The ring of dual numbers over a field $k$ is defined as the quotient
$$k[\varepsilon]/\varepsilon^2.$$
I was reading this question with an interesting answer about some of their basic properties and, if I understand correctly, this ring offers a way to produce approximations of the first order of non-analytic objects defined on $k$.
But the most important [ ;-) ] question remains open: Why are they called dual numbers?