Answer by J.G. for Construction of dual numbers
methods of constructionDual numbers have a matrix representation (in fact more than one), or can be identified with $\Bbb R[X]/X^2$. Either shows duals are equiconsistent with reals. See here for a...
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Out of curiosity, I was seeing about hypercomplex numbers. In that article, the definition says that, "Where possible, it is conventional to choose the basis so that $i_k^2 \in \{ -1, 0, +1 \}$. A...
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