Let 
be an 
-dimensional
linear space over a field 
, and let 
be a quadratic form on 
. A Clifford algebra is then defined
over 
,
where 
is the tensor algebra over 
and 
is a particular ideal of 
.
Clifford algebraists call their higher dimensional numbers hypercomplex even though they do not share all the properties of complex numbers and no classical function theory can be constructed over them.
When 
is Euclidean space, the Clifford algebra is generated
by the standard basis vectors 
with the relations
 |  |  |
(1)
|
 |  |  |
(2)
|
for 
.
The standard Clifford algebra is then generated additively by elements of
the form 
,
where 
,
and so the dimension is 
, where 
is the dimension of 
.
The defining relation in the general case with vectors 
is
 |
(3)
|
where 
denotes the quadratic form, or equivalently,
 |
(4)
|
where 
is the symmetric bilinear form associated
with 
.
Clifford algebras are associative but not commutative.
When 
,
the Clifford algebra becomes exterior algebra.
Clifford algebras are used to define spinors.
