@nprofile1q... the funny thing is that calculus was invented with an informal version of hyperreals in mind, since dy and dx were supposed to be nonzero values so you could "divide by zero", but so small that you could disregard them at the end

@nprofile1q... And derivatives and integrals are defined in terms of infinitesimal quantities dy and dx and the standard part function in place of a limit. If you remember the real number definitions of each, or even just the graphical explanations (slope and area under the curve) you can pretty easily guess what the hyperreal definitions are without ever seeing them.

@nprofile1q... The hyperreal numbers extend the real number line to include infinities (defined in terms of omega, an infinite number strictly greater than every real number) and infinitesimals (defined in terms of epsilon, or 1 over omega).

The closest thing to a limit would be the standard part function st(), which just takes the closest real number to a finite hyperreal number. The hyperreal definition of e is just st( (1+epsilon)omega ).

If you ever wondered why you could treat infinity as a number and it almost always works, you were just using a less formal version of hyperreals. The formal version fixes the cases where it doesn't work.

In the hyperreal domain, 0.9 repeating is not 1, but the standard part of 0.9 repeating is.

@nprofile1q... you don't need limits to define e in the hyperreals either: https://youtu.be/dyjlRi8nuw0?si=I2xEnw9PwVrH91hR&t=570

@nprofile1q... epsilon is cool and we should respect it

@nprofile1q... dy/dx is just a ratio of two infinitesimals and I'm tired of pretending it isn't

in this house, there are infinities upon infinities, and they all have names

in this house, there are more numbers in between any two real numbers than all of the real numbers themselves

in this house, we treat dy and dx as numbers

in this house we don't believe .9 repeating = 1