Some Math fun for the holidays, do ponder.

@ElwoodBlues Yes they are very different from reals. Infinitesimals are an extension of the real number system, just like complex numbers.
Look at this equation:
[i]dy = k * dx[/i]
The product of the infinitesimal [i]dx[/i] and a real number [i]k[/i] will always be infinitesimal, however big [i]k[/i] is. The product will never be >1, so [i]dx[/i] isn't even Archimedean.

@helenS OK, I have now learned about hyperreal and surreal numbers, LOL!!!

@ElwoodBlues It's amazing how easy it is to see if a given real function is continuous at x=x0, using hyperreals: if x is increased by an infinitesimal amount dx, then y(x) will also change by an infinitesimal quantity dy.
Check with y=x^2 if you want to see a trivial example:
y(x+dx) = (x + dx)^2 = x^2 + 2x*dx + (dx)^2;
y(x+dx) – y(x) = 2x*dx + (dx)^2.
2x*dx + (dx)^2 is, of course, infinitesimal, so the function is continuous for all x∈ℝ.
Compare that with Cauchy/Weierstrass epsilontics!

I've never read anything about the "surreals", but I read an introductory textbook about the hyperreals some time ago, and it clicked immediately. Now I'm no longer ashamed to see dy/dx as a fraction, instead of a limit.

Hey it's nice to talk with someone on SW about maths! 🌷