I always thought calculus should work equally well with rationals, because they are continuous (in that you can always get closer...)
Take a derivative? that's a limit. Take an integral? That's a limit. And limits of seemingly rational expressions are sometimes irrational, case and point the limit of
(1 + 1/n)^n
as n approaches infinity is euler's constant, e.
So if you're taking limits with rationals you're going to find yourself with undefined limits (because irrational numbers are no longer defined) breaking things constantly.
Formally, the irrationals are dense but not complete.
That means they're dense, but continuous is something else.
You sort of can't get arbitrarily close. And that's the whole reason calculus doesn't work for rationals.
Rationals fundamentally have "gaps", and correspond 1:1 with integers.
Is there an obvious concrete example? I've never spent proper time studying Real Analysis, and I confess I have the same intuition as OP: That you could use rationals to approximate irrationals to arbitrary precision.
TL;DR: Every real number can be approximated by rationals. GP was being careless with language.
What's special about real numbers is that, if you have a sequence of reals for which the distance between two consecutive elements approaches zero, then there exists a real number that's the limit of the starting sequence.
This isn't the case with rational numbers, eg. the sequence 1.4, 1.41, 1.414 ... (EDIT: these are increasing approximations of sqrt(2)) satisfies the hypothesis but there's no single rational number this sequence approaches.
This property is called completeness. The real numbers are a complete topological space, whereas the rationals aren't.
”There are also representations like
{ 0, 1, 2, 3, … | } = ω
{ 0 | 1, 1/2, 1/4, 1/8, … } = ε
where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number.”
So, ε is a gap just above zero, π-ε a gap just below π, etc.
(The difference is that, when stepping from real to real, you won’t accidentally step on a surreal in the way you’ll frequently hit irrationals when you step from rational to rational)
I’ve never seen derivatives or integrals, but https://math.stackexchange.com/questions/112492/integral-of-... claims one can define them.
Maybe you have to leave those gaps unfilled.
A consequence is that you can divide the rationals cleanly into disconnected parts. For example A = the set of rationals < sqrt(2) and B = the set of rationals > sqrt(2)
https://en.wikipedia.org/wiki/Completeness_of_the_real_numbe...