just like how infinitesimals are useful because they're indiscernible from 0, but have the advantage of being able to be divided by.

the other important number besides 0 is 1. gpt even suggested it as i was typing it.

the monad of 1 is numbers of the form (1 + ε) where ε is an infinitesimal. multiplying by them should be almost as good as multiplying by 1, aka a freebie.

this is just the lie algebra, and is why elements of it are always invertible. by using the infinitesimal version of a clifford algebra on the tangent space, we can use its rotor represention to see how the exponential map gives lie's third theorem.