A number is just a number, it doesn't approach anything. A series can approach something, but a number can't. In any system where 0.99... is valid notation for a number, it doesn't approach anything.
The point I'm making is that the "obvious truth" 0.99... = 1 that we're all talking about depends on the assumption that we're working in the real numbers.
I claim that the real numbers are not something intuitively obvious to every sufficiently intelligent person; instead they are kind of weird and technical. I go on to claim, though I'm more unsure of this, that the real numbers are not even the only way to make calculus work.
Anyway, in the surreal numbers you could probably make up a notation where 0.999... actually denotes 1 - ε or something. But I daresay it might not be very useful because then how do you denote 1 - ε/2 or anything else.
1.000...0 = 1
1.000...05 = 1 + ε/2
1.000...1 = 1 + ε
0.999...8 = 1 - 2ε
0.999...9 = 1 - ε
0.999...98 = 1 - ε/5
.
.
.> 0.999...98 = 1 - ε/5
cough
I don't know whether repeating decimals are useful for testing equality of surreal numbers.
All I'm trying to say is we're all talking about anything and everything except the definition "when are two real numbers equal?"
And then we're saying people who don't understand the consequences of that definition are kind of dummies ... while we continue to not actually say what the definition is.
That claim is certainly true. The proof is that calculus (Analysis) exists for the complex number system too. Although I don’t think that’s what you meant and I doubt the complex number system is “more intuitive.” Just out of curiosity, have you heard of real analysis? How do you define calculus?
I'm eager to be corrected if you can tell me something I said that's wrong. I'm not interested in gradually upping the ante with you until it's clear who really has more math background.
https://en.wikipedia.org/wiki/Nonstandard_calculus
It is based on the hyperreal numbers:
https://en.wikipedia.org/wiki/Hyperreal_number
Practically speaking, I don't think it buys you anything over traditional calculus/analysis. It's just pointing out that there are alternative approaches to formalizing calculus.