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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?

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Abramo
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  • Uninformed guess: because the elements have representations $a + \epsilon b$ as pairs of two ("dual") numbers. Obviously this is not unique to this particular ring. – Ryan Reich Sep 15 '13 at 15:27
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    I agree with Ryan, but I'm not sure. And also I wonder, does $k[\varepsilon]/\varepsilon^{n+1}$ have a name? This ring classifies $n$-jets. – Martin Brandenburg Sep 15 '13 at 19:53

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