Adv. Appl. Cliﬀord Algebras (2018) 28:58
2018 Springer International Publishing AG,
part of Springer Nature
Applied Cliﬀord Algebras
Hybrid Complex Numbers: The Matrix
G. Dattoli, S. Licciardi
, R. M. Pidatella and E. Sabia
Communicated by RafalAblamowicz
Abstract. In this paper we review the notion of hybrid complex num-
bers, recently introduced to provide a comprehensive conceptual and
formal framework to deal with circular, hyperbolic and dual complex.
We exploit the established isomorphism between complex numbers as
abstract entities and as two dimensional matrices in order to derive the
associated algebraic properties. Within such a respect we derive gen-
eralized forms of Euler exponential formula and explore the usefulness
and relevance of operator ordering procedure of the Wei-Norman type.
We also discuss the properties of dual numbers in terms of Pauli ma-
trices. Finally we explore generalized forms of Dirac-like factorization,
emerging from the properties of these numbers.
Keywords. Hybrid complex numbers, Matrix algebra, Quaternions,
Operator ordering, Cliﬀord numbers, Dual numbers.
The theory of complex numbers has many facets. One of its most appealing
features is their framing within the context of Geometric algebra. According
to such a point of view the real number system is completed by the inclusion
of abstract entities, which once squared yield +1 or −1. The completion of
real numbers occurs therefore through the inclusion of the square roots of
+1 and −1 which realize orthogonal directions in higher dimensional spaces
From the algebraic point of view they provide a set of non commuting
quantities and the matrix formalism is a very eﬀective tool to deal with the
The use of the isomorphism between complex numbers of diﬀerent na-
ture and 2 × 2 matrices provides an eﬃcient tool, allowing the uniﬁcation