the ring of dual numbers over a field $k$

Suppose $k$ is a field,then the quotient ring $k[\epsilon]/\epsilon^2$ is called the ring of dual numbers over $k$. I learn this from Hartshorne. I wonder why it has this name(maybe this question is a bit soft,or senseless). Are there any interesting things about this ring?Would someone be kind enough to say something about it?Thank you very much!

Well, one interesting fact about the dual numbers of $\mathbb{R}$: consider its polynomial ring, and specifically identify an object $f(x) = \sum_{i=0}^n a_ix^i , a_i \in \mathbb{R}[\epsilon]/\epsilon^2$. Now evaluating $f(a + b\epsilon), a,b \in \mathbb{R}$ will yield $f(a) + bf'(a)\epsilon$ (hint: binomial theorem) which allows for automatic differentiation and an interesting approach for non-standard analysis.

Working in a more general $k[\epsilon]/\epsilon^2$, since $(a + b\epsilon)(a^{-1} - ba^{-2}\epsilon) = 1,$ we see that for all nonzero $a$, $a + b\epsilon$ is a unit. So our ring of dual numbers over $k$ has a unique maximal ideal $(\epsilon)$ and the ring is local.

On a note more relating to Hartshorne: let $f: X \rightarrow S$ be a morphism of schemes. Using the ring of dual numbers, one can construct the pointed tangent space of $X$ over $S$, but I'm in no means qualified to talk about that.