# Double and dual numbers

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). 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).

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 [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 [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) 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 [1], [3].

#### References

 [1] D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) [2] 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) [3] M. Demazure, A. Grothendieck, "Schémas en groupes I" , Lect. notes in math. , 151–153 , Springer (1970) [4] S. Lichtenbaum, M. Schlessinger, "The cotangent complex of a morphism" Trans. Amer. Math. Soc. , 128 : 1 (1967) pp. 41–70

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.