The rigorous definition of infinitesimals is developed in non-standard analysis: https://en.wikipedia.org/wiki/Non-standard_analysis https://en.wikipedia.org/wi

The rigorous definition of infinitesimals is developed in non-standard analysis: https://en.wikipedia.org/wiki/Non-standard_analysis https://en.wikipedia.org/wiki/Non-standard_analysis IIRC you define e such that e^2 = 0 and then normal algebra just works. There is background work needed to precisely concepts but that's the gist of it (again, IIRC).