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Comment by tom-thistime | Hacker News Reader

Sure. Instead of "0.99..." please substitute lim n->inf sum(1..n)(9 times 10^-n).

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.

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Well, a limit of a series is just a number and doesn't approach anything either. If the series approaches something, we say the limit exists and is equal to that thing.

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.

Oh, I figured it out, kind of. [I don't really know what I'm talking about either.]

  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...8 = 1 - 2ε

> 0.999...98 = 1 - ε/5

cough

Sure, why not? Different notations for different surreal numbers. But it's not actually a good notation; I wouldn't know how to write 1 + 10ε.

We've defined the series we're talking about, and we've defined 0.99... as the limit of that series.

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.

Fair. Maybe the definition is that two real numbers are equal iff there is no real number between them. Which is nonintuitive if you're used to things like natural numbers which have successors. But if you teach people this way of thinking about the real numbers, then arguments like "it's the last number riiight before 1" stop working.

It's smart of you to bring up the surreal numbers: https://thatsmaths.com/2019/01/10/really-0-999999-is-equal-t...

Strictly speaking I brought up the surreal numbers.

Then I guess you're the smart one. My apologies.

Sorry, sorry. I've been trying to bring up nonstandard analysis, and repeatedly getting poked by people saying "what do you think 'calculus' is?" "have you ever heard of a limit?" and so forth. In the process I have apparently become even more ornery than usual.

As we have no idea who you are don’t you think it makes sense we try to figure it out? How one answers the question will vary based on background.

> 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.

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?

A Riemann integral in complex analysis uses the same definitions as one in real analysis. Derivatives ditto. In the thread we're comparing real analysis to nonstandard analysis.

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.

I didn’t really see anything wrong in the thread but you use words like “calculus” and I honestly don’t know exactly what you mean. Do you mean the plug and chug methods often taught in high school? Or do you mean the application of a set of theorems derived from the axioms of a given system of numbers? If you mean the former then perhaps we can clear up some misconceptions which are clarified by the latter.

Take a look at nonstandard calculus:

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.

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