[SideQuark]

You've filled this page with comments on non-standard analysis, but the dual numbers have precisely zero to do with it. Calling people n00bs on a topic you apparently do no understand is silly.

Non-standard analysis deals with fields only, and the dual numbers are not a field, there are infinitely many zero divisors.

You should read the wiki pages on both, then maybe this mathoverflow post explaining it. The clearest way to maybe grasp the difference for you is that in any formulation of non-standard analysis, the square of any infinitesimal is another infinitesimal, and never 0. In the dual numbers, the square of any infinitesimal is always precisely, exactly zero.

They are so fundamentally different that anyone (like you) that claims to be so cognizant of either would never repeat they are the same as loudly and frequently as you are.

https://math.stackexchange.com/questions/341535/is-the-theor...