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An element of a finite-dimensional algebra with a unit element over the field of real numbers  $  \mathbf R $
(formerly known as a hypercomplex system). Historically, hypercomplex numbers arose as a generalization of complex numbers (cf. [[Complex number|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  $  n $
is obtained by introducing a multiplication in the  $  n $-dimensional real space  $  \mathbf R  ^ {n} $
which satisfies the axioms of an algebra over a field. Let 1 be the unit of a hypercomplex system  $  U $
and let  $  1, i _ {1} \dots i _ {n-1} $
be some basis of  $  \mathbf R  ^ {n} $. 
The hypercomplex number

$$ 
\overline \alpha \;  =  a _ {0} - a _ {1} i _ {1} - \dots - a _ {n} i _ {n}  $$

of  $  U $
is said to be the conjugate hypercomplex number of

$$ 
\alpha  =  a _ {0} + a _ {1} i _ {1} + \dots + a _ {n} i _ {n} .
$$

Let  $  U  ^ {( 2)} = \{ u _ {1} + u _ {2} e \} $, 
where  $  u _ {1} , u _ {2} \in U $
and  $  e $
is some new symbol. The set  $  U  ^ {( 2)} $
may be converted into a hypercomplex system by defining addition by

$$ 
( u _ {1} + u _ {2} e) + ( v _ {1} + v _ {2} e)  = \ 
( u _ {1} + v _ {1} ) + ( u _ {2} + v _ {2} ) e
$$

and multiplication by

$$ 
( u _ {1} + u _ {2} e) ( v _ {1} + v _ {2} e)  = \ 
( u _ {1} v _ {1} - \overline{ {v _ {2} }}\, u _ {2} ) +
( v _ {2} u _ {1} + u _ {2} \overline{ {v _ {1} }}\; ) e.
$$

The hypercomplex system  $  U  ^ {( 2)} $
is called the doubling of  $  U $.

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|Quaternion]]; [[Cayley numbers|Cayley numbers]]). Other examples include [[Double and dual numbers|double and dual numbers]], and hypercomplex systems of the form

$$ 
A  =  a _ {0} \cdot 1 +
\sum _ {\gamma = 1 } ^ { {2 ^ {n} - 1 } }
a _  \gamma  i _  \gamma  ,
$$

which, if  $  n = 4 $, 
are known as Clifford–Lipschitz numbers (these hypercomplex numbers are elements of the [[Clifford algebra|Clifford algebra]] of rank  $  2  ^ {n} $). 
An important example of hypercomplex systems are complete matrix algebras over  $  \mathbf R $.

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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.L. Kantor,   A.S. Solodovnikov,   "Hyperkomplexe Zahlen" , Teubner  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. Kaluzhnin,   "Introduction to general algebra" , Moscow  (1973)  (In Russian)</TD></TR></table>