The Silvanus P. Thompson book suggested by the sibling comment is lovely and very clear. For a more algebraic treatment, and its important applications to auto

The Silvanus P. Thompson book suggested by the sibling comment is lovely and very clear. For a more algebraic treatment, and its important applications to automatic differentiation, I'd suggest starting with the relevant wikipedia articles: https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Automatic_differentiation https://en.wikipedia.org/wiki/Automatic_differentiation