What is "dual number"

n. 1 (context grammar English) Grammatical (l/en: number) denoting a quantity of exactly two. 2 (context mathematics English) An element of the (l/en: algebra) of dual numbers, which includes the (l/en: real numbers) and an element ε which satisfies ε² = 0.

In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε = 0 (ε is nilpotent). The collection of dual numbers forms a particular two- dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε where a and b are uniquely determined real numbers. The dual numbers can also be thought of as the exterior algebra of a one dimensional vector space.

The algebra of dual numbers is a ring that is a local ring since the principal ideal generated by ε is its only maximal ideal. Dual numbers form the coefficients of dual quaternions.