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0.999...= 1

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Re: 0.999...= 1

#261

Post by esmi »

tom-thistime wrote: post #140
Scarblac wrote: post #131

Earlier quoted context omitted.

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.

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…

> 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?

Re: 0.999...= 1

#262

Post by thaumasiotes »

chris_wot wrote: post #256
2OEH8eoCRo0 wrote: post #221

Earlier quoted context omitted.

Ask for a number between .9 repeated and 1

So does this mean that an infinitely small number is zero? As in 1/∞ ?

You can read about this in any work on nonstandard analysis. ("Nonstandard" is just the name, much like "imaginary" numbers.)

An infinitely small number is zero when projected onto the real number line. If you introduce an infinitesimal quantity to the reals, then for every number there is a unique real number to which that first number is infinitely close (that is, the difference between them is infinitesimal). You can use that real number as a (good) approximation of all the nonstandard numbers in its halo. (As long as you're comparing it to other real numbers.)

Re: 0.999...= 1

#263

Post by remexre »

tom-thistime wrote: post #190
lmkg wrote: post #137

Earlier quoted context omitted.

It's more fundamental: People seem to have the intuition that the decimal representation of a number is a number . I don't know if it's because decimals resemble the ntural numbers or what, but decimals seem to have a primacy for people that fractions do not. The idea that there's a gap between the symbol for a thing and the thing itself is the stumbling block.

Possibly people are looking at two different symbols and asking "can you show me logically why those are equal." If they're given a definition of "equal" and they still object, that's a different problem. I have this problem every time I play with group theory again. You get the axioms for a group, which say there is some identity but don't explicity require the identity to be unique. You can easily prove that the id…

I used to have the same annoyance about group and category theory; learning some about the type-theory-as-foundations work (particularly homotopy type theory and cubical type theory) has helped w/ this; in that setting, you have several distinct but well-developed notions of equality: propositional equality, which group theory and most math cares about, vs judgemental equality, which is the one that's "obviously true" by the rules of the system.

e.g. 5 = 5 is true under judgemental and propositional equality, whereas x + 2 = 2 + x is only true under propositional

Re: 0.999...= 1

#264

Post by _Donny »

317070 wrote: post #192
dwheeler wrote: post #153

A formally rigorous proof of this (in Metamath) is here: http://us.metamath.org/mpeuni/0.999....html Unlike typical math proofs, which hint at the underlying steps, every step in this proof only uses precisely an axiom or previously-proven theorem, and you can click on the step to see it. The same is true for all the other theorems. In the end it only depends on predicate logic and ZFC set theory. All the proofs have…

It depends on more than just ZFC, also on the definitions of the real/complex numbers. The crux of the proof is that 0.99999... is being constructed within the real/complex numbers, and in that system it is equal to 1. And at the point where students see this, the whole concept of real numbers and infinity is usually ill-defined. I actually understand the scepsis for this theorem and where it comes from. The proof re…

I think this is spot on, at least for me personally.

I am not very good at mathematics, so I never questioned my professors when they said that "You cannot treat infinites as regular numbers".

Perhaps due to that statement, I did not really pursue these kinds of equations. For instance, I do not really see how the algebraic argument on the Wiki is any different from:

  2 * inf = inf
  inf + inf = inf   (subtract inf from both sides)
  inf = 0

Re: 0.999...= 1

#265

Post by tom-thistime »

zests wrote: post #234
tom-thistime wrote: post #156

Earlier quoted context omitted.

Sure. And it's true, 0.99... is equal to 1. All I'm saying is [SOPI below] it's all a little more technical than the junior high school proof. For example if 23+6\epsilon = 23, then how do I define 23 + 6\epsilon - 23? I can choose different approaches here, but "zero" is going to be pretty inconvenient when I go to do an integral. [SOPI] Statement of Personal Ignorance. I don't quite know what I'm talking about. If…

What are you integrating over?

I'm taking the limit of any convergent infinite sum of terms each weighted by \epsilon. Although \epsilon itself equals zero, of course the limit of the sum isn't necessarily zero.

So if we go and define some other funky "infinitesimal" \epsilon != 0 in the surreals, we have to be careful. Apparently people have done that kind of thing successfully, but it took a long time after Cauchy for that to happen.

Re: 0.999...= 1

#266

Post by UncleMeat »

jonsen wrote: post #215
_bxg1 wrote: post #204

Earlier quoted context omitted.

Why not 0.00...1?

This is not an infinite decimal. The digit 1 is somewhere out there.

But this is not a compelling argument to somebody in this situation. While correct, it feels identical to saying "it just is".

Re: 0.999...= 1

#269

Post by CogitoCogito »

ramshorns wrote: post #222
tom-thistime wrote: post #140

Earlier quoted context omitted.

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…

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.

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

Re: 0.999...= 1

#270

Post by naringas »

ping_pong wrote: post #160

My 5 year old stumped me with this, and I had to look it up. He asked me why 1/3 + 1/3 + 1/3 = 1, since it's equal to 0.333... + 0.333... + 0.333... which is 0.999... How can that possibly equal 1.000...? And is 0.66... equal to 0.67000...? I didn't have a good enough answer for him, so I had to look it up and found this page. I tried to explain it to him but since I'm a terrible teacher and he's only 5, it was hard…

> Luckily he has many years before it matters!

and depending on career choices, it might never matter at all.

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