dualnumber

Dual numbers are a simple algebraic extension of a field obtained by adjoining a single nilpotent element ε with ε^2 = 0. A dual number is written as a + bε, where a and b belong to the base field (commonly the real numbers). The set of all dual numbers over F is the quotient ring F[ε]/(ε^2), often denoted F ⊕ Fε, with ε^2 = 0 and ε behaving as an infinitesimal.

Arithmetic in the dual-number system is defined by the usual addition and multiplication, using ε^2 = 0. For dual numbers a + bε and c + dε, addition is (a + bε) + (c + dε) = (a + c) + (b + d)ε, and multiplication is (a + bε)(c + dε) = ac + (ad + bc)ε, since the term ε^2 vanishes. A dual number is invertible precisely when its real part a is nonzero; its inverse is (a + bε)^{-1} = a^{-1} - (b/a^2)ε.

Functions extend to dual numbers via a first-order Taylor expansion. If f is differentiable at a, then f(a + bε) = f(a) + f'(a) b ε, because ε^2 = 0 nullifies higher-order terms. This property underpins forward-mode automatic differentiation, where carrying the pair (f(a), f'(a)·b) through computations yields derivatives alongside function values.

Geometrically, dual numbers encode first-order infinitesimals: the base a tracks a point on the real line, while the coefficient b measures a first-order tangent direction. In this view, the dual number a + bε represents a first-order neighborhood around a, linking algebra with the tangent space at a.

Dual numbers can be defined over any field and play roles in geometry, numerical analysis, and theoretical physics, notably in perturbation theory and automatic differentiation. They are distinct from hyperreal or other infinitesimal number systems, which have different algebraic properties.