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This goes with Multiplicative Calculus, but where as that is mostly summarizing others work, this is original as far as I know, and so I split it out.

Infinitesimals are liked, despite their formal rigor (in most settings), are liked in some settings, like informally solving differential equations, and other applied tasks. (It is interesting to ask why, how they can be made formal, and other questions, but I will refrain from doing so here.) They are, "additive", in a few key ways:

For the first, the intuition for a regular infinitesimal is that is a very small number, smaller than any non-zero real number. If we multiply a real number by an infinitesimal, we always get another infinitesimal, never a real number, just like when we multiply any real number by 0, we always get zero. 0 and infinitesimals are "not multiplicative numbers" like the non-zero real numbers, because they represent this "point of no return".

Note that every $d E$, where $E$ is a metavariable, becomes $E[x \mapsto a] - E$, i.e. we subtract $E$ from "almost $E$", but replacing $x$ with $a$ in the first term. (Exactly formalizing this is probably harder, so I won't attempt it.) This is the subtraction I was referring to.

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