One way to view automatic differentiation is to think of it as adjoining an "infinitesimal" element d, such that d²=0, to the reals, ie. forming ℝ[d]/(d²). If f is a polynomial then f(x+d)=f(x)+df'(x) giving a nice way to compute derivatives on a computer - especially as it can be extended to rational and even transcendental functions f. It doesn't form a field though. For example you can't always divide by d.
TIL There is a field, named after Levi-Civita, that generalises ℝ[d]/(d²) quite a bit.
Each element is a "formal" sum ∑aᵢεⁱ where the sum is over some subset S of the rationals which is left-finite, ie. for any z, S has only finitely many elements less than z. Addition and multiplication work in the way you might guess.
This means we can form things like ε^(1/2) or even the "infinite" 1/ε. It's not just a field, it's an ordered field so we have, for example, that 1 > ε^(1/2) > ε > ε² > 0.
You can even construct a Dirac delta-like function δ(x) = ε/π(x²+ε²).
@dpiponi Thanks for this post, it was really interesting (and sent me down the sort of rabbit hole I enjoy). Coincidentally, I had just been asking about usefulness of the hyperreals recently, so it's interesting to see this similar but distinct idea that apparently does have pretty practical uses.
