View source for Double and dual numbers

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Hypercomplex numbers of the form  $  a + be $, 
where  $  a $
and  $  b $
are real numbers, and where the double numbers satisfy the relation  $  e  ^ {2} = 1 $, 
while the dual numbers satisfy the relation  $  e  ^ {2} = 0 $(
cf. [[Hypercomplex number|Hypercomplex number]]). Addition of double and dual numbers is defined by

$$ 
( a _ {1} + b _ {1} e) + ( a _ {2} + b _ {2} e)  = \ 
( a _ {1} + a _ {2} ) + ( b _ {1} + b _ {2} ) e.
$$

Multiplication of double numbers is defined by

$$ 
( a _ {1} + b _ {1} e) ( a _ {2} + b _ {2} e )  = \ 
( a _ {1} a _ {2} + b _ {1} b _ {2} ) + ( a _ {1} b _ {2} +
a _ {2} b _ {1} ) e ,
$$

and that of dual numbers by

$$ 
( a _ {1} + b _ {1} e )( a _ {2} + b _ {2} e )  =  a _ {1} a _ {2} +
( a _ {1} b _ {2} + a _ {2} b _ {1} ) e .
$$

Complex numbers, double numbers and dual numbers are also called complex numbers of hyperbolic, elliptic and parabolic types, respectively. These numbers are sometimes used to represent motions in the three-dimensional spaces of Lobachevskii, Riemann and Euclid (see, for instance, [[Helical calculus|Helical calculus]]).

Both double and dual numbers form two-dimensional (with base 1 and  $  e $) 
associative-commutative algebras over the field of real numbers. As distinct from the field of complex numbers, these algebras comprise zero divisors, all these having the form  $  a \pm  ae $
in the algebra of double numbers. The algebra of double numbers may be split into a direct sum of two real number fields. Hence yet another name for double numbers — splitting complex numbers. Double numbers have yet another appellation — paracomplex numbers. The algebra of dual numbers is considered not only over the field  $  \mathbf R $
of real numbers, but also over an arbitrary field or commutative ring. Let  $  A $
be a commutative ring and let  $  M $
be an  $  A $-
module. The direct sum of  $  A $-
modules  $  A \oplus M $
equipped with the multiplication

$$ 
( a , m ) ( a  ^  \prime  , m  ^  \prime  )  =  ( aa  ^  \prime  , am  ^  \prime  +
a  ^  \prime  m )
$$

is a commutative  $  A $-
algebra and is denoted by  $  I _ {A} ( M) $. 
It is known as the algebra of dual numbers with respect to the module  $  M $. 
The  $  A $-
module  $  M $
is identical with the ideal of the algebra  $  I _ {A} ( M) $
which is the kernel of the augmentation homomorphism

$$ 
\epsilon :  I _ {A} ( M)  \rightarrow  A \  ( ( a , m ) \rightarrow a ) .
$$

The square  $  M  ^ {2} $
of this ideal is zero, while  $  I _ {A} ( M) / M \simeq A $. 
If  $  A $
is a regular ring the converse is also true: If  $  B $
is an  $  A $-
algebra and  $  M $
is an ideal in  $  B $
such that  $  M  ^ {2} = 0 $
and  $  B/M \simeq A $, 
then  $  B \simeq I _ {A} ( M) $, 
where  $  M $
is regarded as an  $  A $-
module [[#References|[4]]].

If  $  M = A $, 
the algebra  $  I _ {A} ( M) $(
then denoted by  $  I _ {A} $) 
is isomorphic to the quotient algebra of the algebra of polynomials  $  A( T) $
by the ideal  $  T ^ { 2 } $. 
Many properties of an  $  A $-
module may be formulated as properties of the algebra  $  I _ {A} ( M) $; 
as a result, many problems on  $  A $-
modules can be reduced to corresponding problems in the theory of rings [[#References|[2]]].

Let  $  B $
be an arbitrary  $  A $-
algebra, let  $  \phi :  B \rightarrow A $
be a homomorphism and let  $  \partial  :  B \rightarrow M $
be a derivation (cf. [[Derivation in a ring|Derivation in a ring]]) of  $  B $
with values in the  $  A $-
module  $  M $, 
regarded as a  $  B $-
module with respect to the homomorphism  $  \phi $. 
The mapping  $  \overline \partial \; :  B \rightarrow I _ {A} ( M) $(
$  b \rightarrow ( \phi ( b), \partial  ( b)) $) 
will then be a homomorphism of  $  A $-
algebras. Conversely, for any homomorphism of  $  A $-
algebras  $  f :  B \rightarrow I _ {A} ( M) $
the composition  $  \epsilon  ^  \prime  \circ f :  B \rightarrow M $, 
where  $  \epsilon  ^  \prime  :  I _ {A} ( M) \rightarrow M $
is the projection of  $  I _ {A} ( M) $
onto  $  M $, 
is an  $  A $-
derivation of  $  B $
with values in  $  M $, 
regarded as a  $  B $-
module with respect to the homomorphism  $  \epsilon \circ f :  B \rightarrow A $. 
This property of double and dual numbers is utilized for the description of the tangent space to an arbitrary functor in the category of schemes [[#References|[1]]], [[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Mumford,   "Lectures on curves on an algebraic surface" , Princeton Univ. Press  (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Fossum,   P.A. Griffith,   I. Reiten,   "Trivial extensions of Abelian categories. Homological algebra of trivial extensions of Abelian categories with applications to ring theory" , Springer  (1975)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Demazure,   A. Grothendieck,   "Schémas en groupes I" , ''Lect. notes in math.'' , '''151–153''' , Springer  (1970)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Lichtenbaum,   M. Schlessinger,   "The cotangent complex of a morphism"  ''Trans. Amer. Math. Soc.'' , '''128''' :  1  (1967)  pp. 41–70</TD></TR></table>

====Comments====
An old-fashioned term for an associative algebra  $  A $
with unit element over  $  \mathbf R $
is system of hypercomplex numbers, and an element of  $  A $
is then called a hypercomplex number. There are (up to isomorphism) precisely three of these algebras of dimension 2: the complex numbers, the dual numbers and the double numbers.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Demazure,   P. Gabriel,   "Groupes algébriques" , '''1''' , Masson  (1970)</TD></TR></table>