Dual number

In algebra, the dual numbers are a quadratic algebra 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 ε 2 = 0 {\displaystyle \varepsilon ^{2}=0} with ε ≠ 0 {\displaystyle \varepsilon \neq 0} . Dual numbers can be added component-wise, and multiplied by the formula ( a + b ε ) ( c + d ε ) = a c + ( a d + b c ) ε , {\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.