Dual number

Dual number has two very different meanings. This article will deal with the mathematical sense. For the linguistic sense, 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 over any field, not just over the field of real numbers.

The word "dual" in mathematics is used in several other meanings as well, see for instance dual space and dual polyhedron.