Dual numbers
From Wacklepedia - The Free Encyclopedia
In
abstract algebra, the
dual numbers are a particular two-
dimensional commutative associative algebra over the
real numbers, arising from the reals
R by adjoining one new element ε with the property ε
2 = 0. Every dual number has the form
a +
bε with
a and
b uniquely determined real numbers.
This construction can be carried out more generally: for a commutative ring R one can define the dual numbers over R as the quotient R[X]/(X2): the image of X then has square equal to zero. This ring and its generalisations play an important part in the algebraic theory of derivations and differential forms.
The dual numbers over any field form a commutative local ring; the maximal ideal consists of classes of the form a + bX where a ≠ 0.
The word "dual" in
mathematics is used in several other meanings as well, see for instance
dual space and
dual polyhedron.