus bicomplex numbers are “complex numbers with complex coefficients”, which explains the name of bicomplex, and in what follows we will try to emphasize the similarities between the properties of complex and bicomplex numbers. As one might expect, although the bicomplex numbers share some structures and properties of the complex numbers, there are many deep and even striking differences between these two types of numbers.