Dual number - Definition and Overview

A variety of dualities in mathematics are listed at duality (mathematics).
For the dual grammatical number found in some languages see dual grammatical number.

In abstract algebra, the dual numbers are a particular two-dimensional commutative unital associative algebra over the real numbers, arising from the reals 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 of the polynomial ring R[X] by the ideal (X²): the image of X then has square equal to zero and corresponds to the element ε from above. This ring and its generalisations play an important part in the algebraic theory of derivations and Kähler differentials (purely algebraic differential forms).

Over any ring R, the dual number a + bε is a unit (i.e. multiplicatively invertible) if and only if a is a unit in R. In this case, the inverse of a + bε is a^-1 - ba^-2ε. As a consequnce, we see that the dual numbers over any field (or any commutative local ring) form a local ring.