The grandparent I'm responding to sure uses a very sloppy presentation of things. Not everyone here is a trained mathematician though, so you may want to give people some slack.
Obviously, if h² = 0, then h = 0, so this statement made no sense. What the author probably tried to convey, is that one can reason with infinitely small values as symbols, and perform automatic differentiation with that.
No, there’s an abstract algebra extension of real numbers to have an extra symbol h such that h^2=0. This is not a real number so you cannot apply the argument h^2=0 implies h=0, much like complex numbers don’t obey all properties of real numbers.
(For example for real numbers, x!=0 implies x^2>0 but i^2=-1)
a^2 = 1, first base vector is a regular one
b^2 = -1, second base vector is "imaginary"
ab = 0, base vectors are orthogonal
(a+b)^2 = a^2 + 2ab + b^2 = 1 + 2\*0 + (-1) = 0
Trick is taken from conformal geometric algebra [1].
Obviously, if h² = 0, then h = 0, so this statement made no sense. What the author probably tried to convey, is that one can reason with infinitely small values as symbols, and perform automatic differentiation with that.