An element of a finite-dimensional algebra with a
unit element over the field of real numbers
(formerly known as a
hypercomplex system).
Historically, hypercomplex numbers arose as a generalization of complex numbers (cf.
Complex number).
Operations on complex numbers correspond to geometrical transformations
of the plane (translation, rotation, dilation, and combinations
of such operations). In trying to construct
numbers whose role with respect to three-dimensional space
corresponds to the role played by complex numbers with respect to
the plane, it became clear that a full analogy is
not possible; this gave rise to the development
of the theory of systems of hypercomplex numbers.
A
hypercomplex system
of rank
is obtained by introducing a multiplication in the
-dimensional
real space
which satisfies the axioms of an algebra over a field.
Let 1 be the unit of a hypercomplex system
and let
be some basis of
.
The hypercomplex number
of
is said to be the
conjugate hypercomplex number
of
Let
,
where
and
is some new symbol. The set
may be converted into a hypercomplex system by defining addition by
and multiplication by
The hypercomplex system
is called the
doubling
of
.
Examples of hypercomplex systems are: the real numbers, the complex numbers,
the quaternions, and the Cayley numbers (in this list each
successive system is obtained by doubling the preceding one, cf.
Quaternion;
Cayley numbers).
Other examples include
double and dual numbers,
and hypercomplex systems of the form
which, if
,
are known as
Clifford–Lipschitz numbers
(these hypercomplex numbers are elements of the
Clifford algebra
of rank
).
An important example of hypercomplex systems are complete matrix algebras over
.
The definition of a system of hypercomplex numbers may include
the requirement of associativeness of multiplication; one also identifies
the concepts of an algebra and a hypercomplex system.