Dual number
A variety of dualities in mathematics are listed at duality (mathematics).
For an article about the dual grammatical number found in some languages see dual grammatical number.
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 ε2 = 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 Kähler differentials (purely algebraic 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 is zero, since the units consist of the classes with a ≠ 0.
