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0.999...= 1

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Re: 0.999...= 1

#281

Post by CogitoCogito »

ColinWright wrote: post #252
CogitoCogito wrote: post #196

Earlier quoted context omitted.

Here is a comment where I tried to clarify, but yes you seem to basically understand my point: https://news.ycombinator.com/item?id=23007600

My personal experience is that people want to argue from intuition about what 0.9999... means, and when you try to make it precise they say that it's obvious. Then they derive all sorts of nonsense and conclude that mathematics is all rubbish. If someone really wants to understand it then I'll explain current mathematical thinking, including non-standard analysis and the surreals. But most people don't want to put in…

Someone else here brought up the surreal numbers and my intuition says that it's right to do that. The various arithmetic proofs thrown around here don't explicitly make use of completeness. As such they should be correct proofs in the surreal numbers as well. But they basically are not. Here is a blog post about it:

https://thatsmaths.com/2019/01/10/really-0-999999-is-equal-t...

I don't quite know how to formalize it, but I'm pretty certain that if these proofs logically worked (in the "theory of proofs sense"), then they should work in the surreals as well.

Anyway it's just intuition. My main point in this thread is that I don't really accept the proofs of this that don't use completeness as a step. Though I do suspect that proofs not making use of it are actually incorrect proofs in their own right. If I were curious enough I'd think back about formal proofs and models and all that jazz, but I probably already have spent more time in this thread than I should. :)

edit: The more I think about it I feel like someone actually explained to me this (i.e. why this proof is wrong using surreals as reasoning) a long time ago and I'm just remembering echos of it in my mind. Wish I could remember something more useful...or that I were a logician...

Re: 0.999...= 1

#282

Post by tom-thistime »

CogitoCogito wrote: post #269
ramshorns wrote: post #222

Earlier quoted context omitted.

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...

Strictly speaking I brought up the surreal numbers.

Re: 0.999...= 1

#283

Post by saalweachter »

ramshorns wrote: post #243
ramshorns wrote: post #222

Earlier quoted context omitted.

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

Re: 0.999...= 1

#284

Post by phaemon »

thenewwazoo wrote: post #251
TomGullen wrote: post #167

Earlier quoted context omitted.

I find the algebraic way convinces most people: x = 0.9999.. 10x = 9.9999... 10x - x = 9 9x = 9 x = 1

Because I wondered, it’s not a trick: x = 0.444444... 10x = 4.444444... 10x - x = 4 9x = 4 x = 4/9 = 0.444444...

All the single digit repeating decimals are x/9.

0.111... == 1/9

0.222... == 2/9

...

0.888... == 8/9

0.999... == 9/9 :)

Re: 0.999...= 1

#285

Post by Recursing »

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…

Very cool page!

The only interesting step is step 32, which is just an application of http://us.metamath.org/mpeuni/geoisum1c.html, whose only interesting step is step 21 which is just an application of http://us.metamath.org/mpeuni/geoisum1.html.

They key steps for that are http://us.metamath.org/mpeuni/geolim2.html and http://us.metamath.org/mpeuni/isumclim.html , which is indeed the crux of the issue

Re: 0.999...= 1

#286

Post by CogitoCogito »

dahart wrote: post #257
CogitoCogito wrote: post #218

Earlier quoted context omitted.

> The proof hinges on the definition of infinity. 0.9bar7 is a completely nonsensical number precisely because of the definition of infinity. Yes, I think you’re failing to accept the definition of infinity. You’re rejecting a proof that many other PhDs in math accept along with some notable mathematicians like Euler. I reject your appeal to authority; having an advanced degree doesn’t mean you’re somehow automatical…

Okay, I understand what you’re trying to say. I accept that the proof does not define what “...” means formally, and that is the problem you have with it. The infinity is understood implicitly to have the property that the 9s never end. In that sense, I think the proof does make use of limits, it just relies on definitions not written as part of the proof. Isn’t that okay, doesn’t that actually happen very often? The…

See my comment here since it's related to this: https://news.ycombinator.com/item?id=23008366

Assuming that all my suspicions in that comment are correct and these proofs actually are invalid proofs (not the results which are true), then the question might become: does it matter if the proof of a fact is incorrect if the fact itself is correct? That is a philosophical question and I'm honestly not sure how I'd answer it...

Re: 0.999...= 1

#287

Post by __s »

calibas wrote: post #271

And this is why I prefer hyperreals. 0.999... = 1 - 1/∞ We talk about infinity all the time in mathematics, teachers use the concept to introduce calculus in a way that people can more easily understand, but using infinity directly is almost universally banned within classrooms. Nonstandard analysis is a much more intuitive way of understanding calculus, it's the whole "infinite number of infinitely small pieces" con…

I think what's important here is that if you're making that claim,

0.333... = 1/3 - 1/∞

Which implies 3/∞ = 1/∞

Re: 0.999...= 1

#289

Post by GolDDranks »

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/∞ ?

In real numbers, there doesn't exist such a thing as "infinitely small number" that is apart from zero. Yes, there exists infinitely many numbers between any minisculely small number and zero, but the way they are defined, every single number you can grasp, is finitely small. The "infinitely" small gap is inaccessible. In some other number systems it isn't, but in the standard reals it is.

That means that the "infinitely small" doesn't exist; "smallest apart from zero" doesn't exist either.

Re: 0.999...= 1

#290

Post by sethammons »

TomGullen wrote: post #167
undecisive wrote: post #115

There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't hav…

I find the algebraic way convinces most people: x = 0.9999.. 10x = 9.9999... 10x - x = 9 9x = 9 x = 1

I was prepared to blow my then 6th or 7th grade daughter's mind with this algebraic proof. I started by asking if 0.999... = 1, to which she said "no." I rephrased it and said it is equal, do you know why? She thought for a moment and said "1/9 is 0.111..., so 9/9 is 0.999... and 9/9 is 1." And I had to admit she had a far better solution than I did.
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