In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form a + bε, where a and b are real numbers, and ε is a symbol taken to satisfy ${\displaystyle \varepsilon ^{2}=0}$ with ${\displaystyle \varepsilon \neq 0}$.

Dual numbers can be added component-wise, and multiplied by the formula

${\displaystyle (a+b\varepsilon )(c+d\varepsilon )=ac+(ad+bc)\varepsilon ,}$

which follows from the property ε² = 0 and the fact that multiplication is a bilinear operation.

The dual numbers form a commutative algebra of dimension two over the reals, and also an Artinian local ring. They are one of the simplest examples of a ring that has nonzero nilpotent elements.